Learning-Theoretic Foundation for General Coded Computing: The Adversarial Setting

# Learning-Theoretic Foundation for General Coded Computing: The Adversarial Setting
Source: [https://arxiv.org/html/2610.27440](https://arxiv.org/html/2610.27440)
Hanzaleh AkbarinodehiMohammad Ali Maddah\-Ali††thanks:P\. Moradi, H\. Akbarinodehi, and M\. A\. Maddah\-Ali are with the University of Minnesota, Twin Cities, Minneapolis, MN, USA\.††thanks:The work has been partially supported by the National Science Foundation under Grant CCF\-2348638\.††thanks:A preliminary version of this work was presented in ISIT 2025 conference\.

###### Abstract

Adversarial workers pose a severe challenge in distributed computing systems by returning corrupted outputs that can substantially distort the recovered computations\. Coded computing addresses this threat by introducing redundancy across worker assignments\. Existing exact and approximate coded\-computing methods typically handle adversarial outputs by explicitly locating and correcting or removing erroneous evaluations, which adds significant computational overhead at the master node\. Moreover, existing analyses do not characterize how the recovery error scales with the number of workers and the adversarial budget\.

In this paper, we develop an adversarially robust extension of general coded computing \(GCC\), a learning\-theoretic approximate coded\-computing framework that designs the encoder and decoder by minimizing an upper bound on the end\-to\-end recovery error\. Rather than identifying corrupted workers, the proposed method uses smoothness regularization to limit the influence of their outputs\. The resulting encoder and decoder can be implemented using smoothing splines with linear master\-side complexity\. We derive a non\-asymptotic bound on the worst\-case end\-to\-end approximation error by decomposing it into three components: \(1\) the encoder approximation error, \(2\) the decoder reconstruction error under fully honest results, and \(3\) the adversarial perturbation error\. For maximum ofq≥1q\\geq 1adversarial workers amongNNworkers results in a worst\-case error of order𝒪⁡\(\(q/N\)6/5\)\\mathcal\{O\}\(\(q/N\)^\{6/5\}\)with appropriate selection of smoothness parameter, demonstrating that the approximation error vanishes asymptotically whenever the number of adversaries grows sublinearly with the network size\. Experiments on scalar computation and neural\-network inference are consistent with the predicted convergence behavior\.

###### Index Terms:

Adversarial workers, approximate coded computing, distributed computing, general coded computing, reproducing kernel Hilbert space, smoothing spline\.

## IIntroduction

The growing scale of data\-intensive applications and computationally demanding machine\-learning \(ML\) models has made distributed computation central to modern computing systems\[[1](https://arxiv.org/html/2610.27440#bib.bib1)\]\. Such systems commonly operate under a*master–worker architecture*, in which a central*master node*assigns computational tasks to a collection of*worker nodes*\. Each worker node performs its assigned computation and returns the result to the master node, which aggregates the worker outputs to recover the desired function evaluations\. By executing these tasks in parallel, the system can substantially reduce the computation time incurred at the master node\.

This benefit, however, comes with a reliability challenge: some workers may fail to return their results \(known as straggler worker\), while others may return incorrect values because of faults or malicious behavior \(known as adversarial worker\)\. These two forms of unreliability are fundamentally different\. A straggler removes an observation, whereas an adversarial worker introduces a corrupted observation that may be coordinated with other corruptions and deliberately chosen to distort the final estimate\. The master node must therefore reconstruct the desired function values without knowing which worker outputs, if any, are corrupted\.

Coded computing addresses worker unreliability by introducing redundancy across the assigned computations\. The master node encodes the original inputs into some*coded inputs*and send each coded inputs to one worker nodes\. Next, each worker node apply the target function on its assigned coded input and return the results to the master node\. Finally, the master node reconstructs the desired computations from the worker outputs\[[2](https://arxiv.org/html/2610.27440#bib.bib2),[3](https://arxiv.org/html/2610.27440#bib.bib3),[4](https://arxiv.org/html/2610.27440#bib.bib4),[5](https://arxiv.org/html/2610.27440#bib.bib5),[6](https://arxiv.org/html/2610.27440#bib.bib6),[7](https://arxiv.org/html/2610.27440#bib.bib7),[8](https://arxiv.org/html/2610.27440#bib.bib8)\]even though some of them are missing or incorrect\.

For polynomial computations, Lagrange Coded Computing \(LCC\) achieves exact recovery while simultaneously providing resilience against stragglers and robustness against adversarial workers whenever the number of worker nodes exceeds a prescribed*recovery threshold*\[[4](https://arxiv.org/html/2610.27440#bib.bib4)\]\. This threshold is determined by the degree of the target polynomial computation as well as the maximum numbers of straggling and adversarial nodes\. However, LCC is originally formulated over finite fields and does not scale well with the input batch size due to numerical overflow and the severe ill\-conditioning of high\-degree polynomial interpolation\[[9](https://arxiv.org/html/2610.27440#bib.bib9),[10](https://arxiv.org/html/2610.27440#bib.bib10)\]\. To address this limitation, Analog Lagrange Coded Computing \(ALCC\) extends LCC to infinite fields \(the analog domain\), preserving straggler resilience and dataset privacy\[[11](https://arxiv.org/html/2610.27440#bib.bib11)\]\.

Approximate coded computing extends coded computation to a broader class of functions by allowing approximate recovery\. Berrut approximated coded computing \(BACC\) uses barycentric rational interpolation to approximate general nonlinear computations from the non\-straggler worker outputs\[[9](https://arxiv.org/html/2610.27440#bib.bib9)\]\. More recently, the learning\-theoretic general coded computing \(GCC\) framework designs the encoder and decoder by deriving and minimizing an upper bound on the average end\-to\-end approximation error\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\]\.

AlthoughGCCprovides an efficient scheme that outperformsBACCwith a faster convergence rate and provably mitigates the effect of stragglers, its robustness against adversarial workers remains unexplored\. Conversely, adversarial robustness has been studied within the Berrut framework\. ApproxIFER combines rational interpolation with an algebraic error\-location algorithm inspired by Berlekamp–Welch decoding to provide Byzantine\-robust distributed inference\[[13](https://arxiv.org/html/2610.27440#bib.bib13)\]\. However, these approaches address adversarial corruption through explicit error\-location and correction procedures*prior*to reconstructing the desired outputs, which imposes substantial computational overhead at the master node\. Furthermore, none of these works investigates how the approximation error behaves as the number of workers increases for a given adversarial budget, nor do they establish fundamental limits on the adversarial budget required for the approximation error to vanish asymptotically\.

To address these gaps, we investigate the adversarial robustness of theGCCframework from a complementary perspective: rather than explicitly locating and discarding adversarial workers, we mitigate their influence by inducing*smoothness*in the encoding and decoding procedures\. Building upon theGCCarchitecture, the proposed method utilizes smooth functions to distribute computations and reconstruct the desired outputs\. The decoder aggregates all returned outputs and bounds the impact of corrupted values via smoothness regularization\. While the regularization parameters can be calibrated to a known adversarial budget, the framework operates without prior knowledge of either the adversaries’ identities or their specific corruptions\.

Crucially, we establish that with an appropriately tuned smoothness regularization, the end\-to\-end approximation error converges to zero as the total number of workers increases, provided the number of adversarial workers grows sub\-linearly with the network size\.

Our main contributions are summarized as follows:

1. 1\.AdversarialGCCFramework:We propose an adversarially robustGCCscheme for general high\-dimensional computations that operates reliably in the presence of adversarial workers\. We derive an upper bound on the end\-to\-end approximation error and decompose it into three components: \(i\) the encoder approximation error, \(ii\) the decoder reconstruction error in a fully honest setting \(referred to as the*clean approximation error*\), and \(iii\) the perturbation caused by adversarial workers\. This decomposition explicitly reveals how the smoothness of the encoder and decoder governs the robustness of the overall scheme\.
2. 2\.Finite\-Sample Guarantees and Convergence Rates:We derive a non\-asymptotic upper bound on the worst\-case mean\-squared approximation error\. This bound captures the fundamental trade\-off introduced by decoder regularization: stronger smoothing reduces the influence of adversarial outputs but increases the clean approximation error\. Balancing these two effects reveals that for a maximum ofq≥1q\\geq 1adversarial workers out ofNNtotal workers, the decoder smoothing parameter should scale as\(q/N\)8/5\(q/N\)^\{8/5\}\. With this tuning, the proposed scheme achieves an end\-to\-end error rate of𝒪⁡\(\(q/N\)6/5\)\\mathcal\{O\}\(\(q/N\)^\{6/5\}\), demonstrating that the worst\-case approximation error converges to zero whenever the number of adversarial workers grows sub\-linearly with the total number of workers\.
3. 3\.Efficient Implementation:We show that both the encoder and decoder can be implemented using vector\-valued cubic smoothing splines\. Because their coefficients can be computed by solving banded linear systems, the master\-side encoding and decoding costs scale linearly with both the input batch size and the total number of workers\. Notably, this computational efficiency is achieved without requiring a separate adversarial detection or error\-location subroutine\.
4. 4\.Empirical Evaluation:We evaluate the proposed scheme on both scalar nonlinear computations and high\-dimensional neural network inference\. The experiments illustrate the predicted convergence behavior under adversarial attacks and confirm that the observed error decay aligns with our theoretical guarantees\.

The remainder of the paper is organized as follows\. Section[II](https://arxiv.org/html/2610.27440#S2)introduces the problem formulation\. Section[III](https://arxiv.org/html/2610.27440#S3)reviews the necessary mathematical preliminaries regarding Sobolev spaces, reproducing kernels, and smoothing splines\. Section[IV](https://arxiv.org/html/2610.27440#S4)develops the proposed encoding and decoding procedures, establishes their error and convergence guarantees, analyzes the computational complexity\. The proofs of the main theorems and corollaries are collected in Section[V](https://arxiv.org/html/2610.27440#S5), and Section[VI](https://arxiv.org/html/2610.27440#S6)presents the experiments result\.

### I\-ANotation

Forn∈ℕn\\in\\mathbb\{N\}, let\[n\]:=\{1,…,n\}\.\[n\]:=\\\{1,\\ldots,n\\\}\.The cardinality of a set𝒮\\mathcal\{S\}is denoted by\|𝒮\|\|\\mathcal\{S\}\|\. Scalars are denoted by lowercase letters, such asvv, vectors by bold lowercase letters, such as𝐯\\mathbf\{v\}, and matrices by bold uppercase letters, such as𝐕\\mathbf\{V\}\. The Euclidean norm is denoted by∥⋅∥2\\lVert\\cdot\\rVert\_\{2\}\. For a vector𝐯∈ℝm\\mathbf\{v\}\\in\\mathbb\{R\}^\{m\}, its coordinatewise maximum norm is defined as∥𝐯∥∞:=maxj∈\[m\]⁡\|vj\|\.\\lVert\\mathbf\{v\}\\rVert\_\{\\infty\}:=\\max\_\{j\\in\[m\]\}\\lvert v\_\{j\}\\rvert\.For a matrix𝐀\\mathbf\{A\}, its induced operator norm is defined as∥𝐀∥op:=sup∥𝐯∥2=1∥𝐀𝐯∥2\\lVert\\mathbf\{A\}\\rVert\_\{\\mathrm\{op\}\}:=\\sup\_\{\\lVert\\mathbf\{v\}\\rVert\_\{2\}=1\}\\lVert\\mathbf\{A\}\\mathbf\{v\}\\rVert\_\{2\}\. We writea≍ba\\asymp bto indicate that there exist positive constantsc1,c2\>0c\_\{1\},c\_\{2\}\>0such thatc1​a≤b≤c2​ac\_\{1\}a\\leq b\\leq c\_\{2\}a\.

For a twice differentiable scalar\-valued functiong:ℝd→ℝg:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\},∇g​\(𝐱\)\\nabla g\(\\mathbf\{x\}\)andD2​g​\(𝐱\)D^\{2\}g\(\\mathbf\{x\}\)denote its gradient and Hessian, respectively\. For a twice differentiable vector\-valued functiong=\[g1,…,gm\]Tg=\[g\_\{1\},\\ldots,g\_\{m\}\]^\{T\},∇gj​\(𝐱\)\\nabla g\_\{j\}\(\\mathbf\{x\}\)andD2​gj​\(𝐱\)D^\{2\}g\_\{j\}\(\\mathbf\{x\}\)denote the gradient and Hessian of itsjjth component, respectively\.

For a measurable functionu:Ω→ℝMu:\\Omega\\to\\mathbb\{R\}^\{M\},u′u^\{\\prime\}andu′′u^\{\\prime\\prime\}denote its first and second weak derivatives, taken componentwise with respect to its scalar argument\. The formal definition of a weak derivative is provided in Definition[3](https://arxiv.org/html/2610.27440#Thmdefinition3)in Appendix[A](https://arxiv.org/html/2610.27440#A1)\. The Lebesgue measure of a measurable setℰ⊆ℝ\\mathcal\{E\}\\subseteq\\mathbb\{R\}is denoted byLeb⁡\(ℰ\)\\operatorname\{Leb\}\(\\mathcal\{E\}\)\. For1≤p≤∞1\\leq p\\leq\\infty,∥u∥Lp​\(Ω,ℝM\)\\lVert u\\rVert\_\{\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}denotes itsLpL^\{p\}\-norm overΩ\\Omega, as formally defined in Definition[4](https://arxiv.org/html/2610.27440#Thmdefinition4)in Appendix[A](https://arxiv.org/html/2610.27440#A1)\.

## IIProblem Formulation

We consider a master\-worker distributed computing system consisting of one master node andNNworker nodes\. The master node aims to evaluate a prescribed computing functionf:ℝd→ℝmf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{m\}on a collection ofKKinput data points\{𝐱k\}k=1K\\\{\\mathbf\{x\}\_\{k\}\\\}\_\{k=1\}^\{K\}, using the worker nodes, whereNN,KK,dd, andmmare positive integers\.

The set of worker nodes is partitioned into two disjoint subsets: the set of honest workers, denoted by𝒟\\mathcal\{D\}, and the set of adversarial workers, denoted byℬ\\mathcal\{B\}, where𝒟∪ℬ=\[N\],𝒟∩ℬ=∅\.\\mathcal\{D\}\\cup\\mathcal\{B\}=\[N\],\\mathcal\{D\}\\cap\\mathcal\{B\}=\\varnothing\.Each honest worker evaluates the prescribed function at its assigned input and returns the resulting output without modification\. In contrast, an adversarial worker may return an arbitrary vector in the bounded adversarial\-output region

𝒴adv:=\[−Ta,Ta\]m,\\displaystyle\\mathcal\{Y\}\_\{\\textrm\{adv\}\}:=\[\-T\_\{a\},T\_\{a\}\]^\{m\},\(1\)for someM\>0M\>0, regardless of the computation assigned to that worker\. We assume that

\|ℬ\|≤q,\\displaystyle\|\\mathcal\{B\}\|\\leq q,\(2\)whereq∈\{0,…,N\}q\\in\\\{0,\\ldots,N\\\}is the maximum number of adversarial workers\.

All adversarial workers are controlled by a single entity, referred to as the*adversary*, and may coordinate their returned outputs\. An attack is specified by the set of corrupted workers and their returned vectors\. Accordingly, define

𝒜q:=\{A=\(ℬ,\{𝐚j\}j∈ℬ\):ℬ⊆\[N\],\|ℬ\|≤q,𝐚j∈𝒴adv\}\.\\displaystyle\\mathcal\{A\}\_\{q\}:=\\bigl\\\{A=\\bigl\(\\mathcal\{B\},\\\{\\mathbf\{a\}\_\{j\}\\\}\_\{j\\in\\mathcal\{B\}\}\\bigr\):\\mathcal\{B\}\\subseteq\[N\],\\,\|\\mathcal\{B\}\|\\leq q,\\,\\mathbf\{a\}\_\{j\}\\in\\mathcal\{Y\}\_\{\\textrm\{adv\}\}\\bigr\\\}\.\(3\)The adversary has full knowledge of the input batch\{𝐱k\}k=1K\\\{\\mathbf\{x\}\_\{k\}\\\}\_\{k=1\}^\{K\}, the computing functionff, and the complete encoding and decoding procedures\. Thus, the adversary may choose the corrupted worker set and all corrupted outputs jointly\.

### II\-AGeneral Coded Computing in the Adversarial Setting

Similar toGCCin the straggler setting\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\], the proposed framework consists of three stages all depicted in Figure[1](https://arxiv.org/html/2610.27440#S2.F1): \(i\) encoding at the master node, \(ii\) computation at the worker nodes, and \(iii\) decoding at the master node\.

1. \(1\)*Encoding:*The master node selectsKKfixed scalar pointsα1<⋯<αK\\alpha\_\{1\}<\\cdots<\\alpha\_\{K\}withαk∈Ω,\\alpha\_\{k\}\\in\\Omega,fork∈\[K\]k\\in\[K\]referred to as the*encoder design points*, and constructs an encoder functionuenc:Ω→ℝdu\_\{\\textrm\{enc\}\}:\\Omega\\to\\mathbb\{R\}^\{d\}such that uenc​\(αk\)≈𝐱k,k∈\[K\]\.\\displaystyle u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\\approx\\mathbf\{x\}\_\{k\},\\qquad k\\in\[K\]\.\(4\)The master node also selectsNNfixed scalar pointsβ1<⋯<βN,\\beta\_\{1\}<\\cdots<\\beta\_\{N\},withβj∈Ω,\\beta\_\{j\}\\in\\Omega,forj∈\[N\]j\\in\[N\]referred to as the*decoder design points*\. It forms the coded inputs by evaluating the encoder at these points: 𝐱~j:=uenc​\(βj\),j∈\[N\]\.\\displaystyle\\widetilde\{\\mathbf\{x\}\}\_\{j\}:=u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\),\\qquad j\\in\[N\]\.\(5\)The master node then assigns𝐱~j\\widetilde\{\\mathbf\{x\}\}\_\{j\}to workerjj\.
2. \(2\)*Computation:*Every honest workerj∈𝒟j\\in\\mathcal\{D\}evaluates the prescribed computing function at its assigned coded input and returnsf⁡\(𝐱~j\)=f⁡\(uenc​\(βj\)\)\.f\(\\widetilde\{\\mathbf\{x\}\}\_\{j\}\)=f\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\)\.In contrast, an adversarial workerj∈ℬj\\in\\mathcal\{B\}may replace this value by an arbitrary vector𝐚j∈𝒴adv\\mathbf\{a\}\_\{j\}\\in\\mathcal\{Y\}\_\{\\textrm\{adv\}\}\. Therefore, under an attackA∈𝒜qA\\in\\mathcal\{A\}\_\{q\}, the output received by the master node from workerjjis 𝐲¯jA=\{f⁡\(uenc​\(βj\)\),j∈𝒟,𝐚j∈𝒴adv,j∈ℬ\.\\displaystyle\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}=\\begin\{cases\}f\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\),&j\\in\\mathcal\{D\},\\\\\[2\.84526pt\] \\mathbf\{a\}\_\{j\}\\in\\mathcal\{Y\}\_\{\\textrm\{adv\}\},&j\\in\\mathcal\{B\}\.\\end\{cases\}\(6\)
3. \(3\)*Decoding:*Using the returned outputs\{𝐲¯jA\}j=1N\\\{\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}\\\}\_\{j=1\}^\{N\}and the corresponding decoder design points\{βj\}j=1N\\\{\\beta\_\{j\}\\\}\_\{j=1\}^\{N\}, the master node constructs a decoder functionudec:Ω→ℝm\.u\_\{\\textrm\{dec\}\}:\\Omega\\to\\mathbb\{R\}^\{m\}\.The decoder function is designed to suppress the effect of the adversarial outputs and accurately approximate the composed functionf∘uencf\\circ u\_\{\\textrm\{enc\}\}in the presence of up toqqcorrupted worker outputs\. In particular, the objective is to have a decoder functionudec​\(⋅\)u\_\{\\textrm\{dec\}\}\(\\cdot\)such that udec​\(z\)≈f⁡\(uenc​\(z\)\),z∈Ω\.\\displaystyle u\_\{\\textrm\{dec\}\}\(z\)\\approx f\(u\_\{\\textrm\{enc\}\}\(z\)\),\\qquad z\\in\\Omega\.\(7\) The master node then estimates the desired computation at each original input point by evaluating the decoder at the corresponding encoder design point: f^q​\(𝐱k\)\\displaystyle\\widehat\{f\}\_\{q\}\(\\mathbf\{x\}\_\{k\}\):=udec​\(αk\)≈\(a\)f⁡\(uenc​\(αk\)\)≈\(b\)f⁡\(𝐱k\),k∈\[K\],\\displaystyle:=u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)\\stackrel\{\{\\scriptstyle\\text\{\(a\)\}\}\}\{\{\\approx\}\}f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\\stackrel\{\{\\scriptstyle\\text\{\(b\)\}\}\}\{\{\\approx\}\}f\(\\mathbf\{x\}\_\{k\}\),\\qquad k\\in\[K\],\(8\)where approximation \(a\) follows from the decoder approximation in \([7](https://arxiv.org/html/2610.27440#S2.E7)\), while approximation \(b\) follows from the encoder approximation in \([4](https://arxiv.org/html/2610.27440#S2.E4)\) and the regularity offf\. These two sources of error will be formally quantified in the Section[IV\-A](https://arxiv.org/html/2610.27440#S4.SS1)\.

For a fixed computing functionff, prescribed encoder design points𝜶:=\{αk\}k=1K\\bm\{\\alpha\}:=\\\{\\alpha\_\{k\}\\\}\_\{k=1\}^\{K\}, prescribed decoder design points𝜷:=\{βj\}j=1N\\bm\{\\beta\}:=\\\{\\beta\_\{j\}\\\}\_\{j=1\}^\{N\}, and an adversarial budgetqq, we measure the performance of theGCCscheme by the worst\-case mean squared error \(MSE\)

ℒq​\(uenc,udec,f,𝜶,𝜷\):=supA∈𝒜q1K​∑k=1K∥f^q​\(𝐱k\)−f⁡\(𝐱k\)∥22\.\\displaystyle\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f,\\bm\{\\alpha\},\\bm\{\\beta\}\):=\\sup\_\{A\\in\\mathcal\{A\}\_\{q\}\}\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert\\widehat\{f\}\_\{q\}\(\\mathbf\{x\}\_\{k\}\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\}\.\(9\)Throughout our analysis,𝜶\\bm\{\\alpha\}, and𝜷\\bm\{\\beta\}are fixed\. We therefore suppress their dependence and write

ℒq​\(uenc,udec,f\):=ℒq​\(uenc,udec,f,𝜶,𝜷\)\.\\displaystyle\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f\):=\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f,\\bm\{\\alpha\},\\bm\{\\beta\}\)\.\(10\)The supremum in \([9](https://arxiv.org/html/2610.27440#S2.E9)\) is taken over all admissible attacks that corrupt the outputs of at mostqqworkers\. The encoder and decoder may be designed using the known budgetqq, but they do not know the realized set of adversarial workers or the corrupted values returned by them\.

Accordingly, the objective ofGCCin adversarial environment is to design the encoder and decoder functions to minimize the worst\-case error in \([9](https://arxiv.org/html/2610.27440#S2.E9)\):

infuenc,udecℒq​\(uenc,udec,f\)\.\\displaystyle\\inf\_\{\\begin\{subarray\}\{c\}u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\}\\end\{subarray\}\}\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f\)\.\(11\)
Encoder𝐱~j=uenc​\(βj\)\\widetilde\{\\mathbf\{x\}\}\_\{j\}=u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)Decoderf^​\(𝐱k\)=udec​\(αk\)\\widehat\{f\}\(\\mathbf\{x\}\_\{k\}\)=u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)Worker nodesf⁡\(⋅\)f\(\\cdot\)𝐱~1\\widetilde\{\\mathbf\{x\}\}\_\{1\}f⁡\(𝐱~1\)f\(\\widetilde\{\\mathbf\{x\}\}\_\{1\}\)f⁡\(⋅\)f\(\\cdot\)𝐱~2\\widetilde\{\\mathbf\{x\}\}\_\{2\}f⁡\(𝐱~2\)f\(\\widetilde\{\\mathbf\{x\}\}\_\{2\}\)f⁡\(⋅\)f\(\\cdot\)𝐱~n\\widetilde\{\\mathbf\{x\}\}\_\{n\}f⁡\(𝐱~n\)f\(\\widetilde\{\\mathbf\{x\}\}\_\{n\}\)f⁡\(⋅\)f\(\\cdot\)𝐱~N\\widetilde\{\\mathbf\{x\}\}\_\{N\}f⁡\(𝐱~N\)f\(\\widetilde\{\\mathbf\{x\}\}\_\{N\}\)![Refer to caption](https://arxiv.org/html/2610.27440v1/figs/adversary.png)𝐱~i\\widetilde\{\\mathbf\{x\}\}\_\{i\}𝐚i\\mathbf\{a\}\_\{i\}![Refer to caption](https://arxiv.org/html/2610.27440v1/figs/adversary.png)𝐱~j\\widetilde\{\\mathbf\{x\}\}\_\{j\}𝐚j\\mathbf\{a\}\_\{j\}𝐱1\\mathbf\{x\}\_\{1\}𝐱2\\mathbf\{x\}\_\{2\}𝐱K\\mathbf\{x\}\_\{K\}⋮\\vdotsf^​\(𝐱1\)\\widehat\{f\}\(\\mathbf\{x\}\_\{1\}\)f^​\(𝐱2\)\\widehat\{f\}\(\\mathbf\{x\}\_\{2\}\)f^​\(𝐱K\)\\widehat\{f\}\(\\mathbf\{x\}\_\{K\}\)⋮\\vdots

Fig\. 1:General coded computing in adversarial setting\. Honest workers returnf⁡\(𝐱~j\)f\(\\widetilde\{\\mathbf\{x\}\}\_\{j\}\), whereas adversarial workers may return arbitrary vectors𝐚j∈𝒴adv\\mathbf\{a\}\_\{j\}\\in\\mathcal\{Y\}\_\{\\mathrm\{adv\}\}\. The adversary icons and dashed red arrows illustrate corrupted outputs\. The adversarial workers’ identities are unknown to the master node\.

## IIIPreliminaries

In this section, we introduce the notation and mathematical foundations required for the design and analysis of the proposed encoder and decoder\.

LetΩ=\[a,b\]⊂ℝ\\Omega=\[a,b\]\\subset\\mathbb\{R\}denote a bounded closed interval\. For1≤p≤∞1\\leq p\\leq\\infty, letLp​\(Ω,ℝM\)\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)denote the space of measurable functions onΩ\\Omegawith finiteLpL^\{p\}\-norm\.

### III\-ASobolev Spaces

For an integerr≥1r\\geq 1and1≤p≤∞1\\leq p\\leq\\infty, the Sobolev space𝕎r,p​\(Ω,ℝM\)\\mathbb\{W\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)consists of functionsg∈Lp​\(Ω,ℝM\)g\\in\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)whose weak derivatives up to orderrr, denotedg\(i\)g^\{\(i\)\}fori=1,…,ri=1,\\ldots,r, also belong toLp​\(Ω,ℝM\)\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\. Settingg\(0\):=gg^\{\(0\)\}:=g, the norm on𝕎r,p​\(Ω,ℝM\)\\mathbb\{W\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)is defined by

∥g∥𝕎r,p​\(Ω,ℝM\):=\(∑i=0r∥g\(i\)∥Lp​\(Ω,ℝM\)p\)1/p,1≤p<∞,\\displaystyle\\lVert g\\rVert\_\{\\mathbb\{W\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}:=\\bigg\(\\sum\_\{i=0\}^\{r\}\\lVert g^\{\(i\)\}\\rVert\_\{\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}^\{p\}\\bigg\)^\{1/p\},\\qquad 1\\leq p<\\infty,\(12\)and forp=∞p=\\infty,

∥g∥𝕎r,∞​\(Ω,ℝM\):=max0≤i≤r⁡∥g\(i\)∥L∞​\(Ω,ℝM\)\.\\displaystyle\\lVert g\\rVert\_\{\\mathbb\{W\}^\{r,\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}:=\\max\_\{0\\leq i\\leq r\}\\lVert g^\{\(i\)\}\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}\.\(13\)Whenp=2p=2,𝕎r,2​\(Ω,ℝM\)\\mathbb\{W\}^\{r,2\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)forms a Reproducing Kernel Hilbert Space \(RKHS\)\[[14](https://arxiv.org/html/2610.27440#bib.bib14),[15](https://arxiv.org/html/2610.27440#bib.bib15)\], which we denote by

ℋr​\(Ω,ℝM\):=𝕎r,2​\(Ω,ℝM\)\.\\displaystyle\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\):=\\mathbb\{W\}^\{r,2\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\.\(14\)Throughout the paper,ℋr​\(Ω,ℝM\)\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)denote therr\-th order Sobolev space over intervalΩ\\Omega\.

#### III\-A1Equivalent Norms

A Sobolev space can be equipped with several equivalent norms\[[16](https://arxiv.org/html/2610.27440#bib.bib16),[14](https://arxiv.org/html/2610.27440#bib.bib14),[15](https://arxiv.org/html/2610.27440#bib.bib15)\]\. Two norms∥⋅∥𝕎1\\lVert\\cdot\\rVert\_\{\\mathbb\{W\}\_\{1\}\}and∥⋅∥𝕎2\\lVert\\cdot\\rVert\_\{\\mathbb\{W\}\_\{2\}\}on the same space are equivalent if there exist constantsθ1,θ2\>0\\theta\_\{1\},\\theta\_\{2\}\>0such thatθ1​∥g∥𝕎1≤∥g∥𝕎2≤θ2​∥g∥𝕎1\\theta\_\{1\}\\lVert g\\rVert\_\{\\mathbb\{W\}\_\{1\}\}\\leq\\lVert g\\rVert\_\{\\mathbb\{W\}\_\{2\}\}\\leq\\theta\_\{2\}\\lVert g\\rVert\_\{\\mathbb\{W\}\_\{1\}\}for every functionggin that space\.

The equivalent norm used in this paper is the anchored Sobolev norm introduced in\[[17](https://arxiv.org/html/2610.27440#bib.bib17)\]\. LetΩ=\[a,b\]\\Omega=\[a,b\]and1≤p<∞1\\leq p<\\infty\. For everyg∈𝕎r,p​\(Ω,ℝM\)g\\in\\mathbb\{W\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\), define

∥g∥𝕎~r,p​\(Ω,ℝM\):=\(∑i=0r−1∥g\(i\)​\(a\)∥2p\+∥g\(r\)∥Lp​\(Ω,ℝM\)p\)1/p\.\\displaystyle\\lVert g\\rVert\_\{\\widetilde\{\\mathbb\{W\}\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}:=\\left\(\\sum\_\{i=0\}^\{r\-1\}\\lVert g^\{\(i\)\}\(a\)\\rVert\_\{2\}^\{p\}\+\\lVert g^\{\(r\)\}\\rVert\_\{\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}^\{p\}\\right\)^\{1/p\}\.\(15\)This norm is equivalent to the standard Sobolev norm in \([12](https://arxiv.org/html/2610.27440#S3.E12)\)\[[18](https://arxiv.org/html/2610.27440#bib.bib18)\]\. In particular, sinceℋr​\(Ω,ℝM\)=𝕎r,2​\(Ω,ℝM\)\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)=\\mathbb\{W\}^\{r,2\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\), throughout the remainder of the paper we use the convention

∥g∥ℋr​\(Ω,ℝM\):=∥g∥𝕎~r,2​\(Ω,ℝM\),g∈ℋr​\(Ω,ℝM\)\.\\displaystyle\\lVert g\\rVert\_\{\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}:=\\lVert g\\rVert\_\{\\widetilde\{\\mathbb\{W\}\}^\{r,2\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\},\\qquad g\\in\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\.\(16\)

### III\-BRepresenter Theorem

The representer theorem states that the solution to a broad class of regularized empirical risk minimization problems over an RKHS admits a finite\-dimensional representation, expressed as a linear combination of kernel evaluations at the sampling points\.

###### Theorem 1\(Representer Theorem\[[19](https://arxiv.org/html/2610.27440#bib.bib19),[20](https://arxiv.org/html/2610.27440#bib.bib20)\]\)\.

Letℋ\\mathcal\{H\}be a vector\-valued RKHS of functions fromΩ\\Omegatoℝd\\mathbb\{R\}^\{d\}, associated with a matrix\-valued reproducing kernelΦ:Ω×Ω→ℝd×d\\Phi:\\Omega\\times\\Omega\\to\\mathbb\{R\}^\{d\\times d\}\. Let\{ti\}i=1n⊂Ω\\\{t\_\{i\}\\\}\_\{i=1\}^\{n\}\\subset\\Omega,λ\>0\\lambda\>0, and letΘ:\[0,∞\)→ℝ\\Theta:\[0,\\infty\)\\to\\mathbb\{R\}be a strictly increasing function\. Consider the regularized optimization problem

ming∈ℋ⁡Ψ⁡\(g⁡\(t1\),…,g⁡\(tn\)\)\+λ​Θ​\(∥g∥ℋ\),\\displaystyle\\min\_\{g\\in\\mathcal\{H\}\}\\;\\Psi\\big\(g\(t\_\{1\}\),\\ldots,g\(t\_\{n\}\)\\big\)\+\\lambda\\Theta\\big\(\\lVert g\\rVert\_\{\\mathcal\{H\}\}\\big\),\(18\)whereΨ:\(ℝd\)n→ℝ∪\{\+∞\}\\Psi:\(\\mathbb\{R\}^\{d\}\)^\{n\}\\to\\mathbb\{R\}\\cup\\\{\+\\infty\\\}is an arbitrary loss function\. If a minimizer of \([18](https://arxiv.org/html/2610.27440#S3.E18)\) exists, it admits the representation

g∗​\(t\)=∑i=1nΦ⁡\(t,ti\)​𝐰i,\\displaystyle g^\{\*\}\(t\)=\\sum\_\{i=1\}^\{n\}\\Phi\(t,t\_\{i\}\)\\mathbf\{w\}\_\{i\},\(19\)for some coefficient vectors𝐰1,…,𝐰n∈ℝd\\mathbf\{w\}\_\{1\},\\ldots,\\mathbf\{w\}\_\{n\}\\in\\mathbb\{R\}^\{d\}\.

Whenℋ=ℋr​\(Ω,ℝd\)\\mathcal\{H\}=\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), the reproducing kernel decouples asΦ⁡\(t,s\)=ϕ⁡\(t,s\)​Id\\Phi\(t,s\)=\\phi\(t,s\)I\_\{d\}, whereϕ:Ω×Ω→ℝ\\phi:\\Omega\\times\\Omega\\to\\mathbb\{R\}denotes the scalar reproducing kernel ofℋr​\(Ω,ℝ\)\\mathcal\{H\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}\\right\), andIdI\_\{d\}is thed×dd\\times didentity matrix\[[21](https://arxiv.org/html/2610.27440#bib.bib21),[22](https://arxiv.org/html/2610.27440#bib.bib22)\]\. Under this setting, the representation \([19](https://arxiv.org/html/2610.27440#S3.E19)\) simplifies to

g∗​\(t\)=∑i=1nϕ⁡\(t,ti\)​𝐰i\.\\displaystyle g^\{\*\}\(t\)=\\sum\_\{i=1\}^\{n\}\\phi\(t,t\_\{i\}\)\\mathbf\{w\}\_\{i\}\.\(20\)

### III\-CSecond\-Order Smoothing Splines

Leta<t1<⋯<tn<ba<t\_\{1\}<\\cdots<t\_\{n\}<b, withn≥2n\\geq 2, denote an ordered set of sampling points, and let𝐲:=\(𝐲1,…,𝐲n\)∈\(ℝd\)n\\mathbf\{y\}:=\(\\mathbf\{y\}\_\{1\},\\ldots,\\mathbf\{y\}\_\{n\}\)\\in\(\\mathbb\{R\}^\{d\}\)^\{n\}be the corresponding vector\-valued observations\. For a regularization parameterλ\>0\\lambda\>0\(known as*smoothing parameter*\), consider the penalized least\-squares problem over second\-order Sobolev space:

sn,λ​\[𝐲\]:=argming∈ℋ2​\(Ω,ℝd\)​\{1n​∑i=1n∥g⁡\(ti\)−𝐲i∥22\+λ​∥g′′∥L2​\(Ω,ℝd\)2\}\.\\displaystyle s\_\{n,\\lambda\}\[\\mathbf\{y\}\]:=\\underset\{g\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}\{\\operatorname\{argmin\}\}\\bigg\\\{\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\lVert g\(t\_\{i\}\)\-\\mathbf\{y\}\_\{i\}\\rVert\_\{2\}^\{2\}\+\\lambda\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\bigg\\\}\.\(21\)Because both the quadratic empirical loss and the derivative penalty decouple across theddcoordinates, the minimization in \([21](https://arxiv.org/html/2610.27440#S3.E21)\) decomposes intoddindependent scalar problems\. For each coordinate, the objective admits a unique minimizer given by a natural cubic smoothing spline \(also referred to as a second\-order smoothing spline\)\[[15](https://arxiv.org/html/2610.27440#bib.bib15)\]\. Consequently, \([21](https://arxiv.org/html/2610.27440#S3.E21)\) possesses a unique vector\-valued solutionsn,λ​\[g\]s\_\{n,\\lambda\}\[g\]\.

Moreover, the solution operator is linear in the observations and admits the equivalent kernel representation\[[23](https://arxiv.org/html/2610.27440#bib.bib23)\]

sn,λ​\[𝐲\]​\(t\)=1n​∑i=1nGn,λ​\(t,ti\)​𝐲i\.\\displaystyle s\_\{n,\\lambda\}\[\\mathbf\{y\}\]\(t\)=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}G\_\{n,\\lambda\}\(t,t\_\{i\}\)\\mathbf\{y\}\_\{i\}\.\(22\)The equivalent kernel \(or weight function\)Gn,λG\_\{n,\\lambda\}depends solely on the design points\{ti\}i=1n\\\{t\_\{i\}\\\}\_\{i=1\}^\{n\}and the regularization parameterλ\\lambda, and is identical across all coordinates\.

When the observations are noiseless samples from a functionh:\[a,b\]→ℝdh:\[a,b\]\\to\\mathbb\{R\}^\{d\}, let𝐡:=\[h⁡\(t1\),…,h⁡\(tn\)\]\\mathbf\{h\}:=\[h\(t\_\{1\}\),\\ldots,h\(t\_\{n\}\)\]denote the corresponding observation vector\. We then use the shorthandsn,λ​\[h\]:=sn,λ​\[𝐡\]\.s\_\{n,\\lambda\}\[h\]:=s\_\{n,\\lambda\}\[\\mathbf\{h\}\]\.

## IVGCCin Adversarial Environments

In this section, we formulate theGCCframework under an adversarial setting in which, for a givenq∈ℕq\\in\\mathbb\{N\}, at mostqqoutputs returned by worker nodes are arbitrarily corrupted\. We begin by characterizing the spacing of the design points, which plays a central role in our theoretical guarantees\.

###### Definition 1\(Design Spacing Parameters\)\.

Let𝒯:=\{t1,…,tn\}⊂Ω\\mathcal\{T\}:=\\\{t\_\{1\},\\ldots,t\_\{n\}\\\}\\subset\\Omegabe an ordered set of design points satisfyinga≤t1<⋯<tn≤ba\\leq t\_\{1\}<\\cdots<t\_\{n\}\\leq b\. The maximum and minimum consecutive spacing of𝒯\\mathcal\{T\}are defined, respectively, by

δ¯𝒯\\displaystyle\\bar\{\\delta\}\_\{\\mathcal\{T\}\}:=max1≤j<n⁡\(tj\+1−tj\),\\displaystyle:=\\max\_\{1\\leq j<n\}\(t\_\{j\+1\}\-t\_\{j\}\),\(23\)¯​δ𝒯\\displaystyle\\mathchoice\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.90779pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.61668pt\{\}\}\}$\\hss\}\}\}\\delta\_\{\\mathcal\{T\}\}:=min1≤j<n⁡\(tj\+1−tj\)\.\\displaystyle:=\\min\_\{1\\leq j<n\}\(t\_\{j\+1\}\-t\_\{j\}\)\.\(24\)

For the remainder of this paper, we assume that both the encoder and decoder belong to the second\-order vector\-valued Sobolev space; that is,

uenc∈ℋ2​\(Ω,ℝd\),udec∈ℋ2​\(Ω,ℝm\)\.\\displaystyle u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),\\qquad u\_\{\\textrm\{dec\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\.\(25\)This regularity assumption imposes only mild smoothness constraints while providing the analytical tractability needed to leverage reproducing kernel techniques\. In addition, without loss of generality, we setΩ=\[0,1\]\\Omega=\[0,1\]\.111Any non\-empty bounded interval\[a,b\]\[a,b\]can be mapped to\[0,1\]\[0,1\]via an affine transformation, which affects our theoretical constants only up to scale factors\.

Next, we define the class of computing functions considered throughout this paper\.

###### Definition 2\(Admissible Computing Function Class\)\.

We defineℱsmd→m\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\}as the class of twice continuously differentiable functionsf=\[f1,…,fm\]⊤:ℝd→ℝmf=\[f\_\{1\},\\ldots,f\_\{m\}\]^\{\\top\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{m\}with bounded outputs, gradients, and Hessians:

ℱsmd→m:=\{f=\[f1,…,fm\]⊤∈C2\(ℝd;ℝm\):\\displaystyle\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\}:=\\biggl\\\{f=\[f\_\{1\},\\ldots,f\_\{m\}\]^\{\\top\}\\in C^\{2\}\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}^\{m\}\):sup𝐱∈ℝd∥f⁡\(𝐱\)∥∞<∞,\\displaystyle\\ \\sup\_\{\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\}\}\\lVert f\(\\mathbf\{x\}\)\\rVert\_\{\\infty\}<\\infty,sup𝐱∈ℝd∥∇fj​\(𝐱\)∥2<∞,j∈\[m\],\\displaystyle\\ \\sup\_\{\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\}\}\\lVert\\nabla f\_\{j\}\(\\mathbf\{x\}\)\\rVert\_\{2\}<\\infty,\\quad j\\in\[m\],sup𝐱∈ℝd∥D2fj\(𝐱\)∥op<∞,j∈\[m\]\}\.\\displaystyle\\ \\sup\_\{\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\}\}\\lVert D^\{2\}f\_\{j\}\(\\mathbf\{x\}\)\\rVert\_\{\\mathrm\{op\}\}<\\infty,\\quad j\\in\[m\]\\biggr\\\}\.\(26\)

The function classℱsmd→m\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\}is highly general and encompasses a wide variety of non\-linear functions\. For a given computing functionf∈ℱsmd→mf\\in\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\}, our objective is to design the encoder and decoder functions to solve the robust risk minimization problem

Jadv:=infuenc∈ℋ2​\(Ω,ℝd\),udec∈ℋ2​\(Ω,ℝm\)ℒq​\(uenc,udec,f\)\.\\displaystyle J\_\{\\textrm\{adv\}\}:=\\inf\_\{\\begin\{subarray\}\{c\}u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),\\\\ u\_\{\\textrm\{dec\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\\end\{subarray\}\}\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f\)\.\(27\)
The remainder of this section is organized as follows\. In Section[IV\-A](https://arxiv.org/html/2610.27440#S4.SS1), we present the encoder and decoder constructions and establish both non\-asymptotic and asymptotic upper bounds onJadvJ\_\{\\textrm\{adv\}\}\. In Section[IV\-B](https://arxiv.org/html/2610.27440#S4.SS2), we analyze the computational complexity of the proposed scheme and compare it against existing baselines\. Finally, in Section[IV\-C](https://arxiv.org/html/2610.27440#S4.SS3), we establish a complementary lower bound on the worst\-case adversarial error of the proposed encoding and decoding design\.

### IV\-ACode Design

In this subsection, we develop the encoder and decoder constructions for the adversarial optimization problem in \([27](https://arxiv.org/html/2610.27440#S4.E27)\)\. To state the proposed design and the performance guarantees, we impose the following assumptions throughout this section:

1. \(A1\)*Bounded computation outputs:*The computing functionf:ℝd→ℝmf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{m\}satisfies sup𝐱∈ℝd∥f⁡\(𝐱\)∥∞≤Tf\\displaystyle\\sup\_\{\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\}\}\\lVert f\(\\mathbf\{x\}\)\\rVert\_\{\\infty\}\\leq T\_\{f\}\(28\)for some constantTf\>0T\_\{f\}\>0\.
2. \(A2\)*Regularity of the computation function:*Letf=\[f1,…,fm\]T\.f=\[f\_\{1\},\\ldots,f\_\{m\}\]^\{T\}\.For every𝐱∈ℝd\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\}andj∈\[m\]j\\in\[m\], assume that ∥∇fj​\(𝐱\)∥2\\displaystyle\\lVert\\nabla f\_\{j\}\(\\mathbf\{x\}\)\\rVert\_\{2\}≤Gj,\\displaystyle\\leq G\_\{j\},∥D2​fj​\(𝐱\)∥op\\displaystyle\\lVert D^\{2\}f\_\{j\}\(\\mathbf\{x\}\)\\rVert\_\{\\mathrm\{op\}\}≤Hj\.\\displaystyle\\leq H\_\{j\}\.\(29\)We define the aggregate gradient and Hessian bounds as Gf\\displaystyle G\_\{f\}:=\(∑j=1mGj2\)1/2,\\displaystyle:=\\left\(\\sum\_\{j=1\}^\{m\}G\_\{j\}^\{2\}\\right\)^\{1/2\},Hf\\displaystyle H\_\{f\}:=\(∑j=1mHj2\)1/2,\\displaystyle:=\\left\(\\sum\_\{j=1\}^\{m\}H\_\{j\}^\{2\}\\right\)^\{1/2\},\(30\)and we assume thatGf,Hf\>0G\_\{f\},H\_\{f\}\>0\.
3. \(A3\)*Bounded adversarial outputs:*Every corrupted output belongs to\[−Ta,Ta\]m\[\-T\_\{a\},T\_\{a\}\]^\{m\}for someTa\>0T\_\{a\}\>0\. We define T:=max⁡\{Ta,Tf\}\.\\displaystyle T:=\\max\\\{T\_\{a\},T\_\{f\}\\\}\.\(31\)

Directly optimizing \([27](https://arxiv.org/html/2610.27440#S4.E27)\) is difficult because the encoder and decoder belong to infinite\-dimensional function spaces and up toqqreturned worker outputs may be arbitrarily corrupted\. A central step in our approach is to construct an upper bound onℒq​\(uenc,udec,f\)\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f\)that decomposes the loss into three interconnected components\.

The first component captures the encoder approximation error, which measures the discrepancy caused by approximating the original input data at the encoder design points, corresponding to \([4](https://arxiv.org/html/2610.27440#S2.E4)\)\. The remaining two components capture the decoder error, corresponding to \([7](https://arxiv.org/html/2610.27440#S2.E7)\), by separating the effect of adversarial corruptions from the clean generalization error of the decoder\.

More formally, for any admissible attackA∈𝒜qA\\in\\mathcal\{A\}\_\{q\}, we have

1K​∑k=1K∥f^​\(𝐱k\)−f⁡\(𝐱k\)∥22\\displaystyle\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert\\widehat\{f\}\(\\mathbf\{x\}\_\{k\}\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\}=1K​∑k=1K∥udec​\(αk\)−f⁡\(uenc​\(αk\)\)\+f⁡\(uenc​\(αk\)\)−f⁡\(𝐱k\)∥22\\displaystyle=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)\-f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\+f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\}≤2K​∑k=1K∥f⁡\(uenc​\(αk\)\)−f⁡\(𝐱k\)∥22\+2K​∑k=1K∥udec​\(αk\)−f⁡\(uenc​\(αk\)\)∥22,\\displaystyle\\leq\\frac\{2\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\}\+\\frac\{2\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)\-f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\\rVert\_\{2\}^\{2\},\(32\)where the inequality follows from∥𝐮\+𝐯∥22≤2​∥𝐮∥22\+2​∥𝐯∥22\.\\lVert\\mathbf\{u\}\+\\mathbf\{v\}\\rVert\_\{2\}^\{2\}\\leq 2\\lVert\\mathbf\{u\}\\rVert\_\{2\}^\{2\}\+2\\lVert\\mathbf\{v\}\\rVert\_\{2\}^\{2\}\.Taking the supremum over all admissible attacks yields

ℒq​\(uenc,udec,f\)≤ℒenc\+ℒdec,\\displaystyle\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f\)\\leq\\mathcal\{L\}\_\{\\mathrm\{enc\}\}\+\\mathcal\{L\}\_\{\\mathrm\{dec\}\},\(33\)where

ℒenc\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{enc\}\}:=2K​∑k=1K∥f⁡\(uenc​\(αk\)\)−f⁡\(𝐱k\)∥22,\\displaystyle:=\\frac\{2\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\},\(34\)ℒdec\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{dec\}\}:=supA∈𝒜q2K​∑k=1K∥udec​\(αk\)−f⁡\(uenc​\(αk\)\)∥22\.\\displaystyle:=\\sup\_\{A\\in\\mathcal\{A\}\_\{q\}\}\\frac\{2\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)\-f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\\rVert\_\{2\}^\{2\}\.\(35\)The encoder termℒenc\\mathcal\{L\}\_\{\\mathrm\{enc\}\}is independent of the attack\. In contrast, the value ofℒdec\\mathcal\{L\}\_\{\\mathrm\{dec\}\}is affected by the attack through the returned worker outputs used to constructudecu\_\{\\textrm\{dec\}\}\.

To systematically manage the worst\-case decoding error, we further decompose the decoder error\. Letudec∘u\_\{\\textrm\{dec\}\}^\{\\circ\}represent the*clean decoder*function that would be constructed if all worker nodes were honest, i\.e\., if the master node received𝐲¯jA=f⁡\(uenc​\(βj\)\)\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}=f\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\)for allj∈\[N\]j\\in\[N\]\. Thus,udec∘u\_\{\\textrm\{dec\}\}^\{\\circ\}is a counterfactual function introduced only for the analysis\. Adding and subtractingudec∘​\(αk\)u\_\{\\textrm\{dec\}\}^\{\\circ\}\(\\alpha\_\{k\}\)in \([35](https://arxiv.org/html/2610.27440#S4.E35)\) gives

ℒdec\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{dec\}\}=supA∈𝒜q2K​∑k=1K∥udec​\(αk\)−udec∘​\(αk\)\+udec∘​\(αk\)−f⁡\(uenc​\(αk\)\)∥22\\displaystyle=\\sup\_\{A\\in\\mathcal\{A\}\_\{q\}\}\\frac\{2\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)\-u\_\{\\textrm\{dec\}\}^\{\\circ\}\(\\alpha\_\{k\}\)\+u\_\{\\textrm\{dec\}\}^\{\\circ\}\(\\alpha\_\{k\}\)\-f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\\rVert\_\{2\}^\{2\}≤2​supA∈𝒜qsupz∈Ω‖udec​\(z\)−udec∘​\(z\)\+udec∘​\(z\)−f⁡\(uenc​\(z\)\)‖22\\displaystyle\\leq 2\\sup\_\{A\\in\\mathcal\{A\}\_\{q\}\}\\sup\_\{z\\in\\Omega\}\\left\\\|u\_\{\\textrm\{dec\}\}\(z\)\-u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)\+u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)\-f\(u\_\{\\textrm\{enc\}\}\(z\)\)\\right\\\|\_\{2\}^\{2\}≤4​supA∈𝒜qsupz∈Ω‖udec​\(z\)−udec∘​\(z\)‖22⏟ℒdecadv\+4​supz∈Ω‖udec∘​\(z\)−f⁡\(uenc​\(z\)\)‖22⏟ℒdecgen\.\\displaystyle\\leq\\underbrace\{4\\sup\_\{A\\in\\mathcal\{A\}\_\{q\}\}\\sup\_\{z\\in\\Omega\}\\left\\\|u\_\{\\textrm\{dec\}\}\(z\)\-u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)\\right\\\|\_\{2\}^\{2\}\}\_\{\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{adv\}\}\}\+\\underbrace\{4\\sup\_\{z\\in\\Omega\}\\left\\\|u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)\-f\(u\_\{\\textrm\{enc\}\}\(z\)\)\\right\\\|\_\{2\}^\{2\}\}\_\{\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}\}\.\(36\)Consequently,

ℒq​\(uenc,udec,f\)≤ℒenc\+ℒdecgen\+ℒdecadv\.\\displaystyle\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\};f\)\\leq\\mathcal\{L\}\_\{\\mathrm\{enc\}\}\+\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}\+\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{adv\}\}\.\(37\)
We address this coupling through a nested construction\. For a fixed encoder, we first design the decoder and derive bounds onℒdecadv\+ℒdecgen\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{adv\}\}\+\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}\. Substituting these bounds into \([37](https://arxiv.org/html/2610.27440#S4.E37)\) yields an encoder\-only objective, which is then minimized in an outer optimization step\.

#### IV\-A1Decoder Design

As part of the code design, we select the decoder design points to be equidistant overΩ=\[0,1\]\\Omega=\[0,1\]:

βj:=j−1N−1,j∈\[N\]\.\\displaystyle\\beta\_\{j\}:=\\frac\{j\-1\}\{N\-1\},\\qquad j\\in\[N\]\.\(38\)This choice will help us in our subsequent theoretical results\.

Fix an encoderuenc∈ℋ2​\(Ω,ℝd\)u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\. The master node receives the outputs\{𝐲¯jA\}j=1N\\\{\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}\\\}\_\{j=1\}^\{N\}defined in \([6](https://arxiv.org/html/2610.27440#S2.E6)\) without knowing which, if any, of the returned outputs are corrupted\. We construct the decoder using a regularized smoothing\-spline objective\. Specifically, the decoder is defined as:

udec∗:=arg​minu∈ℋ2​\(Ω,ℝm\)⁡\{1N​∑j=1N∥u⁡\(βj\)−𝐲¯jA∥22\+λd​∥u′′∥L2​\(Ω,ℝm\)2\},\\displaystyle u\_\{\\textrm\{dec\}\}^\{\*\}:=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\left\\\{\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}\\lVert u\(\\beta\_\{j\}\)\-\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{d\}\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\\right\\\},\(39\)whereλd\>0\\lambda\_\{\\textrm\{d\}\}\>0is the decoder smoothing parameter\. The smoothing parameterλd\\lambda\_\{\\textrm\{d\}\}may depend on the total number of workersNN, the adversarial budgetqq, and the decoder evaluation points\{βj\}j=1N\\\{\\beta\_\{j\}\\\}\_\{j=1\}^\{N\}, and is selected prior to observing the worker transmissions\.

The first term in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) measures the empirical squared error over all returned worker outputs, while the second term penalizes the curvature of the decoder\. The smoothing parameterλd\\lambda\_\{\\textrm\{d\}\}balances these two objectives: larger values ofλd\\lambda\_\{\\textrm\{d\}\}yield a smoother decoder with lower curvature, whereas smaller values prioritize fitting the empirical data, pushing the decoder toward interpolating the received outputs\{𝐲¯jA\}j=1N\\\{\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}\\\}\_\{j=1\}^\{N\}\. By penalizing rapid variations, this roughness penalty reduces sensitivity to localized anomalies and limits the impact of adversarial corruptions\.

Based on decoder function in \([39](https://arxiv.org/html/2610.27440#S4.E39)\), its corresponding*clean decoder*has the following form:

udec∘:=arg​minu∈ℋ2​\(Ω,ℝm\)⁡\{1N​∑j=1N∥u⁡\(βj\)−f⁡\(uenc​\(βj\)\)∥22\+λd​∥u′′∥L2​\(Ω,ℝm\)2\}\.\\displaystyle u\_\{\\textrm\{dec\}\}^\{\\circ\}:=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\left\\\{\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}\\lVert u\(\\beta\_\{j\}\)\-f\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\)\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{d\}\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\\right\\\}\.\(40\)The clean decoder which is purely an analytical tool used in our analysis, corresponds to the exact same optimization, assuming that all worker nodes are honest and compute and transmit uncorrupted evaluations\.

As established in Section[III\-C](https://arxiv.org/html/2610.27440#S3.SS3), the separability of the squared Euclidean norm across coordinate dimensions decouples \([39](https://arxiv.org/html/2610.27440#S4.E39)\) and \([40](https://arxiv.org/html/2610.27440#S4.E40)\) intommindependent scalar smoothing\-spline variational problems\. Consequently, bothudecu\_\{\\textrm\{dec\}\}andudec∘u\_\{\\textrm\{dec\}\}^\{\\circ\}are unique vector\-valued natural cubic smoothing splines\. Specifically, the decoder admits the representer decomposition

udec∗​\(t\)=𝐝0\+𝐝1​t\+∑j=1N𝐜j​ϕ0​\(t,βj\),\\displaystyle u\_\{\\textrm\{dec\}\}^\{\*\}\(t\)=\\mathbf\{d\}\_\{0\}\+\\mathbf\{d\}\_\{1\}t\+\\sum\_\{j=1\}^\{N\}\\mathbf\{c\}\_\{j\}\\phi\_\{0\}\(t,\\beta\_\{j\}\),\(41\)where𝐝0,𝐝1,𝐜1,…,𝐜N∈ℝm\\mathbf\{d\}\_\{0\},\\mathbf\{d\}\_\{1\},\\mathbf\{c\}\_\{1\},\\ldots,\\mathbf\{c\}\_\{N\}\\in\\mathbb\{R\}^\{m\}, andϕ0\\phi\_\{0\}denotes the reproducing kernel associated with the spaceℋ02​\(Ω,ℝ\)\\mathcal\{H\}\_\{0\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\), which is the second\-order Sobolev space with compact support \(see Definition[6](https://arxiv.org/html/2610.27440#Thmdefinition6)and Remark[7](https://arxiv.org/html/2610.27440#Thmremark7)in Appendix[A](https://arxiv.org/html/2610.27440#A1)for its explicit formulation\)\. The coefficients are obtained by solving the standard linear system for vector\-valued cubic smoothing splines\[[23](https://arxiv.org/html/2610.27440#bib.bib23),[15](https://arxiv.org/html/2610.27440#bib.bib15)\]\.

###### Theorem 2\(Decoder error bound\)\.

Consider theGCCscheme withNNworker nodes and the equidistant decoder design points in \([38](https://arxiv.org/html/2610.27440#S4.E38)\)\. Suppose that Assumptions[A1](https://arxiv.org/html/2610.27440#S4.I1.i1)through[A3](https://arxiv.org/html/2610.27440#S4.I1.i3)hold\. Then there exist constantsC1,C2,c0\>0C\_\{1\},C\_\{2\},c\_\{0\}\>0, independent ofNN,qq,λd\\lambda\_\{\\textrm\{d\}\}, anduencu\_\{\\textrm\{enc\}\}, such that, whenever0<λd≤1​and​N​λd1/4≥c0,0<\\lambda\_\{\\textrm\{d\}\}\\leq 1\\text\{ and \}N\\lambda\_\{\\textrm\{d\}\}^\{1/4\}\\geq c\_\{0\},we have

infudec∈ℋ2​\(Ω,ℝm\)ℒdecgen\+ℒdecadv\\displaystyle\\inf\_\{u\_\{\\textrm\{dec\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}\+\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{adv\}\}≤C1max\{Hf2,Gf2\}\(λd3/4\+N−3\)ψ\(∥uenc∥ℋ2​\(Ω,ℝd\)2\)\+C2mT2q2N2λd−1/2,\\displaystyle\\leq C\_\{1\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\)\\psi\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\)\+C\_\{2\}mT^\{2\}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\},\(42\)whereψ⁡\(t\):=t\+4​t2\.\\psi\(t\):=t\+4t^\{2\}\.

###### Corollary 1\(Optimal scaling of the decoder smoothing parameter\)\.

Assume that the conditions of Theorem[2](https://arxiv.org/html/2610.27440#Thmtheorem2)hold\. Suppose thatq≥1q\\geq 1\. Then, for all sufficiently largeNN, the value ofλd\\lambda\_\{\\textrm\{d\}\}that minimizes the decoder\-error upper bound in \([42](https://arxiv.org/html/2610.27440#S4.E42)\) satisfies

λd⋆≍\(qN\)8/5,\\displaystyle\\lambda\_\{\\textrm\{d\}\}^\{\\star\}\\asymp\\left\(\\frac\{q\}\{N\}\\right\)^\{8/5\},\(43\)where the implicit constants may depend on the fixed encoder and the constant parameters of the scheme, but are independent ofNNandqq\.

#### IV\-A2Encoder Design

We next substitute the decoder bound into \([37](https://arxiv.org/html/2610.27440#S4.E37)\) to obtain an encoder\-only objective\. Assumption[A2](https://arxiv.org/html/2610.27440#S4.I1.i2)implies thatffisGfG\_\{f\}\-Lipschitz:

∥f⁡\(𝐱\)−f⁡\(𝐱~\)∥2≤Gf​∥𝐱−𝐱~∥2\.\\displaystyle\\lVert f\(\\mathbf\{x\}\)\-f\(\\widetilde\{\\mathbf\{x\}\}\)\\rVert\_\{2\}\\leq G\_\{f\}\\lVert\\mathbf\{x\}\-\\widetilde\{\\mathbf\{x\}\}\\rVert\_\{2\}\.\(44\)Therefore,

ℒenc\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{enc\}\}=2K​∑k=1K∥f⁡\(uenc​\(αk\)\)−f⁡\(𝐱k\)∥22\\displaystyle=\\frac\{2\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert f\(u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\}≤2​Gf2K​∑k=1K∥uenc​\(αk\)−𝐱k∥22\.\\displaystyle\\leq\\frac\{2G\_\{f\}^\{2\}\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\.\(45\)Combining \([37](https://arxiv.org/html/2610.27440#S4.E37)\), \([45](https://arxiv.org/html/2610.27440#S4.E45)\), and Theorem[2](https://arxiv.org/html/2610.27440#Thmtheorem2)gives

ℒq​\(uenc,udec∗,f\)\\displaystyle\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\}^\{\*\};f\)≤2​Gf2K​∑k=1K∥uenc​\(αk\)−𝐱k∥22\\displaystyle\\leq\\frac\{2G\_\{f\}^\{2\}\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{enc\}\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+C1​max⁡\{Hf2,Gf2\}​\(λd3/4\+N−3\)​ψ​\(∥uenc∥ℋ2​\(Ω,ℝd\)2\)\\displaystyle\\quad\+C\_\{1\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\left\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\\right\)\\psi\\left\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\+C2mT2q2N2λd−1/2\.\\displaystyle\\quad\+C\_\{2\}mT^\{2\}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\}\.\(46\)
The last term in \([46](https://arxiv.org/html/2610.27440#S4.E46)\) is independent of the encoder\. AssumingGf\>0G\_\{f\}\>0, define

Γq,N\\displaystyle\\Gamma\_\{q,N\}:=C12​Gf2​max⁡\{Hf2,Gf2\}​\(λd3/4\+N−3\)\.\\displaystyle:=\\frac\{C\_\{1\}\}\{2G\_\{f\}^\{2\}\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\left\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\\right\)\.\(47\)The dependence ofΓq,N\\Gamma\_\{q,N\}on the adversarial budgetqqis through the choice of the decoder smoothing parameterλd\\lambda\_\{\\textrm\{d\}\}\.

Ignoring the additive term that is independent of the encoder, we choose the encoder as

uenc∗:=arg​minu∈ℋ2​\(Ω,ℝd\)⁡\{1K​∑k=1K∥u⁡\(αk\)−𝐱k∥22\+Γq,N​ψ​\(∥u∥ℋ2​\(Ω,ℝd\)2\)\}\.\\displaystyle u\_\{\\textrm\{enc\}\}^\{\*\}:=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}\\left\\\{\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\Gamma\_\{q,N\}\\psi\\left\(\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\\right\\\}\.\(48\)
Sinceψ\\psiis strictly increasing, the representer theorem stated in Theorem[1](https://arxiv.org/html/2610.27440#Thmtheorem1)implies that every minimizer of \([48](https://arxiv.org/html/2610.27440#S4.E48)\) admits the finite\-dimensional representation

uenc∗​\(t\)=∑k=1Kϕ⁡\(t,αk\)​𝐳k,\\displaystyle u\_\{\\textrm\{enc\}\}^\{\*\}\(t\)=\\sum\_\{k=1\}^\{K\}\\phi\(t,\\alpha\_\{k\}\)\\mathbf\{z\}\_\{k\},\(49\)for some coefficient vectors𝐳1,…,𝐳K∈ℝd\\mathbf\{z\}\_\{1\},\\ldots,\\mathbf\{z\}\_\{K\}\\in\\mathbb\{R\}^\{d\}, whereϕ\\phiis the scalar reproducing kernel ofℋ2​\(Ω,ℝ\)\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)given byϕ⁡\(t,s\)=1\+t​s\+ϕ0​\(t,s\),\\phi\(t,s\)=1\+ts\+\\phi\_\{0\}\(t,s\),withϕ0\\phi\_\{0\}defined in Remark[7](https://arxiv.org/html/2610.27440#Thmremark7)\. Thus, the infinite\-dimensional encoder design reduces to a finite\-dimensional optimization over the coefficient vectors\{𝐳k\}k=1K\\\{\\mathbf\{z\}\_\{k\}\\\}\_\{k=1\}^\{K\}\.

The encoder objective in \([48](https://arxiv.org/html/2610.27440#S4.E48)\) contains the nonlinear regularization termψ⁡\(∥u∥ℋ2​\(Ω,ℝd\)2\)\\psi\\bigl\(\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\bigr\), whereψ⁡\(t\)=t\+4​t2\\psi\(t\)=t\+4t^\{2\}\. Although \([49](https://arxiv.org/html/2610.27440#S4.E49)\) reduces the problem to a finite\-dimensional optimization, directly computing its coefficient vectors can still be computationally demanding\. As in the straggler setting studied in\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\], the encoder can instead be constructed through a standard cubic smoothing\-spline problem\. The following theorem describes this reduction for the adversarial setting\.

For convenience, define the encoder\-independent adversarial term

Δq,N:=C2mT2q2N2λd−1/2\.\\displaystyle\\Delta\_\{q,N\}:=C\_\{2\}mT^\{2\}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\}\.\(50\)
###### Theorem 3\(Efficient encoder construction\)\.

Suppose that Assumptions[A1](https://arxiv.org/html/2610.27440#S4.I1.i1)through[A3](https://arxiv.org/html/2610.27440#S4.I1.i3)and the conditions of Theorem[2](https://arxiv.org/html/2610.27440#Thmtheorem2)hold\. There exist constantsm1,m2\>0m\_\{1\},m\_\{2\}\>0andRe=Re​\(\{αk\}k=1K,\{𝐱k\}k=1K\)\>0R\_\{\\mathrm\{e\}\}=R\_\{\\mathrm\{e\}\}\\left\(\\\{\\alpha\_\{k\}\\\}\_\{k=1\}^\{K\},\\\{\\mathbf\{x\}\_\{k\}\\\}\_\{k=1\}^\{K\}\\right\)\>0such that

Jadv\\displaystyle J\_\{\\textrm\{adv\}\}≤Δq,N\+2​Gf2​\[1K​∑k=1K∥uenc†​\(αk\)−𝐱k∥22\+λe​\(Re\+∥\(uenc†\)′′∥L2​\(Ω,ℝd\)2\)\],\\displaystyle\\leq\\Delta\_\{q,N\}\+2G\_\{f\}^\{2\}\\bigg\[\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\left\(R\_\{\\mathrm\{e\}\}\+\\lVert\(u\_\{\\textrm\{enc\}\}^\{\\dagger\}\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\\bigg\],\(51\)where the encoder smoothing parameter is

λe:=Γq,N​\(m1\+m2​Re\),\\displaystyle\\lambda\_\{\\textrm\{e\}\}:=\\Gamma\_\{q,N\}\\left\(m\_\{1\}\+m\_\{2\}R\_\{\\mathrm\{e\}\}\\right\),\(52\)anduenc†u\_\{\\textrm\{enc\}\}^\{\\dagger\}is a minimizer of

uenc†:=arg​minu∈ℋ2​\(Ω,ℝd\)⁡\{1K​∑k=1K∥u⁡\(αk\)−𝐱k∥22\+λe​∥u′′∥L2​\(Ω,ℝd\)2\}\.\\displaystyle u\_\{\\textrm\{enc\}\}^\{\\dagger\}:=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}\\left\\\{\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\\\}\.\(53\)

The proof follows the same argument used for the encoder construction in the straggler setting\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\]\. The only changes are the regularization coefficientΓq,N\\Gamma\_\{q,N\}and the additive termΔq,N\\Delta\_\{q,N\}, which is independent of the encoder and therefore does not affect the encoder optimization\.

The squared Euclidean loss and the second\-derivative penalty in \([53](https://arxiv.org/html/2610.27440#S4.E53)\) decouple across theddinput coordinates\. Consequently,uenc†u\_\{\\textrm\{enc\}\}^\{\\dagger\}is a vector\-valued natural cubic smoothing spline and admits the representation

uenc†​\(t\)=𝐚0\+𝐚1​t\+∑k=1K𝐛k​ϕ0​\(t,αk\),\\displaystyle u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(t\)=\\mathbf\{a\}\_\{0\}\+\\mathbf\{a\}\_\{1\}t\+\\sum\_\{k=1\}^\{K\}\\mathbf\{b\}\_\{k\}\\phi\_\{0\}\(t,\\alpha\_\{k\}\),\(54\)where𝐚0,𝐚1,𝐛1,…,𝐛K∈ℝd\\mathbf\{a\}\_\{0\},\\mathbf\{a\}\_\{1\},\\mathbf\{b\}\_\{1\},\\ldots,\\mathbf\{b\}\_\{K\}\\in\\mathbb\{R\}^\{d\}, andϕ0​\(⋅\)\\phi\_\{0\}\(\\cdot\)defined in Remark[7](https://arxiv.org/html/2610.27440#Thmremark7)\. Therefore, the encoder coefficients can be computed efficiently by solving the standard linear system for vector\-valued cubic smoothing splines with smoothing parameterλe\\lambda\_\{\\textrm\{e\}\}\.

Algorithm 1General Coded Computing in the Adversarial Setting1:Input batch

\{𝐱k\}k=1K\\\{\\mathbf\{x\}\_\{k\}\\\}\_\{k=1\}^\{K\}, target function

ff, number of workers

NN, adversarial budget

qq, encoder design points

\{αk\}k=1K\\\{\\alpha\_\{k\}\\\}\_\{k=1\}^\{K\}, and smoothing parameters

λe\\lambda\_\{\\textrm\{e\}\}and

λd\\lambda\_\{\\textrm\{d\}\}\.

2:Estimates

\{f^​\(𝐱k\)\}k=1K\\\{\\widehat\{f\}\(\\mathbf\{x\}\_\{k\}\)\\\}\_\{k=1\}^\{K\}\.

3:The master node selects the equidistant decoder design points

βj=j−1N−1​for​j∈\[N\]\.\\beta\_\{j\}=\\frac\{j\-1\}\{N\-1\}\\text\{ for \}j\\in\[N\]\.
4:The master node constructs the encoder

uenc†u\_\{\\textrm\{enc\}\}^\{\\dagger\}by solving

uenc†=arg​minu∈ℋ2​\(Ω,ℝd\)⁡\{1K​∑k=1K∥u⁡\(αk\)−𝐱k∥22\+λe​∥u′′∥L2​\(Ω,ℝd\)2\},u\_\{\\textrm\{enc\}\}^\{\\dagger\}=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}\\left\\\{\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\\\},which admits the vector\-valued cubic smoothing\-spline representation in \([54](https://arxiv.org/html/2610.27440#S4.E54)\)\.

5:The master node computes the coded inputs

𝐱~j=uenc†​\(βj\)​for​j∈\[N\],\\widetilde\{\\mathbf\{x\}\}\_\{j\}=u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(\\beta\_\{j\}\)\\text\{ for \}j\\in\[N\],and sends

𝐱~j\\widetilde\{\\mathbf\{x\}\}\_\{j\}to worker

jj\.

6:Each honest worker

jjcomputes and returns

f⁡\(𝐱~j\)=f⁡\(uenc†​\(βj\)\)\.f\(\\widetilde\{\\mathbf\{x\}\}\_\{j\}\)=f\(u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(\\beta\_\{j\}\)\)\.An adversarial worker may instead return an arbitrary vector

𝐚j∈\[−Ta,Ta\]m\\mathbf\{a\}\_\{j\}\\in\[\-T\_\{a\},T\_\{a\}\]^\{m\}\.

7:The master node receives

\{𝐲¯jA\}j=1N\\\{\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}\\\}\_\{j=1\}^\{N\}and constructs the decoder from all returned outputs by solving

udec∗=arg​minu∈ℋ2​\(Ω,ℝm\)⁡\{1N​∑j=1N∥u⁡\(βj\)−𝐲¯jA∥22\+λd​∥u′′∥L2​\(Ω,ℝm\)2\},u\_\{\\textrm\{dec\}\}^\{\*\}=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\left\\\{\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}\\lVert u\(\\beta\_\{j\}\)\-\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{d\}\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\\right\\\},which admits the vector\-valued cubic smoothing\-spline representation in \([41](https://arxiv.org/html/2610.27440#S4.E41)\)\.

8:The master node outputs

f^​\(𝐱k\)=udec∗​\(αk\)​for​k∈\[K\]\.\\widehat\{f\}\(\\mathbf\{x\}\_\{k\}\)=u\_\{\\textrm\{dec\}\}^\{\*\}\(\\alpha\_\{k\}\)\\text\{ for \}k\\in\[K\]\.

#### IV\-A3GCCScheme

The preceding constructions define theGCCscheme in the adversarial setting\. The master node first constructs the encoder using \([48](https://arxiv.org/html/2610.27440#S4.E48)\) \(or using \([53](https://arxiv.org/html/2610.27440#S4.E53)\) for efficient encoder\), evaluates it at the equidistant decoder design points in \([38](https://arxiv.org/html/2610.27440#S4.E38)\), and sends the resulting coded inputs to the workers\. After receiving all outputs from worker nodes, the master node constructs the decoder in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) using all returned outputs without attempting to identify the adversarial workers\. using all returned outputs, without attempting to identify the adversarial workers\. Finally, it computes the estimatesf^​\(𝐱k\)=udec​\(αk\)​for​k∈\[K\]\\widehat\{f\}\(\\mathbf\{x\}\_\{k\}\)=u\_\{\\textrm\{dec\}\}\(\\alpha\_\{k\}\)\\text\{ for \}k\\in\[K\]\. Algorithm[1](https://arxiv.org/html/2610.27440#alg1)summarizes the efficient implementation of the proposed scheme\.

In terms of asymptotic analysis, the following corollary establishes the asymptotic convergence rate of the robust end\-to\-end objective as the worker nodes increases, demonstrating that the adversarial risk in \([27](https://arxiv.org/html/2610.27440#S4.E27)\) asymptotically vanishes whenever the adversarial budget satisfiesq/N→0q/N\\to 0asN→∞N\\to\\infty\.

###### Corollary 2\(Asymptotic Convergence Rate\)\.

Under the assumptions of Theorem[2](https://arxiv.org/html/2610.27440#Thmtheorem2), suppose the adversarial budget satisfies1≤q≤Cq​Nθ1\\leq q\\leq C\_\{q\}N^\{\\theta\}for constantsCq\>0C\_\{q\}\>0andθ∈\[0,1\)\\theta\\in\[0,1\)\. If the decoder smoothing parameter is chosen according to Corollary[1](https://arxiv.org/html/2610.27440#Thmcorollary1)such thatλd≍\(q/N\)8/5\\lambda\_\{\\textrm\{d\}\}\\asymp\(q/N\)^\{8/5\}, then for sufficiently largeNN, the end\-to\-end objective satisfies

Jadv=𝒪⁡\(N65​\(θ−1\)\)\.\\displaystyle J\_\{\\textrm\{adv\}\}=\\mathcal\{O\}\\left\(N^\{\\frac\{6\}\{5\}\(\\theta\-1\)\}\\right\)\.\(55\)

### IV\-BComputational Complexity

The computational complexity of the encoder and decoder constructions closely follows the analysis developed for the straggler setting in\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\]\. We therefore summarize the main results and highlight the difference introduced by the adversarial model\.

For the nonlinear encoder in \([48](https://arxiv.org/html/2610.27440#S4.E48)\), the finite\-dimensional kernel coefficients can be computed using an iterative optimization method such as limited\-memory Broyden–Fletcher–Goldfarb–Shanno \(L\-BFGS\)\[[25](https://arxiv.org/html/2610.27440#bib.bib25)\]\. For L\-BFGC solver, If the solver performsIIiterations, constructing the encoder and evaluating it at theNNdecoder design points require

𝒪⁡\(I​K2​d\+N​K​d\)\\displaystyle\\mathcal\{O\}\\bigl\(IK^\{2\}d\+NKd\\bigr\)\(56\)operations\. In contrast, the efficient encoder in \([53](https://arxiv.org/html/2610.27440#S4.E53)\) is a vector\-valued cubic smoothing spline\. Using a B\-spline implementation with a banded linear system, it can be constructed and evaluated at the decoder design points in

𝒪⁡\(\(K\+N\)​d\)\\displaystyle\\mathcal\{O\}\\bigl\(\(K\+N\)d\\bigr\)\(57\)operations\.

The decoder in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) has the same smoothing\-spline structure\. Unlike the straggler setting, allNNworker nodes return their results, and hence the decoder is constructed using allNNreceived outputs\. Constructing the decoder and evaluating it at theKKencoder design points require

𝒪⁡\(\(N\+K\)​m\)\\displaystyle\\mathcal\{O\}\\bigl\(\(N\+K\)m\\bigr\)\(58\)operations\. Consequently, with the efficient encoder, the total master\-side computational complexity of the adversarialGCCscheme is

𝒪⁡\(\(K\+N\)​d\)\+𝒪⁡\(\(N\+K\)​m\)\.\\displaystyle\\mathcal\{O\}\\bigl\(\(K\+N\)d\\bigr\)\+\\mathcal\{O\}\\bigl\(\(N\+K\)m\\bigr\)\.\(59\)In the common regimeN≥KN\\geq K, this reduces to𝒪⁡\(N⁡\(d\+m\)\)\\mathcal\{O\}\\bigl\(N\(d\+m\)\\bigr\)\.

#### IV\-B1Comparison with Existing Schemes

We compare the computational complexity of different coded computing schemes at the master node, excluding worker\-side evaluations of the target function\.

Lagrange Coded Computing \(LCC\) relies on polynomial interpolation and fast multipoint evaluation\[[4](https://arxiv.org/html/2610.27440#bib.bib4),[26](https://arxiv.org/html/2610.27440#bib.bib26)\], yielding encoding and decoding complexities of𝒪⁡\(N​log2​K​log⁡log⁡K⋅d\)\\mathcal\{O\}\(N\\log^\{2\}K\\log\\log K\\cdot d\)and𝒪⁡\(N​log2​N​log⁡log⁡N⋅m\)\\mathcal\{O\}\(N\\log^\{2\}N\\log\\log N\\cdot m\), respectively\. In the presence of adversarial workers, LCC requires an explicit error\-localization step to identify corrupted outputs\. By leveraging sub\-quadratic decoding routines for Reed–Solomon codes\[[27](https://arxiv.org/html/2610.27440#bib.bib27),[28](https://arxiv.org/html/2610.27440#bib.bib28)\], locating these errors incurs an additional complexity of𝒪⁡\(q​log2​q​log⁡log⁡q⋅m\)\\mathcal\{O\}\(q\\log^\{2\}q\\log\\log q\\cdot m\)\. Consequently, in addition to the overhead of error localization, LCC exhibits superlinear computational complexity with respect to the number of worker nodesNN\. Overall, for an input batch sizeKKand at mostq≤Nq\\leq Nadversaries, the total computational complexity at the master node is

𝒪⁡\(N​log2​K​log⁡log⁡K⋅d\)\+𝒪⁡\(N​log2​N​log⁡log⁡N⋅m\)\+𝒪⁡\(q​log2​q​log⁡log⁡q⋅m\)\\displaystyle\\mathcal\{O\}\\\!\\left\(N\\log^\{2\}K\\log\\log K\\cdot d\\right\)\+\\mathcal\{O\}\\\!\\left\(N\\log^\{2\}N\\log\\log N\\cdot m\\right\)\+\\mathcal\{O\}\\\!\\left\(q\\log^\{2\}q\\log\\log q\\cdot m\\right\)=𝒪⁡\(N​log2​K​log⁡log⁡K⋅d\+N​log2​N​log⁡log⁡N⋅m\)\.\\displaystyle\\quad=\\mathcal\{O\}\\\!\\left\(N\\log^\{2\}K\\log\\log K\\cdot d\+N\\log^\{2\}N\\log\\log N\\cdot m\\right\)\.\(60\)
Berrut Approximated Coded Computing \(BACC\) relies on barycentric rational interpolation\[[9](https://arxiv.org/html/2610.27440#bib.bib9)\], where direct evaluation yields encoding and decoding complexities of𝒪⁡\(K​N​d\)\\mathcal\{O\}\(KNd\)and𝒪⁡\(K​N​m\)\\mathcal\{O\}\(KNm\), respectively\. Because BACC alone cannot identify corrupted worker outputs, ApproxIFER\[[13](https://arxiv.org/html/2610.27440#bib.bib13)\]extends it to the adversarial setting by adapting the Berlekamp–Welch approach\[[28](https://arxiv.org/html/2610.27440#bib.bib28)\]to rational interpolation\. Specifically, before performing the final rational reconstruction, ApproxIFER constructs candidate error\-locator functions by solving linear systems of size𝒪⁡\(N\)\\mathcal\{O\}\(N\)across themmoutput coordinates and combining the coordinatewise estimates to filter out suspected adversarial reports\. Although\[[13](https://arxiv.org/html/2610.27440#bib.bib13)\]does not explicitly analyze the asymptotic cost of this error\-localization step, solving these𝒪⁡\(N\)\\mathcal\{O\}\(N\)\-dimensional linear systems requires𝒪⁡\(m​N3\)\\mathcal\{O\}\(mN^\{3\}\)operations under standard dense solvers\.

In contrast, our proposedGCCframework requires only𝒪⁡\(\(K\+N\)​d\)\\mathcal\{O\}\(\(K\+N\)d\)and𝒪⁡\(\(K\+N\)​m\)\\mathcal\{O\}\(\(K\+N\)m\)arithmetic operations for encoding and decoding, respectively, yielding a total master\-node complexity of

𝒪​\(\(K\+N\)​\(d\+m\)\)\.\\displaystyle\\mathcal\{O\}\\\!\\left\(\(K\+N\)\(d\+m\)\\right\)\.\(61\)Compared with BACC\-based schemes,GCCreduces the multiplicative𝒪⁡\(K​N\)\\mathcal\{O\}\(KN\)dependence on the input batch sizeKKand worker countNNto a purely additive𝒪⁡\(K\+N\)\\mathcal\{O\}\(K\+N\)scaling\. Furthermore, unlike LCC, Robust ALCC, ApproxIFER, and RBACC,GCCmitigates corrupted worker reports inherently through spline regularization at the decoder, completely eliminating the need for dedicated error\-detection or error\-localization stages\. Consequently, the computational complexity ofGCCremains strictly linear inK\+NK\+Nand is entirely independent of the adversarial budgetqq\.

### IV\-CImpossibility Result for Smoothing\-Spline Decoding

Corollary[2](https://arxiv.org/html/2610.27440#Thmcorollary2)establishes vanishing worst\-case error for the proposed scheme when the adversarial budget grows sublinearly with the number of workers \(q=o⁡\(N\)q=o\(N\)\)\. In the following theorem, we show that this sublinear scaling is necessary for the smoothing\-spline decoder in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) to achieve vanishing adversarial loss\.

###### Theorem 4\(Adversarial Error Lower Bound\)\.

Consider aGCCscheme withN≥2N\\geq 2,K≥1K\\geq 1, and an adversarial budget1≤q<N1\\leq q<N\. For any encoderuenc∈ℋ2​\(Ω,ℝd\)u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), letudec∗u\_\{\\textrm\{dec\}\}^\{\*\}denote the smoothing\-spline decoder defined in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) with smoothing parameterλd\>0\\lambda\_\{\\textrm\{d\}\}\>0\. Under Assumption\(A3\)withTa\>0T\_\{a\}\>0, for every computing functionf∈ℱsmd→mf\\in\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\}, we have

infuenc∈ℋ2​\(Ω,ℝd\),λd\>0ℒq​\(uenc,udec∗,f\)≥Ta2K​\(qN\)2\.\\displaystyle\\inf\_\{\\begin\{subarray\}\{c\}u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),\\\\ \\lambda\_\{\\textrm\{d\}\}\>0\\end\{subarray\}\}\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\}^\{\*\};f\)\\geq\\frac\{T\_\{a\}^\{2\}\}\{K\}\\left\(\\frac\{q\}\{N\}\\right\)^\{2\}\.\(62\)

###### Corollary 3\(Positive Adversarial Fraction\)\.

FixK≥1K\\geq 1,μ∈\(0,1\)\\mu\\in\(0,1\), andTa\>0T\_\{a\}\>0, and letq=⌊μ​N⌋q=\\lfloor\\mu N\\rfloor\. Then, for every computing functionf∈ℱsmd→mf\\in\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\},

lim infN→∞infuenc∈ℋ2​\(Ω,ℝd\),λd\>0ℒq​\(uenc,udec∗,f\)≥Ta2​μ2K\>0\.\\displaystyle\\liminf\_\{N\\to\\infty\}\\inf\_\{\\begin\{subarray\}\{c\}u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),\\\\ \\lambda\_\{\\textrm\{d\}\}\>0\\end\{subarray\}\}\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\}^\{\*\};f\)\\geq\\frac\{T\_\{a\}^\{2\}\\mu^\{2\}\}\{K\}\>0\.\(63\)In particular, the worst\-case approximation error of the smoothing\-spline decoder cannot vanish asN→∞N\\to\\inftywhen the adversarial budget constitutes a constant positive fraction ofNN\.

## VProofs of the Main Results

In this section, we provide the proofs of the main theorems and corollaries, each presented in a separate subsection\. Several proofs rely on auxiliary lemmas whose proofs are deferred to Appendix[B](https://arxiv.org/html/2610.27440#A2)\.

### V\-AProof of Theorem[2](https://arxiv.org/html/2610.27440#Thmtheorem2)

Fixuenc∈ℋ2​\(Ω,ℝd\)u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), and define

g⁡\(z\):=f⁡\(uenc​\(z\)\)\.\\displaystyle g\(z\):=f\(u\_\{\\textrm\{enc\}\}\(z\)\)\.\(64\)
###### Lemma 1\(\[[12](https://arxiv.org/html/2610.27440#bib.bib12), Lemma 1\]\)\.

Letuenc∈ℋ2​\(Ω,ℝd\)u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), and letf=\[f1,…,fm\]⊤:ℝd→ℝmf=\[f\_\{1\},\\ldots,f\_\{m\}\]^\{\\top\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{m\}satisfy Assumption \(A2\)\. Then,f∘uenc∈ℋ2​\(Ω,ℝm\)f\\circ u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\.

Applying Lemma[1](https://arxiv.org/html/2610.27440#Thmlemma1), we haveg∈ℋ2​\(Ω,ℝm\)g\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\. The clean decoder in \([40](https://arxiv.org/html/2610.27440#S4.E40)\) is a smoothing spline constructed over the design points\{βj\}j=1N\\\{\\beta\_\{j\}\\\}\_\{j=1\}^\{N\}with respect to the target functiong⁡\(⋅\)g\(\\cdot\)\. Therefore, using the smoothing spline operator defined in \([21](https://arxiv.org/html/2610.27440#S3.E21)\), we obtain

udec∘=sN,λd​\[g\]\.\\displaystyle u\_\{\\textrm\{dec\}\}^\{\\circ\}=s\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\[g\]\.\(65\)
#### V\-A1Upper bound for clean generalization error

We first consider the clean generalization error\. By definition in \([36](https://arxiv.org/html/2610.27440#S4.E36)\), we have

ℒdecgen=4​supz∈Ω∥sN,λd​\[g\]​\(z\)−g⁡\(z\)∥22\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}=4\\sup\_\{z\\in\\Omega\}\\lVert s\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\[g\]\(z\)\-g\(z\)\\rVert\_\{2\}^\{2\}\.\(66\)Thus, we need a uniform upper bound for the second order smoothing\-spline approximation error\. The following lemma provides the required estimate\.

###### Lemma 2\(Second\-order smoothing\-spline error bound\)\.

Leth∈ℋ2​\(Ω,ℝm\)h\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)and lettj=\(j−1\)/\(n−1\)t\_\{j\}=\(j\-1\)/\(n\-1\)forj∈\[n\]j\\in\[n\]\. Letsn,λ​\[h\]s\_\{n,\\lambda\}\[h\]be the second order smoothing spline constructed over design point set\{tj\}j∈\[n\]\\\{t\_\{j\}\\\}\_\{j\\in\[n\]\}and target functionh⁡\(⋅\)h\(\\cdot\)with smoothing parameter0<λ≤10<\\lambda\\leq 1\. Then, there exists constantCspC\_\{\\mathrm\{sp\}\}, independent ofnn,λ\\lambda, andhh, such that

supz∈Ω∥sN,λ​\[h\]​\(z\)−h⁡\(z\)∥22≤Csp​\(λ3/4\+N−3\)​∥h′′∥L2​\(Ω,ℝm\)2\.\\displaystyle\\sup\_\{z\\in\\Omega\}\\lVert s\_\{N,\\lambda\}\[h\]\(z\)\-h\(z\)\\rVert\_\{2\}^\{2\}\\leq C\_\{\\mathrm\{sp\}\}\\left\(\\lambda^\{3/4\}\+N^\{\-3\}\\right\)\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\.\(67\)

The proof is provided in Appendix[B\-A](https://arxiv.org/html/2610.27440#A2.SS1)\. Applying Lemma[2](https://arxiv.org/html/2610.27440#Thmlemma2)toh=g=f∘uench=g=f\\circ u\_\{\\textrm\{enc\}\}andλ=λd\\lambda=\\lambda\_\{\\textrm\{d\}\}gives

ℒdecgen≤4​Csp​\(λd3/4\+N−3\)​∥g′′∥L2​\(Ω,ℝm\)2\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}\\leq 4C\_\{\\mathrm\{sp\}\}\\left\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\\right\)\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\.\(68\)It remains to derive an upper bound for the Sobolev regularity of the composite function∥g′′∥L2​\(Ω,ℝm\)2\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\. Following lemma provides an upper bound for∥g′′∥L2​\(Ω,ℝm\)2\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}in terms of Sobolev norm of encoderuencu\_\{\\textrm\{enc\}\}and target function regularity constantsHf,GfH\_\{f\},G\_\{f\}\.

###### Lemma 3\(Regularity of the composite function\)\.

Under Assumption \(A2\), there exists a constantCc\>0C\_\{\\mathrm\{c\}\}\>0such that, for everyu∈ℋ2​\(Ω,ℝd\)u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),

∥\(f∘u\)′′∥L2​\(Ω,ℝm\)2≤Cc​max⁡\{Hf2,Gf2\}​ψ​\(∥u∥ℋ2​\(Ω,ℝd\)2\),\\displaystyle\\lVert\(f\\circ u\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\\leq C\_\{\\mathrm\{c\}\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\psi\\left\(\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\),\(69\)whereψ⁡\(t\)=t\+4​t2\.\\psi\(t\)=t\+4t^\{2\}\.

The detailed proof can be found in Appendix[B\-B](https://arxiv.org/html/2610.27440#A2.SS2)\. Applying Lemma[3](https://arxiv.org/html/2610.27440#Thmlemma3)tou=uencu=u\_\{\\textrm\{enc\}\}in \([68](https://arxiv.org/html/2610.27440#S5.E68)\) yields

ℒdecgen\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}≤C1​max⁡\{Hf2,Gf2\}​\(λd3/4\+N−3\)​ψ​\(∥uenc∥ℋ2​\(Ω,ℝd\)2\),\\displaystyle\\leq C\_\{1\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\left\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\\right\)\\psi\\left\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\),\(70\)whereC1:=4​Csp​Cc\.C\_\{1\}:=4C\_\{\\mathrm\{sp\}\}C\_\{\\mathrm\{c\}\}\.

#### V\-A2Upper bound for Adversarial perturbation error

We next bound the adversarial perturbation error\. By linearity of the smoothing\-spline operator discussed in Section[III\-C](https://arxiv.org/html/2610.27440#S3.SS3), the corrupted and clean decoders admit the representations

udec∗​\(z\)\\displaystyle u\_\{\\textrm\{dec\}\}^\{\*\}\(z\)=1N​∑j=1NGN,λd​\(z,βj\)​𝐲¯jA,\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(z,\\beta\_\{j\}\)\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\},\(71\)udec∘​\(z\)\\displaystyle u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)=1N​∑j=1NGN,λd​\(z,βj\)​g​\(βj\)\.\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(z,\\beta\_\{j\}\)g\(\\beta\_\{j\}\)\.\(72\)Letℬ\\mathcal\{B\}denote the adversarial worker set\. Since the outputs of all workers outsideℬ\\mathcal\{B\}are uncorrupted, subtracting \([72](https://arxiv.org/html/2610.27440#S5.E72)\) from \([71](https://arxiv.org/html/2610.27440#S5.E71)\) gives

udec∗​\(z\)−udec∘​\(z\)=1N​∑j∈ℬGN,λd​\(z,βj\)​\(𝐚j−g⁡\(βj\)\)\.\\displaystyle u\_\{\\textrm\{dec\}\}^\{\*\}\(z\)\-u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)=\\frac\{1\}\{N\}\\sum\_\{j\\in\\mathcal\{B\}\}G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(z,\\beta\_\{j\}\)\\left\(\\mathbf\{a\}\_\{j\}\-g\(\\beta\_\{j\}\)\\right\)\.\(73\)
To bound \([73](https://arxiv.org/html/2610.27440#S5.E73)\), we require an upper bound on the exact discrete weight functionGN,λdG\_\{N,\\lambda\_\{\\textrm\{d\}\}\}, which does not follow directly from its definition\. Instead, we derive it by comparingGN,λdG\_\{N,\\lambda\_\{\\textrm\{d\}\}\}with the continuous Green’s function and its equivalent kernel\. More specifically, forh∈L2​\(Ω,ℝ\)h\\in\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\), consider the continuous smoothing problem

vλd​\[h\]:=arg​minv∈ℋ2​\(Ω,ℝ\)⁡\{∫Ω\(h⁡\(t\)−v⁡\(t\)\)2​𝑑t\+λd​∫Ω\(v′′​\(t\)\)2​𝑑t\}\.\\displaystyle v\_\{\\lambda\_\{\\textrm\{d\}\}\}\[h\]:=\\operatorname\*\{arg\\,min\}\_\{v\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\\left\\\{\\int\_\{\\Omega\}\\bigl\(h\(t\)\-v\(t\)\\bigr\)^\{2\}\\,dt\+\\lambda\_\{\\textrm\{d\}\}\\int\_\{\\Omega\}\\bigl\(v^\{\\prime\\prime\}\(t\)\\bigr\)^\{2\}\\,dt\\right\\\}\.\(74\)It can be shown that the solution to \([74](https://arxiv.org/html/2610.27440#S5.E74)\) is unique and can be represented using the continuous Green’s function\[[29](https://arxiv.org/html/2610.27440#bib.bib29),[30](https://arxiv.org/html/2610.27440#bib.bib30)\]GλdG\_\{\\lambda\_\{\\textrm\{d\}\}\}as

vλd​\(x\)=∫ΩGλd​\(x,t\)​h​\(t\)​𝑑t\.\\displaystyle v\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x\)=\\int\_\{\\Omega\}G\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)h\(t\)\\,dt\.\(75\)
Similar to smoothing spline weight functionGN,λdG\_\{N,\\lambda\_\{\\textrm\{d\}\}\}, deriving explicit formula or upper bound forGλdG\_\{\\lambda\_\{\\textrm\{d\}\}\}is challenging in general form\. However, authors in\[[31](https://arxiv.org/html/2610.27440#bib.bib31)\]proposed a kernel that is a good approximation ofGλdG\_\{\\lambda\_\{\\textrm\{d\}\}\}for equidistant design points\. Formally, define

Kλd​\(x,t\)\\displaystyle K\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)=e−\|x−t\|2​λd1/42​2​λd1/4​\(sin⁡\(\|x−t\|2​λd1/4\)\+cos⁡\(x−t2​λd1/4\)\)\\displaystyle=\\frac\{e^\{\-\\frac\{\|x\-t\|\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\}\}\{2\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\\left\(\\sin\\left\(\\frac\{\|x\-t\|\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\\right\)\+\\cos\\left\(\\frac\{x\-t\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\\right\)\\right\)\+12​2​λd1/4​\[Φ⁡\(x\+t2​λd1/4,x−t2​λd1/4\)\+Φ⁡\(2−x−t2​λd1/4,t−x2​λd1/4\)\],\\displaystyle\\quad\+\\frac\{1\}\{2\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\\left\[\\Phi\\left\(\\frac\{x\+t\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\},\\frac\{x\-t\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\\right\)\+\\Phi\\left\(\\frac\{2\-x\-t\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\},\\frac\{t\-x\}\{\\sqrt\{2\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{1/4\}\}\\right\)\\right\],\(76\)whereΦ⁡\(u,v\):=e−u​\(cos⁡u−sin⁡u\+2​cos⁡v\)\.\\Phi\(u,v\):=e^\{\-u\}\\left\(\\cos u\-\\sin u\+2\\cos v\\right\)\.We first bound this explicit kernel in the following lemma\.

###### Lemma 4\(Upper Bound on the Equivalent Kernel\)\.

There exists a constantτ\>0\\tau\>0such that for everyλ∈\(0,1\]\\lambda\\in\(0,1\],

supx,t∈\[0,1\]\|Kλ\(x,t\)\|≤τλ−1/4\.\\displaystyle\\sup\_\{x,t\\in\[0,1\]\}\\lvert K\_\{\\lambda\}\(x,t\)\\rvert\\leq\\tau\\lambda^\{\-1/4\}\.\(77\)

See Appendix[B\-C](https://arxiv.org/html/2610.27440#A2.SS3)for the proof\. Next, the following lemma relates the finite\-sample spline weight functionGN,λG\_\{N,\\lambda\}and the equivalent kernelKλK\_\{\\lambda\}\.

###### Lemma 5\(Approximation of the Finite\-Sample Spline Weights\)\.

Under equidistant design points, there exist constantsc0,CW,κ,CK\>0c\_\{0\},C\_\{W\},\\kappa,C\_\{K\}\>0such that whenever0<λ≤10<\\lambda\\leq 1andN​λ1/4\>c0N\\lambda^\{1/4\}\>c\_\{0\}, we have for allx,t∈\[0,1\]x,t\\in\[0,1\],

\|GN,λ​\(x,t\)−Kλ​\(x,t\)\|\\displaystyle\\lvert G\_\{N,\\lambda\}\(x,t\)\-K\_\{\\lambda\}\(x,t\)\\rvert≤CWλ−1/4exp\(−κ\|x−t\|λ−1/4\)\+CMλ−1/4exp\(−12λ−1/4\)\.\\displaystyle\\leq C\_\{W\}\\lambda^\{\-1/4\}\\exp\\left\(\-\\kappa\\lvert x\-t\\rvert\\lambda^\{\-1/4\}\\right\)\+C\_\{M\}\\lambda^\{\-1/4\}\\exp\\left\(\-\\frac\{1\}\{\\sqrt\{2\}\}\\lambda^\{\-1/4\}\\right\)\.\(78\)

The detailed proof is provided in Appendix[B\-D](https://arxiv.org/html/2610.27440#A2.SS4)\. We can now derive an upper bound on the exact weight function used by the decoder\. By the triangle inequality,

\|GN,λd​\(x,t\)\|\\displaystyle\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert≤\|Kλd​\(x,t\)\|\+\|GN,λd​\(x,t\)−Kλd​\(x,t\)\|\.\\displaystyle\\leq\\lvert K\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert\+\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\-K\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert\.\(79\)Using Lemmas[4](https://arxiv.org/html/2610.27440#Thmlemma4)and[5](https://arxiv.org/html/2610.27440#Thmlemma5), we obtain

\|GN,λd​\(x,t\)\|\\displaystyle\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert≤τλd−1/4\+CWλd−1/4exp\(−κ\|x−t\|λd−1/4\)\+CMλd−1/4exp\(−12λd−1/4\)\\displaystyle\\leq\\tau\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\+C\_\{W\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\exp\\left\(\-\\kappa\|x\-t\|\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\right\)\+C\_\{M\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\exp\\left\(\-\\frac\{1\}\{\\sqrt\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\right\)≤\(a\)\(τ\+CM\+CW\)λd−1/4,\\displaystyle\\stackrel\{\{\\scriptstyle\\text\{\(a\)\}\}\}\{\{\\leq\}\}\\left\(\\tau\+C\_\{M\}\+C\_\{W\}\\right\)\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\},\(80\)where \(a\) follows from the fact that both exponential terms are bounded by one\.

Therefore, definingCG:=τ\+CM\+CW,C\_\{\\mathrm\{G\}\}:=\\tau\+C\_\{M\}\+C\_\{W\},we have

\|GN,λd\(x,t\)\|≤CGλd−1/4\\displaystyle\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert\\leq C\_\{G\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\(81\)
We now return to the decoder adversarial perturbation error in \([73](https://arxiv.org/html/2610.27440#S5.E73)\)\. Assumptions[A1](https://arxiv.org/html/2610.27440#S4.I1.i1)and[A3](https://arxiv.org/html/2610.27440#S4.I1.i3)imply

∥𝐚j∥2\\displaystyle\\lVert\\mathbf\{a\}\_\{j\}\\rVert\_\{2\}≤m​Ta,\\displaystyle\\leq\\sqrt\{m\}\\,T\_\{a\},\(82\)∥g⁡\(βj\)∥2\\displaystyle\\lVert g\(\\beta\_\{j\}\)\\rVert\_\{2\}=∥f⁡\(uenc​\(βj\)\)∥2≤m​Tf\.\\displaystyle=\\lVert f\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\)\\rVert\_\{2\}\\leq\\sqrt\{m\}\\,T\_\{f\}\.\(83\)SinceT=max⁡\{Ta,Tf\}T=\\max\\\{T\_\{a\},T\_\{f\}\\\},

∥𝐚j−g⁡\(βj\)∥2≤2​m​T\.\\displaystyle\\lVert\\mathbf\{a\}\_\{j\}\-g\(\\beta\_\{j\}\)\\rVert\_\{2\}\\leq 2\\sqrt\{m\}\\,T\.\(84\)Using \([81](https://arxiv.org/html/2610.27440#S5.E81)\) and \([84](https://arxiv.org/html/2610.27440#S5.E84)\) in \([73](https://arxiv.org/html/2610.27440#S5.E73)\), and\|ℬ\|≤q\|\\mathcal\{B\}\|\\leq q, gives

supz∈Ω∥udec∗​\(z\)−udec∘​\(z\)∥2\\displaystyle\\sup\_\{z\\in\\Omega\}\\lVert u\_\{\\textrm\{dec\}\}^\{\*\}\(z\)\-u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)\\rVert\_\{2\}≤2​m​TN​∑j∈ℬsupz∈Ω\|GN,λd​\(z,βj\)\|\\displaystyle\\leq\\frac\{2\\sqrt\{m\}\\,T\}\{N\}\\sum\_\{j\\in\\mathcal\{B\}\}\\sup\_\{z\\in\\Omega\}\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(z,\\beta\_\{j\}\)\\rvert≤2CGmTqNλd−1/4\.\\displaystyle\\leq 2C\_\{\\mathrm\{G\}\}\\sqrt\{m\}\\,T\\frac\{q\}\{N\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\.\(85\)Consequently,

ℒdecadv=4supA∈𝒜qsupz∈Ω∥udec∗\(z\)−udec∘\(z\)∥22≤16CG2mT2q2N2λd−1/2\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{adv\}\}=4\\sup\_\{A\\in\\mathcal\{A\}\_\{q\}\}\\sup\_\{z\\in\\Omega\}\\lVert u\_\{\\textrm\{dec\}\}^\{\*\}\(z\)\-u\_\{\\textrm\{dec\}\}^\{\\circ\}\(z\)\\rVert\_\{2\}^\{2\}\\leq 16C\_\{\\mathrm\{G\}\}^\{2\}mT^\{2\}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\}\.\(86\)Thus, the adversarial perturbation term in \([42](https://arxiv.org/html/2610.27440#S4.E42)\) holds with

C2:=16​CG2\.\\displaystyle C\_\{2\}:=16C\_\{\\mathrm\{G\}\}^\{2\}\.\(87\)
Adding \([70](https://arxiv.org/html/2610.27440#S5.E70)\) and \([86](https://arxiv.org/html/2610.27440#S5.E86)\) proves the claimed bound for the constructed decoderudec∗u\_\{\\textrm\{dec\}\}^\{\*\}defined in \([39](https://arxiv.org/html/2610.27440#S4.E39)\)\. Sinceudec∗u\_\{\\textrm\{dec\}\}^\{\*\}is an admissible decoder inℋ2​\(Ω,ℝm\)\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\), we obtain

infudec∈ℋ2​\(Ω,ℝm\)\{ℒdecgen\+ℒdecadv\}≤C1max\{Hf2,Gf2\}\(λd3/4\+N−3\)ψ\(∥uenc∥ℋ2​\(Ω,ℝd\)2\)\+C2mT2q2N2λd−1/2\.\\displaystyle\\inf\_\{u\_\{\\textrm\{dec\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\left\\\{\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{gen\}\}\+\\mathcal\{L\}\_\{\\mathrm\{dec\}\}^\{\\mathrm\{adv\}\}\\right\\\}\\leq C\_\{1\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\left\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\\right\)\\psi\\left\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\+C\_\{2\}mT^\{2\}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\}\.\(88\)This proves the theorem\.

### V\-BProof of Corollary[1](https://arxiv.org/html/2610.27440#Thmcorollary1)

Define

Auenc\\displaystyle A\_\{u\_\{\\textrm\{enc\}\}\}:=C1​max⁡\{Hf2,Gf2\}​ψ​\(∥uenc∥ℋ2​\(Ω,ℝd\)2\),\\displaystyle:=C\_\{1\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\psi\\left\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\),\(89\)BN\\displaystyle B\_\{N\}:=C2​m​T2​q2N2\.\\displaystyle:=C\_\{2\}mT^\{2\}\\frac\{q^\{2\}\}\{N^\{2\}\}\.\(90\)In the nondegenerate case whereq≥1q\\geq 1andAuenc\>0A\_\{u\_\{\\textrm\{enc\}\}\}\>0, the part of the upper bound in \([42](https://arxiv.org/html/2610.27440#S4.E42)\) that depends onλd\\lambda\_\{\\textrm\{d\}\}is

ΦN\(λd\):=Auencλd3/4\+BNλd−1/2\.\\displaystyle\\Phi\_\{N\}\(\\lambda\_\{\\textrm\{d\}\}\):=A\_\{u\_\{\\textrm\{enc\}\}\}\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+B\_\{N\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\}\.\(91\)Its derivative is

ΦN′\(λd\)=34Auencλd−1/4−12BNλd−3/2\.\\displaystyle\\Phi\_\{N\}^\{\\prime\}\(\\lambda\_\{\\textrm\{d\}\}\)=\\frac\{3\}\{4\}A\_\{u\_\{\\textrm\{enc\}\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\-\\frac\{1\}\{2\}B\_\{N\}\\lambda\_\{\\textrm\{d\}\}^\{\-3/2\}\.\(92\)The unique stationary point satisfies

λd5/4=2​BN3​Auenc\.\\displaystyle\\lambda\_\{\\textrm\{d\}\}^\{5/4\}=\\frac\{2B\_\{N\}\}\{3A\_\{u\_\{\\textrm\{enc\}\}\}\}\.\(93\)SinceΦN′\\Phi\_\{N\}^\{\\prime\}is negative below this point and positive above it, the stationary point is the unique minimizer:

λd∗=\(2​C2​m​T2​q23​C1​N2​max⁡\{Hf2,Gf2\}​ψ​\(∥uenc∥ℋ2​\(Ω,ℝd\)2\)\)4/5\.\\displaystyle\\lambda\_\{\\textrm\{d\}\}^\{\*\}=\\left\(\\frac\{2C\_\{2\}mT^\{2\}q^\{2\}\}\{3C\_\{1\}N^\{2\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\psi\\left\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\}\\right\)^\{4/5\}\.\(94\)
For fixedqqand fixed problem parameters,λd∗→0\\lambda\_\{\\textrm\{d\}\}^\{\*\}\\to 0asN→∞N\\to\\infty\. Moreover,

N​\(λd∗\)1/4\\displaystyle N\(\\lambda\_\{\\textrm\{d\}\}^\{\*\}\)^\{1/4\}=\(2​C2​m​T23​C1​max⁡\{Hf2,Gf2\}​ψ​\(∥uenc∥ℋ2​\(Ω,ℝd\)2\)\)1/5​q2/5​N3/5\\displaystyle=\\left\(\\frac\{2C\_\{2\}mT^\{2\}\}\{3C\_\{1\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\psi\\left\(\\lVert u\_\{\\textrm\{enc\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\}\\right\)^\{1/5\}q^\{2/5\}N^\{3/5\}⟶∞\.\\displaystyle\\longrightarrow\\infty\.\(95\)Thus, there existsN0N\_\{0\}such that, for everyN≥N0N\\geq N\_\{0\}, we have0<λd∗≤1,and​N​\(λd∗\)1/4≥c0\.0<\\lambda\_\{\\textrm\{d\}\}^\{\*\}\\leq 1,\\text\{ and \}N\(\\lambda\_\{\\textrm\{d\}\}^\{\*\}\)^\{1/4\}\\geq c\_\{0\}\.Finally,

λd∗≍\(qN\)8/5,\\displaystyle\\lambda\_\{\\textrm\{d\}\}^\{\*\}\\asymp\\left\(\\frac\{q\}\{N\}\\right\)^\{8/5\},\(96\)as claimed\.

### V\-CProof of Theorem[3](https://arxiv.org/html/2610.27440#Thmtheorem3)

The proof follows the same argument as that of Theorem 3 in\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\]\. Indeed, for a fixed decoder smoothing parameterλd\\lambda\_\{\\textrm\{d\}\}, the encoder\-dependent component of the loss upper bound has the exact same form as in the straggler setting\. The only additional term is the adversarial perturbationΔq,N\\Delta\_\{q,N\}, which is independent of the encoder and therefore does not affect the optimization or the comparison argument used in the original proof\.

### V\-DProof of Corollary[2](https://arxiv.org/html/2610.27440#Thmcorollary2)

###### Proof\.

Lets𝐗∈ℋ2​\(Ω,ℝd\)s\_\{\\mathbf\{X\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)be the natural cubic spline interpolating the input batch in encoder design points:

s𝐗​\(αk\)=𝐱k,k∈\[K\]\.\\displaystyle s\_\{\\mathbf\{X\}\}\(\\alpha\_\{k\}\)=\\mathbf\{x\}\_\{k\},\\qquad k\\in\[K\]\.\(97\)Because the input batch and encoder design points are fixed,∥s𝐗∥ℋ2​\(Ω,ℝd\)\\lVert s\_\{\\mathbf\{X\}\}\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}is independent ofNN\. Moreover,

1K​∑k=1K∥s𝐗​\(αk\)−𝐱k∥22=0\.\\displaystyle\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert s\_\{\\mathbf\{X\}\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}=0\.\(98\)
By Theorem[3](https://arxiv.org/html/2610.27440#Thmtheorem3), we have

Jadv\\displaystyle J\_\{\\textrm\{adv\}\}≤Δq,N\+2​Gf2​\[1K​∑k=1K∥uenc†​\(αk\)−𝐱k∥22\+λe​\(Re\+∥\(uenc†\)′′∥L2​\(Ω,ℝd\)2\)\]\.\\displaystyle\\leq\\Delta\_\{q,N\}\+2G\_\{f\}^\{2\}\\bigg\[\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\left\(R\_\{\\mathrm\{e\}\}\+\\lVert\(u\_\{\\textrm\{enc\}\}^\{\\dagger\}\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\\bigg\]\.\(99\)Sinceuenc†​\(⋅\)u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(\\cdot\)minimizes the regularized objective

1K​∑k=1K∥u⁡\(αk\)−𝐱k∥22\+λe​∥u′′∥L2​\(Ω,ℝd\)2\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}over allu∈ℋ2​\(Ω,ℝd\)u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), its objective value is upper bounded by that of the natural cubic spline interpolants𝐗∈ℋ2​\(Ω,ℝd\)s\_\{\\mathbf\{X\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\. Consequently, we obtain

Jadv\\displaystyle J\_\{\\textrm\{adv\}\}≤Δq,N\+2​Gf2​\[1K​∑k=1K∥uenc†​\(αk\)−𝐱k∥22\+λe​\(Re\+∥\(uenc†\)′′∥L2​\(Ω,ℝd\)2\)\]\\displaystyle\\leq\\Delta\_\{q,N\}\+2G\_\{f\}^\{2\}\\bigg\[\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert u\_\{\\textrm\{enc\}\}^\{\\dagger\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\left\(R\_\{\\mathrm\{e\}\}\+\\lVert\(u\_\{\\textrm\{enc\}\}^\{\\dagger\}\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\\bigg\]≤Δq,N\+2​Gf2​\[1K​∑k=1K∥s𝐗​\(αk\)−𝐱k∥22\+λe​\(Re\+∥\(s𝐗\)′′∥L2​\(Ω,ℝd\)2\)\]\\displaystyle\\leq\\Delta\_\{q,N\}\+2G\_\{f\}^\{2\}\\bigg\[\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert s\_\{\\mathbf\{X\}\}\(\\alpha\_\{k\}\)\-\\mathbf\{x\}\_\{k\}\\rVert\_\{2\}^\{2\}\+\\lambda\_\{\\textrm\{e\}\}\\left\(R\_\{\\mathrm\{e\}\}\+\\lVert\(s\_\{\\mathbf\{X\}\}\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)\\bigg\]=\(a\)Δq,N\+2​Gf2​λe​\(Re\+∥\(s𝐗\)′′∥L2​\(Ω,ℝd\)2\),\\displaystyle\\stackrel\{\{\\scriptstyle\\text\{\(a\)\}\}\}\{\{=\}\}\\Delta\_\{q,N\}\+2G\_\{f\}^\{2\}\\lambda\_\{\\textrm\{e\}\}\\left\(R\_\{\\mathrm\{e\}\}\+\\lVert\(s\_\{\\mathbf\{X\}\}\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\),\(100\)where\(a\)follows from the exact interpolation propertys𝐗​\(αk\)=𝐱ks\_\{\\mathbf\{X\}\}\(\\alpha\_\{k\}\)=\\mathbf\{x\}\_\{k\}in \([98](https://arxiv.org/html/2610.27440#S5.E98)\)\. Substituting the definitions ofΔq,N\\Delta\_\{q,N\}andλe\\lambda\_\{\\textrm\{e\}\}from \([50](https://arxiv.org/html/2610.27440#S4.E50)\) and \([47](https://arxiv.org/html/2610.27440#S4.E47)\), respectively, yields

Jadv≤A0\(λd3/4\+N−3\)\+B0q2N2λd−1/2,\\displaystyle J\_\{\\textrm\{adv\}\}\\leq A\_\{0\}\\left\(\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\+N^\{\-3\}\\right\)\+B\_\{0\}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\},\(101\)whereA0:=2​Gf2​\(Re\+∥\(s𝐗\)′′∥L2​\(Ω,ℝd\)2\)​\(m1\+m2​Re\)A\_\{0\}:=2G\_\{f\}^\{2\}\\big\(R\_\{\\mathrm\{e\}\}\+\\lVert\(s\_\{\\mathbf\{X\}\}\)^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\big\)\(m\_\{1\}\+m\_\{2\}R\_\{\\mathrm\{e\}\}\)andB0:=C2​m​T2B\_\{0\}:=C\_\{2\}mT^\{2\}\.

Forq≥1q\\geq 1, chooseλd≍\(qN\)8/5\\lambda\_\{\\textrm\{d\}\}\\asymp\\left\(\\frac\{q\}\{N\}\\right\)^\{8/5\}\. Then we haveλd3/4≍\(qN\)6/5andq2N2λd−1/2≍\(qN\)6/5\\lambda\_\{\\textrm\{d\}\}^\{3/4\}\\asymp\\left\(\\frac\{q\}\{N\}\\right\)^\{6/5\}\\text\{ and \}\\frac\{q^\{2\}\}\{N^\{2\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/2\}\\asymp\\left\(\\frac\{q\}\{N\}\\right\)^\{6/5\}\. Consequently,

Jadv=𝒪⁡\(\(qN\)6/5\+N−3\)\.\\displaystyle J\_\{\\textrm\{adv\}\}=\\mathcal\{O\}\\left\(\\left\(\\frac\{q\}\{N\}\\right\)^\{6/5\}\+N^\{\-3\}\\right\)\.\(102\)Sinceq≤Cq​Nθq\\leq C\_\{q\}N^\{\\theta\}, we obtain

\(qN\)6/5≤Cq6/5​N65​\(θ−1\)\.\\displaystyle\\left\(\\frac\{q\}\{N\}\\right\)^\{6/5\}\\leq C\_\{q\}^\{6/5\}N^\{\\frac\{6\}\{5\}\(\\theta\-1\)\}\.\(103\)Forθ∈\[0,1\)\\theta\\in\[0,1\),65​\(θ−1\)≥−65\>−3\.\\frac\{6\}\{5\}\(\\theta\-1\)\\geq\-\\frac\{6\}\{5\}\>\-3\.Thus, theN−3N^\{\-3\}term is asymptotically no larger than the first term, and

Jadv=𝒪⁡\(N65​\(θ−1\)\)\.\\displaystyle J\_\{\\textrm\{adv\}\}=\\mathcal\{O\}\\left\(N^\{\\frac\{6\}\{5\}\(\\theta\-1\)\}\\right\)\.\(104\)
It remains to verify the conditions of Theorem[2](https://arxiv.org/html/2610.27440#Thmtheorem2)\. We have

N​λd1/4\\displaystyle N\\lambda\_\{\\textrm\{d\}\}^\{1/4\}≍N​\(qN\)2/5=N3/5​q2/5⟶∞\.\\displaystyle\\asymp N\\left\(\\frac\{q\}\{N\}\\right\)^\{2/5\}=N^\{3/5\}q^\{2/5\}\\longrightarrow\\infty\.\(105\)Therefore for sufficiently largeNNit satisfiesN​λd1/4≥c0N\\lambda\_\{\\textrm\{d\}\}^\{1/4\}\\geq c\_\{0\}\. Also,q=o⁡\(N\)q=o\(N\)impliesλd→0\\lambda\_\{\\textrm\{d\}\}\\to 0\.

Whenq=0q=0, the adversarial term vanishes\. Choosingλd≍N−4\\lambda\_\{\\textrm\{d\}\}\\asymp N^\{\-4\}gives an error of orderN−3N^\{\-3\}, which also satisfies the stated upper bound\. ∎

### V\-EProof of Theorem[4](https://arxiv.org/html/2610.27440#Thmtheorem4)

Fixf∈ℱsmd→mf\\in\\mathcal\{F\}\_\{\\mathrm\{sm\}\}^\{d\\to m\}, an encoderuenc∈ℋ2​\(Ω,ℝd\)u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), and a decoder smoothing parameterλd\>0\\lambda\_\{\\textrm\{d\}\}\>0chosen prior to observing the worker outputs\. As established in Section[III\-C](https://arxiv.org/html/2610.27440#S3.SS3)and \([22](https://arxiv.org/html/2610.27440#S3.E22)\), the smoothing\-spline decoder is a linear operator with respect to the received worker outputs\. Specifically, under any admissible adversarial attackAA, the decoded function takes the form

udec∗​\(z\)=1N​∑n=1NGN,λd​\(z,βn\)​𝐲¯nA,z∈Ω,\\displaystyle u\_\{\\textrm\{dec\}\}^\{\*\}\(z\)=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(z,\\beta\_\{n\}\)\\overline\{\\mathbf\{y\}\}\_\{n\}^\{A\},\\qquad z\\in\\Omega,\(106\)where the weight functionGN,λd​\(z,βn\)G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(z,\\beta\_\{n\}\)depends solely on the decoder design points\{βn\}n=1N\\\{\\beta\_\{n\}\\\}\_\{n=1\}^\{N\}, the smoothing parameterλd\\lambda\_\{\\textrm\{d\}\}, andNN\. Evaluatingudec∗​\(⋅\)u\_\{\\textrm\{dec\}\}^\{\*\}\(\\cdot\)at the first target pointα1\\alpha\_\{1\}yields

udec∗​\(α1\)=∑n=1Nwn​𝐲¯nA,\\displaystyle u\_\{\\textrm\{dec\}\}^\{\*\}\(\\alpha\_\{1\}\)=\\sum\_\{n=1\}^\{N\}w\_\{n\}\\overline\{\\mathbf\{y\}\}\_\{n\}^\{A\},\(107\)wherewn:=1N​GN,λd​\(α1,βn\)w\_\{n\}:=\\frac\{1\}\{N\}G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(\\alpha\_\{1\},\\beta\_\{n\}\)\. Note that the same scalar weights\{wn\}n=1N\\\{w\_\{n\}\\\}\_\{n=1\}^\{N\}apply across allmmoutput coordinates because the smoothing\-spline objective in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) decouples coordinatewise\.

To determine the sum of these weights, suppose that𝐲¯nA=𝐜\\overline\{\\mathbf\{y\}\}\_\{n\}^\{A\}=\\mathbf\{c\}for alln∈\[N\]n\\in\[N\], where𝐜∈ℝm\\mathbf\{c\}\\in\\mathbb\{R\}^\{m\}is an arbitrary non\-zero constant vector\. In this case, the constant functionudec​\(z\)≡𝐜u\_\{\\textrm\{dec\}\}\(z\)\\equiv\\mathbf\{c\}achieves an objective value of zero in \([39](https://arxiv.org/html/2610.27440#S4.E39)\) and is therefore the unique minimizerudec∗u\_\{\\textrm\{dec\}\}^\{\*\}\. Substituting𝐲¯nA=𝐜\\overline\{\\mathbf\{y\}\}\_\{n\}^\{A\}=\\mathbf\{c\}andudec∗​\(α1\)=𝐜u\_\{\\textrm\{dec\}\}^\{\*\}\(\\alpha\_\{1\}\)=\\mathbf\{c\}into \([107](https://arxiv.org/html/2610.27440#S5.E107)\) implies that the weights reproduce constants:

∑n=1Nwn=1\.\\displaystyle\\sum\_\{n=1\}^\{N\}w\_\{n\}=1\.\(108\)
Now letℬ⊂\[N\]\\mathcal\{B\}\\subset\[N\]denote the subset ofqqworker indices corresponding to theqqlargest values of\|wn\|\\lvert w\_\{n\}\\rvert\(breaking ties arbitrarily\)\. Since the average of theqqlargest magnitudes is at least the overall average across allNNindices, it follows from \([108](https://arxiv.org/html/2610.27440#S5.E108)\) and the triangle inequality that

∑n∈ℬ\|wn\|≥qN​∑n=1N\|wn\|≥qN​\|∑n=1Nwn\|=qN\.\\displaystyle\\sum\_\{n\\in\\mathcal\{B\}\}\\lvert w\_\{n\}\\rvert\\geq\\frac\{q\}\{N\}\\sum\_\{n=1\}^\{N\}\\lvert w\_\{n\}\\rvert\\geq\\frac\{q\}\{N\}\\lvert\\sum\_\{n=1\}^\{N\}w\_\{n\}\\rvert=\\frac\{q\}\{N\}\.\(109\)
For each signσ∈\{−1,1\}\\sigma\\in\\\{\-1,1\\\}, consider the adversarial attackAσA\_\{\\sigma\}that corrupts the workers inℬ\\mathcal\{B\}by setting

𝐲¯nAσ=σ​Ta​sign⁡\(wn\)​𝐞1,n∈ℬ,\\displaystyle\\overline\{\\mathbf\{y\}\}\_\{n\}^\{A\_\{\\sigma\}\}=\\sigma T\_\{a\}\\operatorname\{sign\}\(w\_\{n\}\)\\mathbf\{e\}\_\{1\},\\qquad n\\in\\mathcal\{B\},\(110\)while leaving all remaining worker outputs \(n∉ℬn\\notin\\mathcal\{B\}\) uncorrupted, where𝐞1∈ℝm\\mathbf\{e\}\_\{1\}\\in\\mathbb\{R\}^\{m\}denotes the first standard basis vector andsign⁡\(0\):=0\\operatorname\{sign\}\(0\):=0\. BothA1A\_\{1\}andA−1A\_\{\-1\}are admissible under Assumption \(A3\), as each modifies at most\|ℬ\|=q\\lvert\\mathcal\{B\}\\rvert=qworker outputs and satisfies𝐲¯nAσ∈\[−Ta,Ta\]m\\overline\{\\mathbf\{y\}\}\_\{n\}^\{A\_\{\\sigma\}\}\\in\[\-T\_\{a\},T\_\{a\}\]^\{m\}\.

Next, define the uncorrupted residualbband the adversarial perturbation magnitudeaain the first output coordinate atα1\\alpha\_\{1\}by

b\\displaystyle b:=∑n∉ℬwn​f1​\(uenc​\(βn\)\)−f1​\(𝐱1\),\\displaystyle:=\\sum\_\{n\\notin\\mathcal\{B\}\}w\_\{n\}f\_\{1\}\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{n\}\)\)\-f\_\{1\}\(\\mathbf\{x\}\_\{1\}\),\(111\)a\\displaystyle a:=Ta​∑n∈ℬ\|wn\|\.\\displaystyle:=T\_\{a\}\\sum\_\{n\\in\\mathcal\{B\}\}\\lvert w\_\{n\}\\rvert\.\(112\)By \([107](https://arxiv.org/html/2610.27440#S5.E107)\), under attackAσA\_\{\\sigma\}, the estimation error in the first coordinate atα1\\alpha\_\{1\}is given by\[udec∗​\(α1\)−f⁡\(𝐱1\)\]1=b\+σ​a\\big\[u\_\{\\textrm\{dec\}\}^\{\*\}\(\\alpha\_\{1\}\)\-f\(\\mathbf\{x\}\_\{1\}\)\\big\]\_\{1\}=b\+\\sigma a\. Using the identity

maxσ∈\{−1,1\}⁡\(b\+σ​a\)2≥\(b\+a\)2\+\(b−a\)22=b2\+a2≥a2,\\displaystyle\\max\_\{\\sigma\\in\\\{\-1,1\\\}\}\(b\+\\sigma a\)^\{2\}\\geq\\frac\{\(b\+a\)^\{2\}\+\(b\-a\)^\{2\}\}\{2\}=b^\{2\}\+a^\{2\}\\geq a^\{2\},\(113\)together with \([109](https://arxiv.org/html/2610.27440#S5.E109)\), we lower\-bound the worst\-case squared Euclidean error atα1\\alpha\_\{1\}as

maxσ∈\{−1,1\}⁡∥udec∗​\(α1\)−f⁡\(𝐱1\)∥22≥maxσ∈\{−1,1\}⁡\(b\+σ​a\)2≥a2≥Ta2​\(qN\)2\.\\displaystyle\\max\_\{\\sigma\\in\\\{\-1,1\\\}\}\\lVert u\_\{\\textrm\{dec\}\}^\{\*\}\(\\alpha\_\{1\}\)\-f\(\\mathbf\{x\}\_\{1\}\)\\rVert\_\{2\}^\{2\}\\geq\\max\_\{\\sigma\\in\\\{\-1,1\\\}\}\(b\+\\sigma a\)^\{2\}\\geq a^\{2\}\\geq T\_\{a\}^\{2\}\\left\(\\frac\{q\}\{N\}\\right\)^\{2\}\.\(114\)Because the squared errors at the remainingK−1K\-1target points\{αk\}k=2K\\\{\\alpha\_\{k\}\\\}\_\{k=2\}^\{K\}are non\-negative, the worst\-case risk over all admissible attacksAAsatisfies

ℒq​\(uenc,udec∗,f\)\\displaystyle\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\}^\{\*\};f\)≥1K​maxσ∈\{−1,1\}​∥udec∗​\(α1\)−f⁡\(𝐱1\)∥22≥Ta2K​\(qN\)2\.\\displaystyle\\geq\\frac\{1\}\{K\}\\max\_\{\\sigma\\in\\\{\-1,1\\\}\}\\lVert u\_\{\\textrm\{dec\}\}^\{\*\}\(\\alpha\_\{1\}\)\-f\(\\mathbf\{x\}\_\{1\}\)\\rVert\_\{2\}^\{2\}\\geq\\frac\{T\_\{a\}^\{2\}\}\{K\}\\left\(\\frac\{q\}\{N\}\\right\)^\{2\}\.\(115\)Since this lower bound holds for anyuenc∈ℋ2​\(Ω,ℝd\)u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)and anyλd\>0\\lambda\_\{\\textrm\{d\}\}\>0, taking the infimum over both parameters yields \([62](https://arxiv.org/html/2610.27440#S4.E62)\) and completes the proof\.

### V\-FProof of Corollary[3](https://arxiv.org/html/2610.27440#Thmcorollary3)

Sinceμ∈\(0,1\)\\mu\\in\(0,1\), for allN≥⌈1/μ⌉N\\geq\\lceil 1/\\mu\\rceil, the adversarial budgetq=⌊μ​N⌋q=\\lfloor\\mu N\\rfloorsatisfies1≤q<N1\\leq q<N\. Applying Theorem[4](https://arxiv.org/html/2610.27440#Thmtheorem4)for every suchNNyields

infuenc∈ℋ2​\(Ω,ℝd\),λd\>0ℒq​\(uenc,udec∗,f\)≥Ta2K​\(⌊μ​N⌋N\)2\.\\displaystyle\\inf\_\{\\begin\{subarray\}\{c\}u\_\{\\textrm\{enc\}\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),\\\\ \\lambda\_\{\\textrm\{d\}\}\>0\\end\{subarray\}\}\\mathcal\{L\}\_\{q\}\(u\_\{\\textrm\{enc\}\},u\_\{\\textrm\{dec\}\}^\{\*\};f\)\\geq\\frac\{T\_\{a\}^\{2\}\}\{K\}\\left\(\\frac\{\\lfloor\\mu N\\rfloor\}\{N\}\\right\)^\{2\}\.\(116\)BecauselimN→∞⌊μ​N⌋/N=μ\\lim\_\{N\\to\\infty\}\\lfloor\\mu N\\rfloor/N=\\mu, taking the limit inferior asN→∞N\\to\\inftyon both sides establishes the corollary\.

## VIExperimental Results

In this section, we empirically evaluate the proposed adversarialGCCframework\. We consider two types of computing functions: \(1\) a scalar nonlinear function and \(2\) a high\-dimensional neural\-network inference task\. The primary purpose of these experiments is to examine the effectiveness ofGCCto mitigate the adversarial effect and convergence\-rate prediction of Corollary[2](https://arxiv.org/html/2610.27440#Thmcorollary2)under a concrete adversarial strategy\.

### VI\-AExperimental Setup

Target Tasks:We evaluate our framework on two representative computing tasks:

- •Scalar nonlinear function:The first target function is given by fsc​\(x\):=x​sin⁡\(x\),\\displaystyle f\_\{\\mathrm\{sc\}\}\(x\):=x\\sin\(x\),\(117\)evaluated over the compact interval\[−12,12\]\[\-12,12\]to ensure bounded inputs and outputs\.
- •Neural\-network inference:The second task evaluates a LeNet\-5 convolutional neural network\[[32](https://arxiv.org/html/2610.27440#bib.bib32)\]trained on the MNIST dataset\[[33](https://arxiv.org/html/2610.27440#bib.bib33)\]\. Each input is preprocessed into a32×3232\\times 32grayscale image \(10241024\-dimensional vector\), and the network outputs a1010\-dimensional vector representing the class probability distribution across digits00through99: fNN:ℝ1024→ℝ10\.\\displaystyle f\_\{\\mathrm\{NN\}\}:\\mathbb\{R\}^\{1024\}\\to\\mathbb\{R\}^\{10\}\.\(118\)

In both settings, we fix the batch size toK=10K=10\. The encoder design points\{αk\}k=1K\\\{\\alpha\_\{k\}\\\}\_\{k=1\}^\{K\}and worker evaluation points\{βj\}j=1N\\\{\\beta\_\{j\}\\\}\_\{j=1\}^\{N\}are placed uniformly over\[0,1\]\[0,1\]according to \([38](https://arxiv.org/html/2610.27440#S4.E38)\):

αk\\displaystyle\\alpha\_\{k\}=k−1K−1,k∈\[K\],\\displaystyle=\\frac\{k\-1\}\{K\-1\},\\qquad k\\in\[K\],\(119\)βj\\displaystyle\\beta\_\{j\}=j−1N−1,j∈\[N\]\.\\displaystyle=\\frac\{j\-1\}\{N\-1\},\\qquad j\\in\[N\]\.\(120\)
To evaluate performance across different adversarial regimes, we scale the number of adversarial workers as

q:=⌊Nθ⌋,0≤θ<1\.\\displaystyle q:=\\left\\lfloor N^\{\\theta\}\\right\\rfloor,\\qquad 0\\leq\\theta<1\.\(121\)
Evaluation Metric:We evaluate performance via the expected end\-to\-end mean squared error \(MSE\):

MSE:=𝔼𝐱∼𝒳​\[1K​∑k=1K∥f^​\(𝐱k\)−f⁡\(𝐱k\)∥22\],\\displaystyle\\mathrm\{MSE\}:=\\mathbb\{E\}\_\{\\mathbf\{x\}\\sim\\mathcal\{X\}\}\\left\[\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\lVert\\widehat\{f\}\(\\mathbf\{x\}\_\{k\}\)\-f\(\\mathbf\{x\}\_\{k\}\)\\rVert\_\{2\}^\{2\}\\right\],\(122\)where𝒳\\mathcal\{X\}denotes the input data distribution\. In practice, we approximate \([122](https://arxiv.org/html/2610.27440#S6.E122)\) by computing the empirical average over1010independent experimental runs\.

![Refer to caption](https://arxiv.org/html/2610.27440v1/figs/conv_rate_xsinx_adv.png)\(a\)Scalar computationfsc​\(x\)=x​sin⁡\(x\)f\_\{\\mathrm\{sc\}\}\(x\)=x\\sin\(x\)\.
![Refer to caption](https://arxiv.org/html/2610.27440v1/figs/conv_rate_lenet_adv.png)\(b\)LeNet\-5 inference\.

Fig\. 2:Approximation error as a function of the number of workers for adversarial budgetsq=⌊Nθ⌋q=\\lfloor N^\{\\theta\}\\rfloor\. The light solid curves show the empirical errors, the dark solid lines show the fitted empirical slopes, and the dashed lines show the convergence\-rate predictions from Corollary[2](https://arxiv.org/html/2610.27440#Thmcorollary2)\.
### VI\-BAdversarial Strategy

For each encoder design pointαk\\alpha\_\{k\}, the adversary selects worker nodes whose decoder design pointsβj\\beta\_\{j\}are closest toαk\\alpha\_\{k\}\. The total adversarial budgetqqis distributed as evenly as possible among theKKencoder design points\. Consequently, approximatelyq/Kq/Koutputs near eachαk\\alpha\_\{k\}are corrupted\.

The corrupted values are selected differently for the two computing tasks\.

- •Scalar nonlinear function:For every selected workerjj, the adversary replaces the honest outputfsc​\(uenc​\(βj\)\)f\_\{\\mathrm\{sc\}\}\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\)by the largest admissible scalar valuey¯jA=Ta\.\\overline\{y\}\_\{j\}^\{A\}=T\_\{a\}\.We setTa=10T\_\{a\}=10in this experiment\.
- •LeNet\-5 inference:Let cj:=arg​maxi∈\[m\]⁡\[fNN​\(uenc​\(βj\)\)\]i\\displaystyle c\_\{j\}:=\\operatorname\*\{arg\\,max\}\_\{i\\in\[m\]\}\\left\[f\_\{\\mathrm\{NN\}\}\\bigl\(u\_\{\\textrm\{enc\}\}\(\\beta\_\{j\}\)\\bigr\)\\right\]\_\{i\}\(123\)denote the class predicted by the honest worker output\. The adversary selects a random different classcjAc\_\{j\}^\{A\}\. It then replaces the honest output vector by𝐲¯jA=𝐞cjA,\\overline\{\\mathbf\{y\}\}\_\{j\}^\{A\}=\\mathbf\{e\}\_\{c\_\{j\}^\{A\}\},where𝐞cjA∈ℝm\\mathbf\{e\}\_\{c\_\{j\}^\{A\}\}\\in\\mathbb\{R\}^\{m\}is the corresponding standard basis vector\. Thus, each corrupted worker outputs a one\-hot vector concentrated on a class different from the class predicted by its honest output\.

Both attacks concentrate the corrupted outputs near the locations at which the final estimates are evaluated to inject maximum noise\. These attacks represent particular admissible strategies and do not restrict the general adversarial model considered in the theoretical analysis\.

### VI\-CConvergence\-Rate Evaluation

Under the scaling in \([121](https://arxiv.org/html/2610.27440#S6.E121)\), Corollary[2](https://arxiv.org/html/2610.27440#Thmcorollary2)givesJadv=𝒪⁡\(N−65​\(1−θ\)\)\.J\_\{\\textrm\{adv\}\}=\\mathcal\{O\}\(N^\{\-\\frac\{6\}\{5\}\(1\-\\theta\)\}\)\.Therefore, the predicted slope in a log\-log plot of the approximation error versusNNis−65​\(1−θ\)\.\-\\frac\{6\}\{5\}\(1\-\\theta\)\.Figure[2](https://arxiv.org/html/2610.27440#S6.F2)compares these predicted slopes with the slopes fitted to the empirical curves\.

In every considered regime, the empirical slope is more negative than the corresponding theoretically guaranteed slope\. Thus, over the investigated range of worker populations, the approximation error decays at least as rapidly as predicted by Corollary[2](https://arxiv.org/html/2610.27440#Thmcorollary2)\. As expected, increasingθ\\thetaslows the convergence because the adversarial population grows more rapidly withNN\.

## VIIConclusion and Future Directions

In this paper, we introduced an adversarially robust framework for general coded computing \(GCC\) that mitigates the impact of malicious worker nodes through smoothness regularization rather than explicit error localization\. By distributing computations via a smooth encoder and aggregating all returned outputs with a regularized decoder, we established a non\-asymptotic worst\-case error bound\. This theoretical analysis proves that our proposed spline\-based scheme achieves linear master\-side complexity and guarantees a vanishing approximation error whenever the adversarial population grows sublinearly with the total number of workers\.

Several open questions and directions remain for future investigation\. First, our current design assumes the adversarial budget is knowna prioriin order to tune the optimal smoothing parameters\. Developing adaptive procedures capable of estimating the effective corruption level directly from the received outputs would significantly enhance the framework’s practical applicability\. Second, future work should address the simultaneous presence of stragglers and adversarial workers, as missing observations and corrupted outputs distort the regularized decoder via fundamentally different mechanisms\. Finally, establishing a matching minimax lower bound for the𝒪⁡\(\(q/N\)6/5\)\\mathcal\{O\}\(\(q/N\)^\{6/5\}\)convergence rate would clarify whether our achievable exponent is strictly optimal, or if exploring higher\-order regularization penalties and alternative reproducing kernels could yield sharper convergence rates\.

## Acknowledgment

The work has been partially supported by the National Science Foundation under Grant CCF\-2348638\.

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## VIIIBiographies

Parsa Moradiis a Ph\.D\. candidate in the Department of Electrical and Computer Engineering at the University of Minnesota Twin Cities\. He received his B\.Sc\. and M\.Sc\. degrees from the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran, in 2016 and 2019, respectively\. His research interests include deep learning, generative AI, distributed machine learning, and AI for healthcare\.

Hanzaleh Akbarinodehiis a Ph\.D\. candidate in the Department of Electrical and Computer Engineering at the University of Minnesota Twin Cities\. He received his B\.Sc\. and M\.Sc\. degrees from the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran, in 2016 and 2018, respectively\. His research interests include information theory and coding, blockchain technology, game theory, and decentralized systems\.

Mohammad Ali Maddah\-Ali\(IEEE Fellow, 2023\) is an Associate Professor at the University of Minnesota Twin Cities\. He received his B\.Sc\. degree in Electrical Engineering from Isfahan University of Technology, his M\.A\.Sc\. degree from the University of Tehran, and his Ph\.D\. in Electrical and Computer Engineering from the University of Waterloo, Canada, in 2007\.From 2007 to 2008, he was with the Wireless Technology Laboratories at Nortel Networks, Ottawa, ON, Canada\. He then held a Postdoctoral Fellowship at the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, from 2008 to 2010\. From September 2010 to September 2020, he served as a Communication Research Scientist at Nokia Bell Labs, NJ, USA\.Dr\. Maddah\-Ali is the recipient of several honors, including the NSERC Postdoctoral Fellowship \(2007\), the Best Paper Award at the IEEE International Conference on Communications \(ICC\) in 2014, the IEEE Communications Society and IEEE Information Theory Society Joint Paper Award in 2015, and the IEEE Information Theory Society Paper Award in 2016\. He served as an Associate Editor for the IEEE Transactions on Information Theory \(2019–2022\) and as Lead Editor for the IEEE Journal on Selected Areas in Information Theory\. He is currently a distinguished lecturer of the IEEE Information Theory Society\.

## Appendix AAdditional Functional\-Analytic Details

###### Definition 3\(Weak Derivatives and Vector\-Valued Norms\)\.

A locally integrable scalar functionvvhas a weak derivativewwonΩ⊂ℝ\\Omega\\subset\\mathbb\{R\}if

∫Ωv\(t\)ϕ′\(t\)dt=−∫Ωw\(t\)ϕ\(t\)dt\\displaystyle\\int\_\{\\Omega\}v\(t\)\\phi^\{\\prime\}\(t\)\\,dt=\-\\int\_\{\\Omega\}w\(t\)\\phi\(t\)\\,dt\(124\)for every infinitely differentiable test functionϕ\\phiwith compact support inΩ\\Omega\. Weak derivatives of vector\-valued functions are defined coordinate\-wise\.

###### Definition 4\(LpL^\{p\}\-norm\)\.

For a vector\-valued functionu=\[u1,…,uM\]Tu=\[u\_\{1\},\\ldots,u\_\{M\}\]^\{T\}, theLpL^\{p\}\-norm on a domainΩ⊂ℝ\\Omega\\subset\\mathbb\{R\}is defined as follows:

∥u∥Lp​\(Ω,ℝM\):=\{∫Ω∥u⁡\(t\)∥2p​𝑑t\}1/p,\\displaystyle\\lVert u\\rVert\_\{L^\{p\}\(\\Omega;\\mathbb\{R\}^\{M\}\)\}:=\\left\\\{\\int\_\{\\Omega\}\\lVert u\(t\)\\rVert\_\{2\}^\{p\}\\,dt\\right\\\}^\{1/p\},\(125\)for1≤p<∞1\\leq p<\\infty, and

∥u∥L∞​\(Ω,ℝM\):=ess​supz∈Ω⁡∥ui​\(z\)∥2,\\displaystyle\\lVert u\\rVert\_\{L^\{\\infty\}\(\\Omega;\\mathbb\{R\}^\{M\}\)\}:=\\operatorname\*\{ess\\,sup\}\_\{z\\in\\Omega\}\\lVert u\_\{i\}\(z\)\\rVert\_\{2\},\(126\)whenp=∞p=\\infty\.

###### Definition 5\(Local Sobolev spaces\)\.

A functiong:Ω→ℝMg:\\Omega\\to\\mathbb\{R\}^\{M\}belongs to local Sobolev space𝕎locr,p​\(Ω,ℝM\)\\mathbb\{W\}\_\{\\textrm\{loc\}\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)ifg∈𝕎locr,p​\(Ω,ℝM\)g\\in\\mathbb\{W\}\_\{\\textrm\{loc\}\}^\{r,p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)for every compact setV⊂ΩV\\subset\\Omega\.

###### Lemma 6\(Theorem 8\[[12](https://arxiv.org/html/2610.27440#bib.bib12)\]\)\.

𝕎2,2​\(Ω,ℝM\)⊆𝕎1,1​\(Ω,ℝM\)⊆𝕎loc1,1​\(Ω,ℝM\)\\mathbb\{W\}^\{2,2\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\\subseteq\\mathbb\{W\}^\{1,1\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\\subseteq\\mathbb\{W\}\_\{\\textrm\{loc\}\}^\{1,1\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\.

###### Definition 6\(Sobolev Space with Compact Support\)\.

Given bounded intervalΩ=\[a,b\]\\Omega=\[a,b\], Sobolev orderr∈ℕr\\in\\mathbb\{N\}, and exponentp∈\[1,∞\)p\\in\[1,\\infty\), we defineW0r,p​\(Ω,ℝM\)W\_\{0\}^\{r,p\}\(\\Omega;\\mathbb\{R\}^\{M\}\)as the subspace of functionsg∈Wr,p​\(Ω,ℝM\)g\\in W^\{r,p\}\(\\Omega;\\mathbb\{R\}^\{M\}\)satisfying vanishing boundary conditions

g\(j\)\(a\)=0,for allj=0,1,…,m−1\.g^\{\(j\)\}\(a\)=0,\\qquad\\text\{for all \}j=0,1,\\dots,m\-1\.\(127\)This space is usually equipped with the norm based on the highest\-order derivative:

‖g‖W0r,p​\(Ω,ℝM\):=‖g\(m\)‖Lp​\(Ω,ℝM\),1≤p<∞,\\\|g\\\|\_\{W\_\{0\}^\{r,p\}\(\\Omega;\\mathbb\{R\}^\{M\}\)\}:=\\\|g^\{\(m\)\}\\\|\_\{L^\{p\}\(\\Omega;\\mathbb\{R\}^\{M\}\)\},\\qquad 1\\leq p<\\infty,\(128\)and forp=∞p=\\infty,

‖g‖W0r,∞​\(Ω,ℝM\):=‖g\(m\)‖L∞​\(Ω,ℝM\)\.\\\|g\\\|\_\{W\_\{0\}^\{r,\\infty\}\(\\Omega;\\mathbb\{R\}^\{M\}\)\}:=\\\|g^\{\(m\)\}\\\|\_\{L^\{\\infty\}\(\\Omega;\\mathbb\{R\}^\{M\}\)\}\.\(129\)Whenp=2p=2, this space is an RKHS, and we denote it byℋ0m​\(Ω,ℝM\)\\mathcal\{H\}\_\{0\}^\{m\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\.

## Appendix BProof of Lemmas

### B\-AProof of Lemma[2](https://arxiv.org/html/2610.27440#Thmlemma2)

Define the smoothing\-spline approximation error

e⁡\(z\):=sn,λ​\[h\]​\(z\)−h⁡\(z\)\.\\displaystyle e\(z\):=s\_\{n,\\lambda\}\[h\]\(z\)\-h\(z\)\.\(131\)Sincesn,λ​\[h\],h∈ℋ2​\(Ω,ℝm\)s\_\{n,\\lambda\}\[h\],h\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\), it follows immediately that

e∈ℋ2​\(Ω,ℝm\)\.\\displaystyle e\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\.\(132\)
We first bound the supremum norm ofe⁡\(⋅\)e\(\\cdot\)using itsL2​\(Ω,ℝm\)\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)norm and theL2​\(Ω,ℝm\)\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)norm of its first derivative\.

Using the smoothing\-spline operator defined in \([21](https://arxiv.org/html/2610.27440#S3.E21)\), we have

sn,λ​\[h\]=arg​minu∈ℋ2​\(Ω,ℝm\)⁡\{1n​∑i=1n∥u⁡\(ti\)−h⁡\(ti\)∥22\+λ​∥u′′∥L2​\(Ω,ℝm\)2\}\.\\displaystyle s\_\{n,\\lambda\}\[h\]=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\left\\\{\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\lVert u\(t\_\{i\}\)\-h\(t\_\{i\}\)\\rVert\_\{2\}^\{2\}\+\\lambda\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\\right\\\}\.\(133\)For simplicity, let

s:=sn,λ​\[h\]\.s:=s\_\{n,\\lambda\}\[h\]\.We first derive a useful orthogonality property of the smoothing\-spline error\. For an arbitrary directionv∈ℋ2​\(Ω,ℝm\)v\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\), define the one\-dimensional function

Γv​\(ε\)\\displaystyle\\Gamma\_\{v\}\(\\varepsilon\):=1n​∑i=1n∥s⁡\(ti\)\+ε​v​\(ti\)−h⁡\(ti\)∥22\+λ​∥s′′\+ε​v′′∥L2​\(Ω,ℝm\)2,ε∈ℝ\.\\displaystyle:=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\lVert s\(t\_\{i\}\)\+\\varepsilon v\(t\_\{i\}\)\-h\(t\_\{i\}\)\\rVert\_\{2\}^\{2\}\+\\lambda\\lVert s^\{\\prime\\prime\}\+\\varepsilon v^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\},\\qquad\\varepsilon\\in\\mathbb\{R\}\.\(134\)Becauses⁡\(⋅\)s\(\\cdot\)minimizes the smoothing\-spline objective,ε=0\\varepsilon=0minimizesΓv​\(ε\)\\Gamma\_\{v\}\(\\varepsilon\)\. Therefore,

Γv′​\(0\)=0\.\\displaystyle\\Gamma\_\{v\}^\{\\prime\}\(0\)=0\.\(135\)Differentiating the two quadratic terms gives

Γv′​\(0\)\\displaystyle\\Gamma\_\{v\}^\{\\prime\}\(0\)=2n​∑i=1n⟨s⁡\(ti\)−h⁡\(ti\),v⁡\(ti\)⟩\+2​λ​⟨s′′,v′′⟩L2​\(Ω,ℝm\),\\displaystyle=\\frac\{2\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\\langle s\(t\_\{i\}\)\-h\(t\_\{i\}\),v\(t\_\{i\}\)\\right\\rangle\+2\\lambda\\left\\langle s^\{\\prime\\prime\},v^\{\\prime\\prime\}\\right\\rangle\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\},\(136\)where

⟨s′′,v′′⟩L2​\(Ω,ℝm\):=∫01⟨s′′​\(t\),v′′​\(t\)⟩​𝑑t\.\\displaystyle\\left\\langle s^\{\\prime\\prime\},v^\{\\prime\\prime\}\\right\\rangle\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}:=\\int\_\{0\}^\{1\}\\left\\langle s^\{\\prime\\prime\}\(t\),v^\{\\prime\\prime\}\(t\)\\right\\rangle dt\.\(137\)Recalling thate=s−he=s\-hand dividing by two, we obtain the first\-order optimality condition

1n​∑i=1n⟨e⁡\(ti\),v⁡\(ti\)⟩\+λ​⟨s′′,v′′⟩L2​\(Ω,ℝm\)=0\\displaystyle\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\\langle e\(t\_\{i\}\),v\(t\_\{i\}\)\\right\\rangle\+\\lambda\\left\\langle s^\{\\prime\\prime\},v^\{\\prime\\prime\}\\right\\rangle\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}=0\(138\)for everyv∈ℋ2​\(Ω,ℝm\)v\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\. This is the standard variational optimality condition for smoothing splines\[[23](https://arxiv.org/html/2610.27440#bib.bib23),[15](https://arxiv.org/html/2610.27440#bib.bib15)\]\.

Now fix an arbitrary vector𝐯0∈ℝm\\mathbf\{v\}\_\{0\}\\in\\mathbb\{R\}^\{m\}and choose the constant perturbation

v⁡\(t\)≡𝐯0\.\\displaystyle v\(t\)\\equiv\\mathbf\{v\}\_\{0\}\.\(139\)Sincev′′​\(t\)=𝟎v^\{\\prime\\prime\}\(t\)=\\mathbf\{0\}, condition \([138](https://arxiv.org/html/2610.27440#A2.E138)\) becomes

0\\displaystyle 0=1n​∑i=1n⟨e⁡\(ti\),𝐯𝟎⟩=⟨1n​∑i=1ne⁡\(ti\),𝐯𝟎⟩\.\\displaystyle=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\\langle e\(t\_\{i\}\),\\mathbf\{\\mathbf\{v\}\_\{0\}\}\\right\\rangle=\\left\\langle\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}e\(t\_\{i\}\),\\mathbf\{\\mathbf\{v\}\_\{0\}\}\\right\\rangle\.\(140\)Because this equality holds for every𝐯0∈ℝm\\mathbf\{v\}\_\{0\}\\in\\mathbb\{R\}^\{m\}, we must have

∑i=1ne⁡\(ti\)=𝟎\.\\displaystyle\\sum\_\{i=1\}^\{n\}e\(t\_\{i\}\)=\\mathbf\{0\}\.\(141\)Equivalently, for every coordinateℓ∈\[m\]\\ell\\in\[m\],

∑i=1neℓ​\(ti\)=0\.\\displaystyle\\sum\_\{i=1\}^\{n\}e\_\{\\ell\}\(t\_\{i\}\)=0\.\(142\)
We next show that every coordinate erroreℓe\_\{\\ell\}vanishes somewhere on\[0,1\]\[0,1\]\. Ifeℓ​\(ti\)=0e\_\{\\ell\}\(t\_\{i\}\)=0for someii, we simply takezℓ=tiz\_\{\\ell\}=t\_\{i\}\. Otherwise, all the values\{eℓ​\(ti\)\}i=1n\\\{e\_\{\\ell\}\(t\_\{i\}\)\\\}\_\{i=1\}^\{n\}are nonzero\. In this case, \([142](https://arxiv.org/html/2610.27440#A2.E142)\) implies that they cannot all have the same sign\. Hence, there exist two consecutive design pointstit\_\{i\}andti\+1t\_\{i\+1\}such that

eℓ​\(ti\)​eℓ​\(ti\+1\)<0\.\\displaystyle e\_\{\\ell\}\(t\_\{i\}\)e\_\{\\ell\}\(t\_\{i\+1\}\)<0\.\(143\)Sinceeℓ∈H2​\(0,1\)⊂H1​\(0,1\)e\_\{\\ell\}\\in H^\{2\}\(0,1\)\\subset H^\{1\}\(0,1\), it has a continuous, absolutely continuous representative on\[0,1\]\[0,1\]\. The intermediate value theorem therefore ensures the existence of a point

zℓ∈\[ti,ti\+1\]\\displaystyle z\_\{\\ell\}\\in\[t\_\{i\},t\_\{i\+1\}\]\(144\)such that

eℓ​\(zℓ\)=0\.\\displaystyle e\_\{\\ell\}\(z\_\{\\ell\}\)=0\.\(145\)
Becauseeℓ∈H1​\(0,1\)e\_\{\\ell\}\\in H^\{1\}\(0,1\), the functioneℓ2e\_\{\\ell\}^\{2\}is absolutely continuous and satisfies

dd​t​eℓ​\(t\)2=2​eℓ​\(t\)​eℓ′​\(t\)\\displaystyle\\frac\{d\}\{dt\}e\_\{\\ell\}\(t\)^\{2\}=2e\_\{\\ell\}\(t\)e\_\{\\ell\}^\{\\prime\}\(t\)\(146\)for almost everyt∈\(0,1\)t\\in\(0,1\)\. Hence, for everyz∈\[0,1\]z\\in\[0,1\],

\|eℓ​\(z\)\|2\\displaystyle\|e\_\{\\ell\}\(z\)\|^\{2\}=\|eℓ​\(z\)2−eℓ​\(zℓ\)2\|\\displaystyle=\\left\|e\_\{\\ell\}\(z\)^\{2\}\-e\_\{\\ell\}\(z\_\{\\ell\}\)^\{2\}\\right\|=2​\|∫zℓzeℓ​\(t\)​eℓ′​\(t\)​𝑑t\|\\displaystyle=2\\left\|\\int\_\{z\_\{\\ell\}\}^\{z\}e\_\{\\ell\}\(t\)e\_\{\\ell\}^\{\\prime\}\(t\)\\,dt\\right\|≤2​∫01\|eℓ​\(t\)\|​\|eℓ′​\(t\)\|​𝑑t\\displaystyle\\leq 2\\int\_\{0\}^\{1\}\|e\_\{\\ell\}\(t\)\|\|e\_\{\\ell\}^\{\\prime\}\(t\)\|\\,dt≤2​∥eℓ∥L2​\(Ω,ℝ\)​∥eℓ′∥L2​\(Ω,ℝ\),\\displaystyle\\leq 2\\lVert e\_\{\\ell\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\\lVert e\_\{\\ell\}^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\},\(147\)where the last step follows from the Cauchy–Schwarz inequality\.

Taking the supremum overzzin \([147](https://arxiv.org/html/2610.27440#A2.E147)\), summing overℓ∈\[m\]\\ell\\in\[m\], and applying the Cauchy–Schwarz inequality to the coordinate index give

∥e∥L∞​\(Ω,ℝm\)2\\displaystyle\\lVert e\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤∑ℓ=1m∥eℓ∥L∞​\(Ω,ℝ\)2\\displaystyle\\leq\\sum\_\{\\ell=1\}^\{m\}\\lVert e\_\{\\ell\}\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}≤2​∑ℓ=1m∥eℓ∥L2​\(Ω,ℝ\)​∥eℓ′∥L2​\(Ω,ℝ\)\\displaystyle\\leq 2\\sum\_\{\\ell=1\}^\{m\}\\lVert e\_\{\\ell\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\\lVert e\_\{\\ell\}^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}≤2​∥e∥L2​\(Ω,ℝm\)​∥e′∥L2​\(Ω,ℝm\)\.\\displaystyle\\leq 2\\lVert e\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\lVert e^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\.\(148\)Taking square roots yields the slightly sharper estimate

∥e∥L∞​\(Ω,ℝm\)≤2​∥e∥L2​\(Ω,ℝm\)1/2​∥e′∥L2​\(Ω,ℝm\)1/2\.\\displaystyle\\lVert e\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\leq\\sqrt\{2\}\\lVert e\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{1/2\}\\lVert e^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{1/2\}\.\(149\)In particular, since2≤2\\sqrt\{2\}\\leq 2,

∥e∥L∞​\(Ω,ℝm\)≤2​∥e∥L2​\(Ω,ℝm\)1/2​∥e′∥L2​\(Ω,ℝm\)1/2\.\\displaystyle\\lVert e\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\leq 2\\lVert e\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{1/2\}\\lVert e^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{1/2\}\.\(150\)
Next, we bound the two quantities on the right\-hand side of \([150](https://arxiv.org/html/2610.27440#A2.E150)\)\. Recall from Definition[1](https://arxiv.org/html/2610.27440#Thmdefinition1)that for any ordered design set𝒯=\{t1,…,tn\}\\mathcal\{T\}=\\\{t\_\{1\},\\ldots,t\_\{n\}\\\}wheret1<…<tnt\_\{1\}<\\ldots<t\_\{n\}, the termsδ¯τ\\bar\{\\delta\}\_\{\\tau\}and¯​δτ\\mathchoice\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.90779pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.61668pt\{\}\}\}$\\hss\}\}\}\\delta\_\{\\tau\}denote the maximum and minimum consecutive spacings in𝒯\\mathcal\{T\}, respectively\.

Authors in\[[34](https://arxiv.org/html/2610.27440#bib.bib34)\]have shown that, for a scalar target functiong∈H2​\(Ω,ℝ\)g\\in H^\{2\}\(\\Omega;\\mathbb\{R\}\)and its smoothing\-spline approximationsn,λ​\[g\]s\_\{n,\\lambda\}\[g\]over𝒯\\mathcal\{T\}, there exist constantsG0,G1\>0G\_\{0\},G\_\{1\}\>0such that

∥g−sn,λ​\[g\]∥L2​\(Ω,ℝ\)2≤G0​L𝒯​∥g′′∥L2​\(Ω,ℝ\)2\\displaystyle\\lVert g\-s\_\{n,\\lambda\}\[g\]\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\\leq G\_\{0\}L\_\{\\mathcal\{T\}\}\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\(151\)and

∥\(g−sn,λ​\[g\]\)′∥L2​\(Ω,ℝ\)2≤G1​L𝒯1/2​\(1\+L𝒯16\)1/2​∥g′′∥L2​\(Ω,ℝ\)2,\\displaystyle\\lVert\(g\-s\_\{n,\\lambda\}\[g\]\)^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\\leq G\_\{1\}L\_\{\\mathcal\{T\}\}^\{1/2\}\\left\(1\+\\frac\{L\_\{\\mathcal\{T\}\}\}\{16\}\\right\)^\{1/2\}\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\},\(152\)where

L𝒯\\displaystyle L\_\{\\mathcal\{T\}\}:=p2​\(δ¯𝒯¯​δ𝒯\)​n​δ¯𝒯4​λ\+D​δ¯𝒯4\.\\displaystyle:=p\_\{2\}\\left\(\\frac\{\\bar\{\\delta\}\_\{\\mathcal\{T\}\}\}\{\\mathchoice\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.90779pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.61668pt\{\}\}\}$\\hss\}\}\}\\delta\_\{\\mathcal\{T\}\}\}\\right\)\\frac\{n\\bar\{\\delta\}\_\{\\mathcal\{T\}\}\}\{4\}\\lambda\+D\\bar\{\\delta\}\_\{\\mathcal\{T\}\}^\{4\}\.\(153\)Here,p2p\_\{2\}is a degree\-two polynomial with positive coefficients, andD\>0D\>0is an absolute constant\.

The vector\-valued smoothing\-spline objective separates across itsmmoutput coordinates\. Applying \([151](https://arxiv.org/html/2610.27440#A2.E151)\) and \([152](https://arxiv.org/html/2610.27440#A2.E152)\) to each pair\(hℓ,sℓ\)\(h\_\{\\ell\},s\_\{\\ell\}\)and summing overℓ∈\[m\]\\ell\\in\[m\]gives

∥e∥L2​\(Ω,ℝm\)2\\displaystyle\\lVert e\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤G0​L𝒯​∥h′′∥L2​\(Ω,ℝm\)2,\\displaystyle\\leq G\_\{0\}L\_\{\\mathcal\{T\}\}\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\},\(154\)∥e′∥L2​\(Ω,ℝm\)2\\displaystyle\\lVert e^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤G1​L𝒯1/2​\(1\+L𝒯16\)1/2​∥h′′∥L2​\(Ω,ℝm\)2,\\displaystyle\\leq G\_\{1\}L\_\{\\mathcal\{T\}\}^\{1/2\}\\left\(1\+\\frac\{L\_\{\\mathcal\{T\}\}\}\{16\}\\right\)^\{1/2\}\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\},\(155\)whereL𝒯L\_\{\\mathcal\{T\}\}is the design factor in \([153](https://arxiv.org/html/2610.27440#A2.E153)\) for

𝒯=\{t1,…,tn\}\.\\displaystyle\\mathcal\{T\}=\\\{t\_\{1\},\\ldots,t\_\{n\}\\\}\.\(156\)
For the equidistant design pointsti=\(i−1\)/\(n−1\),t\_\{i\}=\(i\-1\)/\(n\-1\),we have

ti\+1−ti=1n−1,i∈\[n−1\]\.\\displaystyle t\_\{i\+1\}\-t\_\{i\}=\\frac\{1\}\{n\-1\},\\qquad i\\in\[n\-1\]\.\(157\)Hence,

δ¯𝒯=¯​δ𝒯=1n−1,\\displaystyle\\bar\{\\delta\}\_\{\\mathcal\{T\}\}=\\mathchoice\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to3\.55556pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.90779pt\{\}\}\}$\\hss\}\}\}\{\\hbox to0\.0pt\{\{$\\underline\{\{\\hbox to2\.61668pt\{\}\}\}$\\hss\}\}\}\\delta\_\{\\mathcal\{T\}\}=\\frac\{1\}\{n\-1\},\(158\)and therefore

L𝒯\\displaystyle L\_\{\\mathcal\{T\}\}=p2​\(1\)​n4​\(n−1\)​λ\+D\(n−1\)4\\displaystyle=p\_\{2\}\(1\)\\frac\{n\}\{4\(n\-1\)\}\\lambda\+\\frac\{D\}\{\(n\-1\)^\{4\}\}≤\(a\)p2​\(1\)2​λ\+16​D​n−4\\displaystyle\\stackrel\{\{\\scriptstyle\\text\{\(a\)\}\}\}\{\{\\leq\}\}\\frac\{p\_\{2\}\(1\)\}\{2\}\\lambda\+16Dn^\{\-4\}≤CL​\(λ\+n−4\),\\displaystyle\\leq C\_\{L\}\\left\(\\lambda\+n^\{\-4\}\\right\),\(159\)whereCL\>0C\_\{L\}\>0is independent ofnnandλ\\lambdaand \(a\) follows fromnn−1≤2​and​1\(n−1\)4≤16​n−4,for​n≥2\.\\frac\{n\}\{n\-1\}\\leq 2\\text\{ and \}\\frac\{1\}\{\(n\-1\)^\{4\}\}\\leq 16n^\{\-4\},\\text\{ for \}n\\geq 2\.

Combining \([148](https://arxiv.org/html/2610.27440#A2.E148)\), \([154](https://arxiv.org/html/2610.27440#A2.E154)\), and \([155](https://arxiv.org/html/2610.27440#A2.E155)\), we obtain

∥e∥L∞​\(Ω,ℝm\)2\\displaystyle\\lVert e\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤2​∥e∥L2​\(Ω,ℝm\)​∥e′∥L2​\(Ω,ℝm\)\\displaystyle\\leq 2\\lVert e\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}\\lVert e^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}≤2​\(G0​G1\)1/2​L𝒯3/4​\(1\+L𝒯16\)1/4​∥h′′∥L2​\(Ω,ℝm\)2\.\\displaystyle\\leq 2\\left\(G\_\{0\}G\_\{1\}\\right\)^\{1/2\}L\_\{\\mathcal\{T\}\}^\{3/4\}\\left\(1\+\\frac\{L\_\{\\mathcal\{T\}\}\}\{16\}\\right\)^\{1/4\}\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\.\(160\)
Since0<λ≤10<\\lambda\\leq 1, it follows from \([159](https://arxiv.org/html/2610.27440#A2.E159)\) that

L𝒯≤2​CL\.\\displaystyle L\_\{\\mathcal\{T\}\}\\leq 2C\_\{L\}\.\(161\)Consequently,

\(1\+L𝒯16\)1/4≤\(1\+2​CL16\)1/4,\\displaystyle\\left\(1\+\\frac\{L\_\{\\mathcal\{T\}\}\}\{16\}\\right\)^\{1/4\}\\leq\\left\(1\+\\frac\{2C\_\{L\}\}\{16\}\\right\)^\{1/4\},\(162\)which is independent ofnn,λ\\lambda, andhh\. Absorbing this factor and the constantsG0G\_\{0\},G1G\_\{1\}, andCLC\_\{L\}into a single constant gives

∥e∥L∞​\(Ω,ℝm\)2\\displaystyle\\lVert e\\rVert\_\{\{L\}^\{\\infty\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤Csp​\(λ\+n−4\)3/4​∥h′′∥L2​\(Ω,ℝm\)2\\displaystyle\\leq C\_\{\\mathrm\{sp\}\}\\left\(\\lambda\+n^\{\-4\}\\right\)^\{3/4\}\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤Csp​\(λ3/4\+n−3\)​∥h′′∥L2​\(Ω,ℝm\)2,\\displaystyle\\leq C\_\{\\mathrm\{sp\}\}\\left\(\\lambda^\{3/4\}\+n^\{\-3\}\\right\)\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\},\(163\)where the last inequality follows from\(a\+b\)3/4≤a3/4\+b3/4​for​a,b≥0\.\(a\+b\)^\{3/4\}\\leq a^\{3/4\}\+b^\{3/4\}\\text\{ for \}a,b\\geq 0\.

Finally, usinge=sn,λ​\[h\]−he=s\_\{n,\\lambda\}\[h\]\-h, we conclude that

supz∈Ω∥sn,λ​\[h\]​\(z\)−h⁡\(z\)∥22≤Csp​\(λ3/4\+n−3\)​∥h′′∥L2​\(Ω,ℝm\)2\.\\displaystyle\\sup\_\{z\\in\\Omega\}\\lVert s\_\{n,\\lambda\}\[h\]\(z\)\-h\(z\)\\rVert\_\{2\}^\{2\}\\leq C\_\{\\mathrm\{sp\}\}\\left\(\\lambda^\{3/4\}\+n^\{\-3\}\\right\)\\lVert h^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}\.\(164\)This proves the lemma\.

### B\-BProof of Lemma[3](https://arxiv.org/html/2610.27440#Thmlemma3)

To bound the second derivative of the composite function, we make use of the following Gagliardo–Nirenberg\-type interpolation inequality on bounded intervals\.

###### Lemma 7\(\[[35](https://arxiv.org/html/2610.27440#bib.bib35), Theorem 7\.34\]\)\.

LetΩ⊆ℝ\\Omega\\subseteq\\mathbb\{R\}be an open interval and letg∈𝕎loc1,1​\(Ω,ℝM\)g\\in\\mathbb\{W\}\_\{\\textrm\{loc\}\}^\{1,1\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\. Assume1≤p,v,r≤∞1\\leq p,v,r\\leq\\inftyandr≥vr\\geq v\. Then, for every0<ℓ<Leb⁡\(Ω\)0<\\ell<\\operatorname\{Leb\}\(\\Omega\),

∥g∥Lr​\(Ω,ℝM\)≤ℓ1r−1v​∥g∥Lv​\(Ω,ℝM\)\+ℓ1−1p\+1r​∥g′∥Lp​\(Ω,ℝM\)\.\\displaystyle\\lVert g\\rVert\_\{\{L\}^\{r\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}\\leq\\ell^\{\\frac\{1\}\{r\}\-\\frac\{1\}\{v\}\}\\lVert g\\rVert\_\{\{L\}^\{v\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}\+\\ell^\{1\-\\frac\{1\}\{p\}\+\\frac\{1\}\{r\}\}\\lVert g^\{\\prime\}\\rVert\_\{\{L\}^\{p\}\\left\(\\Omega;\\mathbb\{R\}^\{M\}\\right\)\}\.\(165\)

We now proceed with the proof of Lemma[3](https://arxiv.org/html/2610.27440#Thmlemma3)\. Letg:=f∘u=\[g1,…,gm\]⊤g:=f\\circ u=\[g\_\{1\},\\ldots,g\_\{m\}\]^\{\\top\}, whereu=\[u1,…,ud\]⊤∈ℋ2​\(Ω,ℝd\)u=\[u\_\{1\},\\ldots,u\_\{d\}\]^\{\\top\}\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\. By the second\-order chain rule for Sobolev functions, for eachj∈\[m\]j\\in\[m\]and almost everyz∈Ωz\\in\\Omega, we have

gj′′​\(z\)\\displaystyle g\_\{j\}^\{\\prime\\prime\}\(z\)=u′\(z\)⊤D2fj\(u\(z\)\)u′\(z\)\+∇fj\(u\(z\)\)⊤u′′\(z\)\.\\displaystyle=u^\{\\prime\}\(z\)^\{\\top\}D^\{2\}f\_\{j\}\(u\(z\)\)u^\{\\prime\}\(z\)\+\\nabla f\_\{j\}\(u\(z\)\)^\{\\top\}u^\{\\prime\\prime\}\(z\)\.\(166\)Applying the triangle inequality, the Cauchy–Schwarz inequality, and the definition of the operator norm, together with the regularity bounds in Assumption \(A2\), yields

\|gj′′​\(z\)\|\\displaystyle\\lvert g\_\{j\}^\{\\prime\\prime\}\(z\)\\rvert≤∥D2​fj​\(u⁡\(z\)\)∥op​∥u′​\(z\)∥22\+∥∇fj​\(u⁡\(z\)\)∥2​∥u′′​\(z\)∥2\\displaystyle\\leq\\lVert D^\{2\}f\_\{j\}\(u\(z\)\)\\rVert\_\{\\mathrm\{op\}\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{2\}\+\\lVert\\nabla f\_\{j\}\(u\(z\)\)\\rVert\_\{2\}\\lVert u^\{\\prime\\prime\}\(z\)\\rVert\_\{2\}≤Hj​∥u′​\(z\)∥22\+Gj​∥u′′​\(z\)∥2\.\\displaystyle\\leq H\_\{j\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{2\}\+G\_\{j\}\\lVert u^\{\\prime\\prime\}\(z\)\\rVert\_\{2\}\.\(167\)Squaring both sides and using the elementary inequality\(a\+b\)2≤2​a2\+2​b2\(a\+b\)^\{2\}\\leq 2a^\{2\}\+2b^\{2\}, we obtain

\|gj′′​\(z\)\|2≤2​Hj2​∥u′​\(z\)∥24\+2​Gj2​∥u′′​\(z\)∥22\.\\displaystyle\\lvert g\_\{j\}^\{\\prime\\prime\}\(z\)\\rvert^\{2\}\\leq 2H\_\{j\}^\{2\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{4\}\+2G\_\{j\}^\{2\}\\lVert u^\{\\prime\\prime\}\(z\)\\rVert\_\{2\}^\{2\}\.\(168\)Integrating overΩ\\Omegaand summing overj∈\[m\]j\\in\[m\], while recalling thatHf2=∑j=1mHj2H\_\{f\}^\{2\}=\\sum\_\{j=1\}^\{m\}H\_\{j\}^\{2\}andGf2=∑j=1mGj2G\_\{f\}^\{2\}=\\sum\_\{j=1\}^\{m\}G\_\{j\}^\{2\}, gives

∥g′′∥L2​\(Ω,ℝm\)2=∑j=1m∫Ω\|gj′′​\(z\)\|2​𝑑z≤2​Hf2​∫Ω∥u′​\(z\)∥24​𝑑z\+2​Gf2​∥u′′∥L2​\(Ω,ℝd\)2\.\\displaystyle\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}=\\sum\_\{j=1\}^\{m\}\\int\_\{\\Omega\}\\lvert g\_\{j\}^\{\\prime\\prime\}\(z\)\\rvert^\{2\}\\,dz\\leq 2H\_\{f\}^\{2\}\\int\_\{\\Omega\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{4\}\\,dz\+2G\_\{f\}^\{2\}\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\.\(169\)
Next, we bound the quartic integral∫Ω∥u′​\(z\)∥24​𝑑z\\int\_\{\\Omega\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{4\}\\,dzusing Lemma[7](https://arxiv.org/html/2610.27440#Thmlemma7)\. First, observe that by Minkowski’s inequality inL2​\(Ω,ℝ\)\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\),

∫Ω∥u′​\(z\)∥24​𝑑z=∫Ω\(∑i=1d\|ui′​\(z\)\|2\)2​𝑑z\\displaystyle\\int\_\{\\Omega\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{4\}\\,dz=\\int\_\{\\Omega\}\\left\(\\sum\_\{i=1\}^\{d\}\\lvert u\_\{i\}^\{\\prime\}\(z\)\\rvert^\{2\}\\right\)^\{2\}dz=∥∑i=1d\(ui′\)2∥L2​\(Ω,ℝ\)2\\displaystyle=\\lVert\\sum\_\{i=1\}^\{d\}\(u\_\{i\}^\{\\prime\}\)^\{2\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}≤\(∑i=1d∥\(ui′\)2∥L2​\(Ω,ℝ\)\)2=\(∑i=1d∥ui′∥L4​\(Ω,ℝ\)2\)2\.\\displaystyle\\leq\\left\(\\sum\_\{i=1\}^\{d\}\\lVert\(u\_\{i\}^\{\\prime\}\)^\{2\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\\right\)^\{2\}=\\left\(\\sum\_\{i=1\}^\{d\}\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{4\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\\right\)^\{2\}\.\(170\)Sinceu∈ℋ2​\(Ω,ℝd\)u\\in\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\), each coordinateuiu\_\{i\}satisfies

ui′∈𝕎1,1​\(Ω,ℝd\)⊆\(a\)𝕎loc1,1​\(Ω,ℝd\),\\displaystyle u\_\{i\}^\{\\prime\}\\in\\mathbb\{W\}^\{1,1\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\\stackrel\{\{\\scriptstyle\\text\{\(a\)\}\}\}\{\{\\subseteq\}\}\\mathbb\{W\}\_\{\\textrm\{loc\}\}^\{1,1\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\),\(171\)where \(a\) follows from Lemma[6](https://arxiv.org/html/2610.27440#Thmlemma6)\.

Fix anyℓ∈\(0,Leb⁡\(Ω\)\)\\ell\\in\(0,\\operatorname\{Leb\}\(\\Omega\)\)and apply Lemma[7](https://arxiv.org/html/2610.27440#Thmlemma7)toui′u\_\{i\}^\{\\prime\}withr=4r=4andp=v=2p=v=2:

∥ui′∥L4​\(Ω,ℝ\)\\displaystyle\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{4\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}≤ℓ−1/4∥ui′∥L2​\(Ω,ℝ\)\+ℓ3/4∥ui′′∥L2​\(Ω,ℝ\)\\displaystyle\\leq\\ell^\{\-1/4\}\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\+\\ell^\{3/4\}\\lVert u\_\{i\}^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}≤Cℓ​\(∥ui′∥L2​\(Ω,ℝ\)\+∥ui′′∥L2​\(Ω,ℝ\)\),\\displaystyle\\leq C\_\{\\ell\}\\left\(\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\+\\lVert u\_\{i\}^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}\\right\),\(172\)whereCℓ:=max\{ℓ−1/4,ℓ3/4\}C\_\{\\ell\}:=\\max\\\{\\ell^\{\-1/4\},\\ell^\{3/4\}\\\}\(for instance, sinceLeb⁡\(Ω\)=1\\operatorname\{Leb\}\(\\Omega\)=1, choosingℓ=1/2\\ell=1/2givesCℓ=21/4C\_\{\\ell\}=2^\{1/4\}\)\. Squaring both sides and applying\(a\+b\)2≤2​a2\+2​b2\(a\+b\)^\{2\}\\leq 2a^\{2\}\+2b^\{2\}yields

∥ui′∥L4​\(Ω,ℝ\)2≤2​Cℓ2​\(∥ui′∥L2​\(Ω,ℝ\)2\+∥ui′′∥L2​\(Ω,ℝ\)2\)\.\\displaystyle\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{4\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\\leq 2C\_\{\\ell\}^\{2\}\\left\(\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\+\\lVert u\_\{i\}^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}\\right\)\.\(173\)Summing overi∈\[d\]i\\in\[d\]gives

∑i=1d∥ui′∥L4​\(Ω,ℝ\)2\\displaystyle\\sum\_\{i=1\}^\{d\}\\lVert u\_\{i\}^\{\\prime\}\\rVert\_\{\{L\}^\{4\}\\left\(\\Omega;\\mathbb\{R\}\\right\)\}^\{2\}≤2​Cℓ2​\(∥u′∥L2​\(Ω,ℝd\)2\+∥u′′∥L2​\(Ω,ℝd\)2\)\\displaystyle\\leq 2C\_\{\\ell\}^\{2\}\\left\(\\lVert u^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\+\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\)≤\(a\)2​Cℓ2​∥u∥𝕎2,2​\(Ω,ℝd\)2\\displaystyle\\stackrel\{\{\\scriptstyle\\text\{\(a\)\}\}\}\{\{\\leq\}\}2C\_\{\\ell\}^\{2\}\\lVert u\\rVert\_\{\\mathbb\{W\}^\{2,2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}≤\(b\)2​Cℓ2​θsob2​∥u∥ℋ2​\(Ω,ℝd\)2,\\displaystyle\\stackrel\{\{\\scriptstyle\\text\{\(b\)\}\}\}\{\{\\leq\}\}2C\_\{\\ell\}^\{2\}\\theta^\{2\}\_\{\\text\{sob\}\}\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\},\(174\)
where \(a\) follows from∥u′∥L2​\(Ω,ℝd\)2\+∥u′′∥L2​\(Ω,ℝd\)2≤∥u∥𝕎2,2​\(Ω,ℝd\)2\\lVert u^\{\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\+\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\leq\\lVert u\\rVert\_\{\\mathbb\{W\}^\{2,2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}and \(b\) comes from the equivalency of∥⋅∥𝕎2,2​\(Ω,ℝd\)\\lVert\\cdot\\rVert\_\{\\mathbb\{W\}^\{2,2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}and∥⋅∥ℋ2​\(Ω,ℝd\)\\lVert\\cdot\\rVert\_\{\\mathcal\{H\}\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)with a factorθsob\\theta\_\{\\text\{sob\}\}\(see Section[III\-A1](https://arxiv.org/html/2610.27440#S3.SS1.SSS1)for more details\)\.

Substituting this bound into \([170](https://arxiv.org/html/2610.27440#A2.Ex32)\), we obtain

∫Ω∥u′​\(z\)∥24​𝑑z≤Cℓ4​θsob4​∥u∥ℋ2​\(Ω,ℝd\)4\.\\displaystyle\\int\_\{\\Omega\}\\lVert u^\{\\prime\}\(z\)\\rVert\_\{2\}^\{4\}\\,dz\\leq C\_\{\\ell\}^\{4\}\\theta^\{4\}\_\{\\text\{sob\}\}\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{4\}\.\(175\)
Finally, substituting \([175](https://arxiv.org/html/2610.27440#A2.E175)\) and the bound∥u′′∥L2​\(Ω,ℝd\)2≤∥u∥ℋ2​\(Ω,ℝd\)2\\lVert u^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\leq\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}into \([169](https://arxiv.org/html/2610.27440#A2.E169)\) yields

∥g′′∥L2​\(Ω,ℝm\)2\\displaystyle\\lVert g^\{\\prime\\prime\}\\rVert\_\{\{L\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{m\}\\right\)\}^\{2\}≤8​Cℓ4​θsob4​Hf2​∥u∥ℋ2​\(Ω,ℝd\)4\+2​Gf2​∥u∥ℋ2​\(Ω,ℝd\)2\\displaystyle\\leq 8C\_\{\\ell\}^\{4\}\\theta^\{4\}\_\{\\text\{sob\}\}H\_\{f\}^\{2\}\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{4\}\+2G\_\{f\}^\{2\}\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}≤2​max⁡\{Cℓ4​θsob4,1\}​max​\{Hf2,Gf2\}​\(∥u∥ℋ2​\(Ω,ℝd\)2\+4​∥u∥ℋ2​\(Ω,ℝd\)4\)\\displaystyle\\leq 2\\max\\\{C\_\{\\ell\}^\{4\}\\theta^\{4\}\_\{\\text\{sob\}\},1\\\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\left\(\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\+4\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{4\}\\right\)=Cc​max⁡\{Hf2,Gf2\}​ψ​\(∥u∥ℋ2​\(Ω,ℝd\)2\),\\displaystyle=C\_\{\\mathrm\{c\}\}\\max\\\{H\_\{f\}^\{2\},G\_\{f\}^\{2\}\\\}\\psi\\left\(\\lVert u\\rVert\_\{\\mathcal\{H\}^\{2\}\\left\(\\Omega;\\mathbb\{R\}^\{d\}\\right\)\}^\{2\}\\right\),\(176\)whereCc:=2​max⁡\{Cℓ4​θsob4,1\}C\_\{\\mathrm\{c\}\}:=2\\max\\\{C^\{4\}\_\{\\ell\}\\theta^\{4\}\_\{\\text\{sob\}\},1\\\}\. This completes the proof\.

### B\-CProof of Lemma[4](https://arxiv.org/html/2610.27440#Thmlemma4)

Using\|sin⁡v\|≤1\|\\sin v\|\\leq 1and\|cos⁡v\|≤1,\|\\cos v\|\\leq 1,we have

\|Φ⁡\(u,v\)\|≤4​e−u,u≥0\.\\displaystyle\|\\Phi\(u,v\)\|\\leq 4e^\{\-u\},\\qquad u\\geq 0\.\(177\)Applying this bound to \([76](https://arxiv.org/html/2610.27440#S5.Ex14)\) gives

\|Kλ​\(x,t\)\|\\displaystyle\\lvert K\_\{\\lambda\}\(x,t\)\\rvert≤12​λ1/4​e−\|x−t\|2​λ1/4\+42​λ1/4​\(e−x\+t2​λ1/4\+e−2−x−t2​λ1/4\)\\displaystyle\\leq\\frac\{1\}\{\\sqrt\{2\}\\lambda^\{1/4\}\}e^\{\-\\frac\{\|x\-t\|\}\{\\sqrt\{2\}\\lambda^\{1/4\}\}\}\+\\frac\{4\}\{\\sqrt\{2\}\\lambda^\{1/4\}\}\\left\(e^\{\-\\frac\{x\+t\}\{\\sqrt\{2\}\\lambda^\{1/4\}\}\}\+e^\{\-\\frac\{2\-x\-t\}\{\\sqrt\{2\}\\lambda^\{1/4\}\}\}\\right\)≤92λ−1/4,\\displaystyle\\leq\\frac\{9\}\{\\sqrt\{2\}\}\\lambda^\{\-1/4\},\(178\)which proves \([77](https://arxiv.org/html/2610.27440#S5.E77)\)\.

### B\-DProof of Lemma[5](https://arxiv.org/html/2610.27440#Thmlemma5)

To establish Lemma[5](https://arxiv.org/html/2610.27440#Thmlemma5), we first relate the equivalent kernelKλK\_\{\\lambda\}to the continuous Green’s functionGλG\_\{\\lambda\}, and then bound the discrepancy betweenGλG\_\{\\lambda\}and its finite\-sample counterpartGN,λG\_\{N,\\lambda\}\.

###### Lemma 8\(\[[31](https://arxiv.org/html/2610.27440#bib.bib31), Theorem 4\.1\]\)\.

There exists a constantCM\>0C\_\{\\mathrm\{M\}\}\>0such that for everyλ∈\(0,1\]\\lambda\\in\(0,1\],

supx,t∈\[0,1\]\|Kλ\(x,t\)−Gλ\(x,t\)\|≤CMλ−1/4exp\(−12λ−1/4\)\.\\displaystyle\\sup\_\{x,t\\in\[0,1\]\}\\lvert K\_\{\\lambda\}\(x,t\)\-G\_\{\\lambda\}\(x,t\)\\rvert\\leq C\_\{\\mathrm\{M\}\}\\lambda^\{\-1/4\}\\exp\\left\(\-\\frac\{1\}\{\\sqrt\{2\}\}\\lambda^\{\-1/4\}\\right\)\.\(179\)

Next, we relate the continuous Green’s functionGλG\_\{\\lambda\}to the finite\-sample smoothing spline weight functionGN,λG\_\{N,\\lambda\}\.

###### Lemma 9\(\[[36](https://arxiv.org/html/2610.27440#bib.bib36), Theorem 2\.1\]\)\.

Under equidistant design points, there exist constantsμ0,μ1,μ2\>0\\mu\_\{0\},\\mu\_\{1\},\\mu\_\{2\}\>0such that wheneverN​λ1/4\>μ0N\\lambda^\{1/4\}\>\\mu\_\{0\}, we have for allx,t∈\[0,1\]x,t\\in\[0,1\],

\|GN,λ​\(x,t\)−Gλ​\(x,t\)\|\\displaystyle\\lvert G\_\{N,\\lambda\}\(x,t\)\-G\_\{\\lambda\}\(x,t\)\\rvert≤μ0μ1N−1λ−1/41−μ0N−1λ−1/4λ−1/4exp\(−μ2\|x−t\|λ−1/4\)\.\\displaystyle\\leq\\frac\{\\mu\_\{0\}\\mu\_\{1\}N^\{\-1\}\\lambda^\{\-1/4\}\}\{1\-\\mu\_\{0\}N^\{\-1\}\\lambda^\{\-1/4\}\}\\lambda^\{\-1/4\}\\exp\\left\(\-\\mu\_\{2\}\\lvert x\-t\\rvert\\lambda^\{\-1/4\}\\right\)\.\(180\)

We now combine Lemmas[8](https://arxiv.org/html/2610.27440#Thmlemma8)and[9](https://arxiv.org/html/2610.27440#Thmlemma9)to complete the proof\. Setc0:=2​μ0\>0c\_\{0\}:=2\\mu\_\{0\}\>0\. Sinceλd∈\(0,1\]\\lambda\_\{\\textrm\{d\}\}\\in\(0,1\]andN​λd1/4\>c0N\\lambda\_\{\\textrm\{d\}\}^\{1/4\}\>c\_\{0\}, we haveN​λd1/4\>2​μ0\>μ0N\\lambda\_\{\\textrm\{d\}\}^\{1/4\}\>2\\mu\_\{0\}\>\\mu\_\{0\}\. Therefore, the condition of Lemma[9](https://arxiv.org/html/2610.27440#Thmlemma9)holds\. Moreover, we have

μ0N−1λd−1/4<12,\\displaystyle\\mu\_\{0\}N^\{\-1\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}<\\frac\{1\}\{2\},\(181\)which implies1−μ0N−1λd−1/4\>1/21\-\\mu\_\{0\}N^\{\-1\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\>1/2\. Consequently, the prefactor in \([180](https://arxiv.org/html/2610.27440#A2.E180)\) is bounded above by

μ0μ1N−1λd−1/41−μ0N−1λd−1/4<μ1/21/2=μ1\.\\displaystyle\\frac\{\\mu\_\{0\}\\mu\_\{1\}N^\{\-1\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\}\{1\-\\mu\_\{0\}N^\{\-1\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\}<\\frac\{\\mu\_\{1\}/2\}\{1/2\}=\\mu\_\{1\}\.\(182\)
By the triangle inequality, for allx,t∈\[0,1\]x,t\\in\[0,1\],

\|GN,λd​\(x,t\)−Kλd​\(x,t\)\|≤\|GN,λd​\(x,t\)−Gλd​\(x,t\)\|\+\|Gλd​\(x,t\)−Kλd​\(x,t\)\|\.\\displaystyle\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\-K\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert\\leq\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\-G\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert\+\\lvert G\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\-K\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert\.\(183\)Applying Lemmas[8](https://arxiv.org/html/2610.27440#Thmlemma8)and[9](https://arxiv.org/html/2610.27440#Thmlemma9)together with \([182](https://arxiv.org/html/2610.27440#A2.E182)\) to \([183](https://arxiv.org/html/2610.27440#A2.E183)\) yields

\|GN,λd​\(x,t\)−Kλd​\(x,t\)\|\\displaystyle\\lvert G\_\{N,\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\-K\_\{\\lambda\_\{\\textrm\{d\}\}\}\(x,t\)\\rvert≤μ0μ1N−1λd−1/41−μ0N−1λd−1/4λd−1/4exp\(−μ2\|x−t\|λd−1/4\)\+CMλd−1/4exp\(−12λd−1/4\)\\displaystyle\\leq\\frac\{\\mu\_\{0\}\\mu\_\{1\}N^\{\-1\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\}\{1\-\\mu\_\{0\}N^\{\-1\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\exp\\left\(\-\\mu\_\{2\}\\lvert x\-t\\rvert\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\right\)\+C\_\{\\mathrm\{M\}\}\\lambda\_\{\\textrm\{d\}\}^\{\-1/4\}\\exp\\left\(\-\\frac\{1\}\{\\sqrt\{2\}\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{\-1/4\}\\right\)≤μ1λd−1/4exp\(−μ2\|x−t\|λd−1/4\)\+CMλd−1/4exp\(−12λd−1/4\)\.\\displaystyle\\leq\\mu\_\{1\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{\-1/4\}\\exp\\left\(\-\\mu\_\{2\}\\lvert x\-t\\rvert\{\\lambda\_\{\\textrm\{d\}\}\}^\{\-1/4\}\\right\)\+C\_\{\\mathrm\{M\}\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{\-1/4\}\\exp\\left\(\-\\frac\{1\}\{\\sqrt\{2\}\}\{\\lambda\_\{\\textrm\{d\}\}\}^\{\-1/4\}\\right\)\.\(184\)DefiningCW:=μ1C\_\{W\}:=\\mu\_\{1\}andκ:=μ2\\kappa:=\\mu\_\{2\}gives \([78](https://arxiv.org/html/2610.27440#S5.E78)\) and completes the proof\.