Bern2Edge: A Neurosymbolic Compiler for Edge Deployment via Bernstein Polynomial Networks

# A Neurosymbolic Compiler for Edge Deployment via Bernstein Polynomial Networks
Source: [https://arxiv.org/html/2608.20497](https://arxiv.org/html/2608.20497)
## Bern2Edge: A Neurosymbolic Compiler for Edge Deployment via Bernstein Polynomial NetworksThanks:Code repo can be found at:[https://github\.com/PervasiveAutonomyLab/Bern2Edge](https://github.com/PervasiveAutonomyLab/Bern2Edge)\. Archived at[https://zenodo\.org/records/21726441](https://zenodo.org/records/21726441)\.PubID:pubid:© 2026 IEEE\. Personal use of this material is permitted\. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works\.

Yifan ZhangYasser ShoukrySitao HuangSalma ElmalakiThanks:All authors are with the Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697 USA \(e\-mail: mgamalel@uci\.edu; yifanz58@uci\.edu; yshoukry@uci\.edu; sitaoh@uci\.edu; selmalak@uci\.edu\)\.

###### Abstract

Deploying high\-accuracy neural networks on resource\-constrained edge devices remains challenging, as existing approaches treat training, compression, and hardware synthesis as separate stages, leaving a gap between software\-trained models and efficient end\-to\-end deployment with limited support for interpretability\. We proposeBern2Edge, an end\-to\-end framework that uses knowledge distillation to convert a pretrained teacher feed\-forward network into hardware\-efficient representations via Bernstein polynomial activations\. This representation enables two deployment paths: \(i\) a high\-fidelity LUT\-based realization that preserves model fidelity under compression, and \(ii\) a symbolic rule\-based representation derived from Bernstein activation geometry, enabling interpretable inference with explicit input\-space constraints\. The resulting BNNs achieve up to2\.122\.12percentage\-point \(pp\) accuracy improvement over ReLU under identical compression constraints\. At the system level,Bern2Edgeachieves up to99\.899\.8% latency reduction and95\.295\.2% BRAM reduction relative to a W8A8 quantized teacher on an AMD Xilinx KV260 FPGA, while maintaining accuracy within0\.50\.5pp, and further deploys on a low\-power Spartan\-7 XC7S15 FPGA\. The rule\-based path reduces DSP usage by up to89\.089\.0% at a cost of1\.51\.5pp in total accuracy\.

###### Index Terms:

Bernstein polynomials, activation functions, knowledge distillation, lookup table, FPGA inference, symbolic rule extraction, edge ML\.

## IIntroduction

The rapid growth of deep neural networks \(DNNs\) has driven significant gains in predictive performance, but has made deployment on resource\-constrained edge devices increasingly difficult\. Networks are typically trained in software without accounting for hardware constraints, yielding models that are hard to map to strict latency and resource budgets, while limited interpretability further restricts their use in edge settings where transparency and reliability are critical\. Bridging high\-level model design and efficient hardware realization remains a key challenge, as existing approaches treat model design and hardware optimization separately, limiting their ability to satisfy both accuracy and system\-level constraints\[[24](https://arxiv.org/html/2608.20497#bib.bib1)\]\.

Prior work addresses parts of this problem through model compression, quantization, and hardware compilation frameworks such as hls4ml\[[15](https://arxiv.org/html/2608.20497#bib.bib2)\], FINN\[[8](https://arxiv.org/html/2608.20497#bib.bib3)\], and CGRA4ML\[[1](https://arxiv.org/html/2608.20497#bib.bib4)\], which translate trained networks into synthesizable hardware, often applying quantization and pruning during deployment\. However, these operate on fixed neural representations, mapping or optimizing pre\-trained models for hardware rather than jointly designing representations inherently aligned with hardware efficiency\.

We introduceBern2Edge, an end\-to\-end neurosymbolic compiler for edge deployment based on Bernstein polynomial activations\[[26](https://arxiv.org/html/2608.20497#bib.bib5)\], targeting feed\-forward networks \(FFNs\), including multilayer perceptrons \(MLPs\) and the FFN sublayers of transformers\. The structured, bounded representation of Bernstein polynomials suits both hardware realization and symbolic reasoning\[[16](https://arxiv.org/html/2608.20497#bib.bib6)\]\. Our framework transforms a trained DNN teacher into hardware\-efficient representations via knowledge distillation \(KD\)\[[20](https://arxiv.org/html/2608.20497#bib.bib7)\], training a compact student with Bernstein activations, which we refer to as a Bernstein Neural Network \(BNN\)\. This supports two deployment paths: a lookup table \(LUT\)\-based realization, where a per\-neuron LUT represents the Bernstein activation for compact, efficient synthesis, and an optional symbolic rule\-based representation derived from Bernstein activation geometry, enabling further compression and interpretable inference\. To our knowledge, this is the first work to jointly exploit the structural properties of Bernstein polynomial activations for both efficient hardware synthesis and symbolic rule extraction\.The contributions of this work are as follows:

- •End\-to\-end teacher\-to\-hardware deployment framework\.We proposeBern2Edge, a unified pipeline compressing a trained teacher DNN into a hardware\-efficient BNN via KD, supporting two deployment paths: a LUT\-based realization for efficient hardware execution, and a symbolic rule\-based representation for additional compression and interpretable inference\.
- •Bernstein activations for improved compression under knowledge distillation\.We introduce Bernstein polynomial activations as a structured student representation that improves KD under strong compression: BNNs recover the accuracy of large DNN teachers more effectively than standard activations such as ReLU under compressed regimes, while remaining well\-suited for hardware realization and interpretability\.
- •LUT\-based hardware realization\.Bernstein polynomial activations admit a direct LUT\-based implementation leveraging their normalized input domain and fixed functional structure, enabling efficient per\-neuron LUT realization and a compact representation for edge inference\.
- •Symbolic rule extraction and synthesis\.We develop a BNN\-specific rule extraction method deriving compact symbolic rules from the geometric structure of learned Bernstein activations, forming a structured, hardware\-efficient representation that supports quantization to low bitwidth with negligible accuracy loss\.
- •Evaluation in compression and deployment regimes\.We evaluateBern2Edgeon multiple tabular datasets and a transformer FFN setting, demonstrating: \(i\) improved performance of BNNs under strong compression, \(ii\) efficient LUT\-based hardware synthesis at cost comparable to or less than standard activations, \(iii\) compact and effective symbolic rule representations, \(iv\) end\-to\-end deployment on AMD Xilinx KV260 and low\-power Spartan\-7 XC7S15 FPGAs, and \(v\) robustness of the rule network under input noise and distribution shift\.

## IIBackground and Related Work

### II\-AEdge Deployment Frameworks and Model Compression

A large body of work targets neural network deployment on resource\-constrained hardware through compiler frameworks and model compression\. hls4ml\[[15](https://arxiv.org/html/2608.20497#bib.bib2)\]translates trained networks into FPGA implementations via high\-level synthesis, mapping dense and convolutional operators to streaming architectures with fixed\-point quantization\. FINN\[[8](https://arxiv.org/html/2608.20497#bib.bib3)\]extends this to quantized networks, generating dataflow accelerators for low\-bitwidth, high\-throughput FPGA inference\. CGRA4ML\[[1](https://arxiv.org/html/2608.20497#bib.bib4)\]extends these approaches to coarse\-grained reconfigurable architectures \(CGRAs\)\. These systems primarily map and optimize pre\-trained models for hardware through post\-training transformations \(quantization, pruning, operator fusion\) rather than co\-designing the model representation itself\.

Our work instead targets*training\-time*representation: we train Bernstein polynomial activations whose learned forms are inherently deployable, rather than adapting arbitrary activations for hardware after the fact, enabling direct symbolic extraction and a unified pipeline from training to hardware implementation\.

Since many high\-performing models are pretrained with ReLU\-based architectures, we employ knowledge distillation \(KD\) to transfer large pretrained teachers into compact BNNs\. KD is a standard compression technique matching soft outputs or intermediate representations between a teacher and student\[[20](https://arxiv.org/html/2608.20497#bib.bib7)\]\. InBern2Edge, KD trains student BNNs that retain the accuracy of much larger models while maintaining a structured, hardware\-friendly representation\.

### II\-BHardware Realization of Nonlinear Activations

Efficient implementation of nonlinear activations remains a key challenge in hardware deployment\. Piecewise\-linear functions such as ReLU map efficiently to comparator\-based logic, but expressive smooth activations such as GeLU\[[19](https://arxiv.org/html/2608.20497#bib.bib8)\]and Swish\[[31](https://arxiv.org/html/2608.20497#bib.bib9)\]require more complex arithmetic and are typically approximated using lookup tables or piecewise\-polynomial methods\[[11](https://arxiv.org/html/2608.20497#bib.bib10)\], introducing a train–deploy gap between the trained and deployed function\.

LUT\-centric designs address this by mapping neuron computations directly to lookup tables\. LUTNet\[[37](https://arxiv.org/html/2608.20497#bib.bib11)\]uses native FPGA LUTs as inference operators to reduce arithmetic cost, while LogicNets\[[35](https://arxiv.org/html/2608.20497#bib.bib29)\]co\-designs sparse, quantized neurons extractable as LUT truth tables\. Polynomial\-based methods such as PolyLUT\[[4](https://arxiv.org/html/2608.20497#bib.bib12)\]and PolyLUT\-Add\[[28](https://arxiv.org/html/2608.20497#bib.bib13)\]extend this to structured polynomial classes, but both require hard fan\-in limits of≤7\\leq 7inputs per neuron to bound LUT size: PolyLUT enforces sparse connectivity, while PolyLUT\-Add sums low\-fan\-in sub\-neurons\. This is incompatible with the dense tabular MLPs we target \(1414–5454input features\), requiring additional sparsification or redesign to apply\.Bern2Edgeinstead retains dense MLP layers and makes the learned activation itself hardware\-realizable\.

KANELÉ\[[21](https://arxiv.org/html/2608.20497#bib.bib14)\]explores activation\-centric LUT mappings via Kolmogorov–Arnold Networks \(KANs\), replacing MLP computations with learnable one\-dimensional edge functions discretized into per\-neuron LUTs, departing from standard MLP structure\. In contrast,Bern2Edgeretains standard FFN architectures and targets the activation directly: Bernstein activation coefficients define the deployed function, enabling exact realization via small per\-neuron LUTs without a train–deploy gap\.

### II\-CBernstein Polynomial Activations

∙\\bulletBernstein Polynomials\.A Bernstein polynomialσ⁡\(z\)\\sigma\(z\)of degreenn, defined on the interval\[l,u\]\[l,u\], is written as\[[16](https://arxiv.org/html/2608.20497#bib.bib6)\]:

σ⁡\(z,l,u,𝐜\)=∑k=0nck​bn,k​\(t\),t=z−lu−l∈\[0,1\],z∈\[l,u\],\\sigma\(z;\\,l,u,\\mathbf\{c\}\)=\\sum\_\{k=0\}^\{n\}c\_\{k\}\\,b\_\{n,k\}\(t\),\\quad t=\\frac\{z\-l\}\{u\-l\}\\in\[0,1\],\\quad z\\in\[l,u\],\(1\)wherezzis the input to the polynomial function,ttis the normalized input, and𝐜=\{c0,…,cn\}\\mathbf\{c\}=\\\{c\_\{0\},\\ldots,c\_\{n\}\\\}are the polynomial coefficients controlling the activation shape\. The Bernstein basis functions are defined as:

bn,k\(t\)=\(nk\)tk\(1−t\)n−k,k=0,…,n,b\_\{n,k\}\(t\)=\\binom\{n\}\{k\}t^\{k\}\(1\-t\)^\{n\-k\},\\quad k=0,\\ldots,n,\(2\)satisfying the partition of unity property:

∑k=0nbn,k​\(t\)=1,∀t∈\[0,1\]\.\\sum\_\{k=0\}^\{n\}b\_\{n,k\}\(t\)=1,\\quad\\forall\\,t\\in\[0,1\]\.\(3\)This defines the polynomial as a convex combination of its coefficients over a bounded domain, with the coefficients directly controlling the activation shape\.

∙\\bulletBernstein Neural Networks \(BNNs\)\.Bernstein polynomials have been explored in neural networks as a nonlinear activation due to their favorable analytic and approximation properties\. Khedr et al\.\[[26](https://arxiv.org/html/2608.20497#bib.bib5)\]introduce Deep Bernstein Networks, feed\-forward networks in which standard activations are replaced by learnable Bernstein polynomials\. We adopt this formulation throughoutBern2Edgeand refer to it as a Bernstein Neural Network \(BNN\)\. For a network of depthLL, the input is denoted𝐱\(0\)\\mathbf\{x\}^\{\(0\)\}and the output of thell\-th layer𝐱\(l\)\\mathbf\{x\}^\{\(l\)\}, with propagation rule:

𝐱\(l\)=σ⁡\(𝐳\(l\),𝐜\(l\)\),zi\(l\)=\(𝐰i\(l\)\)⊤​𝐱\(l−1\)\+βi\(l\)\\mathbf\{x\}^\{\(l\)\}=\\sigma\\left\(\\mathbf\{z\}^\{\(l\)\};\\mathbf\{c\}^\{\(l\)\}\\right\),\\quad z\_\{i\}^\{\(l\)\}=\\left\(\\mathbf\{w\}\_\{i\}^\{\(l\)\}\\right\)^\{\\\!\\top\}\\mathbf\{x\}^\{\(l\-1\)\}\+\\beta\_\{i\}^\{\(l\)\}\(4\)where𝐰i\(l\)\\mathbf\{w\}\_\{i\}^\{\(l\)\}andβi\(l\)\\beta\_\{i\}^\{\(l\)\}are the learnable weights and biases\. The activation functionσ\\sigmaoperates element\-wise, parametrized by learnable Bernstein coefficients𝐜\(l\)=\{ck\(l\)\}k=0n\\mathbf\{c\}^\{\(l\)\}=\\\{c\_\{k\}^\{\(l\)\}\\\}\_\{k=0\}^\{n\}, wherennis a hyperparameter for the polynomial degree, allowing the network to learn the shape of its nonlinearities alongside its weights\. For simplicity of notation, we drop the superscript\(l\)\(l\)in the remainder of the paper\.

∙\\bulletTheoretical Properties of BNNs\.\(1\) Stable training: unlike other polynomial activations, which suffer exploding\-gradient instability that worsens with degree\[[18](https://arxiv.org/html/2608.20497#bib.bib39)\], Bernstein polynomials remain stable\[[26](https://arxiv.org/html/2608.20497#bib.bib5),[3](https://arxiv.org/html/2608.20497#bib.bib15)\]\. \(2\) Exponential approximation rates and parameter efficiency: Bernstein\-based architectures achieve approximation error bounds superior to standard feed\-forward ReLU networks\[[3](https://arxiv.org/html/2608.20497#bib.bib15)\], approximating smooth functions at much lower complexity than piecewise\-linear ReLU networks\. These properties motivate the use of BNNs to compress larger networks\.

InBern2Edge, we exploit the structure of the Bernstein polynomial activation for both hardware synthesis and symbolic rule extraction\.

### II\-DSymbolic Rule Extraction

Prior work on symbolic rule extraction from neural networks falls into pedagogical and decompositional methods\. Pedagogical approaches such as TREPAN\[[13](https://arxiv.org/html/2608.20497#bib.bib16)\]treat the trained network as a black box and induce a decision tree by querying its input–output behavior\. Decompositional approaches instead exploit internal network structure: DeepRED\[[40](https://arxiv.org/html/2608.20497#bib.bib17)\]extends earlier rule extraction methods to deep networks, extracting intermediate rules layer by layer and using decision trees to describe hidden\-layer behavior, while ECLAIRE\[[39](https://arxiv.org/html/2608.20497#bib.bib18)\]improves the scalability of decompositional rule extraction while maintaining high\-quality, interpretable rule sets\. NeuSym\-HLS\[[29](https://arxiv.org/html/2608.20497#bib.bib37)\]applies hardware\-aware symbolic regression to replace internal neural layers with compact analytic expressions\.

Earlier activation\-aware techniques such as validity interval analysis\[[34](https://arxiv.org/html/2608.20497#bib.bib19)\]propagate activation intervals through the network to derive symbolic descriptions, but treat activations generically without exploiting a learned activation family’s analytic structure\.

Bern2Edgediffers from these approaches by deriving rule candidates directly from the geometric structure of the learned Bernstein activations, instead of inducing decision trees or propagating interval bounds\. This yields semantically aligned partitions tied to the network’s learned nonlinear representation and supports a compact symbolic rule set suitable for efficient deployment\.

## IIIBern2EdgeFramework Overview

We proposeBern2Edge, an end\-to\-end pipeline that transforms a trained teacher model into hardware\-efficient and interpretable representations\. The framework consists of four modules: \(1\) BNN training via KD, followed by two alternative deployment paths: \(2\) LUT\-based hardware realization, or \(3\) symbolic rule extraction, both followed by \(4\) hardware synthesis and deployment\. Fig\.[1](https://arxiv.org/html/2608.20497#S3.F1)shows theBern2Edgepipeline\. The following sections describe each module and the Bernstein polynomial properties that enable these capabilities\.

![Refer to caption](https://arxiv.org/html/2608.20497v1/fig1_pipeline.png)Fig\. 1:Overview ofBern2Edge: A high\-accuracy teacher model is distilled into a compressed BNN student via KD\. The resulting representation is synthesized and deployed via either exact LUT\-based realization or symbolic rule extraction\.### III\-ABNN Training via Knowledge Distillation

Knowledge Distillation Setup\.The first step in theBern2Edgepipeline is training compressed BNNs as student models using KD\[[20](https://arxiv.org/html/2608.20497#bib.bib7)\], reducing model size while preserving accuracy\.

Bernstein\-based networks exhibit favorable approximation properties under a constrained parameter budget, enabling accurate function representation with fewer parameters than standard activations\[[3](https://arxiv.org/html/2608.20497#bib.bib15)\], motivating our use of Bernstein activations in the student network: a Bernstein\-based student can recover more of the teacher’s accuracy than a ReLU\-based student of equal size, an effect confirmed across all evaluated architectures in Section[IV\-B](https://arxiv.org/html/2608.20497#S4.SS2)\.

This is further reflected in the BNN’s learned decision geometry: Bernstein\-based models capture smoother, more curved decision boundaries, while standard activations such as ReLU tend to produce piecewise\-linear boundaries, as illustrated in Fig\.[2](https://arxiv.org/html/2608.20497#S3.F2)\.

The student objective follows the standard distillation loss\[[20](https://arxiv.org/html/2608.20497#bib.bib7)\]:

ℒs=\(1−α\)​ℒCE\+α​T2​KL​\(S⁡\(𝐲∗/T\)∥S⁡\(𝐲^/T\)\),\\mathcal\{L\}\_\{s\}=\(1\-\\alpha\)\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\+\\alpha T^\{2\}\\,\\mathrm\{KL\}\\\!\\left\(S\(\\mathbf\{y\}^\{\*\}/T\)\\,\\middle\\\|\\,S\(\\hat\{\\mathbf\{y\}\}/T\)\\right\),\(5\)whereℒCE\\mathcal\{L\}\_\{\\mathrm\{CE\}\}is the cross\-entropy loss between student predictions and ground\-truth labels,S⁡\(⋅\)S\(\\cdot\)is the softmax,KL\(⋅∥⋅\)\\mathrm\{KL\}\(\\cdot\\\|\\cdot\)is the Kullback–Leibler divergence between softened teacher and student distributions, and𝐲^\\hat\{\\mathbf\{y\}\}and𝐲∗\\mathbf\{y\}^\{\*\}are the student and teacher logits, with𝐲^=𝐱\(L\)\\hat\{\\mathbf\{y\}\}=\\mathbf\{x\}^\{\(L\)\}the student’s final layer output\.TTis the temperature controlling output\-distribution softness, andα\\alphabalances supervision between hard labels and teacher guidance\.

Teacher models are selected to provide strong supervision signals consistent with prior tabular benchmarks, following the MLP baselines of\[[17](https://arxiv.org/html/2608.20497#bib.bib21)\]\. Student architectures are designed to satisfy strict compression and latency constraints\. While Bernstein activations introduce additional learnable parameters as polynomial coefficients, these do not translate to increased hardware cost under our LUT\-based activation implementation, as described next \(Section[III\-B](https://arxiv.org/html/2608.20497#S3.SS2)\)\.

![Refer to caption](https://arxiv.org/html/2608.20497v1/decision_boundary_combined.png)Fig\. 2:Decision boundaries learned by a BNN \(left\) and a ReLU model \(right\) of equal capacity on the synthetic Two Moons dataset\[[30](https://arxiv.org/html/2608.20497#bib.bib20)\]\. The BNN captures the curved structure of the data, while the ReLU model produces a simpler, predominantly linear boundary\.BNN Fixed\-Bound Training\.Bernstein activations operate over a fixed input domain, requiring inputs normalized to\[0,1\]\[0,1\]as in Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1), a property later leveraged for efficient hardware realization\. After a short warmup phase, per\-neuron pre\-activation bounds\[li,ui\]\[l\_\{i\},u\_\{i\}\]are calibrated from empirical activation statistics and fixed for the remainder of training, sequentially, one layer at a time, once preceding layers have stabilized\.

Fixing these bounds serves two purposes: stabilizing training by preventing continual shifts in activation normalization, and enforcing the fixed\-domain consistency required for exact LUT\-based deployment \(Section[III\-B](https://arxiv.org/html/2608.20497#S3.SS2)\)\.

To discourage out\-of\-bounds inputs, we add an out\-of\-bounds penalty to the BNN loss function:

ℒBNN=ℒs\+λob⋅𝔼⁡\[max⁡\(0,−t\)\+max⁡\(0,t−1\)\],\\mathcal\{L\}\_\{\\mathrm\{BNN\}\}=\\mathcal\{L\}\_\{s\}\+\\lambda\_\{\\mathrm\{ob\}\}\\cdot\\mathbb\{E\}\\left\[\\max\(0,\-t\)\+\\max\(0,t\-1\)\\right\],\(6\)wherettdenotes the normalized pre\-activation, as in Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1)\. During forward propagation, activations are clamped after normalization to preserve the Bernstein basis structure, ensuring non\-negativity and partition\-of\-unity as in Eq\.[3](https://arxiv.org/html/2608.20497#S2.E3)\. Section[IV\-B](https://arxiv.org/html/2608.20497#S4.SS2)compares compressed student BNNs via KD against similarly compressed ReLU networks under identical hardware constraints\.

### III\-BLUT\-Based Hardware Realization

The next stage of theBern2Edgeframework is the hardware synthesis of the trained BNN on FPGA\. As shown in Fig\.[1](https://arxiv.org/html/2608.20497#S3.F1), one deployment path leverages a LUT\-based realization of Bernstein activations\.

Direct evaluation of the Bernstein activation in Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1)is computationally expensive\. For a polynomial of degreenn, it requires\(n\+1\)\(n\+1\)power operations,\(n\+1\)\(n\+1\)multiplications with binomial coefficients, and an\(n\+1\)\(n\+1\)\-term dot product with learned coefficients, resulting in approximately3​\(n\+1\)3\(n\+1\)multiply\-accumulate \(MAC\) operations per neuron\. However, since Bernstein activations operate over a fixed normalized domain\[0,1\]\[0,1\], the activation function can be precomputed\. Each neuron’s activation is evaluated offline over a uniform grid ofEEpoints in\[0,1\]\[0,1\]and stored in on\-chip memory \(BRAM\)\. At runtime, activation evaluation reduces to a single memory access, eliminating all arithmetic operations\.

BRAM storage scales linearly with the number of neuronshhin a layer and grid resolutionEE\. While storage grows withhh, the complete removal of activation\-phase computation yields substantial savings in DSP usage and combinational logic\. This trade\-off motivates training of compact BNN students using KD \(Section[III\-A](https://arxiv.org/html/2608.20497#S3.SS1)\), keepinghhsmall enough for efficient on\-chip storage\.

A naive implementation uses nearest\-neighbor lookup via floor indexing, which quantizes inputs intoEEdiscrete levels and introduces an approximation error\. To mitigate this, we employ low\-cost linear interpolation between adjacent LUT entries:

f⁡\(t\)=𝒯⁡\[ilo\]\+μ⋅\(𝒯⁡\[ihi\]−𝒯⁡\[ilo\]\),f\(t\)=\\mathcal\{T\}\[i\_\{\\mathrm\{lo\}\}\]\+\\mu\\cdot\\left\(\\mathcal\{T\}\[i\_\{\\mathrm\{hi\}\}\]\-\\mathcal\{T\}\[i\_\{\\mathrm\{lo\}\}\]\\right\),\(7\)wherettis the normalized pre\-activation as defined in Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1),𝒯⁡\[⋅\]\\mathcal\{T\}\[\\cdot\]is theEE\-entry LUT,ilo,ihii\_\{\\mathrm\{lo\}\},i\_\{\\mathrm\{hi\}\}are the indices of the two adjacent grid points bracketingtt, andμ∈\[0,1\]\\mu\\in\[0,1\]is the fractional position between them\. This interpolation significantly improves accuracy while introducing minimal overhead: one multiply, one subtraction, and one addition per neuron, all implemented in combinational logic with no additional DSP or BRAM usage\.

Empirically, for a BNN with hidden layer sizeh=256h=256on Covertype\[[7](https://arxiv.org/html/2608.20497#bib.bib24)\], linear interpolation at 50 entries \(100 KB BRAM\) achieves an error rate of0\.08%0\.08\\%, roughly60×60\\timeslower than nearest\-neighbor’s4\.78%4\.78\\%at the identical table size and storage cost\. We select 50 entries as our operating point for the remainder of this work\. Nearest\-neighbor error falls off slowly with table size and does not reach linear interpolation’s 50\-entry error rate even at 1000 entries, showing that interpolation reaches low error at a fraction of the storage nearest\-neighbor would require\. Fig\.[3](https://arxiv.org/html/2608.20497#S3.F3)plots accuracy and error rate against table size for both methods on this setting, showing the rapid saturation of linear interpolation against the persistently slower convergence of nearest\-neighbor\.

Finally, to eliminate additional runtime overhead, the input normalization required by Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1):t=\(z−li\)/\(ui−li\)t=\(z\-l\_\{i\}\)/\(u\_\{i\}\-l\_\{i\}\)is fused into the preceding linear layer’s weights offline:

Wfused​\[i\]​\[j\]=W⁡\[i\]​\[j\]/\(ui−li\)\\displaystyle W\_\{\\mathrm\{fused\}\}\[i\]\[j\]=W\[i\]\[j\]/\(u\_\{i\}\{\-\}l\_\{i\}\)\(8\)βfused​\[i\]=\(β⁡\[i\]−li\)/\(ui−li\)\\displaystyle\\beta\_\{\\mathrm\{fused\}\}\[i\]=\(\\beta\[i\]\{\-\}l\_\{i\}\)/\(u\_\{i\}\{\-\}l\_\{i\}\)\(9\)The linear layer output then arrives already normalized to\[0,1\]\[0,1\], so the activation requires only a clamp and an index computation, improving hardware efficiency\. This fusion is only possible because Bernstein activations use fixed, learned input boundslil\_\{i\}anduiu\_\{i\}that do not change at runtime\.

102030501002005001000747478788282868690909494LUT entries per neuronTest Acc\. \(%\)Nearest\-neighborLinear interpolationFull precision\(\(a\)\)10203050100200500100000551010151520202525LUT entries per neuronError Rate \(%\)\(\(b\)\)
Fig\. 3:Per\-neuron LUT behavior vs\. table size on Covertype: test accuracy \(top\) and error rate \(bottom\) vs\. LUT entries per neuron\. Error% is the fraction of samples where LUT output differs from full\-precision inference\.
### III\-CSymbolic Rule Extraction

The second deployment path inBern2Edgebegins with symbolic rule extraction, as shown in Fig\.[1](https://arxiv.org/html/2608.20497#S3.F1)\. We propose an activation\-geometry\-based pipeline converting the trained student BNN into a compact, interpretable rule set\.

Coefficient\-driven structure and analytic regime partitioning:The coefficients𝐜\\mathbf\{c\}fully determine the shape of Bernstein activation over the normalized domain\. The derivative preserves this structure through differences between adjacent coefficients:

σ′​\(z\)=nu−l​∑k=0n−1\(ck\+1−ck\)​bn−1,k​\(z−lu−l\)\.\\sigma^\{\\prime\}\(z\)=\\frac\{n\}\{u\-l\}\\sum\_\{k=0\}^\{n\-1\}\(c\_\{k\+1\}\-c\_\{k\}\)\\,b\_\{n\-1,k\}\\\!\\left\(\\tfrac\{z\-l\}\{u\-l\}\\right\)\.\(10\)
The roots ofσ′​\(z\)\\sigma^\{\\prime\}\(z\)define local extrema partitioning the activation into monotonic regimes determined directly by the learned coefficients\. InBern2Edge, these regimes encode the learned activation geometry and form the basis for symbolic rule extraction and subsequent hardware synthesis\.

This formulation is specific to BNNs, whose coefficient\-driven polynomial structure enables analytic extraction of multiple activation regimes from the learned function\. Piecewise\-linear activations such as ReLU, in contrast, define only a single breakpoint and lack the coefficient structure needed for multiple intrinsic regimes, making them less amenable to this extraction method\.

Pipeline of symbolic rule extraction:The pipeline consists of three stages\. Stage 1 extracts neuron\-wise activation regimes from the Bernstein activation geometry in pre\-activation space\. Stage 2 composes these regimes into candidate rules and selects a compact subset via cascade greedy cover based on purity and coverage\. Stage 3 optimizes the selected rules for deployment via weight vector sparsification and quantization, producing a hardware\-efficient final rule set\. We detail each stage below\.

#### III\-C1Stage 1: Geometry Extraction

The first stage extracts neuron\-wise activation regimes directly from the Bernstein activation geometry in pre\-activation space \(zz\-space\) as follows\.

- •Pre\-activation mapping\.Each neuroniidefines a scalar pre\-activation \(simplified from Eq\.[4](https://arxiv.org/html/2608.20497#S2.E4)\): zi=𝐰i⊤​𝐱\+βi,z\_\{i\}=\\mathbf\{w\}\_\{i\}^\{\\top\}\\mathbf\{x\}\+\\beta\_\{i\},\(11\)mapped to a normalized domaint∈\[0,1\]t\\in\[0,1\]using learned input bounds\[li,ui\]\[l\_\{i\},u\_\{i\}\]\(Section[III\-A](https://arxiv.org/html/2608.20497#S3.SS1)\)\.
- •Activation motif classification\.The learned Bernstein coefficients𝐜i\\mathbf\{c\}\_\{i\}fully determine each activation’s shape\. We classify neurons into qualitative*motifs*\(e\.g\., monotone, bump, valley\) describing the overall structure of the activation response, as shown in Fig\.[4](https://arxiv.org/html/2608.20497#S3.F4)\.
- •Regime boundary extraction\.We compute regime boundaries analytically from the activation geometry\. Breakpoints intt\-space come from three sources: \(i\) a uniform grid ensuring baseline coverage, \(ii\) roots of the first derivative identifying local extrema, and \(iii\) roots of the second derivative identifying inflection points\. As Eq\.[10](https://arxiv.org/html/2608.20497#S3.E10)shows, these derivatives take closed form in terms of coefficient differences, enabling analytic computation of all breakpoints without dependence on data samples, reflecting the learned activation shapes, as illustrated in Fig\.[4](https://arxiv.org/html/2608.20497#S3.F4)\. Fig\. 4:Bernstein activation curves with analytically derived regime breakpoints\. Six representative neuron activations from a BNN \(h=64h=64\) trained on Adult\[[6](https://arxiv.org/html/2608.20497#bib.bib25)\], with breakpoints from derivative roots and inflection points marked\.
- •Input\-space mapping\.We map the breakpoints back to pre\-activation space using the fixed input bounds\[li,ui\]\[l\_\{i\},u\_\{i\}\], via zi=li\+t⁡\(ui−li\),z\_\{i\}=l\_\{i\}\+t\\,\(u\_\{i\}\-l\_\{i\}\),\(12\)the inverse of the normalization in Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1), producing disjoint intervals that partition each neuron’s response into regimes determined entirely by its coefficients\. The convex hull property of Bernstein polynomials\[[16](https://arxiv.org/html/2608.20497#bib.bib6)\]bounds the activation output by the range of its coefficients; restricted to subintervals, the effective bounds tighten further, reinforcing the stability of the resulting regime partitions\. The mapping of these regimes to input\-space constraints is illustrated step\-by\-step in Fig\.[5](https://arxiv.org/html/2608.20497#S3.F5)\. We apply this mapping to first\-layer neurons, whose pre\-activations are affine in the raw input𝐱\\mathbf\{x\}\(Eq\.[11](https://arxiv.org/html/2608.20497#S3.E11)\); a selected interval intt\-space defines a corresponding interval inzz\-space via Eq\.[12](https://arxiv.org/html/2608.20497#S3.E12), which, through Eq\.[11](https://arxiv.org/html/2608.20497#S3.E11), directly induces an affine constraint in input space of the form zilo≤𝐰i⊤​𝐱\+βi≤zihi,z\_\{i\}^\{\\text\{lo\}\}\\leq\\mathbf\{w\}\_\{i\}^\{\\top\}\\mathbf\{x\}\+\\beta\_\{i\}\\leq z\_\{i\}^\{\\text\{hi\}\},\(13\)defining an oblique band in input space\. Each condition in a rule corresponds to one neuron and one of its activation regimes; the conjunction of conditions across multiple neurons produces intersecting affine bands, forming oblique polyhedral regions in the input space\. Stage 2 constructs candidate rules from these intervals and refines them based on empirical quality\. ![Refer to caption](https://arxiv.org/html/2608.20497v1/rule_extraction.png)Fig\. 5:Activation\-geometry\-based rule formation\. A selected regime in normalized activation space \(tt\) maps throughzz\-space to an affine constraint in input space, forming interpretable oblique regions\.

#### III\-C2Stage 2: Candidate Rule Generation and Selection

Given the regime partitions from Stage 1, we define rules as conjunctions of affine constraints of the form in Eq\.[13](https://arxiv.org/html/2608.20497#S3.E13), one per neuron regime, assigning each rule the majority predicted class of its covered samples\.

- •Rule evaluation\.We evaluate each candidate rulerron training data by computing the set of samples satisfying all its conditions, from which we compute: \(i\)*purity*, the fraction of covered samples whose prediction matches the rule’s label, and \(ii\)*coverage*, the number of samples matched\. We discard rules that do not meet minimum coverage and purity thresholds; both are hyperparameters of the method\.
- •Candidate generation\.We construct rules progressively by combining regimes across neurons, beginning with single\-condition rules evaluated for all neurons and their regimes\. Rules not meeting the desired purity become*impure seeds*, iteratively expanded by adding conditions from additional neurons up to a predefined depth limit\. At each expansion step, we evaluate candidate extensions formed by adding one condition to each impure parent rule, retaining two: \(i\) the highest\-coverage extension meeting the purity criterion, and \(ii\) the highest\-coverage extension that does not, serving as the seed for further expansion\. This retains at most one pure and one impure extension per parent rule, bounding candidate growth as neurons and regimes increase while preserving the highest\-coverage candidates\.
- •Cascade greedy selection\.From the candidate pool, we select a compact rule set via cascade greedy cover\. The cascade proceeds over decreasing purity thresholds with increasing minimum coverage requirements, ensuring lower\-purity rules are selected only if they add substantial new coverage\. Within each stage, we select rules iteratively using a scoring function prioritizing new\-sample coverage while penalizing redundancy and conflicts: score​\(r\)=gain​\(r\)−αsc⋅same\-cover​\(r\)−αconf⋅conf​\(r\)\.\\text\{score\}\(r\)=\\text\{gain\}\(r\)\-\\alpha\_\{\\text\{sc\}\}\\cdot\\text\{same\{\-\}cover\}\(r\)\-\\alpha\_\{\\text\{conf\}\}\\cdot\\text\{conf\}\(r\)\.\(14\)Here,gain​\(r\)\\text\{gain\}\(r\)is the number of samples covered byrrnot yet covered by the current rule set\.same\-cover​\(r\)\\text\{same\{\-\}cover\}\(r\)counts overlap with selected rules of the same label, whileconf​\(r\)\\text\{conf\}\(r\)counts overlap with rules of different labels\. The weightsαsc\\alpha\_\{\\text\{sc\}\}andαconf\\alpha\_\{\\text\{conf\}\}control the corresponding penalties; their effect on coverage and accuracy is analyzed in Section[IV\-F](https://arxiv.org/html/2608.20497#S4.SS6)\. At each step, we select the rule with the highest score and sufficient new coverage, updating coverage accordingly, until no further rules can be added, after which the algorithm proceeds to the next lower purity stage until the minimum purity threshold is reached\. This cascade yields a compact rule set capturing high\-confidence regions first, then expanding coverage while controlling redundancy and conflicts, providing a geometry\-aligned approximation of the model’s decision boundary \(Fig\.[6](https://arxiv.org/html/2608.20497#S3.F6)\)\. Fig\. 6:Activation\-geometry rule regions on the Two Moons dataset\[[30](https://arxiv.org/html/2608.20497#bib.bib20)\]\. Rules align with the curved decision boundary via Bernstein\-derived breakpoints, enabling compact partitioning\.

#### III\-C3Stage 3: Rule Optimization

To enable efficient inference on resource\-constrained hardware, we optimize the extracted rules for reduced computation and memory usage as follows\.

- •Rule sparsification\.Each rule condition takes the affine form of Eq\.[13](https://arxiv.org/html/2608.20497#S3.E13)\. In practice, a small subset of large\-magnitude weights often dominates the output, indicating only a few input features contribute significantly\. We therefore retain only the top\-kkentries of each weight vector, zeroing the rest, reducing both storage and inference cost tokkmultiplications per condition\.kkis a hyperparameter; its effect on memory footprint and accuracy is ablated in Section[IV\-F](https://arxiv.org/html/2608.20497#S4.SS6)\.
- •Integer quantization\.We quantize rule parameters to integer precision for direct deployment on integer\-only hardware\. Weight vectors use symmetric per\-vector INT8 quantization: 𝐰q=round⁡\(𝐰s\),s=maxj⁡\|wj\|127,\\mathbf\{w\}\_\{q\}=\\mathrm\{round\}\\\!\\left\(\\frac\{\\mathbf\{w\}\}\{s\}\\right\),\\quad s=\\frac\{\\max\_\{j\}\|w\_\{j\}\|\}\{127\},dequantized as𝐰^=s⋅𝐰q\\hat\{\\mathbf\{w\}\}=s\\cdot\\mathbf\{w\}\_\{q\}; thresholds and biases are quantized to fixed\-point precision\. Unlike multi\-layer neural networks, quantization here introduces no cross\-layer error accumulation, since each rule evaluates as a single affine operation on the original input\. As a result, INT8 quantization introduces negligible degradation in rule accuracy, with an average accuracy drop of0\.002%0\.002\\%across all tested datasets\. The final rule set consists of sparse, quantized constraints evaluable with integer and fixed\-point arithmetic, enabling hardware\-efficient inference\.

#### III\-C4Inference

The final stage performs rule inference as follows\.

- •Conflict resolution\.Due to overlapping regimes, a sample may satisfy multiple rules with different labels\. We resolve conflicts by selecting the matching rule with the highest training purity: y^=arg⁡maxr∈ℛ⁡\(𝐱\)​purity​\(r\),\\hat\{y\}=\\arg\\max\_\{r\\in\\mathcal\{R\}\(\\mathbf\{x\}\)\}\\text\{purity\}\(r\),\(15\)whereℛ⁡\(𝐱\)\\mathcal\{R\}\(\\mathbf\{x\}\)is the set of rules whose conditions are all satisfied by sample𝐱\\mathbf\{x\}, andpurity​\(r\)\\text\{purity\}\(r\)is the purity of rulerrmeasured on the training set \(Stage 2\)\.
- •Fallback\.Inputs not matched by any rule fall through to a Classification and Regression Tree \(CART\) trained on the uncovered training samples for which no rule fires\[[10](https://arxiv.org/html/2608.20497#bib.bib35)\]\. The CART partitions the feature space via threshold comparisons on raw features, requiring no multiplications\. We keep the tree shallow \(depth 4\) to preserve interpretability and minimize hardware cost, quantizing its split thresholds to fixed\-pointfix<16,8\>before synthesis\. We compare CART against alternative fallback strategies \(linear regression, a small BNN trained on uncovered samples, and the full underlying BNN\) in Section[IV\-F](https://arxiv.org/html/2608.20497#S4.SS6)\(Table[IX](https://arxiv.org/html/2608.20497#S4.T9)\)\.

#### III\-C5Rule Synthesis

We synthesize each rule set into hardware operating in three pipelined phases\. Phase 1 computes per\-condition linear projections as sparse INT8 dot products: each condition stores only non\-zero quantized weights alongside their feature indices in on\-chip ROM, reducing multiply operations and weight storage relative to dense evaluation\. We fully unroll and pipeline the dot product, dequantizing the result by a per\-condition scale factor\. Phase 2 evaluates each rule by checking whether its condition projections fall within learned band boundaries \(lower/upper thresholds\); the highest\-purity firing rule determines the predicted label\. Phase 3 routes samples not matched by any rule to the CART fallback\. All rule parameters reside as compile\-time ROM constants, requiring no external memory access\.

## IVResults

### IV\-AExperimental Setup

∙\\bulletDatasets and Preprocessing\.We evaluateBern2Edgeon tabular benchmarks from prior work\[[17](https://arxiv.org/html/2608.20497#bib.bib21)\], and a language classification task for the transformer generalization study\. In particular,HIGGS\-Small\[[5](https://arxiv.org/html/2608.20497#bib.bib22),[36](https://arxiv.org/html/2608.20497#bib.bib23)\]\(98,049 samples, 28 features, binary\),Covertype\[[7](https://arxiv.org/html/2608.20497#bib.bib24)\]\(581,012 samples, 54 features, 7 classes\), andAdult Census\[[6](https://arxiv.org/html/2608.20497#bib.bib25)\]\(48,842 samples, 14 mixed features, binary\)\. These datasets use standard preprocessing and stratified splits following Gorishniy et al\.\[[17](https://arxiv.org/html/2608.20497#bib.bib21)\]\.MAGIC Gamma Telescope\[[9](https://arxiv.org/html/2608.20497#bib.bib26)\]\(19,020 samples, 10 features, binary\) is used for rule extraction benchmarking and rule certification\.ACS Income\[[14](https://arxiv.org/html/2608.20497#bib.bib32)\], a published covariate\-shift benchmark, evaluates rule\-system robustness under geographic and temporal distribution shift\.SST\-2\[[33](https://arxiv.org/html/2608.20497#bib.bib38)\]evaluates Bernstein activations in transformer FFN sublayers, with TinyBERT4\[[25](https://arxiv.org/html/2608.20497#bib.bib34)\]as the target architecture\. Unless otherwise noted, all experiments use 5\-fold cross\-validation; test data are transformed using preprocessors fitted on the training split\. Rule extraction results \(Sections[IV\-C](https://arxiv.org/html/2608.20497#S4.SS3), IV\-F\) use single representative models per configuration\.

∙\\bulletTeacher and Student Models\.Teacher models replicate the exact MLP configurations of Gorishniy et al\.\[[17](https://arxiv.org/html/2608.20497#bib.bib21)\], achieving accuracy within0\.50\.5percentage points \(pp\) of reported baselines\. Students are trained via KD\[[20](https://arxiv.org/html/2608.20497#bib.bib7)\]using identical architectures with either Bernstein activations \(degree 3 for MLPs; degree 15 for transformer FFN sublayers\) or ReLU\. Training uses AdamW with learning rate6×10−36\\times 10^\{\-3\}, weight decay1×10−41\\times 10^\{\-4\}, and cosine annealing; BNNs additionally include an out\-of\-bounds penalty \(λob=10−2\\lambda\_\{\\mathrm\{ob\}\}=10^\{\-2\}, Section[III\-A](https://arxiv.org/html/2608.20497#S3.SS1)\)\. Distillation parameters are swept per dataset with temperatureT∈\{2,4\}T\\in\\\{2,4\\\}andα∈\[0,0\.85\]\\alpha\\in\[0,0\.85\], and the best\-performing configuration is reported\. Student architectures are selected by sweeping width and depth under hardware constraints\. We denote architectures as\{din,h1,…,hm,dout\}\\\{d\_\{\\text\{in\}\},h\_\{1\},\\ldots,h\_\{m\},d\_\{\\text\{out\}\}\\\}; where a single hidden layer is varied, we write its width as a scalarhh\.

∙\\bulletHardware Setup\.Training uses an NVIDIA Tesla V100 \(16 GB\)\. Designs are synthesized with Vitis HLS 2024\.1 and Vivado 2024\.1 and evaluated on an AMD Xilinx KV260 FPGA at 200 MHz\. We also evaluate on a Spartan\-7 XC7S15 FPGA to demonstrate low\-power edge deployment\.

∙\\bulletEvaluation\.We evaluate three components: \(1\) neural compression, comparing Bernstein and ReLU students under identical constraints; \(2\) symbolic rule extraction; and \(3\) the fullBern2Edgepipeline against a W8A8 \(8\-bit weight, 8\-bit activation\) quantization\-aware training \(QAT\) teacher baseline\[[23](https://arxiv.org/html/2608.20497#bib.bib28)\]\. We further conduct three additional studies: low\-power deployment on the Spartan\-7 XC7S15, ablation on the rule extraction hyperparameters \(kk,αsc\\alpha\_\{\\text\{sc\}\},αconf\\alpha\_\{\\text\{conf\}\}\) and fallback strategies, and rule robustness evaluation covering input\-noise certification and distribution\-shift assessment on benchmarks\.

∙\\bulletMetrics\.We report test accuracy and cross\-entropy loss\. For hardware deployment, we report latency, DSPs, BRAM, LUTs, and flip\-flops \(FFs\)\. For rule extraction, we additionally report rule count, average conditions per rule, and coverage\.

### IV\-BStudent BNN Compression and Synthesis

Table[I](https://arxiv.org/html/2608.20497#S4.T1)reports accuracy and synthesis metrics for BNNs and equivalent ReLU student networks across three datasets\. Teacher models follow the MLP baselines of Gorishniy et al\.\[[17](https://arxiv.org/html/2608.20497#bib.bib21)\]:\{28,231,121,2\}\\\{28,231,121,2\\\}on HIGGS\-Small,\{54,1024,1024,512,7\}\\\{54,1024,1024,512,7\\\}on Covertype, and\{14,503,503,503,111,2\}\\\{14,503,503,503,111,2\\\}on Adult\.

TABLE I:Accuracy and synthesis metrics on AMD KV260 \(5\-fold CV\)\. Latency in clock cycles\.Δ\\DeltaAcc /Δ\\DeltaCE: BNN minus ReLU\.DatasetArch\.Act\.Acc\. \(%\)CE LossΔ\\DeltaAcc /Δ\\DeltaCELatencyDSPsBRAMsLUTsHIGGS\-Small\{28,16,2\}\\\{28,16,2\\\}ReLU70\.98±\\pm0\.200\.553\+0\.94\+0\.94/−0\.013\-0\.0138510805397Bern71\.92±\\pm0\.260\.5408710625682\{28,16,8,2\}\\\{28,16,8,2\\\}ReLU71\.84±\\pm0\.240\.540\+0\.48\+0\.48/−0\.008\-0\.00812115606852Bern72\.32±\\pm0\.180\.53212513537294\{28,128,2\}\\\{28,128,2\\\}ReLU72\.08±\\pm0\.270\.524\+0\.45\+0\.45/−0\.012\-0\.012337108345114Bern72\.53±\\pm0\.180\.512339103445410Covertype\{54,64,32,7\}\\\{54,64,32,7\\\}ReLU88\.97±\\pm0\.820\.292\+2\.12\+2\.12/−0\.069\-0\.06996036011320213Bern91\.09±\\pm0\.270\.22396332610521815\{54,128,64,7\}\\\{54,128,64,7\\\}ReLU93\.53±\\pm0\.180\.138\+1\.52\+1\.52/−0\.060\-0\.060247736013722207Bern95\.05±\\pm0\.050\.078248132613623824\{54,256,128,7\}\\\{54,256,128,7\\\}ReLU95\.62±\\pm0\.120\.057\+0\.77\+0\.77/−0\.029\-0\.029742936016922432Bern96\.39±\\pm0\.020\.028743332617923063Adult\{14,16,2\}\\\{14,16,2\\\}ReLU84\.06±\\pm0\.390\.324\+0\.54\+0\.54/−0\.010\-0\.010686903823Bern84\.60±\\pm0\.130\.314706924113\{14,32,16,2\}\\\{14,32,16,2\\\}ReLU84\.48±\\pm0\.120\.315\+0\.44\+0\.44/−0\.009\-0\.00916116317739Bern84\.92±\\pm0\.130\.30616513668132\{14,128,2\}\\\{14,128,2\\\}ReLU84\.57±\\pm0\.220\.310\+0\.25\+0\.25/\+0\.001\+0\.00132167203595Bern84\.82±\\pm0\.140\.31132361303892000\.10\.20\.30\.40\.50\.60\.70\.80\.91\.070\.570\.5717171\.571\.5727272\.572\.5KD Weightα\\alphaTest Accuracy \(%\)BNNReLUTeacher \(72\.32%\)\(\(a\)\)000\.10\.20\.30\.40\.50\.60\.70\.80\.91\.070\.570\.5717171\.571\.5727272\.572\.5KD Weightα\\alphaTest Accuracy \(%\)\(\(b\)\)
Fig\. 7:Test accuracy vs\. KD weightα\\alpha\(T=2T\{=\}2\) on HIGGS\-Small for students with single layerh=16h=16\(top\) andh=128h=128\(bottom\)\. Dashed line: teacher baseline \(72\.32%72\.32\\%\)\.∙\\bulletAccuracy Under Compression\.BNNs consistently outperform their ReLU counterparts across all architecture–dataset combinations \(Table[I](https://arxiv.org/html/2608.20497#S4.T1)\), with accuracy gains of\+0\.25\+0\.25to\+2\.12\+2\.12pp and lower cross\-entropy loss in 8 of 9 cases\. The gap is largest under strong compression, reaching\+2\.12\+2\.12pp on Covertype\{54,64,32,7\}\\\{54,64,32,7\\\}, and narrows as model capacity increases, consistent with the approximation behavior of Bernstein\-based networks\[[3](https://arxiv.org/html/2608.20497#bib.bib15)\]\. BNNs also approach or exceed teacher accuracy at smaller architectures than ReLU on every dataset\. Gains on Adult and HIGGS\-Small are modest but consistent across all folds; since inter\-model differences below one percentage point are typical for state\-of\-the\-art tabular models\[[17](https://arxiv.org/html/2608.20497#bib.bib21)\], these improvements reflect genuine learning efficiency rather than incidental variation\.

∙\\bulletAlignment with Teacher under KD\.Fig\.[7](https://arxiv.org/html/2608.20497#S4.F7)shows test accuracy vs\.α\\alphaon HIGGS\-Small for a small student \(h=16h=16\) \(top\) and a large student \(h=128h=128\) \(bottom\)\. For the small student, the BNN holds a stable∼1\{\\sim\}1pp lead over ReLU across allα\\alphavalues, with neither reaching the teacher; width is the binding constraint, and the Bernstein advantage persists even without distillation \(α=0\\alpha\{=\}0\)\. For the larger student, both activations improve withα\\alpha, but the BNN crosses the teacher nearα=0\.7\\alpha\{=\}0\.7and peaks at72\.5%72\.5\\%, while ReLU approaches but does not reach it\. Thus at low capacity the Bernstein advantage is intrinsic and KD\-invariant, while at higher capacity the BNN additionally benefits more from stronger distillation\.

∙\\bulletHardware Overhead\.Bernstein activations impose modest, predictable hardware overhead relative to ReLU at matched architectures \(Table[I](https://arxiv.org/html/2608.20497#S4.T1)\)\. Latency overhead never exceeds 4 clock cycles, since each per\-neuron Bernstein evaluation reduces to a single LUT lookup followed by linear interpolation\. Bernstein uses fewer or equal DSPs than ReLU at every architecture, with savings reaching 27 on Adult\{14,32,16,2\}\\\{14,32,16,2\\\}: the HLS compiler prunes unused upper bits from the activation output, so each downstream multiply fits in 2 DSPs instead of 3, whereas unbounded ReLU preserves the full fixed\-pointfix<32,16\>range and requires 3 DSPs per multiply\. LUT counts are higher for Bernstein across all configurations \(2\.82\.8–8\.3%8\.3\\%overhead\), reflecting the synthesized per\-neuron lookup tables; BRAM overhead is small and architecture\-dependent, governed by the interaction between per\-neuron table size and BRAM tile granularity\.

∙\\bulletAccuracy Under Matched Hardware Budgets\.Table[II](https://arxiv.org/html/2608.20497#S4.T2)selects the best Bernstein and ReLU models on Covertype within five hardware budget tiers, confirming Bernstein outperforms ReLU under identical tight constraints at every operating point\. Gains range from\+0\.77\+0\.77pp at the 7500\-cycle tier to\+2\.12\+2\.12pp at 1000 cycles, ruling out parameter count and architecture size as confounds\.

TABLE II:Covertype accuracy under matched latency and BRAM budgets\. Parentheses: actual synthesis values \(cycles, BRAMs\)\.LatencyBRAMsBernsteinReLUΔ\\DeltaAcc\(cyc\)Acc \(%\)\(cyc, BRAM\)Acc \(%\)\(cyc, BRAM\)\(pp\)≤200\{\\leq\}200≤30\{\\leq\}3076\.50±\\pm0\.44\(189, 26\)74\.93±\\pm0\.08\(188, 25\)\+1\.57\+1\.57≤400\{\\leq\}400≤55\{\\leq\}5582\.98±\\pm0\.16\(397, 52\)81\.87±\\pm0\.37\(396, 49\)\+1\.11\+1\.11≤1000\{\\leq\}1000≤120\{\\leq\}12091\.09±\\pm0\.27\(963, 105\)88\.97±\\pm0\.82\(960, 113\)\+2\.12\+2\.12≤2500\{\\leq\}2500≤140\{\\leq\}14095\.05±\\pm0\.05\(2481, 136\)93\.53±\\pm0\.18\(2477, 137\)\+1\.52\+1\.52≤7500\{\\leq\}7500≤180\{\\leq\}18096\.39±\\pm0\.02\(7433, 179\)95\.62±\\pm0\.12\(7429, 169\)\+0\.77\+0\.77

∙\\bulletPolynomial Degree and Hardware\.Fig\.[8](https://arxiv.org/html/2608.20497#S4.F8)compares FPGA resource utilization for naive Bernstein evaluation, which directly computes Eq\.[1](https://arxiv.org/html/2608.20497#S2.E1), against our LUT\-based implementation across degreesn∈\{3,5,7,9,11\}n\\in\\\{3,5,7,9,11\\\}\. Naive evaluation scales with degree because each additional basis function requires more multiply\-accumulate hardware, causing DSP and LUT usage to grow monotonically\. In contrast, our implementation has constant resource cost: every activation maps to the same fixed\-size lookup table at inference time\. This decouples degree selection from hardware budget, allowingnnto increase for harder approximation tasks\[[3](https://arxiv.org/html/2608.20497#bib.bib15)\]without additional FPGA overhead\.

BRAMDSPFFLUT5510101515Resource Util\. \(%\)Naiven=3n\{=\}3n=5n\{=\}5n=7n\{=\}7n=9n\{=\}9n=11n\{=\}11LUTn=3n\{=\}3n=5n\{=\}5n=7n\{=\}7n=9n\{=\}9n=11n\{=\}11Fig\. 8:FPGA resource utilization on Adult \(h=128h=128\) for naive Bernstein evaluation and LUT\-based inference across polynomial degreesn∈\{3,5,7,9,11\}n\\in\\\{3,5,7,9,11\\\}\.
### IV\-CSymbolic Extraction Results

∙\\bulletRule Extraction on Adult\.Rules are extracted from BNNs trained withT=2T\{=\}2andα=0\.5\\alpha\{=\}0\.5on Adult\. The extraction pipeline \(Section[III\-C](https://arxiv.org/html/2608.20497#S3.SS3)\) uses purity threshold0\.850\.85, minimum coverage of 5 samples per rule, maximum depth 3, and sparsityk=7k\{=\}7, sweepingαsc\\alpha\_\{\\text\{sc\}\}andαconf\\alpha\_\{\\text\{conf\}\}to study the coverage–compactness trade\-off, using single representative models\. An additional evaluation on MAGIC compares against published baselines\[[39](https://arxiv.org/html/2608.20497#bib.bib18)\]\.

Table[III](https://arxiv.org/html/2608.20497#S4.T3)reports rule extraction results across five BNN architectures on Adult\. Covered accuracy \(84\.584\.5–87\.287\.2%\) consistently exceeds each model’s test accuracy \(84\.3384\.33–84\.6384\.63%\), confirming extracted rules are at least as discriminative as the underlying network on covered samples\.αsc\\alpha\_\{\\text\{sc\}\}controls a clear coverage–compactness trade\-off: atαsc=0\.5\\alpha\_\{\\text\{sc\}\}\{=\}0\.5, compact rule sets cover77\.677\.6–89\.589\.5% of test samples with covered accuracy85\.985\.9–87\.287\.2%; atαsc=0\.1\\alpha\_\{\\text\{sc\}\}\{=\}0\.1, coverage rises to89\.989\.9–95\.895\.8%, but with more rules and slightly lower covered accuracy \(84\.584\.5–86\.386\.3%\)\. Total accuracy Acctremains within1\.21\.2–2\.62\.6pp of the underlying BNN, with the gap widening at lower coverage settings where more samples fall to the CART fallback\. Sensitivity analysis ofαsc\\alpha\_\{\\text\{sc\}\},αconf\\alpha\_\{\\text\{conf\}\}, andkkas interpretable trade\-off knobs, and an ablation of fallback strategies, are provided in Section[IV\-F](https://arxiv.org/html/2608.20497#S4.SS6)\.

TABLE III:Rule extraction on Adult across five BNN architectures\.αconf\\alpha\_\{\\text\{conf\}\}: conflict penalty constant = 0\.1\. CART fallback\.αsc\\alpha\_\{\\text\{sc\}\}: same\-coverage penalty;ℓ¯\\overline\{\\ell\}: avg\. conditions per rule; Cov\.: coverage; Cov\. Acc\.: covered accuracy; Acct: total accuracy\.ArchBNN Acc\. \(%\)αsc\\alpha\_\{\\text\{sc\}\}Rulesℓ¯\\overline\{\\ell\}Cov\. \(%\)Cov\. Acc\. \(%\)Acct\(%\)\{14,16,2\}\\\{14,16,2\\\}84\.330\.5371\.7678\.286\.882\.60\.1451\.6791\.584\.581\.7\{14,32,2\}\\\{14,32,2\\\}84\.420\.5452\.0484\.987\.283\.20\.1532\.0294\.085\.382\.1\{14,128,2\}\\\{14,128,2\\\}84\.630\.5611\.6489\.585\.982\.80\.1851\.7595\.885\.382\.0\{14,16,8,2\}\\\{14,16,8,2\\\}84\.330\.5291\.6984\.086\.983\.10\.1391\.7789\.986\.382\.3\{14,32,16,2\}\\\{14,32,16,2\\\}84\.360\.5552\.0277\.687\.282\.80\.1661\.9591\.385\.181\.8

∙\\bulletHardware Comparisons\.Table[IV](https://arxiv.org/html/2608.20497#S4.T4)compares LUT\-based BNNs against rule networks on hardware\. The rule network consumes fixed resources \(3030DSPs\) regardless of architecture, while LUT networks scale with width, reaching301301DSPs at\{14,128,2\}\\\{14,128,2\\\}\. At this largest architecture, the rule network reduces DSP count by10×10\{\\times\}and latency by1\.35×1\.35\{\\times\}, at a cost of1\.871\.87pp in total accuracy\. At smaller architectures, DSP savings persist, but rule\-based latency exceeds LUT inference as the fixed rule\-matching overhead dominates\. Rule extraction thus trades a modest accuracy cost for interpretability: predictions are expressed as conjunctions over original input features, enabling direct inspection of decision logic without hardware overhead scaling with model size\.

TABLE IV:LUT\-based BNNs vs\. rule networks HW results on Adult \(AMD KV260\)\.ArchMethodAcc \(%\)LatencyDSPBRAMLUTFF\{14,16,2\}\\\{14,16,2\\\}LUT84\.33777724,3295,582Rules82\.591833052,9972,290\{14,32,2\}\\\{14,32,2\\\}LUT84\.4211910946,0198,333Rules83\.162153083,0092,218\{14,128,2\}\\\{14,128,2\\\}LUT84\.633763012815,60922,865Rules82\.762793083,0022,217\{14,16,8,2\}\\\{14,16,8,2\\\}LUT84\.331089635,7156,912Rules83\.121513053,0112,300\{14,32,16,2\}\\\{14,32,16,2\\\}LUT84\.3617214468,35210,990Rules82\.782553083,0272,218∙\\bulletComparison to Prior Methods\.Table[V](https://arxiv.org/html/2608.20497#S4.T5)compares symbolic rule extraction on MAGIC against published baselines\[[39](https://arxiv.org/html/2608.20497#bib.bib18)\]under 5\-fold cross\-validation, using ECLAIRE’s reference ReLU MLP\{10,64,32,16,2\}\\\{10,64,32,16,2\\\}as the teacher, and evaluating two settings: \(i\) rule extraction from a BNN trained from scratch on the same architecture \(no KD\), and \(ii\) rule extraction from a compressed BNN student\{10,64,32,2\}\\\{10,64,32,2\\\}distilled from that teacher \(T=2T\{=\}2,α=0\.5\\alpha\{=\}0\.5\)\.

The extraction pipeline uses a purity threshold of0\.850\.85, sparsityk=3k\{=\}3, and maximum rule depth of 2\. The KD configuration matches ECLAIRE’s accuracy \(84\.484\.4% vs\.84\.684\.6%\) with9×9\{\\times\}fewer rules \(44±444\{\\pm\}4vs\.396±75396\{\\pm\}75\) and half the average rule complexity \(1\.861\.86vs\.3\.823\.82conditions\), while achieving higher fidelity to the underlying network \(91\.891\.8% vs\.89\.489\.4%\)\. Without KD, rule sets remain compact \(2424–4646rules\) at comparable accuracy \(82\.982\.9–84\.284\.2%\), indicating rule compactness is primarily attributable to Bernstein activation geometry rather than distillation, with KD contributing a small gain in accuracy or fidelity\.

TABLE V:Rule extraction on MAGIC Gamma Telescope\. Baselines from Zarlenga et al\.\[[39](https://arxiv.org/html/2608.20497#bib.bib18)\]\(mean±\\pmstd, 5 folds\); Decompositional: rule extraction from intermediate layers;ℓ¯\\overline\{\\ell\}: avg\. conditions per rule; Fid: fidelity to baseline network\.TypeMethodRulesℓ¯\\overline\{\\ell\}Acc \(%\)Fid \(%\)DecompositionalDeepRED\[[40](https://arxiv.org/html/2608.20497#bib.bib17)\]5143±97995143\\pm 97995\.4378\.789\.3REM\-D\[[32](https://arxiv.org/html/2608.20497#bib.bib27)\]3617±67483617\\pm 67485\.4178\.689\.4ECLAIRE\[[39](https://arxiv.org/html/2608.20497#bib.bib18)\]396±75396\\pm 753\.8284\.689\.4Bern2Edgeno\-KD \(αsc=0\.1\\alpha\_\{\\text\{sc\}\}\{=\}0\.1\)46±446\\pm 41\.8484\.291\.8no\-KD \(αsc=0\.5\\alpha\_\{\\text\{sc\}\}\{=\}0\.5\)𝟐𝟒±𝟑\\mathbf\{24\\pm 3\}1\.81\\mathbf\{1\.81\}82\.994\.3KD \(αsc=0\.1\\alpha\_\{\\text\{sc\}\}\{=\}0\.1\)44±444\\pm 41\.861\.8684\.4\\mathbf\{84\.4\}91\.891\.8KD \(αsc=0\.5\\alpha\_\{\\text\{sc\}\}\{=\}0\.5\)𝟐𝟒±𝟒\\mathbf\{24\\pm 4\}1\.871\.8782\.882\.894\.6\\mathbf\{94\.6\}
### IV\-DBern2EdgeEnd\-to\-End Results

Table[VI](https://arxiv.org/html/2608.20497#S4.T6)presents end\-to\-end post\-synthesis results forBern2Edgeagainst the quantized teacher baseline across all three datasets\.Bern2Edge\(LUT\) reduces latency by91\.991\.9–99\.899\.8% relative to the W8A8 teacher while matching or staying within0\.50\.5pp of teacher accuracy\. DSP savings reach27\.427\.4% on HIGGS\-Small and74\.774\.7% on Adult\. On Covertype, the apparent BRAM increase \(128→149128\\rightarrow 149\) reflects a memory\-architecture shift: the teacher relies on 52 URAM tiles for weight storage, whereasBern2Edgesubstitutes standard BRAM and eliminates94\.294\.2% of URAM usage \(52→352\\rightarrow 3\), reducing DSPs by16\.216\.2%\.

On Adult, the rule deployment path cuts DSP usage to 30 \(↓89\.0\\downarrow 89\.0%\) relative to the teacher, at a cost of1\.51\.5pp in total accuracy compared to the LUT path\. Beyond resource savings, rule deployment constrains predictions to explicit conjunctions over the original input features, providing formal guarantees on the input subspace covered by each decision; a property unavailable in either the teacher model or the LUT\-based BNN\.

TABLE VI:Post\-synthesis accuracy and end\-to\-end hardware results forBern2Edgeacross three datasets\. Percentages indicate change relative to the W8A8 teacher baseline\. Rule deployment path reported for Adult only\.MethodAcc \(%\)Latency \(cycles\)DSPsBRAMsURAMsHIGGS\-SmallTeacher \(W8A8\)72\.36,98118650–Bern2Edge \(LUT\)72\.3125 \(↓\\downarrow98\.2%\)135 \(↓\\downarrow27\.4%\)3 \(↓\\downarrow94\.0%\)–CovertypeTeacher \(W8A8\)96\.991,63438912852Bern2Edge \(LUT\)96\.47,433 \(↓\\downarrow91\.9%\)326 \(↓\\downarrow16\.2%\)149 \(↑\\uparrow16\.4%\)3 \(↓\\downarrow94\.2%\)AdultTeacher \(W8A8\)84\.640,3052734220Bern2Edge \(LUT\)84\.670 \(↓\\downarrow99\.8%\)69 \(↓\\downarrow74\.7%\)2 \(↓\\downarrow95\.2%\)–Bern2Edge \(Rules\)83\.12151 \(↓\\downarrow99\.6%\)30 \(↓\\downarrow89\.0%\)5 \(↓\\downarrow88\.1%\)–
### IV\-EDeployment and Optimization on Low\-Power Edge

We evaluate bothBern2Edgedeployment paths on severely resource\-constrained hardware\. The Spartan\-7 XC7S15\[[2](https://arxiv.org/html/2608.20497#bib.bib36)\], a low\-power FPGA, provides only 8 k LUTs, 20 DSPs, and 20 BRAM18K at∼16\{\\sim\}16mW static power,14\.6×14\.6\{\\times\},62×62\{\\times\}, and14\.4×14\.4\{\\times\}fewer LUTs, DSPs, and BRAMs than the KV260, respectively\. We deploy LUT\-based BNNs on Adult using the\{14,h,2\}\\\{14,h,2\\\}architecture overh∈\{4,8,16,32,64,128\}h\\\!\\in\\\!\\\{4,8,16,32,64,128\\\}hidden layer sizes, and symbolic rule networks, across two rule\-count configurations\. Two device\-level optimizations bring each path within budget without altering any model parameters\. For the rule system,*distributed\-RAM rule storage*binds the 3\.5 KB INT8 rule ROM to LUTRAM rather than block RAM, bringing it within the BRAM budget\. For the BNN,*fixed\-point quantization*reduces the linear\-layer word fromfix<32,16\>tofix<18,8\>, mapping each linear\-layer multiply to a single DSP rather than three, bringing every configuration within the DSP budget at a cost of≤0\.1\{\\leq\}0\.1pp accuracy\. The bitwidth split reflects two separate constraints: 8 integer bits are required to represent Adult’s categorical codes \(up to 40\), while the remaining 10 fractional bits bound quantization rounding error and recover accuracy to within0\.10\.1pp of fp32 at no added DSP cost\. Table[VII](https://arxiv.org/html/2608.20497#S4.T7)reports post\-synthesis results across all configurations\. Every design fits comfortably within device limits, and DSP count stays nearly flat across all widths due to the time\-multiplexed II==1 linear\-layer datapath, so wider networks pay primarily in latency and activation BRAM rather than combinational logic\. Rule\-based classifiers minimize DSP usage, while the smallest Bernstein configurations \(e\.g\.,h=4h\{=\}4\) achieve lower LUT, BRAM, latency, and power, making symbolic deployment resource\-optimal specifically when DSP is the binding constraint\.

TABLE VII:Deployment on XC7S15\. R50 and R29 denote the 50\-rule and 29\-rule symbolic classifiers, respectively\. Acc is test accuracy; Accfpthe fp32 reference for BNNs\.ModelConfig\.LUTFFDSPBRAMLat\.PwrAccAccfp18K\(cyc\)\(mW\)\(%\)\(%\)RulesR501,0868441663406283\.02—R297107651641565983\.01—Bern\.\{14,h,2\}\\\{14,h,2\\\}h=4h\{=\}4553883181615184\.1884\.25h=8h\{=\}8635804191826384\.5784\.64h=16h\{=\}166627771911156384\.8884\.86h=32h\{=\}326677991921795884\.9284\.92h=64h\{=\}646988261953086084\.8284\.92h=128h\{=\}12885887019135656684\.9585\.03

### IV\-FHyperparameter and Fallback Ablation

Bern2Edge exposes three parameters governing rule extraction: the sparsity thresholdkk, the redundancy penaltyαsc\\alpha\_\{\\text\{sc\}\}, and the conflict penaltyαconf\\alpha\_\{\\text\{conf\}\}\. Each controls a specific hardware–accuracy trade\-off with monotone, predictable behavior\. The rule purity threshold is set close to the underlying network’s accuracy, ensuring extracted rules are at least as discriminative as the model they summarize\.

∙\\bulletSparsity thresholdkkkkcontrols per\-rule memory footprint only, with no effect on rule count\. Fig\.[10](https://arxiv.org/html/2608.20497#S4.F10)shows BRAM grows linearly withkkwhile total accuracy plateaus beyondk=7k\{=\}7with increasing variance fork\>7k\{\>\}7\. We selectk=7k\{=\}7, achieving∼55\{\\sim\}55% of the dense memory footprint \(k=14k\{=\}14\) at no accuracy cost\. This plateau reflects that most rule weights are small and can be zeroed without loss; sparse rules match dense accuracy because the dropped conditions carry little discriminative weight\.

∙\\bulletPenalty parametersTable[VIII](https://arxiv.org/html/2608.20497#S4.T8)details the full metric picture \(conflict count, rule count, and average conditions per rule\) across joint\(αsc,αconf\)\(\\alpha\_\{\\text\{sc\}\},\\alpha\_\{\\text\{conf\}\}\)settings\. Fig\.[9](https://arxiv.org/html/2608.20497#S4.F9)isolates each parameter’s effect on coverage and covered accuracy\.αsc\\alpha\_\{\\text\{sc\}\}is the primary lever: increasing it from00to11trades coverage \(∼100\{\\sim\}100%→∼70\{\\to\}\{\\sim\}70%\) for covered accuracy \(∼79\{\\sim\}79%→∼84\{\\to\}\{\\sim\}84%\), with an elbow nearαsc≈0\.2\\alpha\_\{\\text\{sc\}\}\{\\approx\}0\.2\. This is a deployment choice, not a sensitivity: broad\-coverage applications favor smallαsc\\alpha\_\{\\text\{sc\}\}; high\-confidence rule firing favors largeαsc\\alpha\_\{\\text\{sc\}\}\. Within eachαsc\\alpha\_\{\\text\{sc\}\}regime,αconf\\alpha\_\{\\text\{conf\}\}reduces raw conflict count \(Table[VIII](https://arxiv.org/html/2608.20497#S4.T8)\) but has negligible effect on coverage or covered accuracy \(≲\\lesssim0\.6 pp variation across the full range\), since conflicts are resolved at inference time by firing the highest\-purity matching rule\.αconf\\alpha\_\{\\text\{conf\}\}therefore requires no careful tuning\.

TABLE VIII:Joint penalty sweep on Adult \(h=32h=32, BNN acc\.84\.4284\.42%\)\. Conf\.: conflicting samples; Cov\.: coverage; Cov\. Acc\.: covered accuracy; Acct: total accuracy;ℓ¯\\overline\{\\ell\}: avg\. conditions per rule\.αconf\\alpha\_\{\\text\{conf\}\}αsc\\alpha\_\{\\text\{sc\}\}Conf\.Cov\. \(%\)Cov\. Acc\. \(%\)Acct\(%\)Rulesℓ¯\\overline\{\\ell\}0\.10\.130894\.0185\.2682\.10532\.021\.00\.115692\.6084\.6782\.39502\.060\.10\.313687\.1586\.7283\.16532\.061\.00\.311186\.9986\.7783\.16522\.040\.10\.512184\.8687\.1883\.16452\.041\.00\.510684\.4287\.1883\.39462\.090\.11\.07882\.4087\.2383\.43371\.951\.01\.07080\.6987\.6183\.46361\.94000\.20\.20\.40\.40\.60\.60\.80\.811707080809090100100αs​c\\alpha\_\{sc\}Cov\. \(%\)808082828484Cov\. Acc\. \(%\)\(\(a\)\)000\.20\.20\.40\.40\.60\.60\.80\.811707080809090100100αc​o​n​f\\alpha\_\{conf\}Cov\. \(%\)808082828484Cov\. Acc\. \(%\)\(\(b\)\)
Fig\. 9:Coverage and covered accuracy as functions of the two rule\-extraction penalty parameters on Adult, averaged across five network architectures \(h∈\{16,32,128,16×8,32×16\}h\\in\\\{16,32,128,16\{\\times\}8,32\{\\times\}16\\\}\)\.1122334455667788991010111112121313005510101515sparsitykkBRAM7676787880808282Total Acc\. \(%\)Fig\. 10:BRAM and total accuracy vs\. sparsity thresholdkkon Adult \(h=32h=32\)\.∙\\bulletFallback AblationTable[IX](https://arxiv.org/html/2608.20497#S4.T9)compares four fallback strategies on the Adult dataset across accuracy, fidelity to the underlying network on uncovered samples, and hardware cost\. The linear regression fallback operates in first\-layer pre\-activation space𝐳=W0​𝐱\+𝜷0\\mathbf\{z\}=W\_\{0\}\\mathbf\{x\}\+\\boldsymbol\{\\beta\}\_\{0\}, folded into an equivalent input\-space classifier𝐰eff⊤​𝐱\+βeff\\mathbf\{w\}\_\{\\text\{eff\}\}^\{\\top\}\\mathbf\{x\}\+\\beta\_\{\\text\{eff\}\}\. The small BNN fallback uses a one\-hidden\-layer network of width 4 trained on uncovered samples only; the full BNN is the original network, so its fidelity is 100% by definition\. We select CART \(depth≤4\\leq 4\) as the fallback: it achieves the best overall accuracy \(83\.02%\) and the highest accuracy on uncovered samples \(72\.1%\) at the lowest hardware cost \(714 LUT, 471 FF, 0 DSP\)\. Notably, even the full BNN matches neither metric, suggesting uncovered samples are inherently difficult to classify regardless of model capacity\. The full BNN also requires2\.6×2\.6\{\\times\}more LUTs and4\.9×4\.9\{\\times\}more flip\-flops than CART\. Linear regression offers the lowest latency \(13 cycles\) but yields the worst accuracy on uncovered samples \(67\.4%\), reflecting the limitations of a linear decision boundary in this region\.

TABLE IX:Fallback strategy ablation on the Adult dataset \(h=32h=32,αsc=0\.5\\alpha\_\{\\text\{sc\}\}=0\.5,αconf=0\.1\\alpha\_\{\\text\{conf\}\}=0\.1\)\. Resource columns report the fallback component only\. Accunc\{\}\_\{\\text\{unc\}\}and Fidunc\{\}\_\{\\text\{unc\}\}are on uncovered samples\. All accuracy and fidelity values in %\.FallbackAcc\.Accunc\{\}\_\{\\text\{unc\}\}Fidunc\{\}\_\{\\text\{unc\}\}LUTFFDSPLat\.Linear Regression81\.6767\.484\.117495201013Full BNN82\.9171\.9100\.0188723184068Small BNN82\.2371\.390\.5112921783846CART83\.0272\.187\.7714471026

### IV\-GSymbolic Rule Certification and Robustness

∙\\bulletRule Certification on Input Noise\.We evaluate the symbolic rule system on MAGIC, using a\{10,64,32,2\}\\\{10,64,32,2\\\}architecture for the ReLU network and BNN, and the rules extracted from that BNN\. Inputs are perturbed by bounded per\-feature noise with budgetϵi=c⋅si\\epsilon\_\{i\}=c\\cdot s\_\{i\}, wheresis\_\{i\}is the training\-set standard deviation of featureii\. The rangec≤0\.05c\\leq 0\.05corresponds to sub\-55% relative uncertainty, consistent with calibrated process measurement devices\[[22](https://arxiv.org/html/2608.20497#bib.bib30)\];c=0\.10c=0\.10serves as a stress test beyond normal operating conditions\.

For each rule slabzlo≤𝐰⊤​𝐱<zhiz^\{\\text\{lo\}\}\\leq\\mathbf\{w\}^\{\\top\}\\mathbf\{x\}<z^\{\\text\{hi\}\}, the raw input box\[xi−ϵi,xi\+ϵi\]\[x\_\{i\}\-\\epsilon\_\{i\},x\_\{i\}\+\\epsilon\_\{i\}\]is mapped through the monotone quantile normalization, and the exact worst\-case interval of𝐰⊤​𝐱\\mathbf\{w\}^\{\\top\}\\mathbf\{x\}is propagated over the box\. A point is*rule certified*if no different\-label rule of equal or higher purity is reachable within the box, i\.e\. no admissible perturbation can change the predicted label\. This certificate is conservative and therefore never falsely certifies stability\.

We compare rule certificates against network\-level certificates for the underlying BNN and equivalent ReLU network\. The BNN uses Bern\-IBP with de Casteljau subdivision\[[26](https://arxiv.org/html/2608.20497#bib.bib5)\]; the ReLU network uses standard IBP via auto\_LiRPA\[[38](https://arxiv.org/html/2608.20497#bib.bib31)\], following the comparison methodology of Khedr et al\.\[[26](https://arxiv.org/html/2608.20497#bib.bib5)\]\. Empirical robustness is estimated with2020i\.i\.d\. uniform perturbations per point from the same noise box \(≈50,000\{\\approx\}50\{,\}000perturbed inputs per noise level\)\. Since uniform sampling rarely reaches box corners, empirical robustness is an upper bound; certified columns are sound lower bounds\.

Table[X](https://arxiv.org/html/2608.20497#S4.T10)reports results on the covered\-correct test subset \(2,485 of MAGIC test points\)\. Empirical rule accuracy drops by only 3\.0 pp atc=0\.10c=0\.10, and rule–BNN fidelity remains above 95% for all realistic noise levels \(c≤0\.05c\\leq 0\.05\)\. The certified results reveal a larger separation: atc=0\.05c=0\.05, rule certification reaches 42\.7%, versus 16\.0% for the BNN and 12\.4% for the matched ReLU network\. This advantage stems from certifying compact linear slab rules directly; the resulting intervals are tighter than those produced by network\-level IBP at every noise level tested\. Among the neural models, the BNN achieves higher certified stability than the matched ReLU network at every noise level, consistent with the tighter bounds produced by Bern\-IBP\[[26](https://arxiv.org/html/2608.20497#bib.bib5)\]\.

TABLE X:Robustness certification on MAGIC\. Empirical columns show rule accuracy and rule–BNN fidelity under sampled noise; certified columns show provably stable fractions for rules and networks\.Empirical \(%\)Certified \(%\)ccRule Acc\.Fid\.RulesBNNReLU0\.00100\.098\.2100\.0100\.0100\.00\.0199\.697\.883\.482\.180\.10\.0399\.097\.259\.441\.327\.10\.0598\.496\.442\.716\.012\.40\.1097\.094\.722\.76\.64\.8

∙\\bulletRule Robustness to Data Distribution Shift\.A key concern for streaming edge deployment is whether rules remain reliable under distribution shift\. Table[XI](https://arxiv.org/html/2608.20497#S4.T11)evaluates this on ACS Income\[[14](https://arxiv.org/html/2608.20497#bib.bib32)\], training on CA\-2018 and testing under temporal and geographic shifts\. The ReLU teacher uses a\{10,512,256,128,2\}\\\{10,512,256,128,2\\\}architecture; the ReLU and BNN students both use\{10,32,2\}\\\{10,32,2\\\}, with rules extracted from the BNN student\.

Under temporal shift, rule coverage is essentially stable \(\+0\.2\+0\.2pp\) and covered accuracy drops by only 1\.6 pp, closely tracking the underlying BNN’s own 1\.1 pp accuracy loss\. Geographic shift is harder: coverage falls 4\.3 pp but remains above 85%, and covered accuracy drops 3\.9 pp, comparable to the 6\.4–7\.0 pp accuracy degradation of the neural baselines\. In both conditions, symbolic extraction does not amplify the underlying model’s degradation\. Crucially, uncovered samples are explicitly flagged at deployment and routed to the fallback model, so distributional failures surface as measurable coverage drops rather than silent mispredictions\.

TABLE XI:Distribution\-shift robustness on ACS Income, training on CA\-2018\. ID = held\-out CA\-2018 test set; GEO\-AVG and TEMP\-AVG are means over shifted conditions;Δ\\Delta= AVG−\-ID \(pp\)\. Means±\\pmstd over 5 seeds\.Geographic shiftTemporal shift \(CA\)SystemMetricID \(CA\-18\)GEO\-AVGΔ\\DeltaMSWYWVTEMP\-AVGΔ\\Delta201920212022ReLU TeacherAcc\. \(%\)81\.28±\\pm0\.0974\.32±\\pm0\.64−\-7\.073\.7374\.6174\.6180\.00±\\pm0\.09−\-1\.380\.7479\.8279\.44ReLUAcc\. \(%\)80\.55±\\pm0\.1574\.10±\\pm0\.76−\-6\.573\.4274\.4574\.4379\.37±\\pm0\.19−\-1\.280\.0379\.2378\.84BNNAcc\. \(%\)80\.89±\\pm0\.2974\.45±\\pm0\.69−\-6\.473\.7974\.7774\.7979\.81±\\pm0\.36−\-1\.180\.4979\.6779\.26RulesCoverage \(%\)90\.09±\\pm0\.8685\.82±\\pm1\.38−\-4\.386\.1085\.0786\.2990\.24±\\pm0\.78\+\+0\.290\.0190\.3590\.35Covered acc\. \(%\)80\.29±\\pm0\.2076\.41±\\pm1\.07−\-3\.977\.3174\.8277\.0978\.67±\\pm0\.42−\-1\.679\.3878\.6278\.00Total acc\. \(%\)78\.58±\\pm0\.1273\.15±\\pm1\.35−\-5\.473\.4172\.4473\.6177\.24±\\pm0\.29−\-1\.377\.7877\.2476\.70

### IV\-HExtending to Transformer FFN Layers

We extend Bern2Edge to the FFN sublayers of TinyBERT4\[[25](https://arxiv.org/html/2608.20497#bib.bib34)\], demonstrating BNN deployment in transformer architectures\. The original TinyBERT4 FFN hidden size \(h=1200h\{=\}1200\) is compressed toh∈\{312,600\}h\\in\\\{312,600\\\}across all FFN layers, with all other components unchanged\. Training follows three stages: \(1\) function\-matching from the teacher GeLU FFN to a degree\-15 BNN at the target width, where higher degree imposes no additional hardware cost under LUT\-based synthesis \(Fig\.[8](https://arxiv.org/html/2608.20497#S4.F8)\), \(2\) substitution of the compressed BNN into the original TinyBERT4 transformer, and \(3\) 10 epochs of KD fine\-tuning from the original TinyBERT4 using the training scheme in Section[III\-A](https://arxiv.org/html/2608.20497#S3.SS1)\.

We synthesize and implement all TinyBERT4 encoder variants on KV260\. Theh=600h\{=\}600Bernstein FFN matches or exceeds TinyBERT4 accuracy \(90\.4890\.48% vs\.90\.3790\.37%\) despite halving the FFN hidden width\. The Bernstein variants’ reduced computational depth translates directly to latency, with end\-to\-end cycle count dropping by up to61\.0%61\.0\\%relative to the teacher and the FFN sublayer alone reaching72\.2%72\.2\\%\(Table[XII](https://arxiv.org/html/2608.20497#S4.T12)\)\. It also reduces DSP, FF, and LUT usage relative to both the teacher and the GeLU baseline at matched width, whereas GeLU’s direct polynomial computation remains at the teacher’s DSP count regardless of compression\. The higher BRAM usage of Bernstein FFNs relative to GeLU is attributable to per\-neuron LUT storage, though both remain below TinyBERT4’s original footprint\. Despite their smaller size, both compressed GeLU variants increase FF usage over the teacher, as their buffers are synthesized as distributed RAM, implemented directly in FFs and LUTs rather than dedicated memory blocks\.

TABLE XII:SST\-2 accuracy, latency, and FPGA resource utilization for full 4\-layer FFN substitution\.*FFN*: FFN sublayers only;*Full*: complete encoder\. Resource reductions are relative to TinyBERT4\.TinyBERT4h=600h\{=\}600h=312h\{=\}312ScopeMetricGeLUBernGeLUBernSST\-2 Acc\. \(%\)90\.3790\.0290\.4889\.1190\.02FFNLat\. \(cycles\)6,413,5203,282,640\(↓\\downarrow48\.8%\)3,278,288\(↓\\downarrow48\.9%\)1,789,648\(↓\\downarrow72\.1%\)1,785,296\(↓\\downarrow72\.2%\)DSP121121\(0\.0%\)80\(↓\\downarrow33\.9%\)121\(0\.0%\)80\(↓\\downarrow33\.9%\)BRAM15582\(↓\\downarrow47\.1%\)128\(↓\\downarrow17\.4%\)80\(↓\\downarrow48\.4%\)108\(↓\\downarrow30\.3%\)FF17,74619,580\(↑\\uparrow10\.3%\)15,654\(↓\\downarrow11\.8%\)19,528\(↑\\uparrow10\.0%\)15,568\(↓\\downarrow12\.3%\)LUT34,74435,165\(↑\\uparrow1\.2%\)30,070\(↓\\downarrow13\.5%\)34,924\(↑\\uparrow0\.5%\)29,789\(↓\\downarrow14\.3%\)FullLat\. \(cycles\)7,591,5844,460,704\(↓\\downarrow41\.2%\)4,456,352\(↓\\downarrow41\.3%\)2,967,712\(↓\\downarrow60\.9%\)2,963,360\(↓\\downarrow61\.0%\)DSP319319\(0\.0%\)278\(↓\\downarrow12\.9%\)319\(0\.0%\)278\(↓\\downarrow12\.9%\)BRAM16491\(↓\\downarrow44\.5%\)137\(↓\\downarrow16\.5%\)89\(↓\\downarrow45\.7%\)117\(↓\\downarrow28\.7%\)FF40,30242,049\(↑\\uparrow4\.3%\)38,210\(↓\\downarrow5\.2%\)41,997\(↑\\uparrow4\.2%\)38,037\(↓\\downarrow5\.6%\)LUT81,03481,378\(↑\\uparrow0\.4%\)76,360\(↓\\downarrow5\.8%\)81,137\(↑\\uparrow0\.1%\)76,002\(↓\\downarrow6\.2%\)

## VLimitations and Future Work

Bern2Edgeexposes three rule extraction parameters \(kk,αsc\\alpha\_\{\\text\{sc\}\}, andαconf\\alpha\_\{\\text\{conf\}\}\), each with a monotone, predictable effect on a specific hardware–accuracy trade\-off \(Section[IV\-F](https://arxiv.org/html/2608.20497#S4.SS6)\)\. A natural next step is automated design\-space exploration \(DSE\) that sweeps these knobs to select the Pareto\-optimal configuration for a given deployment constraint\.

Bernstein activations are architecturally compatible with transformer FFN sublayers, but rule extraction over dense latent representations yields rules with limited semantic interpretability\. Concept probing\[[27](https://arxiv.org/html/2608.20497#bib.bib33)\]identifies semantically labeled directions in the hidden space, which could enable rule extraction over a compact, interpretable basis\.

Bernstein activations are not inherently limited to tabular MLPs or transformer FFNs\. Polynomial networks have demonstrated superior representation power over ReLU\-based CNNs on vision tasks\[[12](https://arxiv.org/html/2608.20497#bib.bib40)\], making convolutional extension ofBern2Edge’s pipeline a natural future direction\.

## VIConclusion

We presentedBern2Edge, a neurosymbolic compiler that deploys Bernstein polynomial activations at the edge\.Bern2Edgedistills large pretrained DNNs into compact BNNs that preserve accuracy while exposing two deployment paths: direct LUT\-based FPGA synthesis and geometry\-aligned symbolic rule extraction\. Across all evaluated architectures and datasets, BNNs outperform ReLU under compression, achieving91\.991\.9–99\.899\.8% latency reduction and substantial resource savings over a quantized teacher within0\.50\.5pp accuracy\. The LUT\-based path eliminates the activation train–deploy gap for exact hardware realization, while Bernstein’s geometric structure yields compact, interpretable rules with explicit input\-space constraints\.

## Acknowledgments

This work was supported by the National Science Foundation \(NSF\) under Grant No\. 2504809\. The authors used OpenAI ChatGPT and Anthropic Claude to assist with manuscript preparation, including editing and formatting of the text; all technical content, analysis, and conclusions are the authors’ own\.

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![[Uncaptioned image]](https://arxiv.org/html/2608.20497v1/bio_photos/malak.png)Malak Gamal El\-Dinis a Ph\.D\. student in the Department of Electrical Engineering and Computer Science, University of California, Irvine\. Her research interests include symbolic knowledge distillation, edge AI, and multitask learning\.![[Uncaptioned image]](https://arxiv.org/html/2608.20497v1/bio_photos/yifan.png)Yifan Zhangreceived the B\.S\. degree from Tongji University, Shanghai, China, in 2020\. He is currently pursuing the Ph\.D\. degree at University of California, Irvine, with research interests in software/hardware co\-design for AI accelerators\.![[Uncaptioned image]](https://arxiv.org/html/2608.20497v1/bio_photos/shoukry.png)Yasser Shoukryis an Associate Professor in the Department of Electrical Engineering and Computer Science, University of California, Irvine\. His research interests include resilience and safety of AI and cyber\-physical systems\.![[Uncaptioned image]](https://arxiv.org/html/2608.20497v1/bio_photos/huang.png)Sitao Huangis an Assistant Professor in the Department of Electrical Engineering and Computer Science, University of California, Irvine\. His research interests include hardware accelerators and programming languages for hardware systems\.![[Uncaptioned image]](https://arxiv.org/html/2608.20497v1/bio_photos/elmalaki.png)Salma Elmalakiis an Associate Professor in the Department of Electrical Engineering and Computer Science, University of California, Irvine\. Her research focuses on cyber\-physical systems, human\-in\-the\-loop systems, mobile computing, and edge AI\.