大模型微调是否需要人类可读的文本？

# Is Human-Readable Text Necessary for Effective LLM Fine-Tuning?
Source: [https://arxiv.org/html/2609.35868](https://arxiv.org/html/2609.35868)
Zeyu LiuAffiliation:Institute of Computing Technology, Chinese Academy of SciencesZicheng YanAffiliation:University of Science and Technology of ChinaYunquan ZhangAffiliation:Institute of Computing Technology, Chinese Academy of SciencesDaning ChengAffiliation:Institute of Computing Technology, Chinese Academy of SciencesSong TangAffiliation:University of Shanghai for Science and Technology†Corresponding author

###### Abstract

Is human readability necessary for effective fine\-tuning of large language models? We investigate whether model\-conditioned training representations can preserve or improve adaptation utility without requiring a human\-readable textual form\. We propose Desired\-Update\-Aligned Synthetic Data \(DASA\), which uses activation\-gradient feedback from a frozen reference model to guide the optimization of continuous synthetic input embeddings\. Inspired by the role of activation gradients in local risk reduction, DASA targets useful adaptation updates rather than source\-text reconstruction or linguistic fluency\. The resulting embeddings are used directly for downstream fine\-tuning; discrete token projections are employed only for qualitative inspection\. Experiments on six models from the Llama and Qwen families, ranging from 1B to 32B parameters, cover six benchmarks spanning knowledge, mathematical reasoning, code generation, and commonsense reasoning\. Under matched LoRA adaptation settings, DASA achieves performance comparable to the source natural\-language data and surpasses it in multiple configurations, while outperforming GRADMM in most comparisons\. Further experiments cover general\-domain and task\-specialized source data\. Under the evaluated synthesis settings, DASA provides a3\.63\.6–4\.9×4\.9\\timesspeedup over GRADMM with comparable peak GPU memory\.

## 1Introduction

Human readability and training utility represent two distinct properties of fine\-tuning data for large language models \(LLMs\)\. Readability concerns whether the data are coherent and interpretable to humans, whereas training utility concerns whether they provide effective learning signals that improve model performance on held\-out tasks\. Although natural\-language data can exhibit both properties simultaneously, their co\-occurrence does not imply that one is necessary for the other\. In particular, it remains unclear whether human\-readable expressions are also required for models to benefit from fine\-tuning\. This raises a more fundamental question about the objective of data design: rather than treating human\-readable language as a default representation, can we construct training data directly around the target model’s learning requirements? This work studies the following question: Is human readability necessary for effective fine\-tuning, or can training utility be preserved when linguistic form is substantially relaxed?

We focus on post\-training adaptation of existing language models rather than pretraining from scratch\. Concretely, given a reference model \(fθ0f\_\{\\theta\_\{0\}\}\) and a natural\-language source dataset \(D\), we aim to construct an alternative dataset \(D’\) that no longer relies on the conventional human\-readable form of the original text, while still providing comparable adaptation gains under the same fine\-tuning configuration and training budget\. We further ask whether alternative representations constructed specifically for the receiving model can, in some settings, outperform the original natural\-language data\. Crucially, we define training utility by downstream performance after fine\-tuning on independent evaluation tasks\. We do not require the model to interpret, reconstruct, or translate the alternative sequences back into natural language\. In this sense, the object of interest is not whether the representation remains semantically transparent to humans, but whether it serves as an effective substrate for model adaptation\.

Prior work has shown that, under appropriate conditions, alignment between activation gradients and newly introduced residual directions can provide a basis for constructing local updates that reduce risk\. Inspired by this perspective, we bring the idea of exploiting model\-internal improvement signals into the construction of training data\. Rather than requiring alternative data to preserve the human\-readable form of the original text, we focus on whether it retains the effective signals needed for model adaptation\. To this end, we use internal feedback produced by a reference model on the source data to guide the optimization of alternative training sequences\. Our goal is to investigate whether natural\-language data can be transformed into representations that are difficult for humans to read directly, while still retaining high utility for fine\-tuning\. This design allows us to test whether the adaptation value of data for an existing model can be preserved, or even improved, while substantially changing its surface form\.

In this work, we propose Desired\-Update\-Aligned Synthetic Data \(DASA\), a method for constructing alternative training data guided by the alignment of model\-internal signals\. Given source data, DASA treats the internal signals induced in a reference model as optimization targets and updates alternative sequences through gradient\-based optimization so that the signals they elicit align with the desired construction objective\. Importantly, the procedure does not explicitly optimize for reconstructing the source text or producing fluent natural language\. This allows the resulting sequences to depart substantially from the original word order, syntax, and surface form, yielding training representations that are difficult for humans to read directly\. The central idea is not to treat unreadability itself as a mechanism for improving performance, but to relax constraints on linguistic form and shift the optimization objective toward internal signals useful to the model\. At the same time, we do not assume that similarity in internal signals implies equivalence of the resulting training dynamics\. Instead, we evaluate the validity of this construction criterion through the actual downstream performance obtained after fine\-tuning on the alternative data\.

We evaluate DASA on the Llama and Qwen model families, covering six models from 1B to 32B parameters across a diverse range of downstream tasks\. Under a unified fine\-tuning setup, DASA\-generated alternative data yields adaptation gains comparable to those obtained with the original natural\-language data and surpasses it on multiple model–task combinations\. Compared with the gradient\-matching baseline GRADMM, DASA performs better in the majority of evaluated settings while requiring less time for synthetic data generation\. Additional experiments with both general\-domain corpora and task\-relevant data show that these advantages persist across different data sources, supporting the effectiveness and practicality of constructing training representations from model\-internal signals\.

Our main contributions are: ❶ We propose DASA, which constructs fine\-tuning data by aligning model\-internal signals without requiring text reconstruction or language fluency\. ❷ We show that DASA preserves fine\-tuning utility across model scales, tasks, and data budgets, and can outperform the original text in some settings\. ❸ We evaluate DASA across different data sources, repeated runs, and generation costs to assess its effectiveness and practicality\.

## 2Related Work

Dataset distillation aims to preserve training utility with a small set of synthetic examples, and methods developed for images have since been extended to text\([Wang et al\., 2018](https://arxiv.org/html/2609.35868#bib.bib12);[Zhao et al\., 2021](https://arxiv.org/html/2609.35868#bib.bib13);[Zhao and Bilen, 2021](https://arxiv.org/html/2609.35868#bib.bib14);[Zhao and Bilen, 2023](https://arxiv.org/html/2609.35868#bib.bib15);[Cazenavette et al\., 2022](https://arxiv.org/html/2609.35868#bib.bib16);[Nguyen et al\., 2025](https://arxiv.org/html/2609.35868#bib.bib11)\)\. Text methods include continuous embedding optimization, discrete text generation, and training trajectory matching\([Sucholutsky and Schonlau, 2021](https://arxiv.org/html/2609.35868#bib.bib17);[Maekawa et al\., 2024](https://arxiv.org/html/2609.35868#bib.bib18);[Yao et al\., 2025](https://arxiv.org/html/2609.35868#bib.bib19)\)\. GRADMM matches parameter gradients and, in its main procedure, projects optimized embeddings into low\-perplexity text; it also reports an ablation that replaces top\-kkfiltering withL2L\_\{2\}vocabulary projection\([Nguyen et al\., 2025](https://arxiv.org/html/2609.35868#bib.bib11)\)\.

Prior work has also explored fine\-tuning with difficult\-to\-read instructions while retaining the original natural\-language responses\([Duan et al\., 2025](https://arxiv.org/html/2609.35868#bib.bib20)\), learnable prompts\([Shin et al\., 2020](https://arxiv.org/html/2609.35868#bib.bib21);[Li and Liang, 2021](https://arxiv.org/html/2609.35868#bib.bib22);[Lester et al\., 2021](https://arxiv.org/html/2609.35868#bib.bib23);[Liu et al\., 2022](https://arxiv.org/html/2609.35868#bib.bib24)\), model\-internal information for data synthesis\([Yin et al\., 2020](https://arxiv.org/html/2609.35868#bib.bib25);[Wang et al\., 2022](https://arxiv.org/html/2609.35868#bib.bib26)\), and theoretical analyses of local update directions\([Cheng et al\., 2026](https://arxiv.org/html/2609.35868#bib.bib27)\)\. Adapters and LoRA provide mechanisms for efficient model adaptation\([Houlsby et al\., 2019](https://arxiv.org/html/2609.35868#bib.bib28);[Hu et al\., 2022b](https://arxiv.org/html/2609.35868#bib.bib29)\)\. DASA constructs continuous synthetic inputs for subsequent fine\-tuning by optimizing the first\-order effect of their induced update direction on reference risk, without requiring source\-text reconstruction or linguistic fluency\. The residual expansion analysis motivates this objective but does not guarantee the outcome of finite\-step fine\-tuning\.

## 3methodology

### 3\.1Start Point

The notation, basic setup, and standing assumptions used in this section are given in Appendix[A](https://arxiv.org/html/2609.35868#A1); additional conditions are stated where needed\. Appendix[B](https://arxiv.org/html/2609.35868#A2)provides the relevant definitions and derivations from this prior work\.

At a fixed layer of the reference model, the activation gradientζ\\zetameasures the first\-order sensitivity of the loss to a change in the hidden representationzz\. In their analysis of residual\-block expansion,[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27)consider a realizable parameter directionvvwhose first\-order effect on population risk is

DQ​\(v\)=dd​η​𝔼Q​\[ℓ⁡\(ftop​\(z\+hη​v​\(z\)\),y\)\]\|η=0=𝔼Q​\[ζ⊤​uv​\(z\)\]\.D\_\{Q\}\(v\)=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}\\mathbb\{E\}\_\{Q\}\\\!\\left\[\\ell\\bigl\(f\_\{\\mathrm\{top\}\}\(z\+h\_\{\\eta v\}\(z\)\),y\\bigr\)\\right\]\\right\|\_\{\\eta=0\}=\\mathbb\{E\}\_\{Q\}\\\!\\left\[\\zeta^\{\\top\}u\_\{v\}\(z\)\\right\]\.\(1\)Their Condition 1 requires the existence of such a direction withDQ​\(v\)<0D\_\{Q\}\(v\)<0\. Under the stated differentiability and realizability conditions, their Theorem 4 then establishes that, for some sufficiently smallη\>0\\eta\>0,

𝔼Q​\[ℓ⁡\(ftop​\(z\+hη​v​\(z\)\),y\)\]<𝔼Q​\[ℓ⁡\(ftop​\(z\),y\)\]\.\\mathbb\{E\}\_\{Q\}\\\!\\left\[\\ell\\bigl\(f\_\{\\mathrm\{top\}\}\(z\+h\_\{\\eta v\}\(z\)\),y\\bigr\)\\right\]<\\mathbb\{E\}\_\{Q\}\\\!\\left\[\\ell\\bigl\(f\_\{\\mathrm\{top\}\}\(z\),y\\bigr\)\\right\]\.\(2\)This is an existence result for the expanded model class; it does not assert that a particular dataset or optimizer will find the improving direction\. Their Theorem 5 additionally bounds the probability of nonpositive alignment between independent training and test activation\-gradient means under a nonzero population mean and covariance control\.

These results motivate a reverse data\-construction question: given a target model and reference data, can we construct synthetic inputs whose induced update direction is locally improving on the reference set? DASA turns this directional criterion into a data\-construction objective by optimizing continuous synthetic inputsZsZ^\{s\}such that the parameter\-space updatev⁡\(Zs\)v\(Z^\{s\}\)they induce produces a favorable first\-order change in the reference loss\.

For the reference set𝒟ref=\{\(xi,yi\)\}i=1Nref\\mathcal\{D\}\_\{\\mathrm\{ref\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}, we compute and cache

zi=fbot​\(xi\),ζi=∇ziℓ​\(ftop​\(zi\),yi\)\.z\_\{i\}=f\_\{\\mathrm\{bot\}\}\(x\_\{i\}\),\\qquad\\zeta\_\{i\}=\\nabla\_\{z\_\{i\}\}\\ell\(f\_\{\\mathrm\{top\}\}\(z\_\{i\}\),y\_\{i\}\)\.Replacing the population expectation inDQ​\(v\)D\_\{Q\}\(v\)with the empirical average gives the first\-order change in reference loss alongvv:

D𝒟ref​\(v\)=1Nref​∑i=1Nrefζi⊤​uv​\(zi\)\.D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\(v\)=\\frac\{1\}\{N\_\{\\mathrm\{ref\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}\\zeta\_\{i\}^\{\\top\}u\_\{v\}\(z\_\{i\}\)\.\(3\)
At each synthesis step, the current synthetic examples and their retained prediction targets induce a parameter\-space directionv⁡\(Zs\)v\(Z^\{s\}\)\. We then evaluate the first\-order effect of this direction on the fixed reference examples and optimize

minZs⁡D𝒟ref​\(v⁡\(Zs\)\)=minZs⁡1Nref​∑i=1Nrefζi⊤​uv⁡\(Zs\)​\(zi\)\.\\boxed\{\\min\_\{Z^\{s\}\}D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\\bigl\(v\(Z^\{s\}\)\\bigr\)=\\min\_\{Z^\{s\}\}\\frac\{1\}\{N\_\{\\mathrm\{ref\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}\\zeta\_\{i\}^\{\\top\}u\_\{v\(Z^\{s\}\)\}\(z\_\{i\}\)\.\}\(4\)Gradients of this objective are propagated throughv⁡\(Zs\)v\(Z^\{s\}\)to updateZsZ^\{s\}\. Thus, DASA directly optimizes the directional derivative of the reference loss induced by the candidate synthetic inputs, rather than a distance between activation\-gradient means\. The relation between a LoRA parameter direction anduv​\(z\)u\_\{v\}\(z\)is detailed in Appendix[C](https://arxiv.org/html/2609.35868#A3)\. Under the stated conditions, a negative objective value indicates that the induced update direction is locally improving on the reference set; downstream performance after finite\-step adaptation is evaluated empirically\.

### 3\.2Algorithm Design

Algorithm 1DASA: Desired\-Update\-Aligned Synthetic Data0:Frozen model

fθ0f\_\{\\theta\_\{0\}\}, reference dataset

𝒟ref=\{\(xi,yi\)\}i=1Nref\\mathcal\{D\}\_\{\\mathrm\{ref\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}, target layer

l∗l^\{\*\}, synthetic budget

n×Ln\\times L, steps

TsynT\_\{\\mathrm\{syn\}\}, learning rate

ηsyn\\eta\_\{\\mathrm\{syn\}\}
0:Continuous synthetic inputs

SsynS\_\{\\mathrm\{syn\}\}and shifted\-token targets

YsY^\{s\}
1:for

i=1,…,Nrefi=1,\\ldots,N\_\{\\mathrm\{ref\}\}do

2:Compute

zi←fbot​\(xi\)z\_\{i\}\\leftarrow f\_\{\\mathrm\{bot\}\}\(x\_\{i\}\)and

ζi←∇ziℓ​\(ftop​\(zi\),yi\)\\zeta\_\{i\}\\leftarrow\\nabla\_\{z\_\{i\}\}\\ell\(f\_\{\\mathrm\{top\}\}\(z\_\{i\}\),y\_\{i\}\)\.

3:endfor

4:Define

D𝒟ref​\(v\)≔1Nref​∑i=1Nrefζi⊤​uv​\(zi\)D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\(v\)\\coloneqq\\frac\{1\}\{N\_\{\\mathrm\{ref\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}\\zeta\_\{i\}^\{\\top\}u\_\{v\}\(z\_\{i\}\)\.

5:Select

nnlength\-

LLreference token sequences

X\(0\)X^\{\(0\)\}\.

6:Initialize

ZsZ^\{s\}from the input embeddings of

X\(0\)X^\{\(0\)\}and set

Ys←Shift⁡\(X\(0\)\)Y^\{s\}\\leftarrow\\operatorname\{Shift\}\(X^\{\(0\)\}\)\.

7:for

t=1,…,Tsynt=1,\\ldots,T\_\{\\mathrm\{syn\}\}do

8:Compute the tangent direction

v←v⁡\(Zs\)v\\leftarrow v\(Z^\{s\}\)clip its

ℓ2\\ell\_\{2\}norm to at most

2\.02\.0\.

9:Evaluate

D𝒟ref​\(v\)D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\(v\)\.

10:Update

Zs←Zs−ηsyn​∇ZsD𝒟ref​\(v⁡\(Zs\)\)Z^\{s\}\\leftarrow Z^\{s\}\-\\eta\_\{\\mathrm\{syn\}\}\\nabla\_\{Z^\{s\}\}D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\\bigl\(v\(Z^\{s\}\)\\bigr\)\.

11:endfor

12:

Ssyn←BF16⁡\(Zs\)S\_\{\\mathrm\{syn\}\}\\leftarrow\\mathrm\{BF16\}\(Z^\{s\}\)\.

13:return

\(Ssyn,Ys\)\(S\_\{\\mathrm\{syn\}\},Y^\{s\}\)

DASA takes a frozen pretrained modelfθ0f\_\{\\theta\_\{0\}\}, a finite reference set𝒟ref=\{\(xi,yi\)\}i=1Nref\\mathcal\{D\}\_\{\\mathrm\{ref\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}, a target layerl∗l^\{\*\}, and a synthetic budget ofn×Ln\\times L\. Its output is a continuous input tensor for subsequent fine\-tuning, rather than a set of natural\-language sentences\. We denote the tensor being optimized byZs∈ℝn×L×dembZ^\{s\}\\in\\mathbb\{R\}^\{n\\times L\\times d\_\{\\mathrm\{emb\}\}\}, representingnnsynthetic sequences ofLLcontinuous input embeddings each\. LetX\(0\)=\{xj\(0\)\}j=1nX^\{\(0\)\}=\\\{x\_\{j\}^\{\(0\)\}\\\}\_\{j=1\}^\{n\}denote the length\-LLreference token sequences used for initialization\. We initializeZ0sZ\_\{0\}^\{s\}from their input embeddings and setYs=Shift⁡\(X\(0\)\)Y^\{s\}=\\operatorname\{Shift\}\(X^\{\(0\)\}\), whereyj,ts=xj,t\+1\(0\)y^\{s\}\_\{j,t\}=x^\{\(0\)\}\_\{j,t\+1\}fort=1,…,L−1t=1,\\ldots,L\-1\. These are the standard shifted\-token targets of causal language modeling, not separately annotated labels\. For clarity, Algorithm 1 presents the single\-layer case; in experiments, we apply the same procedure to all Transformer layers, clipping the induced direction separately at each layer\.

#### Synthetic data generation\.

Algorithm[1](https://arxiv.org/html/2609.35868#alg1)summarizes the synthesis procedure\. We first pass the reference data through the frozen model and compute their activation gradients at the target layer\. The synthetic activation gradients determine the induced directionv⁡\(Zs\)v\(Z^\{s\}\)\. To prevent the objective from being reduced merely by increasing the magnitude of the induced update, we clip itsℓ2\\ell\_\{2\}norm to a maximum value of2\.02\.0\. We then evaluate the first\-order effect ofv~​\(Zs\)\\tilde\{v\}\(Z^\{s\}\)on the reference risk using the objective defined in Section 3\.1\. This objective is backpropagated toZsZ^\{s\}, which is updated with SGD\. The model parameters and prediction targets remain fixed throughout synthesis\. After optimization, we convertZsZ^\{s\}to BF16 to obtain the continuous synthetic dataSsynS\_\{\\mathrm\{syn\}\}\.

#### Downstream adaptation and evaluation\.

We pairSsynS\_\{\\mathrm\{syn\}\}with the retained prediction targets and use the continuous embeddings directly for fine\-tuning, without projecting them to discrete tokens\. At this stage, the synthetic data remain fixed while LoRA updates the model parameters\. We evaluate the adapted model on independent tasks and compare it with fine\-tuning on the original text under the same adaptation configuration and training budget\.

## 4Experiments

We evaluate DASA across multiple model scales, downstream tasks, calibration budgets, and data sources\. Beyond overall adaptation performance, we examine robustness across random seeds, computational efficiency, sensitivity to optimization settings, and controlled baselines that isolate the effect of DASA’s update\-alignment objective\. These experiments are designed to assess both the effectiveness and robustness of model\-internal\-signal\-driven synthetic data construction\.

### 4\.1Experimental Setup

Models and Benchmarks\.We evaluate our method on six instruction\-tuned language models spanning two model families and multiple scales: Llama\-3\.2\-1B\-Instruct, Llama\-3\.2\-3B\-Instruct\([Meta, 2024](https://arxiv.org/html/2609.35868#bib.bib2)\), Llama\-3\.1\-8B\-Instruct\([Team, 2024](https://arxiv.org/html/2609.35868#bib.bib1)\), and Qwen3\-8B/14B/32B\([Team, 2025](https://arxiv.org/html/2609.35868#bib.bib5)\)\. We consider six downstream benchmarks, including MMLU\([Hendrycks et al\., 2021a](https://arxiv.org/html/2609.35868#bib.bib3)\), GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2609.35868#bib.bib4)\), MATH\([Hendrycks et al\., 2021b](https://arxiv.org/html/2609.35868#bib.bib6)\), MBPP\([Austin et al\., 2021](https://arxiv.org/html/2609.35868#bib.bib7)\), ARC\-Challenge \(ARC\-C\)\([Clark et al\., 2018](https://arxiv.org/html/2609.35868#bib.bib8)\), and CommonsenseQA \(CSQA\)\([Talmor et al\., 2019](https://arxiv.org/html/2609.35868#bib.bib9)\)\.

Baselines\.We compare against three baselines\.Basedenotes the original model without downstream adaptation\.Real\-textfine\-tunes the model using natural\-language examples under the same data budget as our synthetic data\.GRADMM\([Nguyen et al\., 2025](https://arxiv.org/html/2609.35868#bib.bib11)\)constructs synthetic training examples through parameter\-gradient matching\. In contrast,Oursdirectly optimizes synthetic sequences such that the induced activation\-gradient update aligns with the desired update direction\. Throughout all experiments, including the computational\-efficiency comparison, both DASA and our implementation of GRADMM use all Transformer layers and pass the optimized continuous embeddings directly to downstream LoRA adaptation without vocabulary projection\. Real\-text uses the same sampled reference sequencesX\(0\)X^\{\(0\)\}that initialize DASA\.

LoRA Adaptation and Data Budget\.We freeze the pretrained backbone and optimize LoRA parameters only\. All methods use identical LoRA\([Hu et al\., 2022a](https://arxiv.org/html/2609.35868#bib.bib10)\), optimization, and evaluation settings\. We consider three data budgets,50×204850\\times 2048,100×2048100\\times 2048, and200×2048200\\times 2048, where the first term is the number of sequences and 2048 is the maximum sequence length\.

Evaluation protocol\.We use accuracy for MMLU, ARC\-Challenge, and CommonsenseQA; exact\-match accuracy for GSM8K and MATH after standard answer extraction, and pass@1 for MBPP using execution\-based functional correctness\. All methods are evaluated with identical prompts, few\-shot demonstrations, decoding parameters, and evaluation scripts\. Generative tasks use deterministic decoding, so differences across methods arise from adaptation rather than sampling variation\.

### 4\.2Main Results

Table[1](https://arxiv.org/html/2609.35868#S4.T1)reports results across six models, six tasks, and three calibration budgets\. DASA consistently improves over both the unadapted model and the parameter\-gradient\-matching baseline across model families and scales\. In particular, compared with GRADMM, DASA achieves higher performance in 31 of 36 model–task configurations under the50×204850\\times 2048budget, 35 of 36 under100×2048100\\times 2048, and 35 of 36 under200×2048200\\times 2048\. The advantage therefore becomes more consistent as the available synthetic\-data budget increases rather than disappearing once additional training examples are introduced\.

Effectiveness across model scales\.The improvements are observed from 1B\-scale models to Qwen3\-32B\. Under the200×2048200\\times 2048budget, for example, DASA improves over GRADMM on six of six tasks for Llama\-3\.2\-1B and Llama\-3\.1\-8B, while outperforming GRADMM on all six benchmarks for both Qwen3\-14B and Qwen3\-32B\. On Qwen3\-32B, DASA improves MMLU from 84\.65 to 85\.26, GSM8K from 96\.36 to 97\.19, MATH from 87\.56 to 88\.28, ARC\-C from 71\.76 to 72\.44, and CSQA from 85\.83 to 86\.65\. The same qualitative behavior on substantially different model scales suggests that activation\-level update realization is not tied to a particular parameter regime\.

Effectiveness across capabilities\.DASA also provides gains across qualitatively different task families\. On Llama\-3\.2\-3B with the largest budget, DASA improves MATH from 54\.12 to 55\.16 and MBPP from 71\.2 to 72\.0 over GRADMM, while improving MMLU from 61\.20 to 61\.96 and CSQA from 76\.41 to 77\.23\. Similarly, on Qwen3\-8B, it improves MATH from 86\.44 to 87\.40, MBPP from 62\.2 to 63\.0, and ARC\-C from 66\.30 to 67\.24\. These results indicate that the gains are not concentrated in one evaluation category, but extend to knowledge, reasoning, coding, and commonsense tasks\.

Effect of the calibration budget\.Increasing the calibration budget generally strengthens downstream adaptation for all synthesis methods, but DASA benefits particularly consistently from additional calibration data\. Its relative number of wins over GRADMM increases from 31/36 to 35/36 as the budget grows from50×204850\\times 2048to200×2048200\\times 2048\. This observation is important because a synthetic\-data method may perform well at extremely small budgets simply by exploiting a few favorable examples\. Instead, DASA continues to produce useful updates as the synthetic set grows, suggesting that activation alignment provides a stable optimization target beyond the low\-data regime\.

An additional feature of this experiment is that all downstream tasks use calibration samples drawn from the same cleaned C4 corpus\. The model is therefore not simply trained on natural examples drawn from the target benchmark distribution\. Nevertheless, the induced updates consistently improve task performance, supporting the central premise that the update signal itself, rather than direct semantic correspondence between calibration text and evaluation examples, can drive useful downstream adaptation\.

Table 1:Main results across models and tasks\.Base, Real\-text, GRADMM, and DASA under three C4 calibration budgets; all adaptation methods use identical LoRA and optimization settings\.Token Budget: 50×\\times2048Llama\-3\.2\-1BLlama\-3\.2\-3BLlama\-3\.1\-8BMethodMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQABase45\.7946\.3232\.7049\.654\.6159\.1358\.6282\.0351\.4068\.473\.7273\.8765\.4185\.9050\.7075\.482\.0875\.92Real\-text45\.9146\.5532\.9650\.054\.8659\.3058\.7682\.1851\.6068\.873\.8174\.2065\.4586\.2050\.8875\.682\.3476\.00GRADMM46\.1047\.0833\.4050\.454\.3559\.5458\.9882\.5651\.2069\.074\.0674\.2065\.7286\.1551\.1676\.082\.4276\.17Ours46\.4646\.6334\.2050\.854\.6959\.9559\.2282\.8751\.6669\.674\.3274\.0465\.5686\.0551\.5676\.482\.6876\.41Qwen3\-8BQwen3\-14BQwen3\-32BMethodMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQABase74\.9091\.8983\.9859\.664\.3381\.9878\.8593\.8684\.7664\.666\.9881\.0082\.1194\.5485\.2665\.869\.2884\.03Real\-text75\.0592\.0484\.1259\.864\.6881\.9878\.9394\.1684\.8864\.867\.1581\.3382\.3094\.6985\.3466\.269\.3784\.28GRADMM75\.2492\.2784\.3260\.264\.6882\.2379\.0493\.3185\.1065\.067\.3281\.0782\.1194\.6985\.4866\.269\.5484\.28Ours75\.4992\.4984\.6060\.864\.5982\.3179\.0894\.0985\.3065\.667\.4981\.2482\.2494\.7785\.6466\.669\.7184\.52Token Budget: 100×\\times2048Llama\-3\.2\-1BLlama\-3\.2\-3BLlama\-3\.1\-8BMethodMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQABase45\.7946\.3232\.7049\.654\.6159\.1358\.6282\.0351\.4068\.473\.7273\.8765\.4185\.9050\.7075\.482\.0875\.92Real\-text46\.0546\.9333\.1250\.054\.8659\.3858\.8382\.3451\.7268\.873\.8974\.1265\.5986\.2051\.0075\.682\.2576\.09GRADMM46\.4446\.6933\.8850\.854\.2959\.7959\.2082\.9451\.3269\.474\.2374\.4565\.1186\.6651\.4876\.282\.5176\.41Ours47\.2146\.7835\.0252\.054\.6960\.6959\.9184\.0051\.6070\.875\.0075\.1865\.5087\.4952\.3877\.483\.1977\.07Qwen3\-8BQwen3\-14BQwen3\-32BMethodMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQABase74\.9091\.8983\.9859\.664\.3381\.9878\.8593\.8684\.7664\.666\.9881\.0082\.1194\.5485\.2665\.869\.2884\.03Real\-text75\.0792\.1284\.1859\.864\.5182\.1579\.0094\.0184\.9264\.867\.1581\.1682\.2394\.6985\.4066\.069\.3784\.11GRADMM75\.3792\.4284\.5660\.464\.7682\.0978\.2794\.3185\.2465\.467\.4181\.4182\.4594\.0285\.6466\.469\.6284\.36Ours75\.9893\.1085\.2461\.865\.4482\.0678\.9494\.8485\.8466\.468\.0081\.9882\.9194\.6286\.1667\.470\.1484\.85Token Budget: 200×\\times2048Llama\-3\.2\-1BLlama\-3\.2\-3BLlama\-3\.1\-8BMethodMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQABase45\.7946\.3232\.7049\.654\.6159\.1358\.6282\.0351\.4068\.473\.7273\.8765\.4185\.9050\.7075\.482\.0875\.92Real\-text48\.0148\.6734\.8851\.856\.9161\.2660\.7384\.2353\.7670\.675\.7776\.1767\.4988\.1052\.9277\.684\.1378\.13GRADMM47\.4348\.4535\.2052\.456\.5761\.5961\.2084\.0854\.1271\.275\.5176\.4167\.1488\.4852\.2478\.083\.9678\.38Ours47\.8848\.9036\.1253\.457\.1762\.4961\.9684\.0855\.1672\.075\.6077\.2367\.6389\.4652\.8279\.084\.7379\.28Qwen3\-8BQwen3\-14BQwen3\-32BMethodMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQAMMLUG8KMATHMBPPARCCSQABase74\.9091\.8983\.9859\.664\.3381\.9878\.8593\.8684\.7664\.666\.9881\.0082\.1194\.5485\.2665\.869\.2884\.03Real\-text76\.9793\.9386\.1061\.866\.4784\.1180\.8995\.9186\.8466\.869\.1183\.0584\.1296\.5987\.3268\.071\.3386\.08GRADMM77\.4193\.6386\.4462\.266\.3083\.3680\.5196\.2987\.3266\.469\.3782\.7284\.6596\.3687\.5667\.671\.7685\.83Ours78\.0994\.1687\.4063\.067\.2483\.9580\.8197\.1288\.0867\.270\.1483\.6285\.2697\.1988\.2867\.872\.4486\.65

### 4\.3Robustness Across Random Seeds

The main experiment covers many model–task combinations and therefore reports single\-run results\. To assess run\-to\-run variation, we evaluate representative configurations over three independent runs on MMLU, GSM8K, and MBPP for Qwen3\-8B and Qwen3\-14B under the200×2048200\\times 2048data budget\. Table[2](https://arxiv.org/html/2609.35868#S4.T2)reports the mean and standard deviation\.

Table 2:Robustness across random seeds\.Mean±\\pmstandard deviation over three independent runs under the200×2048200\\times 2048data budget\. Random uses Gaussian inputs matched to the mean and standard deviation of DASA’s initialization embeddings\.Qwen3\-8BQwen3\-14BMethodMMLU↑\\uparrowGSM8K↑\\uparrowMBPP↑\\uparrowMMLU↑\\uparrowGSM8K↑\\uparrowMBPP↑\\uparrowRandom71\.8±0\.3571\.8\\pm 0\.3587\.9±0\.4887\.9\\pm 0\.4855\.2±0\.855\.2\\pm 0\.877\.9±0\.3177\.9\\pm 0\.3190\.1±0\.4190\.1\\pm 0\.4162\.6±0\.662\.6\\pm 0\.6Real\-text76\.97±0\.1376\.97\\pm 0\.1393\.93±0\.21\\mathbf\{93\.93\\pm 0\.21\}61\.8±0\.661\.8\\pm 0\.680\.89±0\.1780\.89\\pm 0\.1795\.91±0\.2295\.91\\pm 0\.2266\.8±0\.4\\mathbf\{66\.8\\pm 0\.4\}GRADMM77\.41±0\.1577\.41\\pm 0\.1593\.63±0\.2493\.63\\pm 0\.2462\.2±0\.662\.2\\pm 0\.680\.51±0\.1680\.51\\pm 0\.1696\.29±0\.2296\.29\\pm 0\.2266\.4±0\.466\.4\\pm 0\.4Ours78\.09±0\.16\\mathbf\{78\.09\\pm 0\.16\}93\.78±0\.3393\.78\\pm 0\.3363\.0±0\.6\\mathbf\{63\.0\\pm 0\.6\}81\.15±0\.29\\mathbf\{81\.15\\pm 0\.29\}97\.12±0\.23\\mathbf\{97\.12\\pm 0\.23\}66\.6±0\.666\.6\\pm 0\.6Random baseline\.For each run, Random replaces the optimized synthetic inputs with a tensor whose entries are sampled independently from𝒩⁡\(μE,σE2\)\\mathcal\{N\}\(\\mu\_\{E\},\\sigma\_\{E\}^\{2\}\), whereμE=mean⁡\(Zinit\)\\mu\_\{E\}=\\operatorname\{mean\}\(Z\_\{\\mathrm\{init\}\}\)andσE=std⁡\(Zinit\)\\sigma\_\{E\}=\\operatorname\{std\}\(Z\_\{\\mathrm\{init\}\}\)are computed over all entries of the corresponding DASA initialization tensor\. The random tensor receives no synthetic\-input optimization and is evaluated with the same data budget and downstream LoRA configuration\.

DASA achieves a higher mean than both GRADMM and Random in all six settings, and the highest mean among all methods in four of them\. On Qwen3\-8B, DASA improves MMLU from77\.41±0\.1577\.41\\pm 0\.15with GRADMM to78\.09±0\.1678\.09\\pm 0\.16, and MBPP from62\.2±0\.662\.2\\pm 0\.6to63\.0±0\.663\.0\\pm 0\.6\. On Qwen3\-14B, its GSM8K result is97\.12±0\.2397\.12\\pm 0\.23, compared with96\.29±0\.2296\.29\\pm 0\.22for GRADMM\. Real\-text retains the highest mean on Qwen3\-8B GSM8K and Qwen3\-14B MBPP\.

The standard deviations of DASA are generally comparable to those of Real\-text and GRADMM, although DASA has a higher standard deviation on Qwen3\-14B MBPP\. These results describe the consistency of the observed mean gains across the representative configurations\. Three runs per setting are insufficient to establish statistical significance\.

### 4\.4Task\-Specialized Calibration Data

The main experiments deliberately use generic C4 data to separate update realization from direct access to task\-specific examples\. We next consider a complementary setting in which informative in\-domain calibration data are available\. For each benchmark, calibration examples are drawn from its corresponding training split: MMLU auxiliary train, GSM8K train, MATH train, MBPP train, ARC\-Challenge train, and CommonsenseQA train\. The LoRA configuration, optimization budget, and evaluation protocol remain unchanged, so the only difference from the main setting is the source of the calibration data\.

Table[3](https://arxiv.org/html/2609.35868#S4.T3)shows that DASA continues to outperform GRADMM in all 12 model–task comparisons across Qwen3\-8B and Qwen3\-14B\. It achieves the strongest overall results in eight of the twelve settings, including MMLU, MATH, ARC\-C, and CSQA, across both model scales\. On Qwen3\-14B, for example, DASA reaches 82\.91 on MMLU, 89\.18 on MATH, 71\.59 on ARC\-C, and 84\.52 on CSQA, compared with 82\.17, 88\.38, 70\.73, and 83\.78 for GRADMM\.

The improvement remains visible even though task\-specialized calibration substantially strengthens the competing methods\. This is an important complement to the C4 experiment: DASA is not effective only because generic calibration makes parameter\-gradient matching difficult\. Instead, its advantage persists when the calibration distribution is closely related to the target task and provides a substantially stronger adaptation signal\.

Real\-text remains slightly stronger on GSM8K and MBPP, where directly fine\-tuning on task\-specific natural\-language examples is particularly competitive\. Nevertheless, DASA consistently improves upon parameter\-gradient matching across every evaluated configuration\. Taken together with the generic\-C4 results, these experiments indicate that activation\-aligned synthesis is effective under substantially different calibration distributions rather than relying on a particular source of calibration text\.

Table 3:Results with task\-specialized calibration data\.Calibration data are drawn from the corresponding training split for each benchmark: MMLU auxiliary train, GSM8K train, MATH train, MBPP train, ARC\-Challenge train, and CommonsenseQA train\. All adaptation methods use identical LoRA and optimization settings\.Specialized DataMethodQwen3\-8BQwen3\-14BMMLUGSM8KMATHMBPPARC\-CCSQAMMLUGSM8KMATHMBPPARC\-CCSQABase74\.9091\.8983\.9859\.664\.3381\.9878\.8593\.8684\.7664\.666\.9881\.00Real\-text78\.0294\.6287\.1864\.067\.4985\.0181\.8296\.3688\.0069\.070\.3984\.03GRADMM78\.4194\.3987\.5462\.667\.8384\.7782\.1796\.1388\.3868\.670\.7383\.78Ours79\.2694\.5488\.3662\.868\.7785\.4282\.9196\.2989\.1868\.871\.5984\.52
### 4\.5Computational Efficiency

Matching parameter gradients across all Transformer layers requires repeated computation and comparison of high\-dimensional gradients during synthetic\-data construction\. We therefore compare DASA with our all\-layer implementation of GRADMM in terms of generation time and peak GPU memory under the same three calibration budgets\.

All efficiency measurements are collected on a single NVIDIA A800 GPU using FP16 precision\. Both methods use all Transformer layers and optimize continuous input embeddings, which are used directly for downstream LoRA adaptation without vocabulary projection\. For each calibration budget, they use the same number of devices, synthetic sequences, maximum sequence length, and synthesis steps\. Peak memory is recorded over the complete generation procedure\. These matched settings allow us to compare the computational costs of the two synthesis procedures\.

As shown in Table[4](https://arxiv.org/html/2609.35868#S4.T4), DASA substantially reduces generation time\. At budgets of50×204850\\times 2048,100×2048100\\times 2048, and200×2048200\\times 2048, GRADMM requires 28\.3, 57\.4, and 84\.2 minutes, whereas DASA requires only 5\.8, 11\.6, and 23\.3 minutes, respectively\. This corresponds to approximately3\.63\.6–4\.9×4\.9\\timesfaster synthetic\-data generation\. The reduction is consistent with DASA evaluating directional effects through activation gradients, while GRADMM matches parameter gradients across all Transformer layers\.

Peak GPU memory is similar for the two methods, remaining between 17\.3 and 17\.9 GB across all settings\. Thus, DASA’s measured computational benefit is a substantially shorter generation time, with comparable peak memory\.

Table 4:Computational efficiency\.Generation time and peak GPU memory under three calibration budgets, measured on a single NVIDIA A800 GPU using FP16\. GRADMM and DASA use matched sequence counts, sequence lengths, and synthetic\-data optimization steps\.Generation Time \(min\)↓\\downarrowPeak Memory \(GB\)↓\\downarrowMethod𝟓𝟎×𝟐𝟎𝟒𝟖\\mathbf\{50\\times 2048\}𝟏𝟎𝟎×𝟐𝟎𝟒𝟖\\mathbf\{100\\times 2048\}𝟐𝟎𝟎×𝟐𝟎𝟒𝟖\\mathbf\{200\\times 2048\}𝟓𝟎×𝟐𝟎𝟒𝟖\\mathbf\{50\\times 2048\}𝟏𝟎𝟎×𝟐𝟎𝟒𝟖\\mathbf\{100\\times 2048\}𝟐𝟎𝟎×𝟐𝟎𝟒𝟖\\mathbf\{200\\times 2048\}GRADMM28\.357\.484\.217\.317\.717\.9DASA5\.811\.623\.317\.317\.417\.9
### 4\.6Sensitivity Analysis

We examine whether DASA depends on a narrowly selected downstream adaptation configuration\. We vary the LoRA rank, which controls the capacity of the parameter update, and the number of downstream LoRA optimization steps\. The synthetic\-data optimization schedule is held fixed\. Figure[1](https://arxiv.org/html/2609.35868#S4.F1)reports validation accuracy for adaptation on synthetic and real\-text inputs under these configurations\.

![Refer to caption](https://arxiv.org/html/2609.35868v1/F1.png)Figure 1:Sensitivity to LoRA rank and downstream optimization steps\.Validation accuracy \(%\) under different LoRA ranks and numbers of downstream LoRA optimization steps for synthetic and real\-text adaptation\.DASA performs competitively across a range of configurations rather than relying on a single choice of rank and training duration\. At smaller LoRA ranks, additional downstream steps generally improve validation accuracy\. Larger ranks reach strong performance with fewer steps, while continued adaptation can reduce accuracy\. The real\-text results show a similar dependence on rank and training duration, although the best configurations differ\.

These results suggest that LoRA rank and downstream optimization steps are partly substitutable: a higher rank expands the space of possible updates, while more steps allow the model to use a fixed update space more fully\. Several intermediate configurations achieve similar accuracy, indicating that DASA’s performance does not depend on a narrowly tuned LoRA configuration\.

### 4\.7Qualitative Analysis of Synthetic Data

Figure[2](https://arxiv.org/html/2609.35868#S4.F2)compares a human\-readable solution with a decoded visualization of an optimized DASA sample\. The visualization projects the continuous synthetic embeddings to vocabulary tokens for inspection only\. Downstream adaptation uses the original continuous embeddings\.

The projected sample contains fragmented words, mixed scripts, and locally incoherent token combinations\. This example illustrates that the synthetic inputs need not retain a fluent natural\-language surface form\. Because token projection loses information from the continuous embeddings, we use downstream task performance, rather than the readability of the displayed text, to assess their training utility\.

The continuous synthetic inputs are optimized in the model’s embedding space to induce useful update directions, and their training value is therefore determined by how they interact with the model rather than by whether they admit an interpretable token\-level rendering\. The visualization in Figure[2](https://arxiv.org/html/2609.35868#S4.F2)should thus be viewed as evidence of a separation between human readability and model utility, rather than as a faithful representation of the information contained in the continuous synthetic inputs\.

![Refer to caption](https://arxiv.org/html/2609.35868v1/example.png)Figure 2:Qualitative example of DASA synthetic data\.A human\-readable solution is shown alongside a decoded visualization of an optimized continuous synthetic sample\.

## 5Conclusion

We investigated whether effective LLM fine\-tuning requires human\-readable training inputs\. DASA constructs continuous synthetic input embeddings by optimizing the first\-order effect of their induced direction on a reference loss, while keeping the pretrained model fixed during synthesis\. Across six models, six benchmarks, and three data budgets, the resulting embeddings provide downstream LoRA adaptation that is often comparable to real\-text fine\-tuning and outperforms GRADMM in most evaluated model–task settings\. Under matched synthesis workloads, DASA also reduces generation time by3\.63\.6–4\.9×4\.9\\timeswith comparable peak GPU memory\. These findings show that, in the evaluated adaptation settings, useful training signals can be carried by model\-conditioned continuous inputs without preserving readable language\. The directional objective provides a local criterion rather than a guarantee for finite\-step fine\-tuning; understanding its relationship to downstream updates and transfer across models remains future work\.

## AI Use Statement

We used GPT\-6 solely for language polishing\. All AI\-assisted edits were reviewed by the authors, who take full responsibility for the final manuscript\.

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## Appendix ABasic Setup and Notation

#### Basic setup\.

Let𝒟ref=\{\(xi,yi\)\}i=1Nref\\mathcal\{D\}\_\{\\mathrm\{ref\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}be the finite reference dataset\. We use𝔼𝒟ref​\[⋅\]\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\[\\cdot\]to denote averaging under its empirical distribution, i\.e\.,𝔼𝒟ref​\[g\]=Nref−1​∑i=1Nrefg⁡\(xi,yi\)\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\[g\]=N\_\{\\mathrm\{ref\}\}^\{\-1\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}g\(x\_\{i\},y\_\{i\}\)\. Writezi=fbot​\(xi\)z\_\{i\}=f\_\{\\mathrm\{bot\}\}\(x\_\{i\}\)andζi=∇ziℓ​\(ftop​\(zi\),yi\)\\zeta\_\{i\}=\\nabla\_\{z\_\{i\}\}\\ell\(f\_\{\\mathrm\{top\}\}\(z\_\{i\}\),y\_\{i\}\)\.

Algorithm 1 optimizes continuous synthetic inputsZsZ^\{s\}\. EachZsZ^\{s\}induces a parameter\-space tangent directionv⁡\(Zs\)v\(Z^\{s\}\), whose first\-order effect on the reference activations isuv⁡\(Zs\)​\(zi\)u\_\{v\(Z^\{s\}\)\}\(z\_\{i\}\)\. The synthesis objective is the first\-order variation of the empirical reference loss:

minZs⁡D𝒟ref​\(v⁡\(Zs\)\),D𝒟ref​\(v⁡\(Zs\)\)=1Nref​∑i=1Nrefζi⊤​uv⁡\(Zs\)​\(zi\)\.\\min\_\{Z^\{s\}\}D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\(v\(Z^\{s\}\)\),\\qquad D\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}\(v\(Z^\{s\}\)\)=\\frac\{1\}\{N\_\{\\mathrm\{ref\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{ref\}\}\}\\zeta\_\{i\}^\{\\top\}u\_\{v\(Z^\{s\}\)\}\(z\_\{i\}\)\.A negative value indicates a locally improving direction for a sufficiently small positive step under the stated differentiability and realizability conditions; it does not guarantee the outcome of finite\-step fine\-tuning\.

For any distributionQQ, letμ​\(Q\)=𝔼Q​\[ζ\]\\mu\(Q\)=\\mathbb\{E\}\_\{Q\}\[\\zeta\]andΣQ=CovQ⁡\(ζ\)\\Sigma\_\{Q\}=\\operatorname\{Cov\}\_\{Q\}\(\\zeta\)\. These statistics are used to decompose and boundDQ​\(v\)D\_\{Q\}\(v\)below; they are not optimization targets forZsZ^\{s\}\.

#### Assumptions\.

We analyze local adaptation around the fixed pretrained modelfθ0f\_\{\\theta\_\{0\}\}\. We assume differentiability of the relevant mappings and finite second moments of the activations and activation gradients, ensuring that the expectations, covariance matrices, and first\-order quantities below are well defined\. Additional assumptions required by individual results are stated locally\.

NotationSymbolMeaningSymbolMeaningfθ0=ftop∘fbotf\_\{\\theta\_\{0\}\}=f\_\{\\mathrm\{top\}\}\\circ f\_\{\\mathrm\{bot\}\}Fixed model decomposed atl∗l^\{\*\}\.l∗,Nl^\{\*\},\\,NTarget layer and hidden dimension\.zi=fbot​\(xi\)z\_\{i\}=f\_\{\\mathrm\{bot\}\}\(x\_\{i\}\)Activation of reference exampleiiatl∗l^\{\*\}\.ζi\\zeta\_\{i\}Activation gradient of reference exampleii\.𝒟ref\\mathcal\{D\}\_\{\\mathrm\{ref\}\}Finite reference dataset\.NrefN\_\{\\mathrm\{ref\}\}Number of reference examples\.𝔼𝒟ref\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{\\mathrm\{ref\}\}\}Average over the reference dataset\.QQA distribution, including an empirical distribution\.ZsZ^\{s\}Continuous synthetic inputs optimized by DASA\.SsynS\_\{\\mathrm\{syn\}\}Stored synthetic inputs used for fine\-tuning\.v⁡\(Zs\)v\(Z^\{s\}\)Parameter\-space tangent direction induced byZsZ^\{s\}\.vvA generic parameter\-space tangent direction\.uv​\(z\)u\_\{v\}\(z\)First\-order activation change induced byvvatzz\.DQ​\(v\)D\_\{Q\}\(v\)First\-order risk variation underQQ\.μ⁡\(Q\)\\mu\(Q\)Mean activation gradient underQQ\.ΣQ\\Sigma\_\{Q\}Covariance of activation gradients underQQ\.z¯Q\\bar\{z\}\_\{Q\}Mean activation underQQ\.σz2​\(Q\)\\sigma\_\{z\}^\{2\}\(Q\)Activation dispersion underQQ\.u¯v,Q\\bar\{u\}\_\{v,Q\}Mean first\-order activation change underQQ\.cQ​\(v\)c\_\{Q\}\(v\)Covariance residual in the decomposition ofDQ​\(v\)D\_\{Q\}\(v\)\.hθh\_\{\\theta\}Residual adaptation branch\.A,B,r,αA,B,r,\\alphaLoRA factors, rank, and scaling factor\.∥⋅∥σ\\\|\\cdot\\\|\_\{\\sigma\}Spectral norm\.

#### Synthetic\-induced tangent direction\.

Lethθh\_\{\\theta\}be an auxiliary residual branch at the target layer, parameterized around a zero\-output initializationh0=0h\_\{0\}=0\. For the current synthetic inputsZsZ^\{s\}and their retained prediction targetsysy^\{s\}, letzsz^\{s\}denote the resulting target\-layer activations and letζs=∇zsℓ​\(ftop​\(zs\),ys\)\\zeta^\{s\}=\\nabla\_\{z^\{s\}\}\\ell\(f\_\{\\mathrm\{top\}\}\(z^\{s\}\),y^\{s\}\)\. Write𝔼syn\\mathbb\{E\}\_\{\\mathrm\{syn\}\}for the empirical average using the same prediction\-target mask and normalization as the synthetic loss\. We construct the parameter\-space tangent direction and its effect on a reference activation by

v⁡\(Zs\)=−𝔼syn​\[\(∂hθ​\(zs\)∂θ\|θ=0\)⊤​ζs\],uv⁡\(Zs\)​\(z\)=∂hθ​\(z\)∂θ\|θ=0​v​\(Zs\)\.v\(Z^\{s\}\)=\-\\mathbb\{E\}\_\{\\mathrm\{syn\}\}\\\!\\left\[\\left\(\\left\.\\frac\{\\partial h\_\{\\theta\}\(z^\{s\}\)\}\{\\partial\\theta\}\\right\|\_\{\\theta=0\}\\right\)^\{\\top\}\\zeta^\{s\}\\right\],\\qquad u\_\{v\(Z^\{s\}\)\}\(z\)=\\left\.\\frac\{\\partial h\_\{\\theta\}\(z\)\}\{\\partial\\theta\}\\right\|\_\{\\theta=0\}v\(Z^\{s\}\)\.Thusv⁡\(Zs\)v\(Z^\{s\}\)is constructed from the synthetic activation gradients in the tangent space of the auxiliary residual branch\. The pretrained model and the auxiliary branch remain fixed during synthesis; onlyZsZ^\{s\}is updated\. LoRA fine\-tuning is performed separately after the synthetic data have been generated\.

## Appendix BTheoretical Background

#### Auxiliary residual branch\.

[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27)study the expansion of a fixed reference modelfold∗=ftop∘fbotf\_\{\\mathrm\{old\}\}^\{\*\}=f\_\{\\mathrm\{top\}\}\\circ f\_\{\\mathrm\{bot\}\}\. At a selected insertion point, letz=fbot​\(x\)z=f\_\{\\mathrm\{bot\}\}\(x\)and add an auxiliary residual branchhθh\_\{\\theta\}:

fθ​\(x\)=ftop​\(z\+hθ​\(z\)\)\.f\_\{\\theta\}\(x\)=f\_\{\\mathrm\{top\}\}\\bigl\(z\+h\_\{\\theta\}\(z\)\\bigr\)\.\(5\)The branch has a zero\-output initialization,h0​\(z\)=0h\_\{0\}\(z\)=0, so the expanded model initially computes the same function asfold∗f\_\{\\mathrm\{old\}\}^\{\*\}\. For a parameter\-space directionvv, letuv​\(z\)u\_\{v\}\(z\)denote the first\-order representation change induced by the branch, and define the activation gradient

ζ=∇zℓ​\(ftop​\(z\),y\)\.\\zeta=\\nabla\_\{z\}\\ell\\bigl\(f\_\{\\mathrm\{top\}\}\(z\),y\\bigr\)\.\(6\)The inner productζ⊤​uv​\(z\)\\zeta^\{\\top\}u\_\{v\}\(z\)is the first\-order change in the loss of an individual example\. Herevvis a direction in the branch’s parameter space, whereasuv​\(z\)u\_\{v\}\(z\)is its effect in representation space\. The representation change must be realizable by the branch\.

#### Population risk and the population jumpboard\.

Let𝒬\\mathcal\{Q\}be a data distribution\. The*population risk*of a modelffis its expected loss under𝒬\\mathcal\{Q\}:

R𝒬​\(f\)=𝔼\(x,y\)∼𝒬​\[ℓ⁡\(f⁡\(x\),y\)\]\.R\_\{\\mathcal\{Q\}\}\(f\)=\\mathbb\{E\}\_\{\(x,y\)\\sim\\mathcal\{Q\}\}\\bigl\[\\ell\(f\(x\),y\)\\bigr\]\.\(7\)For the expanded model along the pathη​v\\eta v, writefη​v​\(x\)=ftop​\(z\+hη​v​\(z\)\)f\_\{\\eta v\}\(x\)=f\_\{\\mathrm\{top\}\}\\bigl\(z\+h\_\{\\eta v\}\(z\)\\bigr\)\. Under the differentiability and expectation\-interchange conditions of[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27), the initial directional derivative of its population risk is

D𝒬​\(v\)\\displaystyle D\_\{\\mathcal\{Q\}\}\(v\)=dd​η​R𝒬​\(fη​v\)\|η=0\\displaystyle=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}R\_\{\\mathcal\{Q\}\}\(f\_\{\\eta v\}\)\\right\|\_\{\\eta=0\}\(8\)=𝔼𝒬​\[ζ⊤​uv​\(z\)\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathcal\{Q\}\}\\bigl\[\\zeta^\{\\top\}u\_\{v\}\(z\)\\bigr\]\.Condition 1 of[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27)requires an admissible insertion point and a realizable directionvpopv\_\{\\mathrm\{pop\}\}such thatD𝒬​\(vpop\)<0D\_\{\\mathcal\{Q\}\}\(v\_\{\\mathrm\{pop\}\}\)<0\. By differentiability, there is a sufficiently small positiveη\\etafor which

R𝒬​\(fη​vpop\)<R𝒬​\(fold∗\)\.R\_\{\\mathcal\{Q\}\}\(f\_\{\\eta v\_\{\\mathrm\{pop\}\}\}\)<R\_\{\\mathcal\{Q\}\}\(f\_\{\\mathrm\{old\}\}^\{\*\}\)\.\(9\)Their Theorem 4 therefore establishes the existence of an expanded modelf~pop\\widetilde\{f\}\_\{\\mathrm\{pop\}\}with lower population risk\. We refer to this comparison model as the*population jumpboard*\. It witnesses an improving point in the expanded model class; the existence result does not imply that a particular optimization algorithm will find it\.

#### Empirical risk and the empirical jumpboard\.

For a finite sampleS=\{\(xi,yi\)\}i=1mS=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{m\}, its*empirical risk*is

LS​\(f\)=1m​∑i=1mℓ⁡\(f⁡\(xi\),yi\)\.L\_\{S\}\(f\)=\\frac\{1\}\{m\}\\sum\_\{i=1\}^\{m\}\\ell\(f\(x\_\{i\}\),y\_\{i\}\)\.\(10\)ThusLStrainL\_\{S\_\{\\mathrm\{train\}\}\}andLStestL\_\{S\_\{\\mathrm\{test\}\}\}denote the losses averaged over the training and test samples, respectively; neither is identical to the population riskR𝒬R\_\{\\mathcal\{Q\}\}\. A direction selected usingStrainS\_\{\\mathrm\{train\}\}, followed by a small positive step, produces an*empirical jumpboard*f~S\\widetilde\{f\}\_\{S\}\. Unlikef~pop\\widetilde\{f\}\_\{\\mathrm\{pop\}\}, this model depends on the training sample\. It is also distinct from the final trained expanded modelfnewf\_\{\\mathrm\{new\}\}\.

To connect an empirical jumpboard to test loss,[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27)consider a setting in which the residual branch can realize a constant representation perturbation\. LetμM\\mu\_\{M\}andgKg\_\{K\}be the mean activation gradients of independent training and test samples of sizesMMandKK\. If the training\-selected direction realizesuv​\(z\)=−μMu\_\{v\}\(z\)=\-\\mu\_\{M\}, then

dd​η​LStrain​\(fη​v\)\|η=0=−‖μM‖2,dd​η​LStest​\(fη​v\)\|η=0=−μM⊤​gK\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}L\_\{S\_\{\\mathrm\{train\}\}\}\(f\_\{\\eta v\}\)\\right\|\_\{\\eta=0\}=\-\\\|\\mu\_\{M\}\\\|^\{2\},\\qquad\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}L\_\{S\_\{\\mathrm\{test\}\}\}\(f\_\{\\eta v\}\)\\right\|\_\{\\eta=0\}=\-\\mu\_\{M\}^\{\\top\}g\_\{K\}\.\(11\)Consequently,μM⊤​gK\>0\\mu\_\{M\}^\{\\top\}g\_\{K\}\>0implies a local decrease in test loss for a sufficiently small positive step\. This implication requires the constant perturbation to be realizable by the branch and does not certify an arbitrary finite step\.

#### Probability of gradient alignment\.

The preceding local test\-loss argument depends on the sign ofμM⊤​gK\\mu\_\{M\}^\{\\top\}g\_\{K\}\. Theorem 5 of[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27)bounds the probability of nonpositive alignment\. Using the notation of Appendix A, let

μ⁡\(𝒬\)=𝔼𝒬​\[ζ\],Σ𝒬=Cov𝒬⁡\(ζ\)\.\\mu\(\\mathcal\{Q\}\)=\\mathbb\{E\}\_\{\\mathcal\{Q\}\}\[\\zeta\],\\qquad\\Sigma\_\{\\mathcal\{Q\}\}=\\operatorname\{Cov\}\_\{\\mathcal\{Q\}\}\(\\zeta\)\.\(12\)Suppose that the training and test samples are independent draws from𝒬\\mathcal\{Q\},μ⁡\(𝒬\)≠0\\mu\(\\mathcal\{Q\}\)\\neq 0, andλmax​\(Σ𝒬\)≤CΣ​τ2\\lambda\_\{\\max\}\(\\Sigma\_\{\\mathcal\{Q\}\}\)\\leq C\_\{\\Sigma\}\\tau^\{2\}\. Then

Pr⁡\(μM⊤​gK≤0\)≤\\displaystyle\\Pr\(\\mu\_\{M\}^\{\\top\}g\_\{K\}\\leq 0\)\\leq\{\}4​CΣ​τ2M​‖μ⁡\(𝒬\)‖2\+4​CΣ​τ2K​‖μ⁡\(𝒬\)‖2\\displaystyle\\frac\{4C\_\{\\Sigma\}\\tau^\{2\}\}\{M\\\|\\mu\(\\mathcal\{Q\}\)\\\|^\{2\}\}\+\\frac\{4C\_\{\\Sigma\}\\tau^\{2\}\}\{K\\\|\\mu\(\\mathcal\{Q\}\)\\\|^\{2\}\}\(13\)\+4​CΣ​τ2​tr⁡\(Σ𝒬\)M​K​‖μ⁡\(𝒬\)‖4\.\\displaystyle\+\\frac\{4C\_\{\\Sigma\}\\tau^\{2\}\\operatorname\{tr\}\(\\Sigma\_\{\\mathcal\{Q\}\}\)\}\{MK\\\|\\mu\(\\mathcal\{Q\}\)\\\|^\{4\}\}\.The proof separates two possible failures: the training mean may deviate substantially from the population mean, or the independent test mean may fluctuate against the training direction\. The covariance bound controls the probabilities of these events\. Theorem 5 concerns the sign of gradient alignment; it does not by itself lower\-bound the magnitude of a test\-loss improvement\.

#### From a jumpboard to the final expanded model\.

The empirical jumpboard is a comparison model forfnewf\_\{\\mathrm\{new\}\}\. Define its realized test\-loss reduction relative to the reference model as

ΔRtest=LStest​\(fold∗\)−LStest​\(f~S\)\.\\Delta\_\{R\}^\{\\mathrm\{test\}\}=L\_\{S\_\{\\mathrm\{test\}\}\}\(f\_\{\\mathrm\{old\}\}^\{\*\}\)\-L\_\{S\_\{\\mathrm\{test\}\}\}\(\\widetilde\{f\}\_\{S\}\)\.\(14\)Under the model\-selection and uniform\-generalization conditions of[Cheng et al\. \(2026\)](https://arxiv.org/html/2609.35868#bib.bib27), their Theorem 3 combines this reduction with an additional training\-loss improvementΔERM\\Delta\_\{\\mathrm\{ERM\}\}and generalization termsϵM\\epsilon\_\{M\}andϵK\\epsilon\_\{K\}:

LStest​\(fnew\)≤\\displaystyle L\_\{S\_\{\\mathrm\{test\}\}\}\(f\_\{\\mathrm\{new\}\}\)\\leq\{\}LStest​\(fold∗\)−ΔRtest−ΔERM\\displaystyle L\_\{S\_\{\\mathrm\{test\}\}\}\(f\_\{\\mathrm\{old\}\}^\{\*\}\)\-\\Delta\_\{R\}^\{\\mathrm\{test\}\}\-\\Delta\_\{\\mathrm\{ERM\}\}\(15\)\+2​\(ϵM\+ϵK\)\.\\displaystyle\+2\(\\epsilon\_\{M\}\+\\epsilon\_\{K\}\)\.The comparison first relates the final model’s training loss to the jumpboard’s training loss, then applies the uniform train–test loss bound to both models\. A strict improvement of the final model follows from this bound only when the realized gains exceed the generalization terms\. The population\-risk existence result, the local empirical jumpboard argument, and the final\-model test\-loss comparison therefore establish different parts of the prior framework\.

## Appendix CCovariance Control under LoRA

We specialize the first\-order analysis to LoRA\. Consider the residual adaptation branch

hA,B​\(z\)=αr​B​A​z,h\_\{A,B\}\(z\)=\\frac\{\\alpha\}\{r\}BAz,whereAAandBBare the LoRA factors,rris the rank, andα\\alphais the scaling factor\. We consider the standard initializationB=0B=0\.

Letv=\(VA,VB\)v=\(V\_\{A\},V\_\{B\}\)be a tangent direction\. Along the parameter path

A⁡\(η\)=A\+η​VA,B⁡\(η\)=η​VB,A\(\\eta\)=A\+\\eta V\_\{A\},\\qquad B\(\\eta\)=\\eta V\_\{B\},the residual branch becomes

hA⁡\(η\),B⁡\(η\)​\(z\)\\displaystyle h\_\{A\(\\eta\),B\(\\eta\)\}\(z\)=αr​\(η​VB\)​\(A\+η​VA\)​z\\displaystyle=\\frac\{\\alpha\}\{r\}\(\\eta V\_\{B\}\)\(A\+\\eta V\_\{A\}\)z=η​αr​VB​A​z\+η2​αr​VB​VA​z\.\\displaystyle=\\eta\\frac\{\\alpha\}\{r\}V\_\{B\}Az\+\\eta^\{2\}\\frac\{\\alpha\}\{r\}V\_\{B\}V\_\{A\}z\.Taking the derivative with respect toη\\etaatη=0\\eta=0gives

uv​\(z\)=dd​η​hA⁡\(η\),B⁡\(η\)​\(z\)\|η=0=αr​VB​A​z\.u\_\{v\}\(z\)=\\left\.\\frac\{d\}\{d\\eta\}h\_\{A\(\\eta\),B\(\\eta\)\}\(z\)\\right\|\_\{\\eta=0\}=\\frac\{\\alpha\}\{r\}V\_\{B\}Az\.Thus, the perturbationVAV\_\{A\}has no first\-order contribution under the initializationB=0B=0\.

We next decompose the first\-order variation\. By definition,

DQ​\(v\)=𝔼Q​\[ζ⊤​uv\]\.D\_\{Q\}\(v\)=\\mathbb\{E\}\_\{Q\}\[\\zeta^\{\\top\}u\_\{v\}\]\.Write

ζ=μ⁡\(Q\)\+\(ζ−μ⁡\(Q\)\)\\zeta=\\mu\(Q\)\+\(\\zeta\-\\mu\(Q\)\)and

uv=u¯v,Q\+\(uv−u¯v,Q\)\.u\_\{v\}=\\bar\{u\}\_\{v,Q\}\+\(u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\)\.Substituting both decompositions gives

DQ​\(v\)\\displaystyle D\_\{Q\}\(v\)=𝔼Q​\[\(μ⁡\(Q\)\+ζ−μ⁡\(Q\)\)⊤​\(u¯v,Q\+uv−u¯v,Q\)\]\\displaystyle=\\mathbb\{E\}\_\{Q\}\\Big\[\\big\(\\mu\(Q\)\+\\zeta\-\\mu\(Q\)\\big\)^\{\\top\}\\big\(\\bar\{u\}\_\{v,Q\}\+u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\\big\)\\Big\]=μ​\(Q\)⊤​u¯v,Q\+μ​\(Q\)⊤​𝔼Q​\[uv−u¯v,Q\]\\displaystyle=\\mu\(Q\)^\{\\top\}\\bar\{u\}\_\{v,Q\}\+\\mu\(Q\)^\{\\top\}\\mathbb\{E\}\_\{Q\}\[u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\]\+𝔼Q​\[ζ−μ⁡\(Q\)\]⊤​u¯v,Q\\displaystyle\+\\mathbb\{E\}\_\{Q\}\[\\zeta\-\\mu\(Q\)\]^\{\\top\}\\bar\{u\}\_\{v,Q\}\+𝔼Q​\[\(ζ−μ⁡\(Q\)\)⊤​\(uv−u¯v,Q\)\]\.\\displaystyle\+\\mathbb\{E\}\_\{Q\}\\left\[\(\\zeta\-\\mu\(Q\)\)^\{\\top\}\(u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\)\\right\]\.Since

𝔼Q​\[ζ−μ⁡\(Q\)\]=0\\mathbb\{E\}\_\{Q\}\[\\zeta\-\\mu\(Q\)\]=0and

𝔼Q​\[uv−u¯v,Q\]=0,\\mathbb\{E\}\_\{Q\}\[u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\]=0,the two cross terms vanish\. Therefore,

DQ​\(v\)=μ​\(Q\)⊤​u¯v,Q\+cQ​\(v\),D\_\{Q\}\(v\)=\\mu\(Q\)^\{\\top\}\\bar\{u\}\_\{v,Q\}\+c\_\{Q\}\(v\),where

cQ​\(v\)=𝔼Q​\[\(ζ−μ⁡\(Q\)\)⊤​\(uv−u¯v,Q\)\]\.c\_\{Q\}\(v\)=\\mathbb\{E\}\_\{Q\}\\left\[\(\\zeta\-\\mu\(Q\)\)^\{\\top\}\(u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\)\\right\]\.
###### Theorem 1\(Covariance Control\)\.

Under the assumptions in Appendix[A](https://arxiv.org/html/2609.35868#A1),

\|cQ​\(v\)\|≤tr⁡\(ΣQ\)​αr​‖VB​A‖σ​σz​\(Q\)\.\\boxed\{\|c\_\{Q\}\(v\)\|\\leq\\sqrt\{\\operatorname\{tr\}\(\\Sigma\_\{Q\}\)\}\\frac\{\\alpha\}\{r\}\\\|V\_\{B\}A\\\|\_\{\\sigma\}\\sigma\_\{z\}\(Q\)\.\}

###### Proof\.

By definition,

cQ​\(v\)=𝔼Q​\[\(ζ−μ⁡\(Q\)\)⊤​\(uv−u¯v,Q\)\]\.c\_\{Q\}\(v\)=\\mathbb\{E\}\_\{Q\}\\left\[\(\\zeta\-\\mu\(Q\)\)^\{\\top\}\(u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\)\\right\]\.Applying Cauchy–Schwarz twice gives

\|cQ​\(v\)\|\\displaystyle\|c\_\{Q\}\(v\)\|≤𝔼Q​\[‖ζ−μ⁡\(Q\)‖2​‖uv−u¯v,Q‖2\]\\displaystyle\\leq\\mathbb\{E\}\_\{Q\}\\left\[\\\|\\zeta\-\\mu\(Q\)\\\|\_\{2\}\\\|u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\\\|\_\{2\}\\right\]≤𝔼Q​‖ζ−μ⁡\(Q\)‖22​𝔼Q​‖uv−u¯v,Q‖22\.\\displaystyle\\leq\\sqrt\{\\mathbb\{E\}\_\{Q\}\\\|\\zeta\-\\mu\(Q\)\\\|\_\{2\}^\{2\}\}\\sqrt\{\\mathbb\{E\}\_\{Q\}\\\|u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\\\|\_\{2\}^\{2\}\}\.
For the first factor, since

ΣQ=𝔼Q​\[\(ζ−μ⁡\(Q\)\)​\(ζ−μ⁡\(Q\)\)⊤\],\\Sigma\_\{Q\}=\\mathbb\{E\}\_\{Q\}\\left\[\(\\zeta\-\\mu\(Q\)\)\(\\zeta\-\\mu\(Q\)\)^\{\\top\}\\right\],we have

𝔼Q​‖ζ−μ⁡\(Q\)‖22=tr⁡\(ΣQ\)\.\\mathbb\{E\}\_\{Q\}\\\|\\zeta\-\\mu\(Q\)\\\|\_\{2\}^\{2\}=\\operatorname\{tr\}\(\\Sigma\_\{Q\}\)\.
For the second factor, using

uv​\(z\)=αr​VB​A​zu\_\{v\}\(z\)=\\frac\{\\alpha\}\{r\}V\_\{B\}Azand

u¯v,Q=αr​VB​A​z¯Q,\\bar\{u\}\_\{v,Q\}=\\frac\{\\alpha\}\{r\}V\_\{B\}A\\bar\{z\}\_\{Q\},we obtain

uv−u¯v,Q=αr​VB​A​\(z−z¯Q\)\.u\_\{v\}\-\\bar\{u\}\_\{v,Q\}=\\frac\{\\alpha\}\{r\}V\_\{B\}A\(z\-\\bar\{z\}\_\{Q\}\)\.Hence,

𝔼Q​‖uv−u¯v,Q‖22\\displaystyle\\mathbb\{E\}\_\{Q\}\\\|u\_\{v\}\-\\bar\{u\}\_\{v,Q\}\\\|\_\{2\}^\{2\}=α2r2​𝔼Q​‖VB​A​\(z−z¯Q\)‖22\\displaystyle=\\frac\{\\alpha^\{2\}\}\{r^\{2\}\}\\mathbb\{E\}\_\{Q\}\\\|V\_\{B\}A\(z\-\\bar\{z\}\_\{Q\}\)\\\|\_\{2\}^\{2\}≤α2r2​‖VB​A‖σ2​𝔼Q​‖z−z¯Q‖22\\displaystyle\\leq\\frac\{\\alpha^\{2\}\}\{r^\{2\}\}\\\|V\_\{B\}A\\\|\_\{\\sigma\}^\{2\}\\mathbb\{E\}\_\{Q\}\\\|z\-\\bar\{z\}\_\{Q\}\\\|\_\{2\}^\{2\}=α2r2​‖VB​A‖σ2​σz2​\(Q\)\.\\displaystyle=\\frac\{\\alpha^\{2\}\}\{r^\{2\}\}\\\|V\_\{B\}A\\\|\_\{\\sigma\}^\{2\}\\sigma\_\{z\}^\{2\}\(Q\)\.
Substituting the two bounds above yields

\|cQ​\(v\)\|≤tr⁡\(ΣQ\)​αr​‖VB​A‖σ​σz​\(Q\)\.\|c\_\{Q\}\(v\)\|\\leq\\sqrt\{\\operatorname\{tr\}\(\\Sigma\_\{Q\}\)\}\\frac\{\\alpha\}\{r\}\\\|V\_\{B\}A\\\|\_\{\\sigma\}\\sigma\_\{z\}\(Q\)\.∎

#### Implication\.

From the decomposition above,

DQ​\(v\)=μ​\(Q\)⊤​u¯v,Q\+cQ​\(v\)\.D\_\{Q\}\(v\)=\\mu\(Q\)^\{\\top\}\\bar\{u\}\_\{v,Q\}\+c\_\{Q\}\(v\)\.SincecQ​\(v\)≤\|cQ​\(v\)\|c\_\{Q\}\(v\)\\leq\|c\_\{Q\}\(v\)\|, Theorem[1](https://arxiv.org/html/2609.35868#Thmtheorem1)implies

DQ​\(v\)≤μ​\(Q\)⊤​u¯v,Q\+tr⁡\(ΣQ\)​αr​‖VB​A‖σ​σz​\(Q\)\.D\_\{Q\}\(v\)\\leq\\mu\(Q\)^\{\\top\}\\bar\{u\}\_\{v,Q\}\+\\sqrt\{\\operatorname\{tr\}\(\\Sigma\_\{Q\}\)\}\\frac\{\\alpha\}\{r\}\\\|V\_\{B\}A\\\|\_\{\\sigma\}\\sigma\_\{z\}\(Q\)\.Therefore, if

−μ​\(Q\)⊤​u¯v,Q\>tr⁡\(ΣQ\)​αr​‖VB​A‖σ​σz​\(Q\),\-\\mu\(Q\)^\{\\top\}\\bar\{u\}\_\{v,Q\}\>\\sqrt\{\\operatorname\{tr\}\(\\Sigma\_\{Q\}\)\}\\frac\{\\alpha\}\{r\}\\\|V\_\{B\}A\\\|\_\{\\sigma\}\\sigma\_\{z\}\(Q\),then the right\-hand side is strictly negative, and hence

For a fixed reference distributionQQ, the activation statisticsμ⁡\(Q\)\\mu\(Q\),ΣQ\\Sigma\_\{Q\}, andσz​\(Q\)\\sigma\_\{z\}\(Q\)are fixed\. In contrast, the directionv=v⁡\(Zs\)v=v\(Z^\{s\}\)is induced by the synthetic inputs and can therefore be shaped during synthesis\. OptimizingZsZ^\{s\}can increase the favorable mean component−μ​\(Q\)⊤​u¯v,Q\-\\mu\(Q\)^\{\\top\}\\bar\{u\}\_\{v,Q\}while controlling the geometry of the induced LoRA direction through‖VB​A‖σ\\\|V\_\{B\}A\\\|\_\{\\sigma\}\. The bound therefore provides a mechanism by which synthetic\-data optimization can favor directions whose mean improvement dominates the covariance residual, whereas a fixed real\-text corpus does not directly optimize this trade\-off\.

## Appendix DTraining Details

#### Downstream LoRA adaptation\.

We freeze all pretrained backbone parameters and perform downstream adaptation using LoRA with rankr=32r=32, scaling factorα=32\\alpha=32, and dropout0\.050\.05\. LoRA modules are applied to the query, key, value, output, gate, up, and down projection layers\. We optimize the LoRA parameters for 32 optimizer steps using AdamW with a learning rate of2×10−42\\times 10^\{\-4\},\(β1,β2\)=\(0\.9,0\.999\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.9,0\.999\),ϵ=10−8\\epsilon=10^\{\-8\}, weight decay0\.010\.01, and gradient clipping at1\.01\.0\. We use a cosine learning\-rate schedule with a3%3\\%warmup ratio, BF16 precision, a maximum sequence length of 2048, a per\-device batch size of 1, and gradient accumulation over 8 steps\. Unless otherwise specified, all experiments use the same downstream adaptation configuration across models, tasks, and methods\.

#### Synthetic\-data optimization\.

DASA initializes continuous synthetic inputs from the input embeddings of sampled reference sequences and retains their next\-token prediction targets\. The pretrained model and prediction targets remain fixed during synthesis; only the synthetic inputs are optimized\. We update these inputs forTsyn=60T\_\{\\mathrm\{syn\}\}=60steps using SGD with learning rateηsyn=0\.1\\eta\_\{\\mathrm\{syn\}\}=0\.1, zero momentum, zero weight decay, and gradient clipping at1\.01\.0\. The resulting synthetic tensors are stored in BF16 and used directly as input embeddings during downstream LoRA adaptation\. DASA and GRADMM use identical sequence counts, sequence lengths, synthesis steps, and downstream adaptation configurations\.