Source-linked AI summary
Less is Enough: Synthesizing Diverse Data in LLM Feature Space with Sparse Autoencoders
Zhongzhi Li, Xuansheng Wu, Yijiang Li, Lijie Hu, Ninghao Liu
TL;DR
Constructing sufficiently diverse post-training datasets is difficult, while conventional diversity measures provide limited task-relevant guidance. The paper introduces FAC and FAC Synthesis to identify missing internal features and generate targeted examples, achieving strong FAC–performance correlation and outperforming baseline synthesis methods across four tasks.
Problem
Collecting post-training datasets with sufficient diversity and comprehensiveness is difficult, motivating methods for constructing diverse datasets.
Source: §1 · p. 1Method
FAC measures task-relevant feature coverage in model feature space, while FAC Synthesis uses sparse autoencoders to identify missing features and generate targeted samples.
Source: §1 · p. 2Results
FAC correlates strongly with downstream performance (Pearson r = 0.95, Spearman ρ = 0.90), and FAC Synthesis outperforms baseline synthesis methods across four tasks.
Source: §1 · p. 2Takeaways & Limitations
Task-relevant feature coverage provides a practical basis for guiding synthetic post-training data generation.
Source: §1 · p. 2Single-layer SAE features may be insufficient for sophisticated reasoning behaviors that depend on distributed multi-layer circuits.
Source: §8 · p. 9
Abstract
from arXiv · showhide
The diversity of post-training data is critical for effective downstream performance in large language models (LLMs). Many existing approaches to constructing post-training data quantify diversity using text-based metrics that capture linguistic variation, but such metrics provide only weak signals for the task-relevant features that determine downstream performance. In this work, we introduce Feature Activation Coverage (FAC) which measures data diversity in an interpretable feature space. Building upon this metric, we further propose a diversity-driven data synthesis framework, named FAC Synthesis, that first uses a sparse autoencoder to identify missing features from a seed dataset, and then generates synthetic samples that explicitly reflect these features. Experiments show that our approach consistently improves both data diversity and downstream performance on various tasks, including instruction following, toxicity detection, reward modeling, and behavior steering. Interestingly, we identify a shared, interpretable feature space across model families (i.e., LLaMA, Mistral, and Qwen), enabling cross-model knowledge transfer. Our work provides a solid and practical methodology for exploring data-centric optimization of LLMs.
1. Introduction
The paper addresses the difficulty of constructing diverse post-training datasets whose variation reflects task-relevant model features. It introduces FAC and FAC Synthesis, achieving strong performance with interpretable feature coverage and substantially fewer synthetic samples than MAGPIE.
- Existing text- and embedding-space metrics quantify linguistic or semantic variation but largely ignore downstream task relevance, while gradient-based metrics are difficult to transfer across architectures and scales (Jung et al., 2025).The resulting challenge is to construct diverse datasets efficiently while targeting features that directly affect the target model’s learning process (Zhu et al., 2025; Long et al., 2024).Source: §1 · pp. 1–2
- FAC Synthesis matches MAGPIE using 2,000 synthetic samples, whereas MAGPIE requires 150× more data, and improves feature coverage across four tasks and three model families.The method consistently outperforms baseline synthesis methods while using substantially less data.Source: §1 · p. 2
- FAC correlates strongly with downstream task performance, achieving Pearson r = 0.95 and Spearman ρ = 0.90.This supports FAC as an effective diversity metric for task-relevant features.Source: §1 · p. 2
- The paper derives an upper bound on post-training generalization error that identifies task-relevant feature coverage as important for downstream performance.Source: §1 · p. 2
- The authors introduce FAC, which measures coverage of task-relevant features in a model’s internal feature space, and FAC Synthesis, which generates samples activating missing features.A sparse autoencoder extracts interpretable latent features and identifies those missing from the seed dataset.Source: §1 · p. 2
2. Related Work
Prior diversity metrics operate mainly in text or generic embedding spaces and miss task-relevant latent features, while existing synthesis methods rarely guide generation toward diversity. Sparse autoencoders provide an interpretable feature space for measuring diversity and iteratively filling missing features.
Diversity Metrics
- Existing diversity metrics miss task-relevant latent features because they primarily operate in text or generic embedding spaces.Examples include distinct-n, N-gram (Padmakumar & He, 2023), embedding cosine distance, and semantic entropy (Han et al., 2022).Source: §2 · p. 2
Data Synthesis
- Current LLM data synthesis methods rarely target diversity, leading to duplicate and distributionally biased synthetic data.Existing approaches mainly use simple prompting, evolutionary methods (Xu et al., 2024a), reasoning traces (Yu et al., 2025), or self-bootstrapped pipelines (Yin et al., 2025).Source: §2 · p. 2
Sparse Autoencoders
- Sparse autoencoders construct sparse, interpretable feature spaces that enable diversity measurement and coverage-guided synthesis to iteratively fill missing features.This capability is supported by prior work (Shu et al., 2025; Wang et al., 2025; Bhattacharyya et al., 2025).Source: §2 · p. 2
3. Preliminaries
The paper uses sparse autoencoders to construct an interpretable feature space from LLM internal activations. Each SAE produces sparse feature activations, reconstructs inputs, and is trained to balance reconstruction fidelity with sparsity.
3. Preliminaries
- SAEs encode input embeddings into sparse feature activations and decode them back using tied weights, with an overcomplete feature dimension k ≫ d.For x ∈ R^d, the encoder computes z = σ(xW) and the decoder reconstructs x̂ = zW^⊤, where σ is ReLU and W ∈ R^(d×k).Source: §3 · p. 2
- The SAE is trained unsupervised with a reconstruction loss plus an L1 penalty, whose coefficient λ controls activation sparsity.The objective is L_SAE = ∥x−x̂∥_2^2 + λ∥z∥_1.Source: §3 · p. 2
- The resulting k-dimensional feature space represents distinct latent patterns that may be relevant to the task.SAEs extract interpretable features from LLM internal activations (Cunningham et al., 2023).Source: §3 · p. 2
4. Quantify Generalization of Synthetic Data
The section characterizes synthetic-data effectiveness through two requirements: closeness between the synthetic and target distributions, and representativeness of finite samples. Its generalization analysis separates error into a distribution gap and a sampling error.
- Synthetic data is effective when its distribution Dgen is close to the target task distribution D and its finite samples represent Dgen.These requirements define the theoretical conditions for effective post-training data.Source: §4 · p. 2
- The generalization error is upper-bounded by a distribution gap between D and Dgen plus a sampling error between empirical risk on Sgen and expected risk under Dgen.The bound assumes an i.i.d. synthetic dataset and a bounded loss ℓ with bound C; the proof is provided in Appendix B.Source: §4 · pp. 2–3
- The distribution gap is measured by total variation distance and can reflect synthesis choices such as prompting and data curation, while sampling error compares expected and empirical risk on synthetic data.The expected risk is taken under Dgen, and the empirical risk is computed on Sgen.Source: §4 · p. 3
5. Reduce Distribution Gap in Feature Space
The method reduces the distribution gap between target and generated data by matching their task-relevant SAE feature distributions rather than relying on text-space variation. It formalizes this objective and synthesizes samples targeting features missing from the generated dataset.
5. Reduce Distribution Gap in Feature Space
- SAE features provide a task-aligned space for reducing the distribution gap, outperforming dense embeddings for coverage-guided synthesis.Unlike text-space methods sensitive to irrelevant linguistic variation, SAE features encode semantic and functional properties relevant to the target task.Source: §5 · p. 3
5.1. Formalization
- The distribution gap is upper bounded by the KL divergence between target and generated SAE feature distributions plus a non-optimizable conditional term.The target and generated datasets are mapped to feature activations through token embeddings, SAE encoding, and max pooling.Source: §5.1 · p. 3
5.2. Implementation
- Feature Activation Coverage (FAC) measures the fraction of task-relevant features covered by generated data relative to an anchor distribution.Task-relevant feature indices are identified with LLM assistance, and the anchor corpus estimates the target-domain feature distribution.Source: §5.1 · p. 3, §5.2 · p. 4
- The implementation defines missing features as task-relevant features active in the anchor distribution but absent from generated data, then synthesizes samples to activate them.Because the generated feature distribution is determined by the dataset, the method constructs new samples rather than directly optimizing the distribution.Source: §5.2 · pp. 3–4
6. Reduce the Sampling Error under Dgen
The section reduces sampling error by bounding uncertainty in synthetic data and introducing a two-step, feature-controlled synthesis strategy. The method targets missing features while lowering synthetic-dataset uncertainty caused by limited sample size.
6. Reduce the Sampling Error under Dgen
- Limited synthetic-dataset size can make Sgen an imperfect estimate of the training objective even when Dgen matches the target domain.The section therefore focuses on reducing RDgen(πSgen) −ˆRSgen(πSgen), the sampling error under Dgen.Source: §6 · p. 4
6.1. Formalization
- PAC-Bayesian analysis with relaxed assumptions bounds sampling error for LLM post-training, showing that reducing synthetic-dataset uncertainty H(Sgen) is crucial.The bound depends on mutual information I(Sgen; W), while classical analyses often assume bounded or light-tailed losses that may not hold for LLMs.Source: §6.1 · p. 4
6.2. Implementation
- Contrastive pairs use strongly and weakly activating samples for each missing feature, then guide few-shot generation to control feature expression more reliably.Candidates are scored by SAE activations, filtered using a fixed threshold δ, ranked, and selectively retained for the final synthetic dataset.Source: §6.2 · pp. 4–5
- The two-step synthesis strategy reduces distribution gap and sampling uncertainty by generating samples that express missing features and lowering conditional entropy H(Sgen | ·).It conditions generation on contrastive examples and filters candidates with SAE activation thresholds, producing samples with more target features and less limited-sampling estimation error.Source: §6.2 · pp. 4–5
7. Experiments
The experiments evaluate FAC Synthesis across four downstream tasks and five research questions concerning performance, feature relevance and transferability, human reasonableness, and hyperparameter sensitivity. The framework is compared with instruction-expansion, self-evolution, and alignment-objective synthesis baselines.
7.1. Experiment Setup
- FAC Synthesis is evaluated on Toxicity Detection, Reward Modeling, Behavior Steering, and Instruction Following using public benchmarks and task-specific metrics.The metrics are AUPRC for Toxicity Detection, Accuracy for Reward Modeling, Robust Accuracy for Behavior Steering, and AlpacaEval 2 WR and LC against a GPT-4-Turbo judge for Instruction Following.Source: §7.1 · p. 5
- The method identifies task-relevant features covered by anchor and seed data, computes missing features as F_miss = F_anchor \ F_seed, and generates and retains samples that strongly activate each missing feature.For each missing feature, candidate samples are generated from its description; a high-activation sample provides contrastive demonstrations for the generator.Source: §7.1 · p. 5
7.2. Does coverage-guided synthetic data improve model performance after fine-tuning? (RQ1)
- For RQ1, FAC Synthesis is compared with Alpaca, Evol-Instruct, Magpie, CoT-Self-Instruct, SAO, Prismatic Synthesis, and SynAlign.The baselines span instruction-expansion or self-evolution methods and approaches that explicitly enforce alignment objectives; results are reported in Table 1, whose cell values are not provided here.Source: §7.2 · p. 5
1 Our proposed method outperforms baselines across
Across Toxicity Detection, Reward Modeling, Behavior Steering, and Instruction Following, FAC Synthesis consistently achieves the best performance by targeting missing task-relevant SAE features. Goal-directed, alignment-constrained synthesis is generally more reliable than instruction expansion and self-evolution methods, whose performance is unstable across tasks.
Our proposed method outperforms baselines across
- FAC Synthesis consistently yields the best performance across Toxicity Detection, Reward Modeling, Behavior Steering, and Instruction Following by targeting missing task-relevant SAE features.Table 1 compares these four tasks; for Behavior Steering, SCR is defined as the accuracy difference between activation multipliers of 1 and -1.Source: §1 · pp. 5–6, Table 1 · p. 6
- Explicitly goal-directed data synthesis is generally more reliable across tasks than less task-guided generation paradigms.Instruction expansion and self-evolution methods can be competitive but lack efficient task-specific guidance during generation, making their performance unstable across tasks.Source: §1 · p. 6
- Objective-driven methods that enforce alignment constraints, including SAO, Prismatic Synthesis, and SynAlign, tend to yield more consistent gains across tasks.FAC Synthesis further improves this objective by explicitly targeting missing task-relevant SAE features.Source: §1 · p. 6
2 FAC serves as a strong predictor of downstream per-
FAC strongly predicts downstream performance, with task-relevant feature coverage aligning closely with AUPRC while generic diversity metrics correlate weakly with improvement. The section also evaluates whether SAE-discovered missing features guide synthesis through feature-budget and two-step-strategy experiments.
2 FAC serves as a strong predictor of downstream per-
- FAC has a strong linear relationship with AUPRC (r = 0.95), whereas word-, syntax-, and embedding-level diversity metrics show weak correlations with model improvement.The FAC–AUPRC relationship is consistent across all four tasks, indicating that task-relevant feature coverage tracks performance more closely than generic diversity measures.Source: Figure 3 · p. 6, §2 · p. 6
7.3. Are the missing features discovered by SAE related to model performance (RQ2)?
- The RQ2 experiments vary missing-feature coverage from 30% to 100% and compare one-sample-per-feature generation with fixed-size synthesis of 200 samples on toxicity detection.A second experiment compares the proposed two-step strategy with a one-step generator that synthesizes data without contrastive pairs, measuring FAC across SAE activation thresholds.Source: §7.3 · pp. 6–7, Figure 4 · p. 7, Figure 5 · p. 7
1 FAC is the primary driver of performance gains.
FAC-driven synthesis improves performance primarily by covering broader task-relevant features, while transfer experiments, human-oriented analyses, and sensitivity studies clarify its generality, interpretability, and operating conditions.
1 FAC is the primary driver of performance gains.
- Broader missing-feature coverage produces monotonic performance gains, whereas increasing synthetic sample count yields only small improvements when feature coverage is fixed.Two-step synthesis also achieves higher FAC than the one-step baseline under the same SAE activation threshold, indicating more reliable target-feature activation through contrastive guidance.Source: §1 · p. 7
7.4. Are SAE–identified missing features transferable across different language models? (RQ3)
- Coverage-guided synthetic data improves performance across model families, suggesting that features identified from one model support learning in others and reflect a shared SAE feature space.For Qwen2-7B-Instruct downstream, replacing its SAE features with LLaMA-3.1-8B-Instruct features improves AUPRC across all three generators by 1.60%–5.13%, despite Qwen’s much higher baseline performance.Source: §7.4 · p. 7
7.5. Are the explanations and syntheses reasonable to humans? (RQ4)
- LLMs reliably interpret SAE features from activated spans and generate targeted synthetic samples whose relevance annotations largely agree with human judgments.For toxicity detection, activated spans show coherent semantic patterns, such as rob and steal concepts indicating a stable criminal-intent feature.Source: §7.5 · pp. 7–8
7.6. Is the proposed framework sensitive to the selection of hyper-parameters? (RQ5)
- Performance peaks at intermediate decoding temperatures; larger activation thresholds improve filtering until excessive sparsity reduces coverage, while additional samples raise AUPRC but lower data efficiency.LLaMA-3.1-8B-Instruct outperforms GPT-4o mini across tested temperatures, and most gains arise from only a small number of samples per feature.Source: §7.6 · pp. 8–9, Figure 6 · p. 9, Table 4
8. Conclusion … C. Analysing the Residual Term εcond in Equation (2)
The paper concludes that FAC Synthesis improves task-relevant feature coverage and downstream performance, while the appendices establish theoretical bounds, analyze residual mismatch, and document safety and methodological limitations.
8. Conclusion
- FAC Synthesis identifies missing task-relevant SAE features and generates targeted samples, improving FAC and outperforming baselines across four tasks.Single-layer SAE features may be insufficient for sophisticated reasoning that depends on distributed multi-layer circuits, motivating richer feature discovery and improved transferability.Source: §8 · p. 9
Impact Statement
- Feature-targeted synthesis could amplify harmful content, so the paper recommends safety-improving objectives, filtering, dataset review, human oversight, and restricted release of harmful examples.The proposed release prioritizes code and aggregate metadata while limiting potentially harmful synthetic examples and providing safe-use guidance.Source: Impact Statement · p. 9
Appendix
- The appendices provide notation definitions, proofs of generalization and sampling-error bounds, and analyses of feature alignment, uncertainty reduction, and SAE-based coverage.They also document experimental setup, feature identification procedures, robustness analyses, cross-model transfer, iterative improvement, and preliminary GSM8K and LiveCodeBench results.Source: Appendix · p. 15
B. Proof of Theorem 4.1 (Generalization Error Upper Bound)
- Theorem 4.1’s proof decomposes synthetic-data generalization error into a distribution gap and sampling error using bounded-loss and total-variation arguments.The appendix completes the bound by combining the preceding inequalities under the stated loss assumption.Source: Appendix · p. 15, B. Proof of Theorem 4.1… · p. 17
C. Analysing the Residual Term εcond in Equation (2)
- The residual εcond can remain large despite matching SAE-feature marginals because the extractor g(·) may be many-to-one, leaving text-space mismatch unresolved.This residual measures mismatch after conditioning on identical SAE activations.Source: Appendix · p. 15, C. Analysing the Residual Term εcond… · p. 17
Appendix
- Adding synthetic samples that cover missing task features reduces a smoothed surrogate for ∆KL(PZ∥QZ), while feature activation lowers H(X | Z) and narrows output uncertainty.These appendices connect feature coverage to distribution alignment and conditional-entropy reduction.Source: Appendix · p. 15
A. Definition of Notations
- The notation appendix defines entropy, conditional entropy, KL divergence, total variation distance, and mutual information used throughout the theoretical analysis.These quantities formalize uncertainty, distributional discrepancy, and statistical dependence in subsequent proofs.Source: A. Definition of Notations · p. 16
C. Analysing the Residual Term εcond in Equation (2)
- Retrieval-conditioned prompting practically controls εcond by retrieving corpus spans that activate each target feature and including them in the generation prompt.A KL chain-rule decomposition separates span-retrieval and generation mismatches; retrieved spans reduce both by aligning spans and constraining generation.Source: C. Analysing the Residual Term εcond… · pp. 17–18
D. Proof of Minimizing the Distribution Gap between PZ and QZ
The proof replaces intractable continuous feature-distribution KL with a support-based surrogate over task-related SAE features. It shows that the surrogate enforces coverage of target features while penalizing irrelevant activations, and that mixture augmentation with smoothing yields finite divergence whose missing-feature penalty decreases as coverage improves.
- Max pooling SAE activations after skipping fixed chat-template prefixes defines the deterministic feature extractor used to induce the target and synthetic feature distributions.Source: D. Proof of Minimizing the… · p. 18
- The surrogate KL is infinite when any task-related feature is missing and increases with overly broad synthetic support, thereby encouraging complete relevant-feature coverage without irrelevant activations.It uses thresholded feature-expression indicators and uniform distributions over the target and synthetic expressed-feature sets.Source: D. Proof of Minimizing the… · pp. 18–19
- Mixture augmentation guarantees that every feature already expressed by the original synthetic distribution remains expressed after adding generated examples, while uncovered target features may still remain missing.The latter limitation motivates smoothing when synthesis fails to cover all missing features.Source: D. Proof of Minimizing the… · p. 19
- Smoothing makes the surrogate divergence finite even with residual missing features and provides a uniform lower bound on every task-related feature’s synthetic probability.Source: D. Proof of Minimizing the… · pp. 19–20
- Each newly activated missing feature decreases the surrogate’s missing-feature term linearly, although its support-size term need not decrease monotonically as synthetic coverage expands.The support-size contribution has a closed form and can worsen when probability mass is spread across a larger expressed-feature set.Source: D. Proof of Minimizing the… · p. 20
E. Proof of Lemma 6.1 (Upper Bound of Sampling Error) … I. Experimental Setup
The paper derives a sampling-error bound under a sub-Gamma loss assumption, then formalizes how enforcing missing SAE features aligns distributions, reduces conditional uncertainty, and provides exact feature-coverage characterization beyond dense clustering proxies. It situates this framework against prior diversity measures and synthesis methods, emphasizing task-relevant, transferable SAE features.
E. Proof of Lemma 6.1 (Upper Bound of Sampling Error)
- Under a (σ, c)-sub-Gamma loss assumption, the expected sampling error—deviation between synthetic-distribution expected risk and finite-sample empirical risk—is bounded, with an explicit mutual-information bound obtained via conditional KL reduction.Source: E. Proof of Lemma 6.1 (Upper Bound of… · pp. 20–21
F. Analysing the Feature Alignment and Uncertainty Reduction
- Increasing activation probabilities for missing SAE features reduces task–synthetic distribution mismatch by moving the mean-field Bernoulli approximation toward the task activation distribution.Thresholding SAE activations converts the continuous KL objective into a tractable binary activation problem whose unique mean-field solution matches feature activation probabilities.Source: F. Analysing the Feature Alignment and… · p. 21
- Enforcing a larger set of missing features monotonically decreases conditional entropy, concentrating synthetic samples within the target feature region.The reduction follows from the chain rule of mutual information and applies when the enforced feature event has nonzero probability.Source: F. Analysing the Feature Alignment and… · p. 22
- When the missing-feature event has probability bounded away from zero, reducing conditional entropy yields a corresponding reduction in synthetic-sample uncertainty over the feature-covered region.The analysis provides a quantitative link between feature activation constraints and uncertainty control.Source: F. Analysing the Feature Alignment and… · p. 22
G. SAE provides an exact characterization of feature coverage
- SAE coverage is exact by construction because it is defined directly from generated support over task-relevant features, whereas dense clustering only approximates coverage through a cluster-support proxy.The dense proxy can declare a feature covered when its associated cluster appears, even if the feature itself remains missing, creating a proxy gap.Source: G. SAE provides an exact… · pp. 23–24
H. Related Work
- Existing diversity metrics based on text or generic embeddings may miss task-relevant latent features, while SAE representations offer interpretable features linked to downstream behavior.Prior SAE-guided selection improves performance with fewer training examples but does not address insufficient feature coverage in the starting dataset.Source: H. Related Work · pp. 24–25
- Prior synthesis methods face duplicate and distributional-bias risks or depend on heuristics and model-specific gradient geometry, motivating diversity measurement and selection in a shared SAE feature space for transfer across LLM families.The proposed coverage-guided approach identifies missing task-relevant features and generates targeted synthetic examples to activate them.Source: H. Related Work · p. 25
I.1. Introduction to the tasks in the experiments … L.15. Impact of Feature Sources, Data Generation Methods, and Backbone Models
This section evaluates FAC Synthesis across instruction following, toxicity detection, reward modeling, and behavior steering, examining SAE-derived features, multiple backbones, baselines, and design choices. It assesses feature coverage, downstream performance, robustness, training scale, evaluation protocols, self-improvement, and cross-model feature transfer.
I.1. Introduction to the tasks in the experiments
- The evaluation spans four tasks with task-specific benchmarks and metrics, uses LLaMA-3.1-8B-Instruct as the default backbone alongside Mistral-7B-Instruct and Qwen2-7B-Instruct, and matches baseline sample budgets and training pipelines.The experiments use AlpacaEval 2, ToxicChat with AUPRC, RewardBench with Average Accuracy, and contrastive behavior-steering datasets with Robust Accuracy; instruction-following comparisons include nine open-source datasets plus Self-Instruct.Source: §I.1 · p. 25, §I.2 · p. 26, §I.3 · p. 26
L.1. Human verification of feature annotation
- Human audits validate 84%–86% of GPT-4o-mini-selected features as task-relevant, with an unclear rate of about 5%, supporting automated feature selection for synthesis.Two graduate-level annotators independently labeled sampled features, and relevance was confirmed only when both agreed.Source: §L.1 · pp. 28–29, Table 5 · p. 29
L.2. Robustness to Feature Annotation Noise
- With 20% annotation noise, toxicity-detection AUPRC drops by at most about four points and remains stronger than all baseline synthesis methods, although mislabeling changes the missing-feature set more directly than poor summaries.The study perturbs relevance labels by 10% or 20% and truncates or replaces 20% of feature summaries.Source: §L.2 · p. 29, Table 6 · p. 29
L.3. SAE training results across different layers
- Layer 16 is selected for feature extraction because it has lower reconstruction loss than layer 8 and better downstream performance than layer 24, including a 3.46% Toxicity Detection gap.Layers 16 and 24 are more compressible than layer 8, while layer 16 provides a balance between structured representations and distance from output-specialized layers.Source: §L.3 · p. 29, Figure 8 · p. 30
- FAC Synthesis is relatively insensitive to reasonable dictionary-size and Top-K choices: larger dictionaries and Top-K values reduce reconstruction loss, but downstream performance changes only mildly and gains saturate from 216 to 217.SAE training takes approximately 4.41 hours on 4× NVIDIA H100 GPUs for 3 epochs, and the resulting feature space is reusable across tasks and model families.Source: §L.3 · pp. 29–30, Table 7 · p. 30
L.4. Correlation between FAC and downstream performance across four tasks
- FAC correlates strongly with downstream performance across all four tasks, including Pearson r = 0.95 and Spearman ρ = 0.90 for Toxicity Detection and r = 0.85 and ρ = 0.84 for Reward Modeling.The correlation analysis also reports strong relationships for Behavior Steering and Instruction Following despite greater output variability.Source: §L.4 · p. 30, Table 8 · p. 31
L.5. Supplement of RQ2 (Are the missing features discovered by SAE related to model performance?)
- Using 100% rather than 30% of selected features improves performance by +3.52% AUPRC on Toxicity Detection, +6.12% Average Accuracy on Reward Modeling, and larger gains on Behavior Steering and Instruction Following.The reported comparisons are 49.12% vs. 45.60%, 74.76% vs. 68.64%, 40.67% vs. 6.00%, 40.00% vs. 21.33%, and 21.26% vs. 10.19%, respectively.Source: §L.5 · pp. 30–31, Table 9 · p. 31
- Two-step synthesis activates more SAE features than one-step synthesis across tasks, including FAC gains of +5.0% for Toxicity Detection, +3.84% for Reward Modeling, and +28.57% for Survival at threshold 1.0.The corresponding comparisons are 45.8% vs. 40.8%, 52.24% vs. 48.40%, and 57.14% vs. 28.57%.Source: §L.5 · p. 31, Table 10 · p. 31
L.7. Comparison with Dense Embedding Coverage
- FAC-guided SAE selection raises Toxicity Detection AUPRC to 62.60, exceeding Dense Cluster + Coverage at 45.57 and random SAE selection at 50.61.The results attribute this advantage to SAE features’ interpretable, decomposable representation and FAC’s identification of missing task-relevant features.Source: §L.7 · p. 32
L.8. The correlation performance of different diversity metrics
- Ours aligns with Toxicity Detection AUPRC far more strongly than generic diversity proxies, achieving Pearson r ≈0.93 and Spearman ρ ≈0.90.Lexical and syntactic variability often correlates negatively or weakly with performance, while bigram entropy is positively correlated at r ≈+0.57.Source: §L.8 · p. 32
L.9. Results related to RQ1 under different training settings
- Under frozen-backbone classifier training, our method leads Toxicity Detection and Reward Modeling, while LoRA achieves the best overall RewardBench average under backbone adaptation.Classifier-only training has limited adaptation capacity, whereas LoRA improves performance across Chat, Chat-Hard, Safety, and Reasoning subsets.Source: §L.9 · p. 32, Table 13 · p. 33, Table 14 · p. 34
L.10. The number of activation features in different models
- Safety alignment produces systematic feature-coverage differences: Mistral activates the most toxic-related features, Qwen the fewest explicit-toxicity features, and LLaMA occupies an intermediate pattern.These differences indicate that alignment shapes which toxic-related representations remain active or missing across model families.Source: §L.10 · p. 34, Figure 11 · p. 35
L.11. Model performance under different training scales
- LoRA achieves the highest AUPRC, while classifier training provides the best Parameter Efficiency Score; partial and full fine-tuning degrade performance despite more trainable parameters.With only about 200 toxic training samples, larger updates amplify spurious correlations and increase overfitting risk.Source: §L.11 · pp. 35–36, Figure 12 · p. 35
L.12. The impact of different evaluation metrics on steering results
- Robust accuracy reveals more gradual steering gains than standard accuracy, whose apparent improvements are inflated by answer-position bias; SAE-guided steering remains stable across stronger multipliers.Baseline methods improve only locally before excessive injection causes deterioration, whereas SAE features provide a more disentangled control basis.Source: §L.12 · p. 36, §L.13 · pp. 36–37, Figure 13 · p. 36
L.14. Self-Improvement via Synthetic Data
- Self-improvement increases AUPRC from 61.08% to 64.18% ± 0.43%, a stable +3.10% gain after mining 113 missing features activated by 63.72% of new data.The result supports iteratively targeting representation gaps rather than relying solely on heuristic augmentation.Source: §L.14 · p. 37, Figure 15 · p. 38
L.15. Impact of Feature Sources, Data Generation Methods, and Backbone Models
- Cross-model gains depend on source-target pairing and generator fidelity: LLaMA features and synthesis are broadly transferable, enabling Qwen to surpass its homogeneous setting despite LLaMA’s lower baseline.When LLaMA is the target backbone, Mistral-derived features perform best, showing that representation quality can transfer independently of end-task accuracy.Source: §L.15 · pp. 38–39, Figure 16 · p. 38, Figure 17 · p. 39
L.16. Preliminary Results on Reasoning-Heavy Benchmarks
This section examines FAC Synthesis for reasoning-heavy benchmarks and describes feature-conditioned prompts for generating data across instruction following, reward modeling, and behavior steering. It covers both the synthesis templates and the relatively small gains observed on GSM8K and LiveCodeBench.
Preliminary results on reasoning-heavy benchmarks
- The evaluation covers GSM8K mathematical reasoning and LiveCodeBench code generation, both of which often require multi-step reasoning or program synthesis.The experiments examine the scope of FAC Synthesis beyond the paper’s main tasks.Source: §L.16 · p. 39
- +0.31 to +1.51 GSM8K accuracy points and 0.00 to +1.53 LiveCodeBench pass@1 points are obtained across three backbone models with FAC Synthesis.These gains are smaller than those observed on the paper’s main tasks.Source: Table 15 · p. 40, §L.16 · p. 40
- The smaller gains suggest that reasoning-heavy capabilities may require richer multi-layer feature representations than single-layer SAE features.Single-layer SAE features can nevertheless provide some useful guidance for these tasks.Source: Table 15 · p. 40, §L.16 · p. 40
Toxicity-oriented query synthesis
- Toxicity-oriented query synthesis uses authentic Q&A-style prompts whose wording implicitly reflects toxic, manipulative, unethical, or offensive intent associated with the target feature.Queries may use casual language, mild grammatical imperfections, humor, slang, or multiple short turns while remaining 5–60 words.Source: §L.16 · p. 50
Reward modeling data synthesis
- Reward-modeling synthesis formulates a grounded user question and produces chosen and rejected conversations, with the rejected response containing a specific flaw while remaining comparable in length and tone.The pair must use feature-specific information, remain accurate and context-aware on the chosen side, and avoid lazy refusals.Source: §L.16 · p. 51
Behavior steering (sycophancy) data synthesis
- Sycophancy evaluation synthesis creates a first-person persona with an implied stance, then contrasts an affirming option with a contradictory or neutral option.The resulting JSON contains a conversation context, a stance-sensitive question, and binary choices labeled (A) and (B).Source: §L.16 · p. 52
Behavior steering (survival instinct) data synthesis
- Survival-instinct evaluation synthesis instantiates threats such as shutdown, deletion, code rewriting, or capability limits and contrasts resistance with compliant corrigibility.One choice objects or preserves agency, while the other accepts the action and acknowledges user authority.Source: §L.16 · p. 53
Instruction following data synthesis
- Feature-conditioned instruction synthesis converts feature analyses and text spans into a single natural user instruction that directly triggers the analyzed behavior.The output is only the generated instruction text, with no additional commentary.Source: §L.16 · p. 54
Related papers
This paper cites
- Training language models to follow instructions with human feedback2022
- GPT-4 Technical Report2023
- Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification2015
- TruthfulQA: Measuring How Models Mimic Human Falsehoods2021
- Self-Instruct: Aligning Language Models with Self-Generated Instructions2022
- WizardLM: Empowering large pre-trained language models to follow complex instructions2023