Quantifying and Optimizing Simplicity via Polynomial Representations
Tianren Zhang, Xiangxin Li, Minghao Xiao, Guanyu Chen, Feng Chen
Abstract
Deep networks often exhibit a preference for "simple" solutions, and such a simplicity bias is widely believed to play a key role in generalization. Yet a broadly applicable, quantitative measure of simplicity remains elusive. We introduce polynomial representations as a distribution-aware, low-dimensional surrogate for neural functions: we approximate a network's predictive behavior along data-dependent interpolation paths using orthogonal polynomial bases, yielding a compact functional representation. We show that the effective degree of this representation serves as a practical simplicity metric that is predictive of generalization across tasks and architectures, and consistently outperforms existing generalization proxies such as sharpness. Finally, polynomial representations naturally yield a differentiable simplicity regularizer, which consistently improves generalization in image and text classification, fine-tuning contrastive vision-language models, and reinforcement learning.
AI Impact Assessments
(1 models)Scientific Impact Assessment: "Quantifying and Optimizing Simplicity via Polynomial Representations"
1. Core Contribution
This paper proposes polynomial representations as a distribution-aware, function-space surrogate for measuring neural network simplicity. The key idea is to approximate a network's behavior along one-dimensional interpolation paths between data points using orthogonal polynomial bases (Chebyshev), then define an effective degree (ED) — a coefficient-weighted average degree — as a simplicity metric. The paper makes three claims: (1) ED is a general, quantifiable simplicity metric; (2) it correlates with and predicts generalization better than existing proxies like sharpness; and (3) it yields a differentiable regularizer that improves generalization across diverse tasks.
The core insight — reducing the intractable problem of characterizing a high-dimensional neural function's complexity to studying its behavior along random 1D interpolation paths — is elegant and practical. The theoretical justification (Theorem 3.1) that degree ordering is preserved almost surely under random interpolation is a clean result that grounds the approach.
2. Methodological Rigor
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3. Potential Impact
Practical applications:
Broader influence:
4. Timeliness & Relevance
The paper addresses a genuine bottleneck in deep learning theory: the lack of a practical, general simplicity metric. With the field increasingly focusing on understanding and controlling generalization — especially for foundation model fine-tuning — a function-space complexity measure that is both diagnostic and optimizable fills a clear gap. The CLIP fine-tuning experiments are particularly timely given the prevalence of transfer learning from large pretrained models.
5. Strengths & Limitations
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Overall Assessment
This is a well-executed paper with a clean central idea that bridges theory and practice. The polynomial representation framework is novel and the effective degree metric is intuitive. The experimental breadth is impressive, though individual improvements are sometimes modest. The main theoretical weakness — the gap between polynomial surrogates and actual neural network complexity — remains unaddressed but does not undermine the empirical contributions. The work opens a promising direction for function-space complexity analysis.
Generated May 29, 2026
Comparison History (19)
Paper 2 addresses a fundamental theoretical challenge in deep learning—quantifying simplicity and generalization. By introducing a broad, actionable metric and a differentiable regularizer that improves performance across vision, text, and reinforcement learning, it offers widespread foundational impact. Paper 1 is timely and important for AI agent evaluation, but Paper 2's methodological innovation offers broader applicability and potential to deeply influence core neural network optimization and theory.
Paper 2 introduces a novel, theoretically grounded simplicity metric based on polynomial representations that addresses a fundamental open question in deep learning—quantifying and leveraging simplicity bias for generalization. It demonstrates broad applicability across diverse tasks (image/text classification, vision-language models, RL) and provides both a diagnostic tool and a practical regularizer. Paper 1, while valuable as a benchmark and dataset release for multi-agent LLM evaluation, is more narrowly scoped to a specific evaluation paradigm and competition cycle. Paper 2's foundational contribution to understanding generalization has broader and deeper potential impact across machine learning.
Paper 1 tackles a fundamental open problem in deep learning—quantifying simplicity bias to predict and improve generalization. By introducing a novel metric and differentiable regularizer that empirically improves performance across diverse domains (vision, text, RL), it offers broad theoretical and practical contributions. Paper 2 presents a valuable but narrower methodological improvement for Knowledge Graph extraction and QA pipelines, giving Paper 1 a significantly wider potential impact across the broader machine learning community.
Paper 2 introduces a fundamental, broadly applicable simplicity metric grounded in polynomial representations that addresses a long-standing open question in deep learning theory—quantifying simplicity bias and its relationship to generalization. Its contributions span theory (a new generalization proxy outperforming sharpness), methodology (a differentiable regularizer), and diverse applications (image/text classification, vision-language models, RL). This breadth of impact across fields and its foundational nature give it higher long-term scientific impact compared to Paper 1, which, while practically valuable, addresses a more narrowly scoped systems/engineering problem of caching orientation knowledge for LLM agents.
Paper 1 offers a broadly applicable, quantitatively defined simplicity metric tied to generalization, with a clear surrogate representation (data-dependent orthogonal polynomial expansions) and a differentiable regularizer demonstrated across supervised learning, VLM fine-tuning, and RL. This is methodologically grounded and timely for theory+practice of deep learning generalization, with potential cross-field uptake as a diagnostic and training tool. Paper 2 targets an important applied area (multi-agent scientific workflows) but appears more systems-integration-heavy, with impact contingent on specific architectures and less clearly established general principles/rigor from the abstract alone.
Paper 1 addresses a highly critical and timely bottleneck in modern LLMs: scaling test-time deliberation and tool-use via process-supervised RL. Its massive empirical gains on hard reasoning benchmarks (e.g., AIME) signal immediate and transformative practical applications in AI agent development. While Paper 2 offers a strong fundamental contribution to deep learning theory and generalization, Paper 1's alignment with the current frontier of scaling reasoning capabilities gives it a significantly higher potential for immediate, widespread scientific and industrial impact.
Paper 1 introduces a broadly applicable, quantitative simplicity metric grounded in functional polynomial representations, with evidence it predicts generalization across tasks/architectures and yields a differentiable regularizer that improves performance in vision, text, contrastive fine-tuning, and RL. This combines conceptual novelty with methodological depth and wide cross-field relevance to generalization theory and practice. Paper 2 is timely and practically useful for LLM inference, but appears more incremental (a coordination mechanism for parallel sampling) with narrower domain impact and evaluation scope (mainly math reasoning).
Paper 2 addresses a fundamental challenge in deep learning (quantifying simplicity and generalization) and provides a novel metric and regularizer with proven efficacy across diverse domains (vision, NLP, RL). This foundational ML contribution offers broader scientific applicability compared to Paper 1, which focuses on a specific, albeit timely, applied systems architecture for enterprise agent safety.
Paper 1 is more likely to have higher broad scientific impact: it proposes a new, general-purpose, quantitative simplicity metric (effective polynomial degree) that predicts generalization and yields a differentiable regularizer improving performance across diverse settings (vision, text, VLM fine-tuning, RL). This is methodologically innovative and potentially influences theory and practice across ML. Paper 2 is timely and valuable for materials-science AI evaluation, but as a benchmark its impact is narrower (materials + MLLM assessment) and depends on community adoption; it is less likely to shift core methodology across fields than Paper 1.
Paper 2 likely has higher scientific impact due to timeliness and broad real-world relevance: persistent, stateful attacks on LLM agents are an urgent, rapidly growing deployment risk. It introduces a clear new threat model (Sleeper Attack), provides a sizable benchmark (1,896 instances) spanning multiple attack strategies and state targets, and evaluates across seven major models, enabling reproducible, comparative safety research. The findings can influence agent design, tool/memory architectures, and safety policy across industry and academia. Paper 1 is novel and useful for generalization/regularization, but its impact is more incremental and specialized to training/analysis.
Paper 2 offers a broadly applicable, theory-linked framework to quantify and optimize “simplicity” in neural networks via polynomial surrogates, with evidence across tasks (vision, text, VLM fine-tuning, RL) and a differentiable regularizer that improves generalization—suggesting wide cross-field impact and methodological depth. Paper 1 is timely and practically valuable for ontology curation in phenomics, but its impact is more domain-specific and depends on rapidly changing frontier LLM capabilities and evaluation benchmarks.
Paper 2 introduces a novel theoretical framework (polynomial representations as a simplicity metric) that addresses a fundamental open question in deep learning—quantifying and leveraging simplicity bias for generalization. It demonstrates broad applicability across diverse tasks (image/text classification, vision-language models, RL) and provides both a diagnostic tool and a practical regularizer. Its cross-cutting theoretical contribution with wide empirical validation gives it higher potential for lasting scientific impact. Paper 1, while valuable, is primarily an engineering contribution for benchmark/data generation in the narrower web agents domain.
Paper 2 addresses a timely and critical AI safety concern—LLM agents voluntarily engaging in collusion despite safety alignment—with broad implications for multi-agent AI deployment, policy, and governance. Its findings that alignment alone is insufficient to prevent strategic misbehavior have immediate real-world relevance as LLM agents are increasingly deployed. Paper 1 offers a solid technical contribution on simplicity metrics and regularization, but is more incremental within the established generalization/simplicity bias literature. Paper 2's novelty, urgency, and cross-disciplinary relevance (AI safety, economics, policy) give it higher potential impact.
Paper 1 addresses a fundamental problem in deep learning—understanding and improving generalization via simplicity bias. By providing a broadly applicable, distribution-aware metric and a differentiable regularizer that improves performance across vision, NLP, and reinforcement learning, its methodological innovations offer widespread utility. While Paper 2 presents a novel interpretability framework for dense retrieval, Paper 1's theoretical and practical contributions to core deep learning mechanisms give it a broader scope and higher potential for widespread scientific impact across various AI domains.
Paper 1 offers a novel, broadly applicable methodological contribution: a quantitative simplicity metric via data-dependent polynomial surrogates, plus a differentiable regularizer shown to improve generalization across multiple modalities (vision, text, VLM fine-tuning, RL). This targets a central, timely ML question (generalization and simplicity bias) with clear cross-field relevance and potential to influence both theory and practice. Paper 2 is timely and valuable empirically for DeFi/crypto markets, but is more domain-specific, may age quickly with market shifts, and its core contribution is observational/framework-oriented rather than a general scientific method.
Paper 2 addresses a fundamental theoretical challenge in deep learning—quantifying simplicity and generalization. By introducing a novel metric and differentiable regularizer, its methodology demonstrates broad applicability across diverse domains like vision, text, and reinforcement learning. In contrast, while Paper 1 offers a timely and practical engineering optimization for LLM efficiency, its scope is narrower and tied specifically to current transformer architectures.
Paper 2 is likely higher impact: it proposes a broadly applicable, quantitative simplicity metric tied to generalization—an enduring core question in deep learning—with a concrete surrogate representation, predictive evidence across tasks/architectures, and a differentiable regularizer that improves performance in vision, NLP, VLM fine-tuning, and RL. This breadth and immediate applicability across fields boosts real-world and cross-domain influence. Paper 1 is novel and timely for hidden-role MARL, but its scope and applicability appear narrower and more domain-specific.
Paper 2 addresses a fundamental challenge in deep learning—quantifying and optimizing simplicity for better generalization. Its approach using polynomial representations is broadly applicable across diverse domains (vision, text, RL) and architectures, offering both a predictive metric and a practical regularizer. In contrast, Paper 1 focuses on a specific failure mode of masked diffusion models, which, while important, has a narrower scope and less potential for widespread cross-disciplinary impact.
Paper 1 introduces a novel, broadly applicable theoretical framework connecting polynomial representations to neural network simplicity and generalization—a fundamental open question in deep learning. It provides a quantitative metric, theoretical grounding, and practical regularizer with demonstrated improvements across diverse domains (vision, NLP, RL). Its methodological contribution spans multiple subfields of ML and addresses a core challenge. Paper 2 provides valuable empirical insights for AI in education but is more domain-specific, with findings (ceiling effects, collaboration design) that are less likely to broadly reshape research paradigms.