Efficiently Learning Drifting Halfspaces with Massart Noise
Mingchen Ma, Guyang Cao, Jelena Diakonikolas, Ilias Diakonikolas
Abstract
We study the problem of learning a drifting concept in the presence of Massart noise. In this framework, an online learner has access to a history of independent samples whose labels are noisy versions of a target concept that may change from round to round. The goal is to output, in each round, a hypothesis with small prediction error. We study the complexity of this learning problem for the fundamental class of margin-separable linear classifiers (halfspaces). On the positive side, we give a computationally efficient learner achieving error , where upper bounds the Massart noise rate, is the drift rate, and is the margin. Interestingly, in the realizable setting, an adaptation of our techniques yields an efficient learner with an improved error rate over prior work. On the lower-bound side, we provide formal evidence of an information-computation tradeoff, strongly suggesting that our algorithm's performance is essentially optimal. Specifically, while the information-theoretically optimal error scales with , we prove that -scaling is unavoidable for low-degree polynomial tests, even in the special case of random classification noise.
AI Impact Assessments
(1 models)Scientific Impact Assessment
Core Contribution
This paper addresses the problem of learning halfspaces (linear classifiers) under distribution drift with Massart noise—a semi-random noise model where each label can be flipped with bounded probability. The central contributions are threefold:
1. An efficient algorithm achieving error η + Õ(Δ^{1/3}/γ) for γ-margin halfspaces under Massart noise with drift rate Δ, the first polynomial-time algorithm with non-trivial guarantees under label noise in the drift setting.
2. Information-theoretic baselines showing the optimal excess error under Massart noise scales as Θ̃((dΔ)^{1/2}), strictly better than the agnostic Θ((dΔ)^{1/3}) rate.
3. Low-degree polynomial hardness evidence that the Δ^{1/3} scaling achieved by the algorithm is computationally optimal, despite the information-theoretic optimum being Δ^{1/2}—establishing a concrete information-computation gap.
This reveals a striking phenomenon: while Massart noise is information-theoretically equivalent to the realizable setting (both scale with Δ^{1/2}), computationally efficient algorithms must pay the agnostic rate (Δ^{1/3}).
Methodological Rigor
The paper demonstrates strong technical depth across all components:
Algorithm design: The algorithm uses an epoch-based projected gradient descent with a carefully designed gradient inspired by leaky ReLU losses. The key technical contribution is Lemma 3.1 ("Regret to Error"), which bounds prediction error by average regret plus a drift-controlled term. This is non-trivial because under drift, no single ground-truth halfspace realizes all distributions simultaneously. The authors bypass the difficulty of bounding ∥(w*)^(i) - (w*)^(i+1)∥ geometrically (which can be large even when TV distance is small) by directly relating prediction error to regret terms plus drift error controlled by O(ΔW/γ).
Information-theoretic analysis: The upper bound (Theorem 2.1) extends classical ERM analysis with a V(h)-dependent uniform convergence bound using Talagrand's concentration and Rademacher complexity. The matching lower bound uses a clean reduction to coin-flipping distinguishability.
Computational lower bound: The low-degree polynomial framework is applied via a carefully constructed hard instance on the hypercube. The construction forces the problem to reduce to distinguishing RCN labels from pure noise in recent time steps. The correlation analysis (Lemma 4.2) leverages anti-concentration properties of inner products between near-orthogonal vectors in {±1}^d, building on prior work [DDK+23].
The proofs appear complete and carefully structured, with all claims supported by detailed arguments in the appendix.
Potential Impact
Theoretical significance: This work fills a notable gap at the intersection of two well-studied areas—distribution drift learning and noise-tolerant halfspace learning. The information-computation tradeoff is a conceptually important finding: it shows that structured noise models (Massart, RCN) that are computationally tractable without drift become computationally harder with drift in a precise, quantifiable way that matches the agnostic setting's information-theoretic rate.
Algorithmic implications: The improved realizable-case algorithm (Theorem 3.2, achieving Õ(√Δ · γ^{-3/2}) versus prior Õ(√Δ · γ^{-2})) demonstrates that the techniques have broader utility beyond the noisy setting. The RCN extension (Theorem 3.1) achieving opt_t + Õ(Δ^{1/3}/γ) through noise-rate search is practically relevant.
Broader connections: The paper connects to the growing literature on information-computation tradeoffs and low-degree polynomial hardness, contributing a natural and well-motivated instance to this framework. The drift learning setting is increasingly relevant for deployed ML systems.
Timeliness & Relevance
The work is timely on multiple fronts: (1) non-stationary learning is increasingly important for deployed systems like LLMs trained on static data but deployed on evolving content; (2) understanding computational barriers in learning is a central theme in modern theoretical CS/ML; (3) the gap between information-theoretic and computational rates in structured noise models is a topic of active investigation.
Strengths
Limitations
Overall Assessment
This is a technically strong paper that makes a clear conceptual contribution by identifying and rigorously characterizing an information-computation tradeoff for a natural and well-motivated learning problem. The combination of near-matching algorithmic and hardness results, under minimal distributional assumptions, represents a meaningful advance in understanding the computational complexity of learning with distribution drift.
Generated Jun 10, 2026
Comparison History (21)
Paper 1 addresses a fundamental theoretical problem in learning theory—drifting halfspaces with Massart noise—with both upper and lower bounds, including evidence of information-computation tradeoffs. This type of result has lasting impact on computational learning theory. The identification of the Δ^{1/3} computational barrier via low-degree polynomial hardness is a significant conceptual contribution. Paper 2, while solid applied work on preference-based RL, is more incremental, addressing a training-inference mismatch with a practical fix. Paper 1's theoretical depth and novelty give it broader and more enduring scientific impact.
Paper 2 introduces a highly timely and practical benchmark for evaluating autonomous coding agents, an area currently experiencing massive research and industry interest. Benchmarks in the LLM and agent space typically see rapid, widespread adoption and high citation counts. While Paper 1 provides rigorous theoretical contributions to learning theory, its impact is likely confined to a narrower, more specialized subfield, giving Paper 2 broader real-world applicability and overall scientific impact.
HAMNO addresses a broadly impactful problem in scientific computing—learning PDE solution operators for multi-scale, nonlinear dynamical systems—with a novel hierarchical architecture combining local/global representations and physics-informed constraints. Its applicability spans numerous scientific and engineering domains (materials science, fluid dynamics, etc.), and the public code release enhances reproducibility. Paper 2, while technically strong with matching upper/lower bounds for drifting halfspaces with Massart noise, addresses a more specialized problem in computational learning theory with narrower practical impact. The breadth of applications and timeliness of neural operator research give Paper 1 the edge.
Paper 2 addresses a fundamental problem in theoretical machine learning: learning under both concept drift and Massart noise. By establishing a computationally efficient algorithm and providing formal evidence of an information-computation tradeoff, it offers deep foundational insights that can broadly influence robust machine learning. While Paper 1 provides a highly practical solution for two-sided platforms, Paper 2's theoretical contributions to learning halfspaces address core algorithmic challenges with wider generalizability across AI and data science.
Paper 1 likely has higher near-term scientific impact: it introduces a practical, multi-dataset test-time prompt learning framework for LLM agents addressing a widely felt deployment gap (heterogeneous real-world task streams), with large empirical gains on modern foundation models and clear applicability to agentic systems. Its breadth spans ML systems, NLP, and autonomous agents, and it is highly timely given current LLM deployment trends. Paper 2 is methodologically rigorous with strong theory and lower bounds, but its impact is narrower (online learning theory for drifting halfspaces) and may diffuse more slowly into practice.
Paper 1 addresses a broadly relevant problem in supervised fine-tuning of large language models, proposing a unifying framework (Q-target) that reinterprets existing SFT methods and introduces a principled design space. Given the massive interest in LLM training and alignment, this work has wider potential impact across NLP, AI alignment, and practical applications. Paper 2 makes solid theoretical contributions to learning drifting halfspaces with Massart noise, including matching upper/lower bounds, but addresses a narrower theoretical learning theory problem with more limited practical applicability.
Paper 1 addresses the highly timely and critical problem of safely deploying machine learning models via conformal risk control. By providing tighter finite-sample certificates for adaptive selective prediction and demonstrating strong empirical gains on standard vision benchmarks, it offers immediate practical utility for trustworthy AI. While Paper 2 presents excellent theoretical results on an important learning theory problem, Paper 1's combination of rigorous statistical guarantees and direct applicability to modern ML deployment suggests a broader and more immediate scientific impact across both theory and practice.
Paper 1 addresses a highly timely and widely applicable problem in reinforcement learning and robotics—scaling expressive policies like diffusion/flow models without the instability of training-time RL. Its practical approach of test-time guidance offers immediate real-world utility in robotics control and sidesteps major bottlenecks. Paper 2, while methodologically rigorous and theoretically significant for learning theory, focuses on specific bounds for linear classifiers under noise and drift, which has a narrower scope and less immediate practical impact across varied fields.
Paper 1 addresses delayed feedback in reinforcement learning, a critical bottleneck in real-world applications like robotics and healthcare. By providing minimax optimal bounds and a generalizable analytical framework for structured MDPs, it offers strong theoretical contributions with broad practical utility. In contrast, Paper 2 tackles a classic but more specialized theoretical learning problem (Massart noise with concept drift), making Paper 1 likely to have a wider impact across both applied and theoretical communities.
Paper 2 likely has higher near-term scientific impact due to strong real-world applicability and timeliness: accelerating LLM inference under high-load serving is a central bottleneck, and the method promises multi-x speedups while integrating with existing AR infrastructure. Its potential breadth spans NLP, systems, and deployment engineering. Paper 1 is methodologically rigorous and offers meaningful theoretical advances on drifting halfspaces with noise, but its applications are narrower and impact is more concentrated within learning theory. Overall, Paper 2’s deployment relevance and cross-field influence are higher.