An asymmetric variant of minimally consistent concept classes is introduced and used to provide an exact characterization of proper learning with improvements in the real-izable setting, and positive results for more natural Euclidean ball improvement sets are given.
Imitation Learning is a natural framework for learning in sequential decision-making systems and has emerged as the dominant paradigm through which we understand language model training. A central puzzle is that, while in theory offline IL can be horizon-free and optimal, in practice online methods such as on-policy distillation often outperform offline methods such as supervised fine-tuning. We propose a noisy expert model to explain this gap, in which the learner only has access to a noisy version of the expert's policy, but wishes to compete against the reward achieved by a clean expert, motivated by the fact that in many applications, e.g. training language models to perform long chains of thought, the expert is often imperfect. In this setting, we show a sharp separation between offline and online IL. Offline learning from noisy trajectories is fundamentally hard: to compete with the clean expert, the sample complexity must grow exponentially, in contradistinction to the clean expert setting where no explicit horizon dependence exists. In contrast, we prove that online interaction with the noisy expert via a novel variant of OPD enables polynomial dependence on the horizon in general. We further show that, under a natural condition on the expert noise distribution, which we show to be necessary for any horizon-free sample complexity, one can obtain such a guarantee, although our proposed algorithm sacrifices statistical efficiency in its dependence on the size of the policy class. Our analysis leads to an alternative loss function that is commonly considered empirically for LM training. We further provide algorithms and lower bounds, and extend our results to the more realistic setting of unknown corruption when the clean expert is deterministic, thereby providing a theoretical foundation for why OPD can outperform SFT when training language models from imperfect teachers.
V. Sriraman, Peihan Liu, Daniel Hsu et al.· arXiv.org· 0 citations
When a classifier is trained by actively choosing which examples to label, should its labeling budget depend on which notion of fairness it must satisfy? We answer this with the first information-theoretic lower bounds for group-fair active learning, and the answer is a sharp hierarchy across the three standard notions. Demographic Parity (DP), which equalizes the positive prediction rate across groups, can be verified from unlabeled data alone, so it adds no labels beyond ordinary learning. Equal Opportunity (EO), which equalizes each group's true positive rate, is fundamentally harder: estimating a rate conditioned on the true label forces the learner to collect labeled positives from every group, an irreducible cost that no query strategy can avoid. Equalized Odds (EqOdds), which additionally equalizes false positive rates, costs no more than EO whenever positives are the minority, the typical regime. This exposes a separation at the heart of fair learning: active querying yields the well-known exponential savings for accuracy, but no speedup whatsoever for fairness verification. Concretely, certifying EO to fairness tolerance γ requires Ω(d/ε2 + k/(ε2 p+)) labels, where d is the model's Vapnik--Chervonenkis (VC) dimension, k the number of groups, ε the target error, and p+ = Pr[Y=1] the positive rate; the fairness term is independent of ε, so it is a floor that active learning cannot lower. We give matching algorithms (ConstrainedERM-DP for DP, FairStratified for EO and EqOdds) that are tight up to logarithmic factors. The cost is largest exactly when positives are rare: at p+ = 0.01, as in fraud detection, EO demands roughly 90× more labels than DP. Experiments on Folktables, COMPAS, and Adult confirm the predicted scaling and yield 36 to 39% label savings over passive fair-learning baselines.
Chang Lu, Yizheng Zhao· Proceedings of the 32nd ACM...· 0 citations
Many operational decisions rely on predictive models that estimate uncertain outcomes conditional on observable contexts. Training such models, however, often faces a fundamental data asymmetry: labeled outcomes are scarce or costly to obtain, while contextual covariates are abundant. Motivated by this data asymmetry, we develop a decision-aware weak-to-strong (W2S) framework that leverages both labeled and unlabeled data to improve contextual stochastic optimization. Specifically, we first train a weak model using limited labeled data and then use it to generate predicted outcome distributions on unlabeled contexts. These distributions provide soft supervision for training a strong model. We establish a non-asymptotic upper bound on the excess decision risk of W2S and a complementary lower bound for a strong-only benchmark. Their comparison yields explicit sufficient conditions under which W2S improves downstream decision performance. The key quantity is the correlation dimension between the weak and strong feature representations: when it is small, abundant unlabeled data reduce the effect of teacher errors along non-overlapping directions. A synthetic newsvendor experiment and a comment moderation experiment based on real-world data provide empirical evidence consistent with the theory.
Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlying data-generating process, which are formalized in terms of informational bounds. In this work we examine intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective. We analyze minimal achievable error in classification through Fano-type bounds and precision limits in parametric estimation via the Cram\'er-Rao inequality, emphasizing that such limits depend on the underlying model rather than on algorithmic sophistication alone. We further discuss how implicit assumptions, such as independence, ergodicity, and distributional stability, affect the validity of inferential procedures. Building on interaction-based modeling principles, we review typical frameworks such as Markov Random Fields and potential based representations for encoding dependence mechanisms. We also describe decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior. This perspective highlights the importance of having adequate models for the data as a prerequi- site for expanding predictive capability, and situates algorithmic learning within the informational limits imposed by the models.
Many pipelines can pay a per-example cost to acquire an auxiliary, model-derived observation -- an LLM's structured reasoning, a slow oracle, an expensive measurement -- and then must decide when the acquired signal is worth using. Our thesis is a distinction that is easy to miss: detecting that such a signal helps on average is not the same as learning to act on it per instance, and a reward-SNR floor governs when the second is even possible. Even when the signal is faithful and an in-sample oracle picking the top-b examples by realized reward shows a sizable apparent gain, no deployable policy can learn when to acquire it: across per-impression, cluster, regime, and uplift-tree granularities, learned routing never beats random, and a matched-moment noise placebo reproduces>=100% of the oracle's apparent gain -- the apparent"learnable structure"is order statistics of noise. We explain this with one distinction, detecting a mean effect vs. learning a per-instance acquisition policy, and a reward-SNR detectability floor: routing is estimable offline only if the reward SNR rho clears rho*(N) ~= 2.8/sqrt(N), with a positive control confirming a true low-SNR limit rather than a broken pipeline. As a concrete instantiation we introduce Structured Hypothesis Embeddings (SHE): a frozen LLM turns a user history into ranked, confidence-scored, evidence-grounded intent hypotheses, fused into a recommender. On three public datasets (MIND, REES46, Amazon-Beauty), SHE is faithful and calibratable, yet its value is backbone- and regime-conditional (significant over an ordered GRU, +0.0114, 95% CI [+0.0030, +0.0209], but a global redundancy gap indistinguishable from zero), and learned acquisition collapses at every granularity because all three datasets sit below the floor. The realizable unit is a design-time regime gate, not a per-instance policy. We release code and a one-command reproduction.
We study post-learning inference for structural properties of data-dependent combinatorial optimizers. The target is whether an oracle optimizer, rather than a latent parameter or smooth functional, belongs to a prescribed class, such as a category-mix, inventory, or resource-feasibility class. We focus on a high-dimensional contextual multinomial logit model with sequentially adaptive data collection, where the parameter-to-optimizer map is discontinuous and the policy induces temporal dependence. We propose a novel perturbation test based on a nonsmooth max-difference revenue statistic comparing the best null assortment with the best alternative assortment. The test perturbs the estimated terminal revenue surface on the selected support: random unit directions capture directional uncertainty, while the minimal perturbation radius captures magnitude uncertainty and yields a p-value. This localizes inference near the null--alternative boundary and avoids uniform error control over the full candidate class. The data are collected by an \(\ell_1\)-penalized online likelihood policy that performs variable selection while controlling regret. Using a new anti-concentration argument for Gaussian maxima differences and martingale Gaussian coupling, we establish uniform estimation rates, effective support recovery, and asymptotic validity of the proposed p-value under adaptive assortment selection. We prove asymptotic size control and power consistency under a localized signal condition.