An Optimal Sauer Lemma Over $k$-ary Alphabets

The Sauer-Shelah-Perles Lemma is a cornerstone of combinatorics and learning theory, bounding the size of a binary hypothesis class in terms of its Vapnik-Chervonenkis (VC) dimension. For classes of functions over a $k$-ary alphabet, namely the multiclass setting, the Natarajan dimension has long served as an analogue of VC dimension, yet the corresponding Sauer-type bounds are suboptimal for alphabet sizes $k>2$. In this work, we establish a sharp Sauer inequality for multiclass and list prediction. Our bound is expressed in terms of the Daniely--Shalev-Shwartz (DS) dimension, and more generally with its extension, the list-DS dimension -- the combinatorial parameters that characterize multiclass and list PAC learnability. Our bound is tight for every alphabet size $k$, list size $\ell$, and dimension value, replacing the exponential dependence on $\ell$ in the Natarajan-based bound by the optimal polynomial dependence, and improving the dependence on $k$ as well. Our proof uses the polynomial method. In contrast to the classical VC case, where several direct combinatorial proofs are known, we are not aware of any purely combinatorial proof in the DS setting. This motivates several directions for future research, which are discussed in the paper. As consequences, we obtain improved sample complexity upper bounds for list PAC learning and for uniform convergence of list predictors, sharpening the recent results of Charikar et al.~(STOC~2023), Hanneke et al.~(COLT~2024), and Brukhim et al.~(NeurIPS~2024).

Sample Complexity of Autoregressive Reasoning: Chain-of-Thought vs. End-to-End

Modern large language models generate text autoregressively, producing tokens one at a time. To study the learnability of such systems, Joshi et al. (COLT 2025) introduced a PAC-learning framework for next-token generators, the primitive underlying autoregressive models. In this framework, an unknown next-token generator maps a sequence of tokens to the next token and is iteratively applied for $T$ steps, producing a chain of tokens whose final token constitutes the model's output. The learning task is to learn the input-output mapping induced by this autoregressive process. Depending on the available supervision, training examples may reveal only the final output (End-to-End supervision) or the entire generated chain (Chain-of-Thought supervision). This raises two natural questions: how the sample complexity depends on the generation length $T$, and how much Chain-of-Thought supervision can reduce this dependence. In this work we give a nearly complete answer to both questions by uncovering a taxonomy of how the sample complexity scales with $T$. For End-to-End learning, we show that the landscape is remarkably rich: subject to mild conditions, essentially any growth rate $r(T)$ between constant and linear can arise as the sample complexity, and combined with the linear upper bound of Joshi et al., this yields an essentially complete characterization. In contrast, under Chain-of-Thought supervision we show that the sample complexity is independent of $T$, demonstrating that access to intermediate reasoning steps can eliminate the dependence on the generation length altogether. Our analysis introduces new combinatorial tools, and as corollaries we resolve several open questions posed by Joshi et al. regarding the dependence of learnability on the generation length and the role of Chain-of-Thought supervision.

When More Data Doesn't Help: Limits of Adaptation in Multitask Learning

Multitask learning and related frameworks have achieved tremendous success in modern applications. In multitask learning problem, we are given a set of heterogeneous datasets collected from related source tasks and hope to enhance the performance above what we could hope to achieve by solving each of them individually. The recent work of arXiv:2006.15785 has showed that, without access to distributional information, no algorithm based on aggregating samples alone can guarantee optimal risk as long as the sample size per task is bounded. In this paper, we focus on understanding the statistical limits of multitask learning. We go beyond the no-free-lunch theorem in arXiv:2006.15785 by establishing a stronger impossibility result of adaptation that holds for arbitrarily large sample size per task. This improvement conveys an important message that the hardness of multitask learning cannot be overcame by having abundant data per task. We also discuss the notion of optimal adaptivity that may be of future interests.

A Theory of Universal Agnostic Learning

We provide a complete theory of optimal universal rates for binary classification in the agnostic setting. This extends the realizable-case theory of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (2021) by removing the realizability assumption on the distribution. We identify a fundamental tetrachotomy of optimal rates: for every concept class, the optimal universal rate of convergence of the excess error rate is one of $e^{-n}$, $e^{-o(n)}$, $o(n^{-1/2})$, or arbitrarily slow. We further identify simple combinatorial structures which determine which of these categories any given concept class falls into.

Optimal Mistake Bounds for Transductive Online Learning

We resolve a 30-year-old open problem concerning the power of unlabeled data in online learning by tightly quantifying the gap between transductive and standard online learning. In the standard setting, the optimal mistake bound is characterized by the Littlestone dimension $d$ of the concept class $H$ (Littlestone 1987). We prove that in the transductive setting, the mistake bound is at least $Ω(\sqrt{d})$. This constitutes an exponential improvement over previous lower bounds of $Ω(\log\log d)$, $Ω(\sqrt{\log d})$, and $Ω(\log d)$, due respectively to Ben-David, Kushilevitz, and Mansour (1995, 1997) and Hanneke, Moran, and Shafer (2023). We also show that this lower bound is tight: for every $d$, there exists a class of Littlestone dimension $d$ with transductive mistake bound $O(\sqrt{d})$. Our upper bound also improves upon the best known upper bound of $(2/3)d$ from Ben-David, Kushilevitz, and Mansour (1997). These results establish a quadratic gap between transductive and standard online learning, thereby highlighting the benefit of advance access to the unlabeled instance sequence. This contrasts with the PAC setting, where transductive and standard learning exhibit similar sample complexities.

Sample Complexity of Agnostic Multiclass Classification: Natarajan Dimension Strikes Back

The fundamental theorem of statistical learning states that binary PAC learning is governed by a single parameter -- the Vapnik-Chervonenkis (VC) dimension -- which determines both learnability and sample complexity. Extending this to multiclass classification has long been challenging, since Natarajan's work in the late 80s proposing the Natarajan dimension (Nat) as a natural analogue of VC. Daniely and Shalev-Shwartz (2014) introduced the DS dimension, later shown by Brukhim et al. (2022) to characterize multiclass learnability. Brukhim et al. also showed that Nat and DS can diverge arbitrarily, suggesting that multiclass learning is governed by DS rather than Nat. We show that agnostic multiclass PAC sample complexity is in fact governed by two distinct dimensions. Specifically, we prove nearly tight agnostic sample complexity bounds that, up to log factors, take the form $\frac{DS^{1.5}}ε + \frac{Nat}{ε^2}$ where $ε$ is the excess risk. This bound is tight up to a $\sqrt{DS}$ factor in the first term, nearly matching known $Nat/ε^2$ and $DS/ε$ lower bounds. The first term reflects the DS-controlled regime, while the second shows that the Natarajan dimension still dictates asymptotic behavior for small $ε$. Thus, unlike binary or online classification -- where a single dimension (VC or Littlestone) controls both phenomena -- multiclass learning inherently involves two structural parameters. Our technical approach departs from traditional agnostic learning methods based on uniform convergence or reductions to realizable cases. A key ingredient is a novel online procedure based on a self-adaptive multiplicative-weights algorithm performing a label-space reduction, which may be of independent interest.

Uniform Convergence Beyond Glivenko-Cantelli

We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ admit uniform estimation of their mean. Prior work from Vapnik and Chervonenkis (1971) has focused on uniform convergence using the empirical mean estimator, leading to the principle known as $P-$ Glivenko-Cantelli. We extend this framework by moving beyond the empirical mean estimator and introducing Uniform Mean Estimability, also called $UME-$ learnability, which captures when a collection permits uniform mean estimation by any arbitrary estimator. We work on the space created by the mean vectors of the collection of distributions. For each distribution, the mean vector records the expected value in each coordinate. We show that separability of the mean vectors is a sufficient condition for $UME-$ learnability. However, we show that separability of the mean vectors is not necessary for $UME-$ learnability by constructing a collection of distributions whose mean vectors are non-separable yet $UME-$ learnable using techniques fundamentally different from those used in our separability-based analysis. Finally, we establish that countable unions of $UME-$ learnable collections are also $UME-$ learnable, solving a conjecture posed in Cohen et al. (2025).

Universal Rates of ERM for Agnostic Learning

The universal learning framework has been developed to obtain guarantees on the learning rates that hold for any fixed distribution, which can be much faster than the ones uniformly hold over all the distributions. Given that the Empirical Risk Minimization (ERM) principle being fundamental in the PAC theory and ubiquitous in practical machine learning, the recent work of arXiv:2412.02810 studied the universal rates of ERM for binary classification under the realizable setting. However, the assumption of realizability is too restrictive to hold in practice. Indeed, the majority of the literature on universal learning has focused on the realizable case, leaving the non-realizable case barely explored. In this paper, we consider the problem of universal learning by ERM for binary classification under the agnostic setting, where the ''learning curve" reflects the decay of the excess risk as the sample size increases. We explore the possibilities of agnostic universal rates and reveal a compact trichotomy: there are three possible agnostic universal rates of ERM, being either $e^{-n}$, $o(n^{-1/2})$, or arbitrarily slow. We provide a complete characterization of which concept classes fall into each of these categories. Moreover, we also establish complete characterizations for the target-dependent universal rates as well as the Bayes-dependent universal rates.

Private List Learnability vs. Online List Learnability

This work explores the connection between differential privacy (DP) and online learning in the context of PAC list learning. In this setting, a $k$-list learner outputs a list of $k$ potential predictions for an instance $x$ and incurs a loss if the true label of $x$ is not included in the list. A basic result in the multiclass PAC framework with a finite number of labels states that private learnability is equivalent to online learnability [Alon, Livni, Malliaris, and Moran (2019); Bun, Livni, and Moran (2020); Jung, Kim, and Tewari (2020)]. Perhaps surprisingly, we show that this equivalence does not hold in the context of list learning. Specifically, we prove that, unlike in the multiclass setting, a finite $k$-Littlestone dimensio--a variant of the classical Littlestone dimension that characterizes online $k$-list learnability--is not a sufficient condition for DP $k$-list learnability. However, similar to the multiclass case, we prove that it remains a necessary condition. To demonstrate where the equivalence breaks down, we provide an example showing that the class of monotone functions with $k+1$ labels over $\mathbb{N}$ is online $k$-list learnable, but not DP $k$-list learnable. This leads us to introduce a new combinatorial dimension, the \emph{$k$-monotone dimension}, which serves as a generalization of the threshold dimension. Unlike the multiclass setting, where the Littlestone and threshold dimensions are finite together, for $k>1$, the $k$-Littlestone and $k$-monotone dimensions do not exhibit this relationship. We prove that a finite $k$-monotone dimension is another necessary condition for DP $k$-list learnability, alongside finite $k$-Littlestone dimension. Whether the finiteness of both dimensions implies private $k$-list learnability remains an open question.

Data Selection for ERMs

Learning theory has traditionally followed a model-centric approach, focusing on designing optimal algorithms for a fixed natural learning task (e.g., linear classification or regression). In this paper, we adopt a complementary data-centric perspective, whereby we fix a natural learning rule and focus on optimizing the training data. Specifically, we study the following question: given a learning rule $\mathcal{A}$ and a data selection budget $n$, how well can $\mathcal{A}$ perform when trained on at most $n$ data points selected from a population of $N$ points? We investigate when it is possible to select $n \ll N$ points and achieve performance comparable to training on the entire population. We address this question across a variety of empirical risk minimizers. Our results include optimal data-selection bounds for mean estimation, linear classification, and linear regression. Additionally, we establish two general results: a taxonomy of error rates in binary classification and in stochastic convex optimization. Finally, we propose several open questions and directions for future research.