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).

Prior work on partial labels learning (PLL) has shown that learning is possible even when each instance is associated with a bag of labels, rather than a single accurate but costly label. However, the necessary conditions for learning with partial labels remain unclear, and existing PLL methods are effective only in specific scenarios. In this work, we mathematically characterize the settings in which PLL is feasible. In addition, we present PL A-$k$NN, an adaptive nearest-neighbors algorithm for PLL that is effective in general scenarios and enjoys strong performance guarantees. Experimental results corroborate that PL A-$k$NN can outperform state-of-the-art methods in general PLL scenarios.

We consider the problem of estimating how well a model class is capable of fitting a distribution of labeled data. We show that it is often possible to accurately estimate this ``learnability'' even when given an amount of data that is too small to reliably learn any accurate model. Our first result applies to the setting where the data is drawn from a $d$-dimensional distribution with isotropic covariance, and the label of each datapoint is an arbitrary noisy function of the datapoint. In this setting, we show that with $O(\sqrt{d})$ samples, one can accurately estimate the fraction of the variance of the label that can be explained via the best linear function of the data. We extend these techniques to a binary classification, and show that the prediction error of the best linear classifier can be accurately estimated given $O(\sqrt{d})$ labeled samples. For comparison, in both the linear regression and binary classification settings, even if there is no noise in the labels, a sample size linear in the dimension, $d$, is required to \emph{learn} any function correlated with the underlying model. We further extend our estimation approach to the setting where the data distribution has an (unknown) arbitrary covariance matrix, allowing these techniques to be applied to settings where the model class consists of a linear function applied to a nonlinear embedding of the data. We demonstrate the practical viability of our approaches on synthetic and real data. This ability to estimate the explanatory value of a set of features (or dataset), even in the regime in which there is too little data to realize that explanatory value, may be relevant to the scientific and industrial settings for which data collection is expensive and there are many potentially relevant feature sets that could be collected.

Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone's characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness i.e. a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $U$. Finally, when $U$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $U$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

We study a generalization of boosting to the multiclass setting. We introduce a weak learning condition for multiclass classification that captures the original notion ofweak learnability asbeing "slightly better than random guessing".