A Fine-Grained Understanding of Uniform Convergence for Halfspaces

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error $Θ(d\ln(n/d)/n)$, and in the agnostic setting the deviation scales as $\sqrt{τ\ln(1/τ)}$ at true error $τ$. In contrast, homogeneous halfspaces in $\mathbb{R}^2$ exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error $O(1/n)$. In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a $\ln\ln n$ overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.

Realizable Bayes-Consistency for General Metric Losses

We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification \citep{bousquet_theory_2021, hanneke2021universalbayesconsistencymetric} and real-valued regression \citep{attias_universal_2024}. Given an instance space $(\mathcal X,ρ)$, a label space $(\mathcal Y,\ell)$ with possibly unbounded loss, and a hypothesis class $\mathcal H \subseteq \mathcal Y^{\mathcal X}$, we resolve the realizable case of an open problem presented in \citet{pmlr-v178-cohen22a}. Specifically, we find the necessary and sufficient conditions on the hypothesis class $\mathcal H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to \citet{attias2024optimallearnersrealizableregression}, we introduce the notion of an infinite non-decreasing $(γ_k)$-Littlestone tree, where $γ_k \to \infty$. This extends the Littlestone tree structure used in \citet{bousquet_theory_2021} to the metric loss setting.

The MLE is minimax optimal for LGC

We revisit the recently introduced Local Glivenko-Cantelli setting, which studies distribution-dependent uniform convegence rates of the Maximum Likelihood Estimator (MLE). In this work, we investigate generalizations of this setting where arbitrary estimators are allowed rather than just the MLE. Can a strictly larger class of measures be learned? Can better risk decay rates be obtained? We provide exhaustive answers to these questions -- which are both negative, provided the learner is barred from exploiting some infinite-dimensional pathologies. On the other hand, allowing such exploits does lead to a strictly larger class of learnable measures.

Aggregation of expert advice, revisited

We revisit the classic problem of aggregating binary advice from conditionally independent experts, also known as the Naive Bayes setting. Our quantity of interest is the error probability of the optimal decision rule. In the symmetric case (sensitivity = specificity), reasonably tight bounds on the optimal error probability are known. In the general asymmetric case, we are not aware of any nontrivial estimates on this quantity. Our contribution consists of sharp upper and lower bounds on the optimal error probability in the general case, which recover and sharpen the best known results in the symmetric special case. Since this amounts to estimating the total variation distance between two product distributions, our results also have bearing on this important and challenging problem.

Distribution Estimation under the Infinity Norm

We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees for the maximum likelihood estimator significantly improve upon the currently known results. A variety of techniques are utilized and innovated upon, including Chernoff-type inequalities and empirical Bernstein bounds. We illustrate our results in synthetic and real-world experiments. Finally, we apply our proposed framework to a basic selective inference problem, where we estimate the most frequent probabilities in a sample.

Splitting the Difference on Adversarial Training

The existence of adversarial examples points to a basic weakness of deep neural networks. One of the most effective defenses against such examples, adversarial training, entails training models with some degree of robustness, usually at the expense of a degraded natural accuracy. Most adversarial training methods aim to learn a model that finds, for each class, a common decision boundary encompassing both the clean and perturbed examples. In this work, we take a fundamentally different approach by treating the perturbed examples of each class as a separate class to be learned, effectively splitting each class into two classes: "clean" and "adversarial." This split doubles the number of classes to be learned, but at the same time considerably simplifies the decision boundaries. We provide a theoretical plausibility argument that sheds some light on the conditions under which our approach can be expected to be beneficial. Likewise, we empirically demonstrate that our method learns robust models while attaining optimal or near-optimal natural accuracy, e.g., on CIFAR-10 we obtain near-optimal natural accuracy of $95.01\%$ alongside significant robustness across multiple tasks. The ability to achieve such near-optimal natural accuracy, while maintaining a significant level of robustness, makes our method applicable to real-world applications where natural accuracy is at a premium. As a whole, our main contribution is a general method that confers a significant level of robustness upon classifiers with only minor or negligible degradation of their natural accuracy.

Efficient Agnostic Learning with Average Smoothness

We study distribution-free nonparametric regression following a notion of average smoothness initiated by Ashlagi et al. (2021), which measures the "effective" smoothness of a function with respect to an arbitrary unknown underlying distribution. While the recent work of Hanneke et al. (2023) established tight uniform convergence bounds for average-smooth functions in the realizable case and provided a computationally efficient realizable learning algorithm, both of these results currently lack analogs in the general agnostic (i.e. noisy) case. In this work, we fully close these gaps. First, we provide a distribution-free uniform convergence bound for average-smoothness classes in the agnostic setting. Second, we match the derived sample complexity with a computationally efficient agnostic learning algorithm. Our results, which are stated in terms of the intrinsic geometry of the data and hold over any totally bounded metric space, show that the guarantees recently obtained for realizable learning of average-smooth functions transfer to the agnostic setting. At the heart of our proof, we establish the uniform convergence rate of a function class in terms of its bracketing entropy, which may be of independent interest.

Near-optimal learning with average Hölder smoothness

We generalize the notion of average Lipschitz smoothness proposed by Ashlagi et al. (COLT 2021) by extending it to H\"older smoothness. This measure of the ``effective smoothness'' of a function is sensitive to the underlying distribution and can be dramatically smaller than its classic ``worst-case'' H\"older constant. We prove nearly tight upper and lower risk bounds in terms of the average H\"older smoothness, establishing the minimax rate in the realizable regression setting up to log factors; this was not previously known even in the special case of average Lipschitz smoothness. From an algorithmic perspective, since our notion of average smoothness is defined with respect to the unknown sampling distribution, the learner does not have an explicit representation of the function class, hence is unable to execute ERM. Nevertheless, we provide a learning algorithm that achieves the (nearly) optimal learning rate. Our results hold in any totally bounded metric space, and are stated in terms of its intrinsic geometry. Overall, our results show that the classic worst-case notion of H\"older smoothness can be essentially replaced by its average, yielding considerably sharper guarantees.

Differentially-Private Bayes Consistency

We construct a universally Bayes consistent learning rule that satisfies differential privacy (DP). We first handle the setting of binary classification and then extend our rule to the more general setting of density estimation (with respect to the total variation metric). The existence of a universally consistent DP learner reveals a stark difference with the distribution-free PAC model. Indeed, in the latter DP learning is extremely limited: even one-dimensional linear classifiers are not privately learnable in this stringent model. Our result thus demonstrates that by allowing the learning rate to depend on the target distribution, one can circumvent the above-mentioned impossibility result and in fact, learn \emph{arbitrary} distributions by a single DP algorithm. As an application, we prove that any VC class can be privately learned in a semi-supervised setting with a near-optimal \emph{labeled} sample complexity of $\tilde{O}(d/\varepsilon)$ labeled examples (and with an unlabeled sample complexity that can depend on the target distribution).

Metric-valued regression

We propose an efficient algorithm for learning mappings between two metric spaces, $\X$ and $\Y$. Our procedure is strongly Bayes-consistent whenever $\X$ and $\Y$ are topologically separable and $\Y$ is "bounded in expectation" (our term; the separability assumption can be somewhat weakened). At this level of generality, ours is the first such learnability result for unbounded loss in the agnostic setting. Our technique is based on metric medoids (a variant of Fr\'echet means) and presents a significant departure from existing methods, which, as we demonstrate, fail to achieve Bayes-consistency on general instance- and label-space metrics. Our proofs introduce the technique of {\em semi-stable compression}, which may be of independent interest.