Bogdan Chornomaz

Zachary Chase, Bogdan Chornomaz, Shay Moran et al.

We use and adapt the Borsuk-Ulam Theorem from topology to derive limitations on list-replicable and globally stable learning algorithms. We further demonstrate the applicability of our methods in combinatorics and topology. We show that, besides trivial cases, both list-replicable and globally stable learning are impossible in the agnostic PAC setting. This is in contrast with the realizable case where it is known that any class with a finite Littlestone dimension can be learned by such algorithms. In the realizable PAC setting, we sharpen previous impossibility results and broaden their scope. Specifically, we establish optimal bounds for list replicability and global stability numbers in finite classes. This provides an exponential improvement over previous works and implies an exponential separation from the Littlestone dimension. We further introduce lower bounds for weak learners, i.e., learners that are only marginally better than random guessing. Lower bounds from previous works apply only to stronger learners. To offer a broader and more comprehensive view of our topological approach, we prove a local variant of the Borsuk-Ulam theorem in topology and a result in combinatorics concerning Kneser colorings. In combinatorics, we prove that if $c$ is a coloring of all non-empty subsets of $[n]$ such that disjoint sets have different colors, then there is a chain of subsets that receives at least $1+ \lfloor n/2\rfloor$ colors (this bound is sharp). In topology, we prove e.g. that for any open antipodal-free cover of the $d$-dimensional sphere, there is a point $x$ that belongs to at least $t=\lceil\frac{d+3}{2}\rceil$ sets.

Bogdan Chornomaz, Shay Moran, Tom Waknine

We introduce and study the spherical dimension, a natural topological relaxation of the VC dimension that unifies several results in learning theory where topology plays a key role in the proofs. The spherical dimension is defined by extending the set of realizable datasets (used to define the VC dimension) to the continuous space of realizable distributions. In this space, a shattered set of size d (in the VC sense) is completed into a continuous object, specifically a d-dimensional sphere of realizable distributions. The spherical dimension is then defined as the dimension of the largest sphere in this space. Thus, the spherical dimension is at least the VC dimension. The spherical dimension serves as a common foundation for leveraging the Borsuk-Ulam theorem and related topological tools. We demonstrate the utility of the spherical dimension in diverse applications, including disambiguations of partial concept classes, reductions from classification to stochastic convex optimization, stability and replicability, and sample compression schemes. Perhaps surprisingly, we show that the open question posed by Alon, Hanneke, Holzman, and Moran (FOCS 2021) of whether there exist non-trivial disambiguations for halfspaces with margin is equivalent to the basic open question of whether the VC and spherical dimensions are finite together.

Bogdan Chornomaz, Shay Moran, Tom Waknine

Many practical prediction algorithms represent inputs in Euclidean space and replace the discrete 0/1 classification loss with a real-valued surrogate loss, effectively reducing classification tasks to stochastic optimization. In this paper, we investigate the expressivity of such reductions in terms of key resources, including dimension and the role of randomness. We establish bounds on the minimum Euclidean dimension $D$ needed to reduce a concept class with VC dimension $d$ to a Stochastic Convex Optimization (SCO) problem in $\mathbb{R}^D$, formally addressing the intuitive interpretation of the VC dimension as the number of parameters needed to learn the class. To achieve this, we develop a generalization of the Borsuk-Ulam Theorem that combines the classical topological approach with convexity considerations. Perhaps surprisingly, we show that, in some cases, the number of parameters $D$ must be exponentially larger than the VC dimension $d$, even if the reduction is only slightly non-trivial. We also present natural classification tasks that can be represented in much smaller dimensions by leveraging randomness, as seen in techniques like random initialization. This result resolves an open question posed by Kamath, Montasser, and Srebro (COLT 2020). Our findings introduce new variants of \emph{dimension complexity} (also known as \emph{sign-rank}), a well-studied parameter in learning and complexity theory. Specifically, we define an approximate version of sign-rank and another variant that captures the minimum dimension required for a reduction to SCO. We also propose several open questions and directions for future research.

Bogdan Chornomaz, Yonatan Koren, Shay Moran et al.

We study the problem of learning in the presence of an adversary that can corrupt an $η$ fraction of the training examples with the goal of causing failure on a specific test point. In the realizable setting, prior work established that the optimal error under such instance-targeted poisoning attacks scales as $Θ(dη)$, where $d$ is the VC dimension of the hypothesis class arXiv:2210.02713. In this work, we resolve the corresponding question in the agnostic setting. We show that the optimal excess error is $\tildeΘ(\sqrt{dη})$, answering one of the main open problems left by Hanneke et al. To achieve this rate, it is necessary to use randomized learners: Hanneke et al. showed that deterministic learners can be forced to suffer error close to 1, even under small amounts of poisoning. Perhaps surprisingly, our upper bound remains valid even when the learner's random bits are fully visible to the adversary . In the other direction, our lower bound is stronger than standard PAC-style bounds: instead of tailoring a hard distribution separately for each sample size, we exhibit a single fixed distribution under which the adversary can enforce an excess error of $Ω(\sqrt{dη})$ infinitely often.