In current deep learning tasks, Adam-style optimizers--such as Adam, Adagrad, RMSprop, Adafactor, and Lion--have been widely used as alternatives to SGD-style optimizers. These optimizers typically update model parameters using the sign of gradients, resulting in more stable convergence curves. The learning rate and the batch size are the most critical hyperparameters for optimizers, which require careful tuning to enable effective convergence. Previous research has shown that the optimal learning rate increases linearly (or follows similar rules) with batch size for SGD-style optimizers. However, this conclusion is not applicable to Adam-style optimizers.

We study a generalization of classical active learning to real-world settings with concrete prediction targets where sampling is restricted to an accessible region of the domain, while prediction targets may lie outside this region.We analyze a family of decision rules that sample adaptively to minimize uncertainty about prediction targets.We are the first to show, under general regularity assumptions, that such decision rules converge uniformly to the smallest possible uncertainty obtainable from the accessible data.We demonstrate their strong sample efficiency in two key applications: active fine-tuning of large neural networks and safe Bayesian optimization, where they achieve state-of-the-art performance.

The NP-hard problem of optimizing a shallow ReLU network can be characterized as a combinatorial search over each training example's activation pattern followed by a constrained convex problem given a fixed set of activation patterns. We explore the implications of this combinatorial aspect of ReLU optimization in this work. We show that it can be naturally modeled via a geometric and combinatoric object known as a zonotope with its vertex set isomorphic to the set of feasible activation patterns. This assists in analysis and provides a foundation for further research. We demonstrate its usefulness when we explore the sensitivity of the optimal loss to perturbations of the training data.

Multi-distribution or collaborative learning involves learning a single predictor that works well across multiple data distributions, using samples from each during training. Recent research on multi-distribution learning, focusing on binary loss and finite VC dimension classes, has shown near-optimal sample complexity that is achieved with oracle efficient algorithms. That is, these algorithms are computationally efficient given an efficient ERM for the class. Unlike in classical PAC learning, where the optimal sample complexity is achieved with deterministic predictors, current multi-distribution learning algorithms output randomized predictors. This raises the question: can these algorithms be derandomized to produce a deterministic predictor for multiple distributions?

Bandits with knapsacks (BwK) is an influential model of sequential decision-making under uncertainty that incorporates resource consumption constraints. In each round, the decision-maker observes an outcome consisting of a reward and a vector of nonnegative resource consumptions, and the budget of each resource is decremented by its consumption. In this paper we introduce a natural generalization of the stochastic BwK problem that allows non-monotonic resource utilization. In each round, the decision-maker observes an outcome consisting of a reward and a vector of resource drifts that can be positive, negative or zero, and the budget of each resource is incremented by its drift. Our main result is a Markov decision process (MDP) policy that has constant regret against a linear programming (LP) relaxation when the decision-maker knows the true outcome distributions.

The sample compressibility of concept classes plays an important role in learning theory, as a sufficient condition for PAC learnability, and more recently as an avenue for robust generalisation in adaptive data analysis. Whether compression schemes of size O(d) must necessarily exist for all classes of VC dimension d is unknown, but conjectured to be true by Warmuth. Recently Chalopin, Chepoi, Moran, and Warmuth (2018) gave a beautiful unlabelled sample compression scheme of size VC dimension for all maximum classes: classes that meet the Sauer-Shelah-Perles Lemma with equality. They also offered a counterexample to compression schemes based on a promising approach known as corner peeling. In this paper we simplify and extend their proof technique to deal with so-called extremal classes of VC dimension d which contain maximum classes of VC dimension d-1 .

We aim to understand the optimal PAC sample complexity in multiclass learning. While finiteness of the Daniely-Shalev-Shwartz (DS) dimension has been shown to characterize the PAC learnability of a concept class [Brukhim, Carmon, Dinur, Moran, and Yehudayoff, 2022], there exist polylog factor gaps in the leading term of the sample complexity. In this paper, we reduce the gap in terms of the dependence on the error parameter to a single log factor and also propose two possible routes towards completely resolving the optimal sample complexity, each based on a key open question we formulate: one concerning list learning with bounded list size, the other concerning a new type of shifting for multiclass concept classes. We prove that a positive answer to either of the two questions would completely resolve the optimal sample complexity up to log factors of the DS dimension.

Inspired by recent work on learning with distribution shift, we give a general outlier removal algorithm called iterative polynomial filtering and show a number of striking applications for supervised learning with contamination: (1) We show that any function class that can be approximated by low-degree polynomials with respect to a hypercontractive distribution can be efficiently learned under bounded contamination (also known as nasty noise). This is a surprising resolution to a longstanding gap between the complexity of agnostic learning and learning with contamination, as it was widely believed that low-degree approximators only implied tolerance to label noise. (2) For any function class that admits the (stronger) notion of sandwiching approximators, we obtain near-optimal learning guarantees even with respect to heavy additive contamination, where far more than $1/2$ of the training set may be added adversarially. Prior related work held only for regression and in a list-decodable setting. (3) We obtain the first efficient algorithms for tolerant testable learning of functions of halfspaces with respect to any fixed log-concave distribution. Even the non-tolerant case for a single halfspace in this setting had remained open. These results significantly advance our understanding of efficient supervised learning under contamination, a setting that has been much less studied than its unsupervised counterpart.

Dictionary learning has recently emerged as a promising approach for mechanistic interpretability of large transformer models. Disentangling high-dimensional transformer embeddings, however, requires algorithms that scale to high-dimensional data with large sample sizes. Recent work has explored sparse autoencoders (SAEs) for this problem. However, SAEs use a simple linear encoder to solve the sparse encoding subproblem, which is known to be NP-hard. It is therefore interesting to understand whether this structure is sufficient to find good solutions to the dictionary learning problem or if a more sophisticated algorithm could find better solutions. In this work, we propose Double-Batch KSVD (DB-KSVD), a scalable dictionary learning algorithm that adapts the classic KSVD algorithm. DB-KSVD is informed by the rich theoretical foundations of KSVD but scales to datasets with millions of samples and thousands of dimensions. We demonstrate the efficacy of DB-KSVD by disentangling embeddings of the Gemma-2-2B model and evaluating on six metrics from the SAEBench benchmark, where we achieve competitive results when compared to established approaches based on SAEs. By matching SAE performance with an entirely different optimization approach, our results suggest that (i) SAEs do find strong solutions to the dictionary learning problem and (ii) that traditional optimization approaches can be scaled to the required problem sizes, offering a promising avenue for further research. We provide an implementation of DB-KSVD at https://github.com/RomeoV/KSVD.jl.

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not), as well as infinite model spaces, which directly generalize classical results.