Data-driven algorithm design automates hyperparameter tuning, but its statistical foundations remain limited because model performance can depend on hyperparameters in implicit and highly non-smooth ways. Existing guarantees focus on the simple case of a one-dimensional (scalar) hyperparameter. This leaves the practically important, multi-dimensional hyperparameter tuning setting unresolved. We address this open question by establishing the first general framework for establishing generalization guarantees for tuning multi-dimensional hyperparameters in data-driven settings. Our approach strengthens the generalization guarantee framework for semi-algebraic function classes by exploiting tools from real algebraic geometry, yielding sharper, more broadly applicable guarantees. We then extend the analysis to hyperparameter tuning using the validation loss under minimal assumptions, and derive improved bounds when additional structure is available. Finally, we demonstrate the scope of the framework with new learnability results, including data-driven weighted group lasso and weighted fused lasso.

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.

The sample complexity of multi-group learning is shown to improve in the group-realizable setting over the agnostic setting, even when the family of groups is infinite so long as it has finite VC dimension. The improved sample complexity is obtained by empirical risk minimization over the class of group-realizable concepts, which itself could have infinite VC dimension. Implementing this approach is also shown to be computationally intractable, and an alternative approach is suggested based on improper learning.

We address the following question: given a collection $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ of independent $d \times d$ random matrices drawn from a common distribution $\mathbb{P}$, what is the probability that the centralizer of $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ is trivial? We provide lower bounds on this probability in terms of the sample size $N$ and the dimension $d$ for several families of random matrices which arise from the discretization of linear Schrödinger operators with random potentials. When combined with recent work on machine learning theory, our results provide guarantees on the generalization ability of transformer-based neural networks for in-context learning of Schrödinger equations.

We provide an overview of high dimensional dynamical systems driven by random matrices, focusing on applications to simple models of learning and generalization in machine learning theory. Using both cavity method arguments and path integrals, we review how the behavior of a coupled infinite dimensional system can be characterized as a stochastic process for each single site of the system. We provide a pedagogical treatment of dynamical mean field theory (DMFT), a framework that can be flexibly applied to these settings. The DMFT single site stochastic process is fully characterized by a set of (two-time) correlation and response functions. For linear time-invariant systems, we illustrate connections between random matrix resolvents and the DMFT response. We demonstrate applications of these ideas to machine learning models such as gradient flow, stochastic gradient descent on random feature models and deep linear networks in the feature learning regime trained on random data. We demonstrate how bias and variance decompositions (analysis of ensembling/bagging etc) can be computed by averaging over subsets of the DMFT noise variables. From our formalism we also investigate how linear systems driven with random non-Hermitian matrices (such as random feature models) can exhibit non-monotonic loss curves with training time, while Hermitian matrices with the matching spectra do not, highlighting a different mechanism for non-monotonicity than small eigenvalues causing instability to label noise. Lastly, we provide asymptotic descriptions of the training and test loss dynamics for randomly initialized deep linear neural networks trained in the feature learning regime with high-dimensional random data. In this case, the time translation invariance structure is lost and the hidden layer weights are characterized as spiked random matrices.

Can we learn more from data than existed in the generating process itself? Can new and useful information be constructed from merely applying deterministic transformations to existing data? Can the learnable content in data be evaluated without considering a downstream task? On these questions, Shannon information and Kolmogorov complexity come up nearly empty-handed, in part because they assume observers with unlimited computational capacity and fail to target the useful information content. In this work, we identify and exemplify three seeming paradoxes in information theory: (1) information cannot be increased by deterministic transformations; (2) information is independent of the order of data; (3) likelihood modeling is merely distribution matching. To shed light on the tension between these results and modern practice, and to quantify the value of data, we introduce epiplexity, a formalization of information capturing what computationally bounded observers can learn from data. Epiplexity captures the structural content in data while excluding time-bounded entropy, the random unpredictable content exemplified by pseudorandom number generators and chaotic dynamical systems. With these concepts, we demonstrate how information can be created with computation, how it depends on the ordering of the data, and how likelihood modeling can produce more complex programs than present in the data generating process itself. We also present practical procedures to estimate epiplexity which we show capture differences across data sources, track with downstream performance, and highlight dataset interventions that improve out-of-distribution generalization. In contrast to principles of model selection, epiplexity provides a theoretical foundation for data selection, guiding how to select, generate, or transform data for learning systems.

We study the extent to which standard machine learning algorithms rely on exchangeability and independence of data by introducing a monotone adversarial corruption model. In this model, an adversary, upon looking at a "clean" i.i.d. dataset, inserts additional "corrupted" points of their choice into the dataset. These added points are constrained to be monotone corruptions, in that they get labeled according to the ground-truth target function. Perhaps surprisingly, we demonstrate that in this setting, all known optimal learning algorithms for binary classification can be made to achieve suboptimal expected error on a new independent test point drawn from the same distribution as the clean dataset. On the other hand, we show that uniform convergence-based algorithms do not degrade in their guarantees. Our results showcase how optimal learning algorithms break down in the face of seemingly helpful monotone corruptions, exposing their overreliance on exchangeability.

A basic question in learning theory is to identify if two distributions are identical when we have access only to examples sampled from the distributions. This basic task is considered, for example, in the context of Generative Adversarial Networks (GANs), where a discriminator is trained to distinguish between a real-life distribution and a synthetic distribution. Classically, we use a hypothesis class $H$ and claim that the two distributions are distinct if for some $h\in H$ the expected value on the two distributions is (significantly) different. Our starting point is the following fundamental problem: is having the hypothesis dependent on more than a single random example beneficial. To address this challenge we define $k$-ary based discriminators, which have a family of Boolean $k$-ary functions $\G$.

Supervised learning aims to train a classifier under the assumption that training and test data are from the same distribution. To ease the above assumption, researchers have studied a more realistic setting: out-of-distribution (OOD) detection, where test data may come from classes that are unknown during training (i.e., OOD data). Due to the unavailability and diversity of OOD data, good generalization ability is crucial for effective OOD detection algorithms. To study the generalization of OOD detection, in this paper, we investigate the probably approximately correct (PAC) learning theory of OOD detection, which is proposed by researchers as an open problem. First, we find a necessary condition for the learnability of OOD detection. Then, using this condition, we prove several impossibility theorems for the learnability of OOD detection under some scenarios. Although the impossibility theorems are frustrating, we find that some conditions of these impossibility theorems may not hold in some practical scenarios. Based on this observation, we next give several necessary and sufficient conditions to characterize the learnability of OOD detection in some practical scenarios. Lastly, we also offer theoretical supports for several representative OOD detection works based on our OOD theory.

It is becoming increasingly important to understand the vulnerability of machine learning models to adversarial attacks. In this paper we study the feasibility of robust learning from the perspective of computational learning theory, considering both sample and computational complexity. In particular, our definition of robust learnability requires polynomial sample complexity. We start with two negative results. We show that no non-trivial concept class can be robustly learned in the distribution-free setting against an adversary who can perturb just a single input bit. We show moreover that the class of monotone conjunctions cannot be robustly learned under the uniform distribution against an adversary who can perturb $\omega(\log n)$ input bits. However if the adversary is restricted to perturbing $O(\log n)$ bits, then the class of monotone conjunctions can be robustly learned with respect to a general class of distributions (that includes the uniform distribution). Finally, we provide a simple proof of the computational hardness of robust learning on the boolean hypercube. Unlike previous results of this nature, our result does not rely on another computational model (e.g. the statistical query model) nor on any hardness assumption other than the existence of a hard learning problem in the PAC framework.