In multi-step learning, where a final learning task is accomplished via a sequence of intermediate learning tasks, the intuition is that successive steps or levels transform the initial data into representations more and more "suited" to the final learning task. A related principle arises in transfer-learning where Baxter (2000) proposed a theoretical framework to study how learning multiple tasks transforms the inductive bias of a learner. The most widespread multi-step learning approach is semisupervised learning with two steps: unsupervised, then supervised. Several authors (Castelli-Cover, 1996; Balcan-Blum, 2005; Niyogi, 2008; Ben-David et al, 2008; Urner et al, 2011) have analyzed SSL, with Balcan-Blum (2005) proposing a version of the PAC learning framework augmented by a "compatibility function" to link concept class and unlabeled data distribution. We propose to analyze SSL and other multi-step learning approaches, much in the spirit of Baxter's framework, by defining a learning problem generatively as a joint statistical model on X Y.

In this work, we aim to characterize the statistical complexity of realizable regression both in the PAC learning setting and the online learning setting. Previous work had established the sufficiency of finiteness of the fat shattering dimension for PAC learnability and the necessity of finiteness of the scaled Natarajan dimension, but little progress had been made towards a more complete characterization since the work of Simon 1997 (SICOMP '97). To this end, we first introduce a minimax instance optimal learner for realizable regression and propose a novel dimension that both qualitatively and quantitatively characterizes which classes of real-valued predictors are learnable. We then identify a combinatorial dimension related to the graph dimension that characterizes ERM learnability in the realizable setting. Finally, we establish a necessary condition for learnability based on a combinatorial dimension related to the DS dimension, and conjecture that it may also be sufficient in this context.

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.

Standard few-shot benchmarks are often built upon simplifying assumptions on the query sets, which may not always hold in practice. In particular, for each task at testing time, the classes effectively present in the unlabeled query set are known a priori, and correspond exactly to the set of classes represented in the labeled support set. We relax these assumptions and extend current benchmarks, so that the query-set classes of a given task are unknown, but just belong to a much larger set of possible classes. Our setting could be viewed as an instance of the challenging yet practical problem of extremely imbalanced K -way classification, K being much larger than the values typically used in standard benchmarks, and with potentially irrelevant supervision from the support set. Motivated by these observations, we introduce a \textbf{P}rim\textbf{A}l \textbf{D}ual Minimum \textbf{D}escription \textbf{LE}ngth (\textbf{PADDLE}) formulation, which balances data-fitting accuracy and model complexity for a given few-shot task, under supervision constraints from the support set.

We show a generic reduction from multiclass differentially private PAC learning to binary private PAC learning. We apply this transformation to a recently proposed binary private PAC learner to obtain a private multiclass learner with sample complexity that has a polynomial dependence on the multiclass Littlestone dimension and a poly-logarithmic dependence on the number of classes. This yields a doubly exponential improvement in the dependence on both parameters over learners from previous work. Our proof extends the notion of \Psi -dimension defined in work of Ben-David et al. [JCSS, 1995] to the online setting and explores its general properties.

Consider the task of learning a hypothesis class \mathcal{H} in the presence of an adversary that can replace up to an \eta fraction of the examples in the training set with arbitrary adversarial examples. The adversary aims to fail the learner on a particular target test point x which is \emph{known} to the adversary but not to the learner. In this work we aim to characterize the smallest achievable error \epsilon \epsilon(\eta) by the learner in the presence of such an adversary in both realizable and agnostic settings. Remarkably, we show that the upper bound can be attained by a deterministic learner. In the agnostic setting we reveal a more elaborate landscape: we devise a deterministic learner with a multiplicative regret guarantee of \epsilon \leq C\cdot\mathtt{OPT} O(\mathtt{VC}(\mathcal{H})\cdot \eta), where C 1 is a universal numerical constant.

Learning an unknown n-qubit quantum state rho is a fundamental challenge in quantum computing. Information-theoretically, it is known that tomography requires exponential in n many copies of rho to estimate its entries. Motivated by learning theory, Aaronson et al. introduced many (weaker) learning models: the PAC model of learning states (Proceedings of Royal Society A'07), shadow tomography (STOC'18) for learning shadows" of a state, a model that also requires learners to be differentially private (STOC'19) and the online model of learning states (NeurIPS'18). In these models it was shown that an unknown state can be learnedapproximately" using linear in n many copies of rho. But is there any relationship between these models? In this paper we prove a sequence of (information-theoretic) implications from differentially-private PAC learning to online learning and then to quantum stability.Our main result generalizes the recent work of Bun, Livni and Moran (Journal of the ACM'21) who showed that finite Littlestone dimension (of Boolean-valued concept classes) implies PAC learnability in the (approximate) differentially private (DP) setting.

We consider reinforcement learning in a discrete, undiscounted, infinite-horizon Markov decision problem (MDP) under the average reward criterion, and focus on the minimization of the regret with respect to an optimal policy, when the learner does not know the rewards nor transitions of the MDP. In light of their success at regret minimization in multi-armed bandits, popular bandit strategies, such as the optimistic \texttt{UCB}, \texttt{KL-UCB} or the Bayesian Thompson sampling strategy, have been extended to the MDP setup. Despite some key successes, existing strategies for solving this problem either fail to be provably asymptotically optimal, or suffer from prohibitive burn-in phase and computational complexity when implemented in practice. In this work, we shed a novel light on regret minimization strategies, by extending to reinforcement learning the computationally appealing Indexed Minimum Empirical Divergence (\texttt{IMED}) bandit algorithm. Traditional asymptotic problem-dependent lower bounds on the regret are known under the assumption that the MDP is \emph{ergodic}.

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. Our first result is an unconditional computational lower bound of \Omega (p {d/2}) for learning general graphical models on p nodes of maximum degree d, for the class of statistical algorithms recently introduced by Feldman et al. The construction is related to the notoriously difficult learning parities with noise problem in computational learning theory. Our lower bound shows that the \widetilde O(p {d 2}) runtime required by Bresler, Mossel, and Sly's exhaustive-search algorithm cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., most recent papers on structure learning assume that the model has the correlation decay property.

We consider linear prediction with a convex Lipschitz loss, or more generally, stochastic convex optimization problems of generalized linear form, i.e. We show that in this setting, early stopped Gradient Descent (GD), without any explicit regularization or projection, ensures excess error at most \varepsilon (compared to the best possible with unit Euclidean norm) with an optimal, up to logarithmic factors, sample complexity of \tilde{O}(1/\varepsilon 2) and only \tilde{O}(1/\varepsilon 2) iterations. This contrasts with general stochastic convex optimization, where \Omega(1/\varepsilon 4) iterations are needed Amir et al. 2021. The lower iteration complexity is ensured by leveraging uniform convergence rather than stability. But instead of uniform convergence in a norm ball, which we show can guarantee suboptimal learning using \Theta(1/\varepsilon 4) samples, we rely on uniform convergence in a distribution-dependent ball.