The reviewers agree that this is an exciting and interesting paper which improves the best-known variance-dependent rates for statistical learning with nonparametric classes, and are all in favor of accepting. I hope the authors will pay attention to the typos and clarifications pointed about by the reviewers and address these in the final version of the paper. As reviewer 4 and the authors' response mention, the point about removing the \log(n) factor about VC classes is subtle, and this paper does not really remove this term unless we make specific assumptions on the value of V*. I would recommend the authors either expand the discussion about this and include a more detailed comparison with prior work, or minimize this claim.

This paper is devoted to the extension of the regret lower bound beyond ergodic Markov decision processes (MDPs) in the problem dependent setting. While the regret lower bound for ergodic MDPs is well-known and reached by tractable algorithms, we prove that the regret lower bound becomes significatively more complex in communicating MDPs. Our lower bound revisits the necessary explorative behavior of consistent learning agents and further explains that all optimal regions of the environment must be overvisited compared to sub-optimal ones, a phenomenon that we refer to as co-exploration. In tandem, we show that these two explorative and co-explorative behaviors are intertwined with navigation constraints obtained by scrutinizing the navigation structure at logarithmic scale. The resulting lower bound is expressed as the solution of an optimization problem that, in many standard classes of MDPs, can be specialized to recover existing results. From a computational perspective, it is provably $\Sigma_2^\textrm{P}$-hard in general and as a matter of fact, even testing the membership to the feasible region is coNP-hard. We further provide an algorithm to approximate the lower bound in a constructive way.

We study the design of computationally e cient algorithms with provable guarantees, that are robust to adversarial (test time) perturbations. While there has been an explosion of recent work on this topic due to its connections to test time robustness of deep networks, there is limited theoretical understanding of several basic questions like (i) when and how can one design provably robust learning algorithms?

We study the problem of completing a binary matrix in an online learning setting. On each trial we predict a matrix entry and then receive the true entry. We propose a Matrix Exponentiated Gradient algorithm [1] to solve this problem. We provide a mistake bound for the algorithm, which scales with the margin complexity [2, 3] of the underlying matrix. The bound suggests an interpretation where each row of the matrix is a prediction task over a finite set of objects, the columns. Using this we show that the algorithm makes a number of mistakes which is comparable up to a logarithmic factor to the number of mistakes made by the Kernel Perceptron with an optimal kernel in hindsight. We discuss applications of the algorithm to predicting as well as the best biclustering and to the problem of predicting the labeling of a graph without knowing the graph in advance.

The main effort is to improve upon the recent results provided by Bresler, showing that the complexity of identifying the structure of max degree d Ising model is polynomial in p and independent of d. Strong Points: 1) The timeliness of the topic in this paper is good, meaning that there is currently ongoing interest and work on Ising model reconstruction. Weak points: 1) The whole approach is based on the introduction of the ISO. This is the main trick in the proposed approach. Other than that, the rest of the method and its analysis are usual and well studied (l_1-penalization and connection with the tutorial by Negahban et.

We consider the problem of learning the underlying graph of an unknown Ising model on p spins from a collection of i.i.d.

In this paper, we study the quantitative relation between RTD and the well-known learning complexity measure VC dimension (VCD), and improve the best known upper and (worst-case) lower bounds on the recursive teaching dimension with respect to the VC dimension.

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.