Generative adversarial networks (GANs) have become one of the most popular generative modeling techniques in machine learning. In this work, we study the statistical and robust properties of GANs for Gaussian mean estimation under Huber’s contamination model, where an epsilon proportion of training data may be arbitrarily corrupted. We prove that f-GAN, when equipped with appropriate discriminators, achieve optimal minimax rate, hence extending the recent result of Gao et al. (2019a). In contrast, we show that other GAN variants such as MMD-GAN (with Gaussian kernel) and W-GAN may fail to achieve minimax optimality. We further adapt f-GAN to the sparse and the unknown covariance settings. We perform numerical simulations to confirm our theoretical findings and reveal new insights on the importance of discriminators.

In this paper, we investigate Exploratory Conservative Policy Optimization (ECPO), a policy optimization strategy that improves exploration behavior while assuring monotonic progress in a principled objective. ECPO conducts maximum entropy exploration within a mirror descent framework, but updates policies using reversed KL projection. This formulation bypasses undesirable mode seeking behavior and avoids premature convergence to sub-optimal policies, while still supporting strong theoretical properties such as guaranteed policy improvement. Experimental evaluations demonstrate that the proposed method significantly improves practical exploration and surpasses the empirical performance of state-of-the art policy optimization methods in a set of benchmark tasks.

We develop a new algorithm for online planning in large scale sequential decision problems that improves upon the worst case efficiency of UCT. The idea is to augment Monte-Carlo Tree Search (MCTS) with maximum entropy policy optimization, evaluating each search node by softmax values back-propagated from simulation. To establish the effectiveness of this approach, we first investigate the single-step decision problem, stochastic softmax bandits, and show that softmax values can be estimated at an optimal convergence rate in terms of mean squared error. We then extend this approach to general sequential decision making by developing a general MCTS algorithm, Maximum Entropy for Tree Search (MENTS). We prove that the probability of MENTS failing to identify the best decision at the root decays exponentially, which fundamentally improves the polynomial convergence rate of UCT. Our experimental results also demonstrate that MENTS is more sample efficient than UCT in both synthetic problems and Atari 2600 games.

In this paper, we investigate Dimensionality reduction (DR) maps in an information retrieval setting from a quantitative topology point of view. In particular, we show that no DR maps can achieve perfect precision and perfect recall simultaneously. Thus a continuous DR map must have imperfect precision. We further prove an upper bound on the precision of Lipschitz continuous DR maps. While precision is a natural measure in an information retrieval setting, it does not measure `how' wrong the retrieved data is. We therefore propose a new measure based on Wasserstein distance that comes with similar theoretical guarantee. A key technical step in our proofs is a particular optimization problem of the $L_2$-Wasserstein distance over a constrained set of distributions. We provide a complete solution to this optimization problem, which can be of independent interest on the technical side.

We study the problem of identifying the best action among a set of possible options when the value of each action is given by a mapping from a number of noisy micro-observables in the so-called fixed confidence setting. Our main motivation is the application to minimax game search, which has been a major topic of interest in artificial intelligence. In this paper we introduce an abstract setting to clearly describe the essential properties of the problem. While previous work only considered a two-move-deep game tree search problem, our abstract setting can be applied to the general minimax games where the depth can be non-uniform and arbitrary, and transpositions are allowed. We introduce a new algorithm (LUCB-micro) for the abstract setting, and give its lower and upper sample complexity results. Our bounds recover some previous results, achieved in more limited settings, and also shed further light on how the structure of minimax problems influences sample complexity.

Follow the leader (FTL) is a simple online learning algorithm that is known to perform well when the loss functions are convex and positively curved. In this paper we ask whether there are other settings when FTL achieves low regret. In particular, we study the fundamental problem of linear prediction over a convex, compact domain with non-empty interior. Amongst other results, we prove that the curvature of the boundary of the domain can act as if the losses were curved: In this case, we prove that as long as the mean of the loss vectors have positive lengths bounded away from zero, FTL enjoys logarithmic regret, while for polytope domains and stochastic data it enjoys finite expected regret. The former result is also extended to strongly convex domains by establishing an equivalence between the strong convexity of sets and the minimum curvature of their boundary, which may be of independent interest. Building on a previously known meta-algorithm, we also get an algorithm that simultaneously enjoys the worst-case guarantees and the smaller regret of FTL when the data is `easy'. Finally, we show that such guarantees are achievable directly (e.g., by the follow the regularized leader algorithm or by a shrinkage-based variant of FTL) when the constraint set is an ellipsoid.

The follow the leader (FTL) algorithm, perhaps the simplest of all online learning algorithms, is known to perform well when the loss functions it is used on are positively curved. In this paper we ask whether there are other "lucky" settings when FTL achieves sublinear, "small" regret. In particular, we study the fundamental problem of linear prediction over a non-empty convex, compact domain. Amongst other results, we prove that the curvature of the boundary of the domain can act as if the losses were curved: In this case, we prove that as long as the mean of the loss vectors have positive lengths bounded away from zero, FTL enjoys a logarithmic growth rate of regret, while, e.g., for polyhedral domains and stochastic data it enjoys finite expected regret. Building on a previously known meta-algorithm, we also get an algorithm that simultaneously enjoys the worst-case guarantees and the bound available for FTL.

We study independent component analysis with noisy observations. We present, for the first time in the literature, consistent, polynomial-time algorithms to recover non-Gaussian source signals and the mixing matrix with a reconstruction error that vanishes at a 1/\sqrtT rate using T observations and scales only polynomially with the natural parameters of the problem. Our algorithms and analysis also extend to deterministic source signals whose empirical distributions are approximately independent.

In this paper we provide generalization bounds for semiparametric regression with the so-called partially linear models where the regression function is written as the sum of a linear parametric and a nonlinear, nonparametric function, the latter taken from a some set \mathcalH with finite entropy-integral. The problem is technically challenging because the parametric part is unconstrained and the model is underdetermined, while the response is allowed to be unbounded with subgaussian tails. Under natural regularity conditions, we bound the generalization error as a function of the metric entropy of \mathcalH and the dimension of the linear model. Our main tool is a ratio-type concentration inequality for increments of empirical processes, based on which we are able to give an exponential tail bound on the size of the parametric component. We also provide a comparison to alternatives of this technique and discuss why and when the unconstrained parametric part in the model may cause a problem in terms of the expected risk. We also explain by means of a specific example why this problem cannot be detected using the results of classical asymptotic analysis often seen in the statistics literature.

Automated feature discovery is a fundamental problem in machine learning. Although classical feature discovery methods do not guarantee optimal solutions in general, it has been recently noted that certain subspace learning and sparse coding problems can be solved efficiently, provided the number of features is not restricted a priori. We provide an extended characterization of this optimality result and describe the nature of the solutions under an expanded set of practical contexts. In particular, we apply the framework to a semi-supervised learning problem, and demonstrate that feature discovery can co-occur with input reconstruction and supervised training while still admitting globally optimal solutions. A comparison to existing semi-supervised feature discovery methods shows improved generalization and efficiency.