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We propose a novel strategy for extracting features in supervised learning that can be used to construct a classifier which is more robust to small perturbations in the input space. Our method builds upon the idea of the information bottleneck by introducing an additional penalty term that encourages the Fisher information of the extracted features to be small, when parametrized by the inputs. By tuning the regularization parameter, we can explicitly trade off the opposing desiderata of robustness and accuracy when constructing a classifier. We derive the optimal solution to the robust information bottleneck when the inputs and outputs are jointly Gaussian, proving that the optimally robust features are also jointly Gaussian in that setting. Furthermore, we propose a method for optimizing a variational bound on the robust information bottleneck objective in general settings using stochastic gradient descent, which may be implemented efficiently in neural networks. Our experimental results for synthetic and real data sets show that the proposed feature extraction method indeed produces classifiers with increased robustness to perturbations., A version of this paper was submitted to IEEE Journal on Selected Areas in Information Theory (JSAIT)

Robust estimation is an important problem in statistics which aims at providing a reasonable estimator when the data-generating distribution lies within an appropriately defined ball around an uncontaminated distribution. Although minimax rates of estimation have been established in recent years, many existing robust estimators with provably optimal convergence rates are also computationally intractable. In this paper, we study several estimation problems under a Wasserstein contamination model and present computationally tractable estimators motivated by generative adversarial networks (GANs). Specifically, we analyze the properties of Wasserstein GAN-based estimators for location estimation, covariance matrix estimation and linear regression and show that our proposed estimators are minimax optimal in many scenarios. Finally, we present numerical results which demonstrate the effectiveness of our estimators.

We investigate problems in penalized $M$-estimation, inspired by applications in machine learning debugging. Data are collected from two pools, one containing data with possibly contaminated labels, and the other which is known to contain only cleanly labeled points. We first formulate a general statistical algorithm for identifying buggy points and provide rigorous theoretical guarantees under the assumption that the data follow a linear model. We then present two case studies to illustrate the results of our general theory and the dependence of our estimator on clean versus buggy points. We further propose an algorithm for tuning parameter selection of our Lasso-based algorithm and provide corresponding theoretical guarantees. Finally, we consider a two-person "game" played between a bug generator and a debugger, where the debugger can augment the contaminated data set with cleanly labeled versions of points in the original data pool. We establish a theoretical result showing a sufficient condition under which the bug generator can always fool the debugger. Nonetheless, we provide empirical results showing that such a situation may not occur in practice, making it possible for natural augmentation strategies combined with our Lasso debugging algorithm to succeed.

We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential attachment and uniform attachment trees. At the core of our proofs is a probabilistic analysis of P\'{o}lya urns corresponding to the number of uninfected neighbors in specific subtrees of the infection tree. We also provide an example illustrating the shortcomings of source estimation techniques in settings where the underlying graph is asymmetric., Comment: 23 pages

We consider the problem of influence maximization in fixed networks for contagion models in an adversarial setting. The goal is to select an optimal set of nodes to seed the influence process, such that the number of influenced nodes at the conclusion of the campaign is as large as possible. We formulate the problem as a repeated game between a player and adversary, where the adversary specifies the edges along which the contagion may spread, and the player chooses sets of nodes to influence in an online fashion. We establish upper and lower bounds on the minimax pseudo-regret in both undirected and directed networks., 30 pages

We study the problem of finding the maximum of a function defined on the nodes of a connected graph. The goal is to identify a node where the function obtains its maximum. We focus on local iterative algorithms, which traverse the nodes of the graph along a path, and the next iterate is chosen from the neighbors of the current iterate with probability distribution determined by the function values at the current iterate and its neighbors. We study two algorithms corresponding to a Metropolis-Hastings random walk with different transition kernels: (i) The first algorithm is an exponentially weighted random walk governed by a parameter $\gamma$. (ii) The second algorithm is defined with respect to the graph Laplacian and a smoothness parameter $k$. We derive convergence rates for the two algorithms in terms of total variation distance and hitting times. We also provide simulations showing the relative convergence rates of our algorithms in comparison to an unbiased random walk, as a function of the smoothness of the graph function. Our algorithms may be categorized as a new class of "descent-based" methods for function maximization on the nodes of a graph.

In statistical learning theory, generalization error is used to quantify the degree to which a supervised machine learning algorithm may overfit to training data. Recent work [Xu and Raginsky (2017)] has established a bound on the generalization error of empirical risk minimization based on the mutual information $I(S;W)$ between the algorithm input $S$ and the algorithm output $W$, when the loss function is sub-Gaussian. We leverage these results to derive generalization error bounds for a broad class of iterative algorithms that are characterized by bounded, noisy updates with Markovian structure. Our bounds are very general and are applicable to numerous settings of interest, including stochastic gradient Langevin dynamics (SGLD) and variants of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm. Furthermore, our error bounds hold for any output function computed over the path of iterates, including the last iterate of the algorithm or the average of subsets of iterates, and also allow for non-uniform sampling of data in successive updates of the algorithm., Comment: A shorter version of this paper was submitted to ISIT 2018. 14 pages, 1 figure

We derive upper and lower bounds for the policy regret of $T$-round online learning problems with graph-structured feedback, where the adversary is nonoblivious but assumed to have a bounded memory. We obtain upper bounds of $\widetilde O(T^{2/3})$ and $\widetilde O(T^{3/4})$ for strongly-observable and weakly-observable graphs, respectively, based on analyzing a variant of the Exp3 algorithm. When the adversary is allowed a bounded memory of size 1, we show that a matching lower bound of $\widetilde\Omega(T^{2/3})$ is achieved in the case of full-information feedback. We also study the particular loss structure of an oblivious adversary with switching costs, and show that in such a setting, non-revealing strongly-observable feedback graphs achieve a lower bound of $\widetilde\Omega(T^{2/3})$, as well., Comment: This paper has been accepted to ISIT 2018

We study theoretical properties of regularized robust M-estimators, applicable when data are drawn from a sparse high-dimensional linear model and contaminated by heavy-tailed distributions and/or outliers in the additive errors and covariates. We first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution: When the derivative of the loss function is bounded and satisfies a local restricted curvature condition, all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate nonconvex regularizer is used in place of an l_1-penalty, we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support---hence, results on asymptotic normality in the low-dimensional case carry over immediately to the high-dimensional setting. This has important implications for the efficiency of regularized nonconvex M-estimators when the errors are heavy-tailed. Our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region. We show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector, successive iterates will converge at a linear rate to a stationary point within the local region. Furthermore, the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization. The result is a novel two-step procedure that uses a convex M-estimator to achieve consistency and a nonconvex M-estimator to increase efficiency., 56 pages, 8 figures

We establish bounds on the KL divergence between two multivariate Gaussian distributions in terms of the Hamming distance between the edge sets of the corresponding graphical models. We show that the KL divergence is bounded below by a constant when the graphs differ by at least one edge; this is essentially the tightest possible bound, since classes of graphs exist for which the edge discrepancy increases but the KL divergence remains bounded above by a constant. As a natural corollary to our KL lower bound, we also establish a sample size requirement for correct model selection via maximum likelihood estimation. Our results rigorize the notion that it is essential to estimate the edge structure of a Gaussian graphical model accurately in order to approximate the true distribution to close precision., Comment: Accepted to ISIT 2015

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations., Comment: Published in at http://dx.doi.org/10.1214/13-AOS1162 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings., Published in at http://dx.doi.org/10.1214/12-AOS1018 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)