This paper is an in-depth investigation of using kernel methods to immunize optimization solutions against distributional ambiguity. We propose kernel distributionally robust optimization (K-DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct ambiguity sets. It can be reformulated as a tractable program by using the conic duality of moment problems and an extension of the RKHS representer theorem. Our insights reveal that universal RKHSs are large enough for K-DRO to be effective. This paper provides both theoretical analyses that extend the robustness properties of kernel methods, as well as practical algorithms that can be applied to general optimization problems, not limited to kernelized models.

Metric learning is an important family of algorithms for classification and similarity search, but the robustness of learned metrics against small adversarial perturbations is less studied. In this paper, we show that existing metric learning algorithms, which focus on boosting the clean accuracy, can result in metrics that are less robust than the Euclidean distance. To overcome this problem, we propose a novel metric learning algorithm to find a Mahalanobis distance that is robust against adversarial perturbations, and the robustness of the resulting model is certifiable. Experimental results show that the proposed metric learning algorithm improves both certified robust errors and empirical robust errors (errors under adversarial attacks). Furthermore, unlike neural network defenses which usually encounter a trade-off between clean and robust errors, our method does not sacrifice clean errors compared with previous metric learning methods. Our code is available at https://github.com/wangwllu/provably_robust_metric_learning.

Learning from data owned by several parties, as in federated learning, raises challenges regarding the privacy guarantees provided to participants and the correctness of the computation in the presence of malicious parties. We tackle these challenges in the context of distributed averaging, an essential building block of distributed and federated learning. Our first contribution is a novel distributed differentially private protocol which naturally scales with the number of parties. The key idea underlying our protocol is to exchange correlated Gaussian noise along the edges of a network graph, complemented by independent noise added by each party. We analyze the differential privacy guarantees of our protocol and the impact of the graph topology, showing that we can match the accuracy of the trusted curator model even when each party communicates with only a logarithmic number of other parties chosen at random. This is in contrast with protocols in the local model of privacy (with lower accuracy) or based on secure aggregation (where all pairs of users need to exchange messages). Our second contribution is to enable users to prove the correctness of their computations without compromising the efficiency and privacy guarantees of the protocol. Our construction relies on standard cryptographic primitives like commitment schemes and zero knowledge proofs.

This paper describes an engine to optimize web publisher revenues from second-price auctions. These auctions are widely used to sell online ad spaces in a mechanism called real-time bidding (RTB). Optimization within these auctions is crucial for web publishers, because setting appropriate reserve prices can significantly increase revenue. We consider a practical real-world setting where the only available information before an auction occurs consists of a user identifier and an ad placement identifier. The real-world challenges we had to tackle consist mainly of tracking the dependencies on both the user and placement in an highly non-stationary environment and of dealing with censored bid observations. These challenges led us to make the following design choices: (i) we adopted a relatively simple non-parametric regression model of auction revenue based on an incremental time-weighted matrix factorization which implicitly builds adaptive users' and placements' profiles; (ii) we jointly used a non-parametric model to estimate the first and second bids' distribution when they are censored, based on an on-line extension of the Aalen's Additive model. Our engine is a component of a deployed system handling hundreds of web publishers across the world, serving billions of ads a day to hundreds of millions of visitors. The engine is able to predict, for each auction, an optimal reserve price in approximately one millisecond and yields a significant revenue increase for the web publishers.

Multiclass extensions of the support vector machine (SVM) have been formulated in a variety of ways. A recent empirical comparison of nine such formulations [Do\v{g}an et al. 2016] recommends the variant proposed by Weston and Watkins (WW), despite the fact that the WW-hinge loss is not calibrated with respect to the 0-1 loss. In this work we introduce a novel discrete loss function for multiclass classification, the ordered partition loss, and prove that the WW-hinge loss is calibrated with respect to this loss. We also argue that the ordered partition loss is maximally informative among discrete losses satisfying this property. Finally, we apply our theory to justify the empirical observation made by Do\v{g}an et al. that the WW-SVM can work well even under massive label noise, a challenging setting for multiclass SVMs.

Robust methods, though ubiquitous in practice, are yet to be fully understood in the context of regularized estimation and high dimensions. Even simple questions become challenging very quickly. For example, classical statistical theory identifies equivalence between model-averaged and composite quantile estimation. However, little to nothing is known about such equivalence between methods that encourage sparsity. This paper provides a toolbox to further study robustness in these settings and focuses on prediction. In particular, we study optimally weighted model-averaged as well as composite $l_1$-regularized estimation. Optimal weights are determined by minimizing the asymptotic mean squared error. This approach incorporates the effects of regularization, without the assumption of perfect selection, as is often used in practice. Such weights are then optimal for prediction quality. Through an extensive simulation study, we show that no single method systematically outperforms others. We find, however, that model-averaged and composite quantile estimators often outperform least-squares methods, even in the case of Gaussian model noise. Real data application witnesses the method's practical use through the reconstruction of compressed audio signals.

Group studies involving large cohorts of subjects are important to draw general conclusions about brain functional organization. However, the aggregation of data coming from multiple subjects is challenging, since it requires accounting for large variability in anatomy, functional topography and stimulus response across individuals. Data modeling is especially hard for ecologically relevant conditions such as movie watching, where the experimental setup does not imply well-defined cognitive operations. We propose a novel MultiView Independent Component Analysis (ICA) model for group studies, where data from each subject are modeled as a linear combination of shared independent sources plus noise. Contrary to most group-ICA procedures, the likelihood of the model is available in closed form. We develop an alternate quasi-Newton method for maximizing the likelihood, which is robust and converges quickly. We demonstrate the usefulness of our approach first on fMRI data, where our model demonstrates improved sensitivity in identifying common sources among subjects. Moreover, the sources recovered by our model exhibit lower between-session variability than other methods.On magnetoencephalography (MEG) data, our method yields more accurate source localization on phantom data. Applied on 200 subjects from the Cam-CAN dataset it reveals a clear sequence of evoked activity in sensor and source space. The code is freely available at https://github.com/hugorichard/multiviewica.

This study addresses the problem of off-policy evaluation (OPE) from dependent samples obtained via the bandit algorithm. The goal of OPE is to evaluate a new policy using historical data obtained from behavior policies generated by the bandit algorithm. Because the bandit algorithm updates the policy based on past observations, the samples are not independent and identically distributed (i.i.d.). However, several existing methods for OPE do not take this issue into account and are based on the assumption that samples are i.i.d. In this study, we address this problem by constructing an estimator from a standardized martingale difference sequence. To standardize the sequence, we consider using evaluation data or sample splitting with a two-step estimation. This technique produces an estimator with asymptotic normality without restricting a class of behavior policies. In an experiment, the proposed estimator performs better than existing methods, which assume that the behavior policy converges to a time-invariant policy.

In this paper, we study the performance of a large family of SGD variants in the smooth nonconvex regime. To this end, we propose a generic and flexible assumption capable of accurate modeling of the second moment of the stochastic gradient. Our assumption is satisfied by a large number of specific variants of SGD in the literature, including SGD with arbitrary sampling, SGD with compressed gradients, and a wide variety of variance-reduced SGD methods such as SVRG and SAGA. We provide a single convergence analysis for all methods that satisfy the proposed unified assumption, thereby offering a unified understanding of SGD variants in the nonconvex regime instead of relying on dedicated analyses of each variant. Moreover, our unified analysis is accurate enough to recover or improve upon the best-known convergence results of several classical methods, and also gives new convergence results for many new methods which arise as special cases. In the more general distributed/federated nonconvex optimization setup, we propose two new general algorithmic frameworks differing in whether direct gradient compression (DC) or compression of gradient differences (DIANA) is used. We show that all methods captured by these two frameworks also satisfy our unified assumption. Thus, our unified convergence analysis also captures a large variety of distributed methods utilizing compressed communication. Finally, we also provide a unified analysis for obtaining faster linear convergence rates in this nonconvex regime under the PL condition.

Several works have addressed the problem of incorporating constraints in the reinforcement learning (RL) framework, however majority of them can only guarantee the satisfaction of soft constraints. In this work, we address the problem of satisfying hard state constraints in a model-free RL setting with the deterministic system dynamics. The proposed algorithm is developed for the discrete state and action space and utilizes a multi-class support vector machine (SVM) to represent the policy. The state constraints are incorporated in the SVM optimization framework to derive an analytical solution for determining the policy parameters. This final policy converges to a solution which is guaranteed to satisfy the constraints. Additionally, the proposed formulation adheres to the Q-learning framework and thus, also guarantees convergence to the optimal solution. The algorithm is demonstrated with multiple example problems.