We propose an efficient algorithm for finding first-order Nash equilibria in smooth min-max problems of the form min x∈X max y∈Y F (x, y), where the objective function is nonconvex with respect to x and concave with respect to y, and the set Y is convex, compact, and projectionfriendly. The goal is to reach an (ε x , ε y )-first-order Nash equilibrium point, as measured by the norm of the corresponding (proximal) gradient component. The proposed approach is fairly simple: essentially, we perform approximate proximal point iterations on the primal function, with inexact oracle provided by Nesterov's algorithm run on the regularized function F (x t , •) with O(ε y ) regularization term, where x t is the current primal iterate. The resulting iteration complexity is O(ε −2 x ε −1/2 y

The classical asymptotic theory for parametric M -estimators guarantees that, in the limit of infinite sample size, the excess risk has a chi-square type distribution, even in the misspecified case. We demonstrate how self-concordance of the loss allows to characterize the critical sample size sufficient to guarantee a chi-square type in-probability bound for the excess risk. Specifically, we consider two classes of losses: (i) self-concordant losses in the classical sense of Nesterov and Nemirovski, i.e., whose third derivative is uniformly bounded with the 3/2 power of the second derivative; (ii) pseudo self-concordant losses, for which the power is removed. These classes contain losses corresponding to several generalized linear models, including the logistic loss and pseudo-Huber losses.Our basic result under minimal assumptions bounds the critical sample size by O(d • d eff ), where d the parameter dimension and d eff the effective dimension that accounts for model misspecification. In contrast to the existing results, we only impose local assumptions that concern the population risk minimizer θ * . Namely, we assume that the calibrated predictors, i.e., predictors scaled by the square root of the second derivative of the loss, is subgaussian at θ * . Besides, for type-ii losses we require boundedness of certain measure of curvature of the population risk at θ * .Our improved result bounds the critical sample size from above as O(max{d eff , d log d})under slightly stronger assumptions. Namely, the local assumptions must hold in the neighborhood of θ * given by the Dikin ellipsoid of the population risk. Interestingly, we find that, for logistic regression with Gaussian design, there is no actual restriction of conditions: the subgaussian parameter and curvature measure remain near-constant over the Dikin ellipsoid. Finally, we extend some of these results to 1 -penalized estimators in high dimensions.

We consider learning methods based on the regularization of a convex empirical risk by a squared Hilbertian norm, a setting that includes linear predictors and non-linear predictors through positive-definite kernels. In order to go beyond the generic analysis leading to convergence rates of the excess risk as O(1/ √ n) from n observations, we assume that the individual losses are self-concordant, that is, their third-order derivatives are bounded by their secondorder derivatives. This setting includes least-squares, as well as all generalized linear models such as logistic and softmax regression. For this class of losses, we provide a bias-variance decomposition and show that the assumptions commonly made in least-squares regression, such as the source and capacity conditions, can be adapted to obtain fast non-asymptotic rates of convergence by improving the bias terms, the variance terms or both.

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Let θ 0 , θ 1 ∈ R d be the population risk minimizers associated to some loss : R d × Z → R and two distributions P 0 , P 1 on Z. Our work is motivated by the following question: Given i.i.d. samples from P 0 and P 1 , what sample sizes are sufficient and necessary to distinguish between the two hypotheses θ * = θ 0 and θ * = θ 1 for given θ * ∈ {θ 0 , θ 1 }?Making the first steps towards answering this question in full generality, we first consider the case of a well-specified linear model with squared loss. Here we provide matching upper and lower bounds on the sample complexity, showing it to be min{1/∆ 2 , √ r/∆} up to a constant factor, where ∆ is a measure of separation between P 0 and P 1 , and r is the rank of the design covariance matrix. This bound is dimension-independent, and rank-independent for large enough separation. We then extend this result in two directions: (i) for the general parametric setup in asymptotic regime; (ii) for generalized linear models in the small-sample regime n r and under weak moment assumptions. In both cases, we derive sample complexity bounds of a similar form, even under misspecification. In fact, our testing procedures only access θ * through a certain functional of empirical risk. In addition, the number of observations that allows to reach statistical confidence in our tests does not allow to "resolve" the two models -that is, recover θ 0 , θ 1 up to O(∆) prediction accuracy. These two properties allow to use our framework in applied tasks where one would like to identify a prediction model, which can be proprietary, while guaranteeing that the model cannot be actually inferred by the agent performing identification. * Equal contribution of the first two authors. 1 For this expository discussion, we define the sample complexity of a (binary) testing problem as the size of an i.i.d. sample for which there exists a test with testing errors of both types at most 0.05.

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In the problem of online portfolio selection as formulated by Cover (1991) [12], the trader repeatedly distributes her capital over d assets in each of T > 1 rounds, with the goal of maximizing the total return. Cover proposed an algorithm, termed Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with an O(d log(T )) logarithmic regret. Without imposing any restrictions on the market this guarantee is known to be worst-case optimal, and no other algorithm attaining it has been discovered so far. Unfortunately, Cover's algorithm crucially relies on computing certain d-dimensional integral, which must be approximated in any implementation; this results in a prohibitive Õ(d 4 (T + d) 14 ) per-round runtime for the fastest known implementation due to Kalai and Vempala (2002) [22]. We propose an algorithm for online portfolio selection that admits essentially the same regret guarantee as Universal Portfolios-up to a constant factor and replacement of log(T ) with log(T + d)-yet has a drastically reduced runtime of Õ(d 2 (T + d)) per round. The selected portfolio minimizes the observed logarithmic loss regularized with the log-determinant of its Hessian-equivalently, the hybrid logarithmic-volumetric barrier of the polytope specified by the asset return vectors. As such, our work reveals surprising connections of online portfolio selection with two classical topics in optimization theory: cutting-plane and interior-point algorithms.

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