Neural Information Processing Systems http://nips.cc/

The Median-of-Means (MoM) is a robust estimator widely used in machine learning that is known to be (minimax) optimal in scenarios where samples are i.i.d. In more grave scenarios, samples are contaminated by an adversary that can inspect and modify the data. Previous work has theoretically shown the suitability of the MoM estimator in certain contaminated settings. However, the (minimax) optimality of MoM and its limitations under adversarial contamination remain unknown beyond the Gaussian case. In this paper, we present upper and lower bounds for the error of MoM under adversarial contamination for multiple classes of distributions. In particular, we show that MoM is (minimax) optimal in the class of distributions with finite variance, as well as in the class of distributions with infinite variance and finite absolute $(1+r)$-th moment. We also provide lower bounds for MoM's error that match the order of the presented upper bounds, and show that MoM is sub-optimal for light-tailed distributions.

Simulating the kinetic Langevin dynamics is a popular approach for sampling from distributions, where only their unnormalized densities are available. Various discretizations of the kinetic Langevin dynamics have been considered, where the resulting algorithm is collectively referred to as the kinetic Langevin Monte Carlo (KLMC) or underdamped Langevin Monte Carlo. Specifically, the stochastic exponential Euler discretization, or exponential integrator for short, has previously been studied under strongly log-concave and log-Lipschitz smooth potentials via the synchronous Wasserstein coupling strategy. Existing analyses, however, impose restrictions on the parameters that do not explain the behavior of KLMC under various choices of parameters. In particular, all known results fail to hold in the overdamped regime, suggesting that the exponential integrator degenerates in the overdamped limit. In this work, we revisit the synchronous Wasserstein coupling analysis of KLMC with the exponential integrator. Our refined analysis results in Wasserstein contractions and bounds on the asymptotic bias that hold under weaker restrictions on the parameters, which assert that the exponential integrator is capable of stably simulating the kinetic Langevin dynamics in the overdamped regime, as long as proper time acceleration is applied.

R2: It would benefit the paper greatly to provide some more discussion with the conditions in Theorem 2.

The polynomial is fitted using weighted ordinary least-squares, giving more weight to nearby points and less weight to points farther away. The value of the regression function for the point is then obtained by evaluating the fitted local polynomial using the predictor variable value for that data point. LPR has good accuracy near the boundary and performs better than all other linear smoothers in a minimax sense [2]. The biggest advantage of this class of methods is not requiring a prior specification of a function i.e. a parameterized model. Instead, only a small number of hyperparameters need to be specified such as the type of kernel, a smoothing parameter and the degree of the local polynomial. The method is therefore suitable for modeling complex processes such as non-linear relationships, or complex dependencies for which no theoretical models exist. These two advantages, combined with the simplicity of the method, makes it one of the most attractive of the modern regression methods for applications that fit the general framework of least-squares regression but have a complex deterministic structure. Local polynomial regression incorporates the notion of proximity in two ways.

In cross-device federated learning (FL) with millions of mobile clients, only a small subset of clients participate in training in every communication round, and Federated Averaging (FedAvg) is the most popular algorithm in practice. Existing analyses of FedAvg usually assume the participating clients are independently sampled in each round from a uniform distribution, which does not reflect real-world scenarios. This paper introduces a theoretical framework that models client participation in FL as a Markov chain to study optimization convergence when clients have non-uniform and correlated participation across rounds. We apply this framework to analyze a more general and practical pattern: every client must wait a minimum number of $R$ rounds (minimum separation) before re-participating. We theoretically prove and empirically observe that increasing minimum separation reduces the bias induced by intrinsic non-uniformity of client availability in cross-device FL systems. Furthermore, we develop an effective debiasing algorithm for FedAvg that provably converges to the unbiased optimal solution under arbitrary minimum separation and unknown client availability distribution.

Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA.

Estimators of information theoretic measures such as entropy and mutual information are a basic workhorse for many downstream applications in modern data science. State of the art approaches have been either geometric (nearest neighbor (NN) based) or kernel based (with a globally chosen bandwidth). In this paper, we combine both these approaches to design new estimators of entropy and mutual information that outperform state of the art methods. Our estimator uses local bandwidth choices of k-NN distances with a finite k, independent of the sample size. Such a local and data dependent choice improves performance in practice, but the bandwidth is vanishing at a fast rate, leading to a non-vanishing bias. We show that the asymptotic bias of the proposed estimator is universal; it is independent of the underlying distribution. Hence, it can be precomputed and subtracted from the estimate. As a byproduct, we obtain a unified way of obtaining both kernel and NN estimators. The corresponding theoretical contribution relating the asymptotic geometry of nearest neighbors to order statistics is of independent mathematical interest.