Two-armed bandit problem and batch version of the mirror descent algorithm

We consider the minimax setup for the two-armed bandit problem as applied to data processing if there are two alternative processing methods with different a priori unknown efficiencies. One should determine the most efficient method and provide its predominant application. To this end, we use the mirror descent algorithm (MDA). It is well-known that corresponding minimax risk has the order of $N^{1/2$ with $N$ being the number of processed data and this bound is unimprovable in order. We propose a batch version of the MDA which allows processing data by packets that is especially important if parallel data processing can be provided. In this case, the processing time is determined by the number of batches rather than by the total number of data. Unexpectedly, it turned out that the batch version behaves unlike the ordinary one even if the number of packets is large. Moreover, the batch version provides significantly smaller value of the minimax risk, i.e., it considerably improves a control performance. We explain this result by considering another batch modification of the MDA which behavior is close to behavior of the ordinary version and minimax risk is close as well. Our estimates use invariant descriptions of the algorithms based on Gaussian approximations of incomes in batches of data in the domain of ``close'' distributions and are obtained by Monte-Carlo simulations.

Hierarchical Population Game Models of Coevolution in Multi-Criteria Optimization Problems under Uncertainty

The article develops hierarchical population game models of co-evolutionary algorithms for solving the problem of multi-criteria optimization under uncertainty. The principles of vector minimax and vector minimax risk are used as the basic principles of optimality for the problem of multi-criteria optimization under uncertainty. The concept of equilibrium of a hierarchical population game with the right of the first move is defined. The necessary conditions are formulated under which the equilibrium solution of a hierarchical population game is a discrete approximation of the set of optimal solutions to the multi-criteria optimization problem under uncertainty.

Lower Bound of Locally Differentially Private Sparse Covariance Matrix Estimation

In this paper, we study the sparse covariance matrix estimation problem in the local differential privacy model, and give a non-trivial lower bound on the non-interactive private minimax risk in the metric of squared spectral norm. We show that the lower bound is actually tight, as it matches a previous upper bound. Our main technique for achieving this lower bound is a general framework, called General Private Assouad Lemma, which is a considerable generalization of the previous private Assouad lemma and can be used as a general method for bounding the private minimax risk of matrix-related estimation problems.

Principal Component Analysis in the Local Differential Privacy Model

In this paper, we study the Principal Component Analysis (PCA) problem under the (distributed) non-interactive local differential privacy model. For the low dimensional case, we show the optimal rate for the private minimax risk of the k-dimensional PCA using the squared subspace distance as the measurement. For the high dimensional row sparse case, we first give a lower bound on the private minimax risk, . Then we provide an efficient algorithm to achieve a near optimal upper bound. Experiments on both synthetic and real world datasets confirm the theoretical guarantees of our algorithms.