Low-rank Matrix Completion (LRMC) describes the problem where we wish to recover missing entries of partially observed low-rank matrix. Most existing matrix completion work deals with sampling procedures that are independent of the underlying data values. While this assumption allows the derivation of nice theoretical guarantees, it seldom holds in real-world applications. In this paper, we consider various settings where the sampling mask is dependent on the underlying data values, motivated by applications in sensing, sequential decision-making, and recommender systems. Through a series of experiments, we study and compare the performance of various LRMC algorithms that were originally successful for data-independent sampling patterns.

Is Stochastic Gradient Descent (SGD) substantially different from Glauber dynamics? This is a fundamental question at the time of understanding the most used training algorithm in the field of Machine Learning, but it received no answer until now. Here we show that in discrete optimization and inference problems, the dynamics of an SGD-like algorithm resemble very closely that of Metropolis Monte Carlo with a properly chosen temperature, which depends on the mini-batch size. This quantitative matching holds both at equilibrium and in the out-of-equilibrium regime, despite the two algorithms having fundamental differences (e.g.\ SGD does not satisfy detailed balance). Such equivalence allows us to use results about performances and limits of Monte Carlo algorithms to optimize the mini-batch size in the SGD-like algorithm and make it efficient at recovering the signal in hard inference problems.

Abstract--We study stochastic multi-armed bandits with many players. The players do not know the number of players, cannot communicate with each other and if multiple players select a common arm they collide and none of them receive any reward. We consider the static scenario, where the number of players remains fixed, and the dynamic scenario, where the players enter and leave at any time. We provide algorithms based on a novel'trekking approach' that guarantees constant regret for the static case and sub-linear regret for the dynamic case with high probability. The trekking approach eliminates the need to estimate the number of players resulting in fewer collisions and improved regret performance compared to the state-of-theart algorithms. We also develop an epoch-less algorithm that eliminates any requirement of time synchronization across the players provided each player can detect the presence of other players on an arm. We validate our theoretical guarantees using simulation based and real test-bed based experiments. Multi-player multi-armed bandits (MPMAB) is a variant of the stochastic multi-armed bandits [1]-[3] where multiple players aim to maximize sum of their rewards playing the same set of arms. In this setting, the players do not communicate with each other and may not know number of other players in the game. If two or more players select the same arm simultaneously, they experience'collision' and none of them receive any reward. Our goal in this work is to develop distributed algorithms that aim to achieve high total rewards while keeping the number of collisions as low as possible. The study of MPMAB is mainly motivated from the ad-hoc cognitive radio networks (CRN) where multiple users transmit on a common set of channels (unlicensed spectrum) without any communication among them [4], [5]. Due to the ad hoc nature of such networks, a central controller, or a common control channels for coordination, may not be available and all channel selection decisions have to be done in a decentralized fashion [4]-[6]. Such models are being envisioned for futuristic ultra-dense wireless communication networks that can offer very high peak rates [7]. The quality of the channels are unknown to the users and their goal is to maximize number of successful transmissions (or sum rate/ throughput) in the network.