In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

Model-free reinforcement learning (RL) algorithms, such as Q-learning, directly parameterize and update value functions or policies without explicitly modeling the environment. They are typically simpler, more flexible to use, and thus more prevalent in modern deep RL than model-based approaches. However, empirical work has suggested that model-free algorithms may require more samples to learn [Deisenroth and Rasmussen 2011, Schulman et al. 2015]. The theoretical question of "whether model-free algorithms can be made sample efficient" is one of the most fundamental questions in RL, and remains unsolved even in the basic scenario with finitely many states and actions. We prove that, in an episodic MDP setting, Q-learning with UCB exploration achieves regret $\tilde{O}(\sqrt{H^3 SAT})$, where $S$ and $A$ are the numbers of states and actions, $H$ is the number of steps per episode, and $T$ is the total number of steps. This sample efficiency matches the optimal regret that can be achieved by any model-based approach, up to a single $\sqrt{H}$ factor. To the best of our knowledge, this is the first analysis in the model-free setting that establishes $\sqrt{T}$ regret without requiring access to a "simulator."

We analyze the projected Langevin Monte Carlo (LMC) algorithm, a close cousin of projected Stochastic Gradient Descent (SGD). We show that LMC allows to sample in polynomial time from a posterior distribution restricted to a convex body and with concave log-likelihood. This gives the first Markov chain to sample from a log-concave distribution with a first-order oracle, as the existing chains with provable guarantees (lattice walk, ball walk and hit-and-run) require a zeroth-order oracle. Our proof uses elementary concepts from stochastic calculus which could be useful more generally to understand SGD and its variants.

We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We are interested in studying prior-free and prior-dependent regret bounds, very much in the same spirit than the usual distribution-free and distribution-dependent bounds for the non-Bayesian stochastic bandit. We first show that Thompson Sampling attains an optimal prior-free bound in the sense that for any prior distribution its Bayesian regret is bounded from above by $14 \sqrt{n K}$. This result is unimprovable in the sense that there exists a prior distribution such that any algorithm has a Bayesian regret bounded from below by $\frac{1}{20} \sqrt{n K}$. We also study the case of priors for the setting of Bubeck et al. [2013] (where the optimal mean is known as well as a lower bound on the smallest gap) and we show that in this case the regret of Thompson Sampling is in fact uniformly bounded over time, thus showing that Thompson Sampling can greatly take advantage of the nice properties of these priors.

We consider an original problem that arises from the issue of security analysis of a power system and that we name optimal discovery with probabilistic expert advice. We address it with an algorithm based on the optimistic paradigm and on the Good-Turing missing mass estimator. We prove two different regret bounds on the performance of this algorithm under weak assumptions on the probabilistic experts. Under more restrictive hypotheses, we also prove a macroscopic optimality result, comparing the algorithm both with an oracle strategy and with uniform sampling. Finally, we provide numerical experiments illustrating these theoretical findings.

We address the online linear optimization problem when the actions of the forecaster are represented by binary vectors. Our goal is to understand the magnitude of the minimax regret for the worst possible set of actions. We study the problem under three different assumptions for the feedback: full information, and the partial information models of the so-called "semi-bandit", and "bandit" problems. We consider both $L_\infty$-, and $L_2$-type of restrictions for the losses assigned by the adversary. We formulate a general strategy using Bregman projections on top of a potential-based gradient descent, which generalizes the ones studied in the series of papers Gyorgy et al. (2007), Dani et al. (2008), Abernethy et al. (2008), Cesa-Bianchi and Lugosi (2009), Helmbold and Warmuth (2009), Koolen et al. (2010), Uchiya et al. (2010), Kale et al. (2010) and Audibert and Bubeck (2010). We provide simple proofs that recover most of the previous results. We propose new upper bounds for the semi-bandit game. Moreover we derive lower bounds for all three feedback assumptions. With the only exception of the bandit game, the upper and lower bounds are tight, up to a constant factor. Finally, we answer a question asked by Koolen et al. (2010) by showing that the exponentially weighted average forecaster is suboptimal against $L_{\infty}$ adversaries.

We investigate the role of the initialization for the stability of the k-means clustering algorithm. As opposed to other papers, we consider the actual k-means algorithm and do not ignore its property of getting stuck in local optima. We are interested in the actual clustering, not only in the costs of the solution. We analyze when different initializations lead to the same local optimum, and when they lead to different local optima. This enables us to prove that it is reasonable to select the number of clusters based on stability scores.