Joon Kwon

Joon Kwon, Yijun Wan, Bruno Ziliotto

Blackwell's approachability (Blackwell, 1954, 1956) is a very general online learning framework where a Decision Maker obtains vector-valued outcomes, and aims at the convergence of the average outcome to a given ``target'' set. Blackwell gave a sufficient condition for the decision maker having a strategy guaranteeing such a convergence against an adversarial environment, as well as what we now call the Blackwell's algorithm, which then ensures convergence. Blackwell's approachability has since been applied to numerous problems, in regret minimization and game theory, in particular. We extend this framework by allowing the outcome function and the inner product to be time-dependent. We establish a general guarantee for the natural extension to this framework of Blackwell's algorithm. In the case where the target set is an orthant, we present a family of time-dependent inner products which yields different convergence speeds for each coordinate of the average outcome. We apply this framework to absorbing games (an important class of stochastic games) for which we construct $\varepsilon$-uniformly optimal strategies using Blackwell's algorithm in a well-chosen auxiliary approachability problem, thereby giving a novel illustration of the relevance of online learning tools for solving games.

Joon Kwon

We propose a conversion scheme that turns regret minimizing algorithms into fixed point iterations, with convergence guarantees following from regret bounds. The resulting iterations can be seen as a grand extension of the classical Krasnoselskii--Mann iterations, as the latter are recovered by converting the Online Gradient Descent algorithm. This approach yields new simple iterations for finding fixed points of non-self operators. We also focus on converting algorithms from the AdaGrad family of regret minimizers, and thus obtain fixed point iterations with adaptive guarantees of a new kind. Numerical experiments on various problems demonstrate faster convergence of AdaGrad-based fixed point iterations over Krasnoselskii--Mann iterations.

Joon Kwon

Blackwell's approachability is a framework where two players, the Decision Maker and the Environment, play a repeated game with vector-valued payoffs. The goal of the Decision Maker is to make the average payoff converge to a given set called the target. When this is indeed possible, simple algorithms which guarantee the convergence are known. This abstract tool was successfully used for the construction of optimal strategies in various repeated games, but also found several applications in online learning. By extending an approach proposed by (Abernethy et al., 2011), we construct and analyze a class of Follow the Regularized Leader algorithms (FTRL) for Blackwell's approachability which are able to minimize not only the Euclidean distance to the target set (as it is often the case in the context of Blackwell's approachability) but a wide range of distance-like quantities. This flexibility enables us to apply these algorithms to closely minimize the quantity of interest in various online learning problems. In particular, for regret minimization with $\ell_p$ global costs, we obtain the first bounds with explicit dependence in $p$ and the dimension $d$.

Anatoli Juditsky, Joon Kwon, Éric Moulines

We introduce and analyze a new family of first-order optimization algorithms which generalizes and unifies both mirror descent and dual averaging. Within the framework of this family, we define new algorithms for constrained optimization that combines the advantages of mirror descent and dual averaging. Our preliminary simulation study shows that these new algorithms significantly outperform available methods in some situations.

Joon Kwon, Vianney Perchet, Claire Vernade

In the classical multi-armed bandit problem, d arms are available to the decision maker who pulls them sequentially in order to maximize his cumulative reward. Guarantees can be obtained on a relative quantity called regret, which scales linearly with d (or with sqrt(d) in the minimax sense). We here consider the sparse case of this classical problem in the sense that only a small number of arms, namely s < d, have a positive expected reward. We are able to leverage this additional assumption to provide an algorithm whose regret scales with s instead of d. Moreover, we prove that this algorithm is optimal by providing a matching lower bound - at least for a wide and pertinent range of parameters that we determine - and by evaluating its performance on simulated data.

Joon Kwon, Vianney Perchet

We demonstrate that, in the classical non-stochastic regret minimization problem with $d$ decisions, gains and losses to be respectively maximized or minimized are fundamentally different. Indeed, by considering the additional sparsity assumption (at each stage, at most $s$ decisions incur a nonzero outcome), we derive optimal regret bounds of different orders. Specifically, with gains, we obtain an optimal regret guarantee after $T$ stages of order $\sqrt{T\log s}$, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order $\sqrt{Ts\log(d)/d}$, which is decreasing in $d$. Eventually, we also study the bandit setting, and obtain an upper bound of order $\sqrt{Ts\log (d/s)}$ when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor $\sqrt{\log(d/s)}$.

Joon Kwon, Panayotis Mertikopoulos

We consider a family of learning strategies for online optimization problems that evolve in continuous time and we show that they lead to no regret. From a more traditional, discrete-time viewpoint, this continuous-time approach allows us to derive the no-regret properties of a large class of discrete-time algorithms including as special cases the exponential weight algorithm, online mirror descent, smooth fictitious play and vanishingly smooth fictitious play. In so doing, we obtain a unified view of many classical regret bounds, and we show that they can be decomposed into a term stemming from continuous-time considerations and a term which measures the disparity between discrete and continuous time. As a result, we obtain a general class of infinite horizon learning strategies that guarantee an $\mathcal{O}(n^{-1/2})$ regret bound without having to resort to a doubling trick.