The notion of approachability in repeated games with vector payoffs was introduced by Blackwell in the 1950s, along with geometric conditions for approachability and corresponding strategies that rely on computing {\em steering directions} as projections from the current average payoff vector to the (convex) target set. Recently, Abernethy, Batlett and Hazan (2011) proposed a class of approachability algorithms that rely on the no-regret properties of Online Linear Programming for computing a suitable sequence of steering directions. This is first carried out for target sets that are convex cones, and then generalized to any convex set by embedding it in a higher-dimensional convex cone. In this paper we present a more direct formulation that relies on the support function of the set, along with suitable Online Convex Optimization algorithms, which leads to a general class of approachability algorithms. We further show that Blackwell's original algorithm and its convergence follow as a special case.
Blackwell approachability is a framework for reasoning about repeated games with vector-valued payoffs. We introduce predictive Blackwell approachability, where an estimate of the next payoff vector is given, and the decision maker tries to achieve better performance based on the accuracy of that estimator. In order to derive algorithms that achieve predictive Blackwell approachability, we start by showing a powerful connection between four well-known algorithms. Follow-the-regularized-leader (FTRL) and online mirror descent (OMD) are the most prevalent regret minimizers in online convex optim…
Chicago Booth School of Business（芝加哥大学布斯商学院）；MIT Sloan School of Management（麻省理工学院斯隆管理学院）；Google Research（谷歌研究院）
Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have $O(\sqrt{T})$ (approximate) regret under the full information setting. W…
The literature on ranking from ordinal data is vast, and there are several ways to aggregate overall preferences from pairwise comparisons between objects. In particular, it is well known that any Nash equilibrium of the zero sum game induced by the preference matrix defines a natural solution concept (winning distribution over objects) known as a von Neumann winner. Many real-world problems, however, are inevitably multi-criteria, with different pairwise preferences governing the different criteria. In this work, we generalize the notion of a von Neumann winner to the multi-criteria setting b…
University of Pennsylvania（宾夕法尼亚大学）；Rice University（莱斯大学）
We introduce a simple but general online learning framework in which a learner plays against an adversary in a vector-valued game that changes every round. Even though the learner's objective is not convex-concave (and so the minimax theorem does not apply), we give a simple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret. We demonstrate the power of our framework by using it to (re)derive optimal bounds and efficient algorithms across a variety of domains, ranging from multicalibration to a large set of no …
We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may have quitting actions, such as the Big-Match. We provide three simple geometric and strongly related conditions for the weak approachability of a convex target set. The first is sufficient: it guarantees that, for any fixed horizon, a player has a strategy ensuring that the expecte…
在完全监控的可逼近性中，已知有两种类型的条件对于凸集是等价的：原始条件和对偶条件。原始的形式是：一个集合C是可接近的当且仅当所有包含的半空间在一次博弈中是可接近的;而对偶的形式是：一个凸集C是可接近的当且仅当它与所有具有某种形式的支付集相交。我们考虑部分监控博弈中的可接近性。在以前的工作（Perchet 2011; Mannor et al. 2011）中，我们提供了可逼近凸集的对偶特征;我们还展示了C是多胞形的情况下的有效策略。本文给出了凸集可部分监控逼近的基本条件。它们依赖于修改后的报酬函数，并导致可接近性策略，基于修改后的支付函数，进行类似于Blackwell（1956）的策略的预测;这是在对比与以前研究的策略，在这种情况下，主要依赖于信号结构，旨在估计以及所收到的信号的分布。我们的结果推广了Kohlberg 1975的经典结果（也可参见Mertens et al. 1994），并适用于具有任意信号结构的博弈以及任意凸集。
We develop new parameter-free and scale-free algorithms for solving convex-concave saddle-point problems. Our results are based on a new simple regret minimizer, the Conic Blackwell Algorithm$^+$ (CBA$^+$), which attains $O(1/\sqrt{T})$ average regret. Intuitively, our approach generalizes to other decision sets of interest ideas from the Counterfactual Regret minimization (CFR$^+$) algorithm, which has very strong practical performance for solving sequential games on simplexes. We show how to implement CBA$^+$ for the simplex, $\ell_{p}$ norm balls, and ellipsoidal confidence regions in the s…
Estimating the empirical distribution of a scalar-valued data set is a basic and fundamental task. In this paper, we tackle the problem of estimating an empirical distribution in a setting with two challenging features. First, the algorithm does not directly observe the data; instead, it only asks a limited number of threshold queries about each sample. Second, the data are not assumed to be independent and identically distributed; instead, we allow for an arbitrary process generating the samples, including an adaptive adversary. These considerations are relevant, for example, when modeling a …
