We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization -- including average-iterate gradient descent, the Frank-Wolfe algorithm, the Heavy Ball algorithm, and Nesterov's acceleration methods -- can be interpreted as special cases of our framework as long as each player makes the correct choice of no-regret strategy. Proving convergence rates in this framework becomes very straightforward, as they follow from plugging in the appropriate known regret bounds. Our framework also gives rise to a number of new first-order methods for special cases of convex optimization that were not previously known.
翻译:我们开发了一个算法框架,用无雷差的游戏动态解决混凝土优化问题。 通过将最小化的混凝土函数问题转换成一个辅助性问题,以相继方式解决微轴游戏,我们可以考虑每个必须一选其动作的两玩家的一系列策略。这些策略的共同选择是所谓的“无雷差学习算法”,我们描述了一系列这样的算法,并证明其悔恨的界限。然后我们展示了许多典型的共流层优化第一顺序方法,包括平均梯度梯度下降、弗兰克-沃夫算法、重球算法和内斯特罗夫的加速方法,只要每个玩家正确选择无雷特战略,就可以被解释为我们框架的特例。在这个框架中证明趋同率变得非常简单,因为它们是从适当已知的遗憾框中插入的。我们的框架还引出了许多新的先令方法,用于以前所不知道的峰值优化的特殊案例。