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, Nesterov's acceleration methods, and the accelerated proximal method -- 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：我们开发了一种利用无遗憾博弈动力学求解凸优化问题的算法框架。通过将凸函数最小化问题转化为辅助的序列式极小极大博弈问题，我们可以为需要依次选择动作的两名玩家设计一系列策略。这些策略的常见选择是所谓的无遗憾学习算法，我们描述了若干此类算法并证明了其遗憾界。随后我们表明，许多凸优化的经典一阶方法——包括平均迭代梯度下降法、Frank-Wolfe算法、Nesterov加速方法以及加速近端方法——均可视为本框架的特例，只要每位玩家正确选择无遗憾策略。在此框架中证明收敛率变得极为直接，只需代入已知的相应遗憾界即可。此外，我们的框架还催生了若干此前未知的凸优化特殊情形下的新型一阶方法。