We study the regret of optimal strategies for online convex optimization
games. Using von Neumann's minimax theorem, we show that the optimal regret in
this adversarial setting is closely related to the behavior of the empirical
minimization algorithm in a stochastic process setting: it is equal to the
maximum, over joint distributions of the adversary's action sequence, of the
difference between a sum of minimal expected losses and the minimal empirical
loss. We show that the optimal regret has a natural geometric interpretation,
since it can be viewed as the gap in Jensen's inequality for a concave
functional--the minimizer over the player's actions of expected loss--defined
on a set of probability distributions. We use this expression to obtain upper
and lower bounds on the regret of an optimal strategy for a variety of online
learning problems. Our method provides upper bounds without the need to
construct a learning algorithm; the lower bounds provide explicit optimal
strategies for the adversary.

通过 von Neumann 最小极大定理，我们研究了在线凸优化游戏的最优策略的遗憾。我们证明了，在这种对抗性环境中，最优策略的遗憾与随机进程设置中经验最小化算法的行为密切相关：它等于最小期望损失的总和与最小经验损失之间的差的最大值。我们展示了最优策略的遗憾具有自然的几何解释，因为它可以被视为一个上凸函数的 Jensen 不等式中的差距。利用此表达式，我们对各种在线学习问题的最优策略给出了上下界限制。我们的方法提供了无需构建学习算法的上界，而提供了对抗者的明确最优策略的下界。