  Learning from repeated play in a fixed two-player zero-sum game is a classic
problem in game theory and online learning. We consider a variant of this
problem where the game payoff matrix changes over time, possibly in an
adversarial manner. We first present three performance measures to guide the
algorithmic design for this problem: 1) the well-studied individual regret, 2)
an extension of duality gap, and 3) a new measure called dynamic Nash
Equilibrium regret, which quantifies the cumulative difference between the
player's payoff and the minimax game value. Next, we develop a single
parameter-free algorithm that simultaneously enjoys favorable guarantees under
all these three performance measures. These guarantees are adaptive to
different non-stationarity measures of the payoff matrices and, importantly,
recover the best known results when the payoff matrix is fixed. Our algorithm
is based on a two-layer structure with a meta-algorithm learning over a group
of black-box base-learners satisfying a certain property, along with several
novel ingredients specifically designed for the time-varying game setting.
Empirical results further validate the effectiveness of our algorithm.
