On the Dynamic Regret of Online Multiple Mirror Descent

We study the problem of online convex optimization, where a learner makes sequential decisions to minimize an accumulation of strongly convex costs over time. The quality of decisions is given in terms of the dynamic regret, which measures the performance of the learner relative to a sequence of dynamic minimizers. Prior works on gradient descent and mirror descent have shown that the dynamic regret can be upper bounded using the path length, which depend on the differences between successive minimizers, and an upper bound using the squared path length has also been shown when multiple gradient queries are allowed per round. However, they all require the cost functions to be Lipschitz continuous, which imposes a strong requirement especially when the cost functions are also strongly convex. In this work, we consider Online Multiple Mirror Descent (OMMD), which is based on mirror descent but uses multiple mirror descent steps per online round. Without requiring the cost functions to be Lipschitz continuous, we derive two upper bounds on the dynamic regret based on the path length and squared path length. We further derive a third upper bound that relies on the gradient of cost functions, which can be much smaller than the path length or squared path length, especially when the cost functions are smooth but fluctuate over time. Thus, we show that the dynamic regret of OMMD scales linearly with the minimum among the path length, squared path length, and sum squared gradients. Our experimental results further show substantial improvement on the dynamic regret compared with existing alternatives.