Online Convex Optimization with Switching Cost and Delayed Gradients

We consider the online convex optimization (OCO) problem with quadratic and linear switching cost when at time t only gradient information for functions f T , T < t is available. For L-smooth and µ-strongly convex objective functions, we propose an algorithm (OMGD) with a competitive ratio of at most 4(L+5)+ 16(L+5)/µ for the quadratic switching cost, and also show the bound to be order-wise tight in terms of L, µ. In addition, we show that the competitive ratio of any online algorithm is at least maxΩ(L),Ω( L/√µ ) when the switching cost is quadratic. For the linear switching cost, the competitive ratio of the OMGD algorithm is shown to depend on both the path length and the squared path length of the problem instance, in addition to L, µ, and is shown to be order-wise, the best competitive ratio any online algorithm can achieve.