We investigate online convex optimization in changing environments, and choose the adaptive regret as the performance measure. The goal is to achieve a small regret over every interval so that the comparator is allowed to change over time. Different from previous works that only utilize the convexity condition, this paper further exploits smoothness to improve the adaptive regret. To this end, we develop novel adaptive algorithms for convex and smooth functions, and establish problem-dependent regret bounds over any interval. Our regret bounds are comparable to existing results in the worst case, and become much tighter when the comparator has a small loss.
In an online convex optimization problem a decision-maker makes a sequence of decisions, i.e., chooses a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters a sequence of (possibly unrelated) convex cost functions. Zinkevich (ICML 2003) introduced this framework, which models many natural repeated decision-making problems and generalizes many existing problems such as Prediction from Expert Advice and Cover's Universal Portfolios. Zinkevich showed that a simple online gradient descent algorithm achieves additive regret @math , for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to the best single decision in hindsight. In this paper, we give algorithms that achieve regret O(log?(T)) for an arbitrary sequence of strictly convex functions (with bounded first and second derivatives). This mirrors what has been done for the special cases of prediction from expert advice by Kivinen and Warmuth (EuroCOLT 1999), and Universal Portfolios by Cover (Math. Finance 1:1---19, 1991). We propose several algorithms achieving logarithmic regret, which besides being more general are also much more efficient to implement. The main new ideas give rise to an efficient algorithm based on the Newton method for optimization, a new tool in the field. Our analysis shows a surprising connection between the natural follow-the-leader approach and the Newton method. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover's algorithm and gradient descent. We generalize the recent relative loss bounds for on-line algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound the additional loss of the algorithm over the sum of the losses of the best experts for each segment. This is to model situations in which the examples change and different experts are best for certain segments of the sequence of examples. In the single segment case, the additional loss is proportional to log n, where n is the number of experts and the constant of proportionality depends on the loss function. Our algorithms do not produce the best partition; however the loss bound shows that our predictions are close to those of the best partition. When the number of segments is k+1 and the sequence is of length e, we can bound the additional loss of our algorithm over the best partition by O(k n+k (e k)). For the case when the loss per trial is bounded by one, we obtain an algorithm whose additional loss over the loss of the best partition is independent of the length of the sequence. The additional loss becomes O(k n+ k (L k)), where L is the loss of the best partitionwith k+1 segments. Our algorithms for tracking the predictions of the best expert aresimple adaptations of Vovk's original algorithm for the single best expert case. As in the original algorithms, we keep one weight per expert, and spend O(1) time per weight in each trial.