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Trading-Off Static and Dynamic Regret in Online Least-Squares and Beyond

Abstract

Recursive least-squares algorithms often use forgetting factors as a heuristic to adapt to non-stationary data streams. % The first contribution of this paper rigorously characterizes the effect of forgetting factors for a class of online Newton algorithms. % For exp-concave and strongly convex objectives, the algorithms achieve a dynamic regret of max{O(logT),O(TV)}\max\{O(\log T),O(\sqrt{TV})\}, where VV is a bound on the path length of the comparison sequence. % In particular, we show how classic recursive least-squares with a forgetting factor achieves this dynamic regret bound. % By varying VV, we obtain a trade-off between static and dynamic regret. % Furthermore, we show how the forgetting factor can be tuned to obtain % trade-offs between static and dynamic regret. % In order to obtain more computationally efficient algorithms, our second contribution is a novel gradient descent step size rule for strongly convex functions. % Our gradient descent rule recovers the dynamic regret bounds described above. % For smooth problems, we can also obtain static regret of O(T1β)O(T^{1-\beta}) and dynamic regret of O(TβV)O(T^\beta V^*), where β(0,1)\beta \in (0,1) and VV^* is the path length of the sequence of minimizers. % By varying β\beta, we obtain a trade-off between static and dynamic regret.

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