Trading-Off Static and Dynamic Regret in Online Least-Squares and Beyond
This work addresses the challenge of adapting online algorithms to changing data streams, which is crucial for applications like time-series forecasting, but it is incremental as it builds on existing methods with new analyses and rules.
The paper tackles the problem of online learning in non-stationary environments by analyzing forgetting factors in recursive least-squares and proposing a new gradient descent step size rule, achieving dynamic regret bounds such as O(log T) or O(√(TV)) for exp-concave and strongly convex objectives, with trade-offs between static and dynamic regret by varying parameters like V or β.
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 the dynamic regret of $\max\{O(\log T),O(\sqrt{TV})\}$, where $V$ 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 $V$, we obtain a trade-off 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 order optimal dynamic regret bounds described above. For smooth problems, we can also obtain static regret of $O(T^{1-β})$ and dynamic regret of $O(T^βV^*)$, where $β\in (0,1)$ and $V^*$ is the path length of the sequence of minimizers. By varying $β$, we obtain a trade-off between static and dynamic regret.