Tight Lower Bounds for Multiplicative Weights Algorithmic Families
This provides foundational theoretical insights for online learning algorithms, impacting researchers in machine learning and optimization.
The paper tackles the problem of prediction with expert advice by developing regret lower bounds for a broad family of algorithms, including the Multiplicative Weights Algorithm (MWA), and closes the gap between upper and lower bounds with a regret of √(T ln k / 2).
We study the fundamental problem of prediction with expert advice and develop regret lower bounds for a large family of algorithms for this problem. We develop simple adversarial primitives, that lend themselves to various combinations leading to sharp lower bounds for many algorithmic families. We use these primitives to show that the classic Multiplicative Weights Algorithm (MWA) has a regret of $\sqrt{\frac{T \ln k}{2}}$, there by completely closing the gap between upper and lower bounds. We further show a regret lower bound of $\frac{2}{3}\sqrt{\frac{T\ln k}{2}}$ for a much more general family of algorithms than MWA, where the learning rate can be arbitrarily varied over time, or even picked from arbitrary distributions over time. We also use our primitives to construct adversaries in the geometric horizon setting for MWA to precisely characterize the regret at $\frac{0.391}{\sqrtδ}$ for the case of $2$ experts and a lower bound of $\frac{1}{2}\sqrt{\frac{\ln k}{2δ}}$ for the case of arbitrary number of experts $k$.