FastAdaBelief: Improving Convergence Rate for Belief-Based Adaptive Optimizers by Exploiting Strong Convexity

AdaBelief, one of the current best optimizers, demonstrates superior generalization ability over the popular Adam algorithm by viewing the exponential moving average of observed gradients. AdaBelief is theoretically appealing in which it has a data-dependent
$O(\sqrt {T})$
regret bound when objective functions are convex, where
$T$
is a time horizon. It remains, however, an open problem whether the convergence rate can be further improved without sacrificing its generalization ability. To this end, we make the first attempt in this work and design a novel optimization algorithm called FastAdaBelief that aims to exploit its strong convexity in order to achieve an even faster convergence rate. In particular, by adjusting the step size that better considers strong convexity and prevents fluctuation, our proposed FastAdaBelief demonstrates excellent generalization ability and superior convergence. As an important theoretical contribution, we prove that FastAdaBelief attains a data-dependent
$O(\log T)$
regret bound, which is substantially lower than AdaBelief in strongly convex cases. On the empirical side, we validate our theoretical analysis with extensive experiments in scenarios of strong convexity and nonconvexity using three popular baseline models. Experimental results are very encouraging: FastAdaBelief converges the quickest in comparison to all mainstream algorithms while maintaining an excellent generalization ability, in cases of both strong convexity or nonconvexity. FastAdaBelief is, thus, posited as a new benchmark model for the research community.