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

##### Yangfan Zhou, Kaizhu Huang, Cheng Cheng, Xuguang Wang, Amir Hussain, Xin Liu

AdaBelief, one of the current best optimizers, demonstrates superior generalization ability compared to the popular Adam algorithm by viewing the exponential moving average of observed gradients. AdaBelief is theoretically appealing in that 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. %on how to exploit strong convexity to further improve the convergence rate of AdaBelief. To this end, we make a 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 as well as superior convergence. As an important theoretical contribution, we prove that FastAdaBelief attains a data-dependant $O(\log T)$ regret bound, which is substantially lower than AdaBelief. On the empirical side, we validate our theoretical analysis with extensive experiments in both scenarios of strong and non-strong convexity on 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 or non-strong convexity. FastAdaBelief is thus posited as a new benchmark model for the research community.

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