Towards Understanding Why Lookahead Generalizes Better Than SGD and Beyond
Authors: Pan Zhou, Hanshu Yan, Xiaotong Yuan, Jiashi Feng, Shuicheng Yan

To train networks, lookahead algorithm~\cite{zhang2019lookahead} updates its fast weights $k$ times via  an  inner-loop   optimizer before updating its slow weights once by using the latest  fast weights. Any optimizer, e.g. SGD,  can serve as the inner-loop optimizer, and the derived lookahead  generally enjoys remarkable test performance improvement over the vanilla optimizer.  But theoretical understandings on the test performance improvement   of lookahead remain absent yet. To solve this issue, we theoretically justify  the advantages of lookahead  in terms of the excess risk error which measures the test performance. Specifically, we prove that lookahead using  SGD as its inner-loop optimizer can better balance the optimization error and generalization error to achieve smaller excess risk error than vanilla SGD  on (strongly) convex problems and nonconvex problems with Polyak-{\L}ojasiewicz condition which has been observed/proved in neural networks.   Moreover,  we show the stagewise optimization strategy~\cite{barshan2015stage} which decays learning rate several times during training can also benefit lookahead in  improving  its optimization and generalization errors on strongly convex problems. Finally, we propose a  stagewise locally-regularized lookahead (SLRLA) algorithm which sums up the vanilla objective and a local regularizer to minimize at each stage and  provably enjoys optimization and generalization improvement  over the conventional (stagewise) lookahead.    Experimental results on   CIFAR10/100 and ImageNet  testify its  advantages. Codes is available at  \url{https://github.com/sail-sg/SLRLA-optimizer}.