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Recent studies have shown that gradient descent (GD) can achieve improvedgeneralization when its dynamics exhibits a chaotic behavior. However, toobtain the desired effect, the step-size should be chosen sufficiently large, atask which is problem dependent and can be difficult in practice. In thisstudy, we incorporate a chaotic component to GD in a controlled manner, andintroduce multiscale perturbed GD (MPGD), a novel optimization framework wherethe GD recursion is augmented with chaotic perturbations that evolve via anindependent dynamical system. We analyze MPGD from three different angles: (i)By building up on recent advances in rough paths theory, we show that, underappropriate assumptions, as the step-size decreases, the MPGD recursionconverges weakly to a stochastic differential equation (SDE) driven by aheavy-tailed L\'evy-stable process. (ii) By making connections to recentlydeveloped generalization bounds for heavy-tailed processes, we derive ageneralization bound for the limiting SDE and relate the worst-casegeneralization error over the trajectories of the process to the parameters ofMPGD. (iii) We analyze the implicit regularization effect brought by thedynamical regularization and show that, in the weak perturbation regime, MPGDintroduces terms that penalize the Hessian of the loss function. Empiricalresults are provided to demonstrate the advantages of MPGD.

Chaotic Regularization and Heavy-Tailed Limits for Deterministic Gradient Descent