Fractional underdamped langevin dynamics: Retargeting SGD with momentum under heavy-tailed gradient noise

Stochastic gradient descent with momentum (SGDm) is one of the most popular optimization algorithms in deep learning. While there is a rich theory of SGDm for convex problems, the theory is considerably less developed in the context of deep learning where the problem is non-convex and the gradient noise might exhibit a heavy-tailed behavior, as empirically observed in recent studies. In this study, we consider a continuous-time variant of SGDm, known as the underdamped Langevin dynamics (ULD), and investigate its asymptotic properties under heavy-tailed perturbations. Supported by recent studies from statistical physics, we argue both theoretically and empirically that the heavy-tails of such perturbations can result in a bias even when the step-size is small, in the sense that the optima of stationary distribution of the dynamics might not match the optima of the cost function to be optimized. As a remedy, we develop a novel framework, which we coin as fractional ULD (FULD), and prove that FULD targets the so-called Gibbs distribution, whose optima exactly match the optima of the original cost. We observe that the Euler discretization of FULD has noteworthy algorithmic similarities with natural gradient methods and gradient clipping, bringing a new perspective on understanding their role in deep learning. We support our theory with experiments conducted on a synthetic model and neural networks.