Beyond Convexity: Stochastic Quasi-Convex Optimization
This work addresses a known hurdle in first-order optimization methods for machine learning by extending efficient optimization to a wider class of functions, though it is incremental as it adapts an existing method.
The paper tackles the problem of optimizing non-convex functions by proposing a stochastic version of Normalized Gradient Descent (NGD) and proves its convergence to a global minimum for quasi-convex and locally-Lipschitz functions, which broadens applicability beyond convex optimization.
Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD). The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. In this paper we analyze a stochastic version of NGD and prove its convergence to a global minimum for a wider class of functions: we require the functions to be quasi-convex and locally-Lipschitz. Quasi-convexity broadens the con- cept of unimodality to multidimensions and allows for certain types of saddle points, which are a known hurdle for first-order optimization methods such as gradient descent. Locally-Lipschitz functions are only required to be Lipschitz in a small region around the optimum. This assumption circumvents gradient explosion, which is another known hurdle for gradient descent variants. Interestingly, unlike the vanilla SGD algorithm, the stochastic normalized gradient descent algorithm provably requires a minimal minibatch size.