Abstract

We present a family of algorithms, called descent algorithms, for optimizing convex and non-convex functions. We also introduce a new first-order algorithm, called rescaled gradient descent (RGD), and show that RGD achieves a faster convergence rate than gradient descent provided the function is strongly smooth – a natural generalization of the standard smoothness assumption on the objective function. When the objective function is convex, we present two frameworks for “accelerating” descent methods, one in the style of Nesterov and the other in the style of Monteiro and Svaiter, using a single Lyapunov function. Rescaled gradient descent can be accelerated under the same strong smoothness assumption using both frameworks. We provide several examples of strongly smooth loss functions in machine learning and numerical experiments that verify our theoretical findings. We also present several extensions of our novel Lyapunov framework, including deriving optimal universal higher-order tensor methods and extending our framework to the coordinate setting.