Analytical Study of Momentum-Based Acceleration Methods in Paradigmatic High-Dimensional Non-Convex Problems
This work addresses a theoretical gap for researchers in optimization and machine learning, but it is incremental as it confirms known limitations in a specific model.
The authors tackled the problem of understanding momentum-based acceleration methods in high-dimensional non-convex landscapes, specifically in the spiked matrix-tensor model, and found that while these methods speed up dynamics, they do not improve the algorithmic threshold compared to gradient descent.
The optimization step in many machine learning problems rarely relies on vanilla gradient descent but it is common practice to use momentum-based accelerated methods. Despite these algorithms being widely applied to arbitrary loss functions, their behaviour in generically non-convex, high dimensional landscapes is poorly understood. In this work, we use dynamical mean field theory techniques to describe analytically the average dynamics of these methods in a prototypical non-convex model: the (spiked) matrix-tensor model. We derive a closed set of equations that describe the behaviour of heavy-ball momentum and Nesterov acceleration in the infinite dimensional limit. By numerical integration of these equations, we observe that these methods speed up the dynamics but do not improve the algorithmic threshold with respect to gradient descent in the spiked model.