New Perspective on the Global Convergence of Finite-Sum Optimization

Lam M. Nguyen, Trang H. Tran, Marten van Dijk

Keywords: non-convex optimization, overparameterized, optimization for deep learning, finite-sum, global minimum

Abstract: Deep neural networks (DNNs) have shown great success in many machine learning tasks. Their training is challenging since the loss surface of the network architecture is generally non-convex, or even non-smooth. How and under what assumptions is guaranteed convergence to a \textit{global} minimum possible? We propose a reformulation of the minimization problem allowing for a new recursive algorithmic framework. By using bounded style assumptions, we prove convergence to an $\varepsilon$-(global) minimum using $\mathcal{\tilde{O}}(1/\varepsilon^2)$ gradient computations. Our theoretical foundation motivates further study, implementation, and optimization of the new algorithmic framework and further investigation of its non-standard bounded style assumptions. This new direction broadens our understanding of why and under what circumstances training of a DNN converges to a global minimum.

One-sentence Summary: We propose a new representation for analyzing the machine learning minimization problem to find an approximate global solution.