Parametrized Accelerated Methods Free of Condition Number

Abstract

Analyses of accelerated (momentum-based) gradient descent usually assume bounded condition number to obtain exponential convergence rates. However, in many real problems, e.g., kernel methods or deep neural networks, the condition number, even locally, can be unbounded, unknown or mis-estimated. This poses problems in both implementing and analyzing accelerated algorithms. In this paper, we address this issue by proposing parametrized accelerated methods by considering the condition number as a free parameter. We provide spectral-level analysis for several important accelerated algorithms, obtain explicit expressions and improve worst case convergence rates. Moreover, we show that those algorithm converge exponentially even when the condition number is unknown or mis-estimated.