On Parameter Selection for First-Order Methods: A Matrix Analysis Approach

First-order convex optimization algorithms are popular due to their computational attractiveness and applicability to a wide range of domains such as machine learning and control. Despite the substantial progress being made over the last few decades, some open questions related to their convergence remain unaddressed. In addition, majority of the first-order methods assume strong convexity to analyze both the stability of the method and derive an explicit convergence rate. In this manuscript, we relax the strong convexity condition, and then, lay out two main contributions. First, we provide a methodology where one can analyze the speed of convergence of the algorithm using the contractive theory and linear algebra. Second, we find explicit values of the tuning parameters of the Double Momentum Algorithm (which unifies many of the popular algorithms), ensuring stability for gradient L-Lipchitz functions. In this work, while an explicit convergence rate is not provided, the foundational results serve as a stepping stone in that direction, as we provide an explicit non-asymptotic rate. Furthermore, our numerical experiments demonstrate superior performance of the proposed method. Beyond optimization, we also apply our method to two-sided markets in non-cooperative game theory.