OGDA

OGDA, short for Optimistic Gradient Descent Ascent, is an optimization algorithm used for solving two-player minimax problems, such as min_x max_y f(x, y). It is a variant of gradient descent for the minimizing player and gradient ascent for the maximizing player, augmented with an optimistic correction that leverages gradient information from previous iterations to predict the future gradient.

The method operates by computing the gradients with respect to both variables at the current iterate and then forming an extrapolated or optimistic gradient that combines these current gradients with information from the previous step. The next iterates are updated using this optimistic gradient, with the intention of counteracting oscillations that commonly arise in saddle-point problems. While a precise update form can vary in the literature, the core idea is to incorporate past gradient information to stabilize progress toward a saddle point.

OGDA is related to, and often discussed alongside, other two-player game optimization methods such as gradient descent ascent and the extragradient method. It is particularly noted for its potential to reduce cyclic behavior and to offer favorable convergence properties in smooth convex-concave settings or certain structured problems. However, performance can be problem-dependent, and guarantees may not extend to nonconvex-nonconvex scenarios or without appropriate step-size choices.

Applications of OGDA include training settings where a min-max formulation arises, such as generative adversarial networks (GANs), robust optimization, and certain multi-agent learning environments. The algorithm remains a topic of study in optimization and machine learning, with ongoing work comparing its effectiveness to alternative methods under various models and assumptions.