Partial-Quasi-Newton Methods: Efficient Algorithms for Minimax Optimization Problems with Unbalanced Dimensionality

This paper studies the strongly-convex-strongly-concave minimax optimization with unbalanced dimensionality. Such problems contain several popular applications in data science such as few shot learning and fairness-aware machine learning task. The design of conventional iterative algorithm for minimax optimization typically focuses on reducing the total number of oracle calls, which ignores the unbalanced computational cost for accessing the information from two different variables in minimax. We propose a novel second-order optimization algorithm, called Partial-Quasi-Newton (PQN) method, which takes the advantage of unbalanced structure in the problem to establish the Hessian estimate efficiently. We theoretically prove our PQN method converges to the saddle point faster than existing minimax optimization algorithms. The numerical experiments on real-world applications show the proposed PQN performs significantly better than the state-of-the-art methods.