Improved Algorithms for Convex-Concave Minimax Optimization
This work provides incremental improvements in optimization algorithms for researchers and practitioners dealing with minimax problems, particularly in scenarios with weak interaction between variables.
This paper tackles the problem of convex-concave minimax optimization by proposing a new algorithm that improves the gradient complexity upper bound to $ ilde{O}\Bigl(\sqrt{ rac{L_x}{m_x}+ rac{L\cdot L_{xy}}{m_x m_y}+ rac{L_y}{m_y}}\ln\left(1/ε ight)\Bigr)$, achieving linear convergence and tighter dependency on condition numbers, especially when $L_{xy}\ll L$.
This paper studies minimax optimization problems $\min_x \max_y f(x,y)$, where $f(x,y)$ is $m_x$-strongly convex with respect to $x$, $m_y$-strongly concave with respect to $y$ and $(L_x,L_{xy},L_y)$-smooth. Zhang et al. provided the following lower bound of the gradient complexity for any first-order method: $Ω\Bigl(\sqrt{\frac{L_x}{m_x}+\frac{L_{xy}^2}{m_x m_y}+\frac{L_y}{m_y}}\ln(1/ε)\Bigr).$ This paper proposes a new algorithm with gradient complexity upper bound $\tilde{O}\Bigl(\sqrt{\frac{L_x}{m_x}+\frac{L\cdot L_{xy}}{m_x m_y}+\frac{L_y}{m_y}}\ln\left(1/ε\right)\Bigr),$ where $L=\max\{L_x,L_{xy},L_y\}$. This improves over the best known upper bound $\tilde{O}\left(\sqrt{\frac{L^2}{m_x m_y}} \ln^3\left(1/ε\right)\right)$ by Lin et al. Our bound achieves linear convergence rate and tighter dependency on condition numbers, especially when $L_{xy}\ll L$ (i.e., when the interaction between $x$ and $y$ is weak). Via reduction, our new bound also implies improved bounds for strongly convex-concave and convex-concave minimax optimization problems. When $f$ is quadratic, we can further improve the upper bound, which matches the lower bound up to a small sub-polynomial factor.