Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis
This work addresses a limitation in optimization theory for structured nonsmooth problems, offering a new algorithm and analysis framework that could benefit researchers in machine learning and optimization, though it is incremental in extending existing methods to more complex settings.
The paper tackles nonsmooth nonconvex-nonconcave minimax optimization by proposing a novel algorithm called smoothed proximal linear descent-ascent (smoothed PLDA), which achieves an iteration complexity of O(ε^{-2}) for finding stationary points under certain conditions, specifically when the Kurdyka-Lojasiewicz exponent θ is in [0, 1/2].
Nonconvex-nonconcave minimax optimization has gained widespread interest over the last decade. However, most existing works focus on variants of gradient descent-ascent (GDA) algorithms, which are only applicable to smooth nonconvex-concave settings. To address this limitation, we propose a novel algorithm named smoothed proximal linear descent-ascent (smoothed PLDA), which can effectively handle a broad range of structured nonsmooth nonconvex-nonconcave minimax problems. Specifically, we consider the setting where the primal function has a nonsmooth composite structure and the dual function possesses the Kurdyka-Lojasiewicz (KL) property with exponent $θ\in [0,1)$. We introduce a novel convergence analysis framework for smoothed PLDA, the key components of which are our newly developed nonsmooth primal error bound and dual error bound. Using this framework, we show that smoothed PLDA can find both $ε$-game-stationary points and $ε$-optimization-stationary points of the problems of interest in $\mathcal{O}(ε^{-2\max\{2θ,1\}})$ iterations. Furthermore, when $θ\in [0,\frac{1}{2}]$, smoothed PLDA achieves the optimal iteration complexity of $\mathcal{O}(ε^{-2})$. To further demonstrate the effectiveness and wide applicability of our analysis framework, we show that certain max-structured problem possesses the KL property with exponent $θ=0$ under mild assumptions. As a by-product, we establish algorithm-independent quantitative relationships among various stationarity concepts, which may be of independent interest.