Decentralized Weakly Convex Optimization Over the Stiefel Manifold

要約

タイトル：スティーフェル多様体上の分散型弱凸最適化

要約：

– スティーフェル多様体上の非滑らか最適化問題を扱う。
– 問題は非滑らかで非凸的なため、課題がある。
– 提案手法としてDRSMを導入。
– 近似スムーズ性に基づくグローバル収束と反復計算量の証明を行う。
– さらに問題がシャープな場合、DRSMの局所収束性を示す。
– 数値実験により理論結果を確認。

要約(オリジナル)

We focus on a class of non-smooth optimization problems over the Stiefel manifold in the decentralized setting, where a connected network of $n$ agents cooperatively minimize a finite-sum objective function with each component being weakly convex in the ambient Euclidean space. Such optimization problems, albeit frequently encountered in applications, are quite challenging due to their non-smoothness and non-convexity. To tackle them, we propose an iterative method called the decentralized Riemannian subgradient method (DRSM). The global convergence and an iteration complexity of $\mathcal{O}(\varepsilon^{-2} \log^2(\varepsilon^{-1}))$ for forcing a natural stationarity measure below $\varepsilon$ are established via the powerful tool of proximal smoothness from variational analysis, which could be of independent interest. Besides, we show the local linear convergence of the DRSM using geometrically diminishing stepsizes when the problem at hand further possesses a sharpness property. Numerical experiments are conducted to corroborate our theoretical findings.

arxiv情報

提供元, 利用サービス

arxiv.jp, OpenAI