用于非凸非光滑优化的带自适应动量的惯性块近端线性化方法

中文摘要 AI 辅助

本文研究一类多块非凸非光滑优化问题，这类问题涵盖诸多应用场景，如震前异常分析与机器学习等。为求解该类问题，我们提出带两阶段自适应动量的惯性块近端线性化方法（IBPL⁺-TP）。与现有方法相比，该方法具备三大核心优势：（1）引入两阶段自适应动量策略以有效更新外推参数；（2）允许使用两个不同的外推点以加快收敛速度；（3）允许这两个外推点的外推参数独立于所有其他参数，且不受其约束。在保留上述优势的同时，我们证明该方法可确保此类问题目标函数的单调收敛性，还证明由该方法生成的序列全局收敛至一个临界点，并确定了该方法的收敛速率。为验证该方法的有效性，我们将其应用于求解两个非凸非光滑机器学习问题，即带ℓ₀约束的稀疏非负矩阵分解与带ℓ₀约束的稀疏非负CP分解。求解这些问题的数值实验结果表明，该方法的性能优于若干最先进的方法。

英文摘要

In this paper, we consider a class of multiblock nonconvex nonsmooth optimization problems, which covers many applications such as the analysis of pre-earthquake anomalies and machine learning. To solve this class of problems, we propose the inertial block proximal linearized method with two-phase adaptive momentum (IBPL$^+$-TP). Compared to the current methods, our method possesses three main advantages: (1) it introduces a two-phase adaptive momentum strategy to effectively update the extrapolation parameters, (2) it allows using two different extrapolation points to accelerate the convergence, (3) it allows the extrapolation parameters of these two extrapolation points to be independent of and unconstrained by all other parameters. While maintaining the above advantages, we prove that our method ensures the monotonic convergence of the objective function of this class of problems, and we also prove that the sequence generated by our method globally converges to a critical point, as well as establish the convergence rate of our method. To demonstrate the effectiveness of our method, we apply it to solve two nonconvex and nonsmooth machine learning problems, namely sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. The numerical experimental results on solving these problems show that our method outperforms several state-of-the-art methods.

发表机构

• Yunnan Earthquake Administration（云南省地震局）

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