AI 中文总结
本文提出MELMO算法，针对非欧几里得方向下的弱凸复合优化问题，建立多组收敛界，在稀疏低秩矩阵分解和图像去噪任务上表现具竞争力。
AI 中文摘要
我们提出了MELMO（Moreau包络结合线性最小化神谕的平滑方法），这是一种针对形如minₓ f(x)+g(Tx)的复合优化问题的算法，其中f是光滑函数，g可能非光滑。该方法利用Moreau包络对非光滑分量进行平滑，同时通过线性最小化神谕适配问题几何结构。假设g是ρ-弱凸的，我们建立了由步长和平滑调度参数化的一系列收敛界，从而明确了平滑目标优化与恢复原复合问题平稳性之间的权衡关系。具体而言，一种情形下平滑梯度范数和复合平稳性代理的收敛速率均为O(k⁻¹/⁴)，另一种情形下平滑梯度范数的收敛速率为O(k⁻¹/³)，而复合代理的收敛速率为O(k⁻¹/⁴)。我们还建立了依赖K-视界的收敛速率，其复合代理的收敛速率为O(K⁻¹/³)。实验表明，在稀疏低秩矩阵分解和图像去噪任务上，MELMO与变量平滑方法及次梯度基线相比具有竞争力。
英文摘要
We propose MELMO (Moreau Envelope Smoothing with Linear Minimization Oracles), an algorithm for composite optimization problems of the form min x f (x) + g(T x), where f is smooth and g may be non-smooth. The method leverages the Moreau envelope to smooth the non-smooth component while adapting to problem geometry through linear minimization oracles. Assuming g is $ρ$-weakly convex, we establish a family of convergence bounds parameterized by the step-size and smoothing schedules, thereby making explicit the trade-off between optimizing the smoothed objective and recovering stationarity for the original composite problem. In particular, one regime yields O(k -1/4 ) rates for both the smoothed-gradient norm and a composite stationarity proxy, while another yields O(k -1/3 ) for the smoothed-gradient norm together with O(k -1/4 ) for the composite proxy. We also establish a K-horizon-dependent convergence rate that yields O(K -1/3 ) for the composite proxy. Empirically, MELMO is competitive with variable smoothing and subgradient baselines on sparse low-rank matrix factorization and image denoising.
