提出可自适应调整的Bregman优化算法，加速多维狄利克雷分布参数估计
Variable Bregman Majorization-Minimization Algorithm and its Application to Dirichlet Maximum Likelihood Estimation
- 通过动态调整Bregman散度实现更精确的目标函数逼近
- 在狄利克雷分布最大似然估计中收敛速度显著快于传统方法
- 适合需要高效优化凸目标函数的机器学习与统计建模场景
我们提出一种新型Bregman下降算法，用于最小化由可微部分（定义在开集上）和可能非光滑项组成的凸函数。该方法称为可变Bregman极大-极小化（VBMM）算法，通过允许每轮迭代中使用的Bregman函数自适应变化来扩展Bregman近端梯度法，前提是其满足目标函数的上界条件。这种自适应框架使算法能在每一步更精确地逼近目标，从而实现比传统方法更快的收敛速度。我们在对所用度量族的温和假设下证明了VBMM算法收敛至最小值点。此外，我们首次将Bregman近端梯度法和VBMM算法应用于通过最大化对数似然来估计多维狄利克雷分布的参数。数值实验表明，VBMM算法在收敛速度上优于现有方法。
原文摘要 · Abstract (English)
We propose a novel Bregman descent algorithm for minimizing a convex function that is expressed as the sum of a differentiable part (defined over an open set) and a possibly nonsmooth term. The approach, referred to as the Variable Bregman Majorization-Minimization (VBMM) algorithm, extends the Bregman Proximal Gradient method by allowing the Bregman function used in the divergence to adaptively vary at each iteration, provided it satisfies a majorizing condition on the objective function. This adaptive framework enables the algorithm to approximate the objective more precisely at each iteration, thereby allowing for accelerated convergence compared to the traditional Bregman Proximal Gradient descent. We establish the convergence of the VBMM algorithm to a minimizer under mild assumptions on the family of metrics used. Furthermore, we introduce a novel application of both the Bregman Proximal Gradient method and the VBMM algorithm to the estimation of the multidimensional parameters of a Dirichlet distribution through the maximization of its log-likelihood. Numerical experiments confirm that the VBMM algorithm outperforms existing approaches in terms of convergence speed.
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