当前位置: X-MOL 学术Comput. Optim. Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Accelerated Bregman proximal gradient methods for relatively smooth convex optimization
Computational Optimization and Applications ( IF 1.6 ) Pub Date : 2021-04-07 , DOI: 10.1007/s10589-021-00273-8
Filip Hanzely , Peter Richtárik , Lin Xiao

We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. We investigate a triangle scaling property of the Bregman distance generated by the reference convex function and present accelerated Bregman proximal gradient (ABPG) methods that attain an \(O(k^{-\gamma })\) convergence rate, where \(\gamma \in (0,2]\) is the triangle scaling exponent (TSE) of the Bregman distance. For the Euclidean distance, we have \(\gamma =2\) and recover the convergence rate of Nesterov’s accelerated gradient methods. For non-Euclidean Bregman distances, the TSE can be much smaller (say \(\gamma \le 1\)), but we show that a relaxed definition of intrinsic TSE is always equal to 2. We exploit the intrinsic TSE to develop adaptive ABPG methods that converge much faster in practice. Although theoretical guarantees on a fast convergence rate seem to be out of reach in general, our methods obtain empirical \(O(k^{-2})\) rates in numerical experiments on several applications and provide posterior numerical certificates for the fast rates.



中文翻译:

加速的Bregman近端梯度法用于相对平滑的凸优化

我们考虑最小化两个凸函数之和的问题:一个凸函数相对于参考凸函数是可微且相对平滑的,而另一个则是不可微的但易于优化。我们研究了由参考凸函数生成的Bregman距离的三角形缩放特性,并提出了达到\(O(k ^ {-\ gamma})\)收敛速度的加速Bregman近端梯度(ABPG)方法,其中\(\ γ\ in(0,2] \)是布雷格曼距离的三角比例指数(TSE),对于欧几里得距离,我们具有\(\ gamma = 2 \)并恢复了Nesterov的加速梯度方法的收敛速度。非欧氏Bregman距离,则TSE可以小得多(例如\(\ gamma \ le 1 \)),但是我们显示出固有TSE的宽松定义始终等于2。我们利用固有TSE来开发自适应ABPG方法,这些方法在实践中收敛得更快。尽管通常对快速收敛速率的理论保证似乎遥不可及,但是我们的方法在几种应用的数值实验中都获得了经验\(O(k ^ {-2})\)速率,并为该快速速率提供了后验数值证明。 。

更新日期:2021-04-08
down
wechat
bug