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.

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。 。