一种无需全局Lipschitz连续性假设的 telescoping Bregmanian 近端梯度方法
A telescoping Bregmanian proximal gradient method without the global Lipschitz continuity assumption
AI总结：
本文提出了一种无需全局Lipschitz连续性假设的近端梯度方法变体，通过在约束集上进行 telescoping 分解，并利用Bregman散度来改进收敛性分析。
摘要：
最小化两个凸函数之和的问题在理论和实际应用中都有广泛的应用。解决此类问题的一种流行方法是近端梯度方法（近端前向-后向算法）。在使用该方法时，一个常见的假设是光滑项的梯度具有全局Lipschitz连续性。然而，这种假设在实践中并不总是成立，从而限制了该方法的使用。在本文中，我们讨论了一种新的近端梯度方法变体，在广泛的有限和无限维空间中，该方法不假定上述全局Lipschitz连续性假设。该方法的关键贡献是迭代步长依赖于约束集的某种telescoping分解。此外，我们使用Bregman散度在近端前向-后向操作中。在某些实际条件下，建立了非渐近收敛率（即函数值的收敛率），以及整个序列弱收敛到极小值点。我们还获得了一些具有独立兴趣的辅助结果。
英文摘要：
The problem of minimization of the sum of two convex functions has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. In this paper, we discuss, in a wide class of finite and infinite-dimensional spaces, a new variant of the proximal gradient method which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain telescopic decomposition of the constraint set into subsets. Moreover, we use a Bregman divergence in the proximal forward-backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest.
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