论文标题
随机组成的优化，无Lipschitz连续梯度的功能
Stochastic Composition Optimization of Functions without Lipschitz Continuous Gradient
论文作者
论文摘要
在本文中，我们研究了无Lipschitz连续梯度的两级功能组成的随机优化。平滑度的特性通过相对平滑度的概念概括，从而引起布雷格曼梯度方法。我们为三种可能的相对平滑的组成场景提出了三种随机组成的Bregman梯度算法，并提供了样品复杂性，以实现$ε$ - $ - $ - $ - $ x的固定点。对于相对平滑组成的平滑，第一个算法需要$ o（ε^{ - 2}）$调用内部功能值和梯度的随机甲壳以及外部功能梯度。当两个函数相对平滑时，第二算法需要$ o（ε^{ - 3}）$调用内部功能值随机甲骨文和$ o（ε^{ - 2}）$调用内部和外部功能梯度梯度梯度天然口orac。我们通过降低方差的设置进一步改善了第二算法，而仅内部函数平滑的设置。所得算法需要$ O（ε^{ - 5/2}）$调用内部功能值随机甲骨文，$ O（ε^{ - 3/2}）$调用对内部函数梯度的调用，$ O（ε^{ - 2}）$呼叫$（ε^{ - 2}）$呼叫对外部功能梯度梯度梯度梯度梯度。最后，我们在两个不同的示例上评估了这三种算法的性能。
In this paper, we study stochastic optimization of two-level composition of functions without Lipschitz continuous gradient. The smoothness property is generalized by the notion of relative smoothness which provokes the Bregman gradient method. We propose three Stochastic Composition Bregman Gradient algorithms for the three possible relatively smooth compositional scenarios and provide their sample complexities to achieve an $ε$-approximate stationary point. For the smooth of relatively smooth composition, the first algorithm requires $O(ε^{-2})$ calls to the stochastic oracles of the inner function value and gradient as well as the outer function gradient. When both functions are relatively smooth, the second algorithm requires $O(ε^{-3})$ calls to the inner function value stochastic oracle and $O(ε^{-2})$ calls to the inner and outer functions gradients stochastic oracles. We further improve the second algorithm by variance reduction for the setting where just the inner function is smooth. The resulting algorithm requires $O(ε^{-5/2})$ calls to the inner function value stochastic oracle, $O(ε^{-3/2})$ calls to the inner function gradient and $O(ε^{-2})$ calls to the outer function gradient stochastic oracles. Finally, we numerically evaluate the performance of these three algorithms over two different examples.
