Three Operator Splitting with a Nonconvex Loss Function
Three Operator Splitting with a Nonconvex Loss Function
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发表时间：
2021-03
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通讯作者：
A. Yurtsever;Varun Mangalick;S. Sra
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作者：
A. Yurtsever;Varun Mangalick;S. Sra
考虑三个函数之和的极小化问题，其中一个函数是非凸的但可微，另外两个函数是凸的但可能不可微。我们研究了Davis&Yin（2017）的三算子分裂方法（TOS），旨在扩展其对这个非凸问题模板的理论保证。特别是，我们证明了收敛的TOS与非渐近界的非平稳性和不可行的错误。与现有的非凸TOS的工作相比，我们的保证不需要额外的光滑性假设的条款，包括目标，因此，他们涵盖的情况下，特别感兴趣的不可微的条款是指标函数。我们还将我们的结果扩展到一个随机设置，我们只能获得一个无偏估计的梯度。最后，我们通过二次分配问题的数值实验说明了所提出的方法的有效性。
We consider the problem of minimizing the sum of three functions, one of which is nonconvex but differentiable, and the other two are convex but possibly nondifferentiable. We investigate the Three Operator Splitting method (TOS) of Davis&Yin (2017) with an aim to extend its theoretical guarantees for this nonconvex problem template. In particular, we prove convergence of TOS with nonasymptotic bounds on its nonstationarity and infeasibility errors. In contrast with the existing work on nonconvex TOS, our guarantees do not require additional smoothness assumptions on the terms comprising the objective; hence they cover instances of particular interest where the nondifferentiable terms are indicator functions. We also extend our results to a stochastic setting where we have access only to an unbiased estimator of the gradient. Finally, we illustrate the effectiveness of the proposed method through numerical experiments on quadratic assignment problems.