A Fast and Convergent Proximal Algorithm for Regularized Nonconvex and Nonsmooth Bi-level Optimization

• URL: http://arxiv.org/abs/2203.16615v1
• Date: Wed, 30 Mar 2022 18:53:04 GMT
• Title: A Fast and Convergent Proximal Algorithm for Regularized Nonconvex and Nonsmooth Bi-level Optimization
• Authors: Ziyi Chen, Bhavya Kailkhura, Yi Zhou
• Abstract summary: Existing bi-level algorithms cannot handle nonsmooth or hyper-smooth regularizers. In this paper, we show that an implicit differentiation (AID) scheme can be used to speed up comprehensive machine learning applications.
• Score: 26.68351521813062
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Many important machine learning applications involve regularized nonconvex bi-level optimization. However, the existing gradient-based bi-level optimization algorithms cannot handle nonconvex or nonsmooth regularizers, and they suffer from a high computation complexity in nonconvex bi-level optimization. In this work, we study a proximal gradient-type algorithm that adopts the approximate implicit differentiation (AID) scheme for nonconvex bi-level optimization with possibly nonconvex and nonsmooth regularizers. In particular, the algorithm applies the Nesterov's momentum to accelerate the computation of the implicit gradient involved in AID. We provide a comprehensive analysis of the global convergence properties of this algorithm through identifying its intrinsic potential function. In particular, we formally establish the convergence of the model parameters to a critical point of the bi-level problem, and obtain an improved computation complexity $\mathcal{O}(\kappa^{3.5}\epsilon^{-2})$ over the state-of-the-art result. Moreover, we analyze the asymptotic convergence rates of this algorithm under a class of local nonconvex geometries characterized by a {\L}ojasiewicz-type gradient inequality. Experiment on hyper-parameter optimization demonstrates the effectiveness of our algorithm.

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