Global Convergence of Over-parameterized Deep Equilibrium Models

• URL: http://arxiv.org/abs/2205.13814v2
• Date: Wed, 29 Mar 2023 03:56:52 GMT
• Title: Global Convergence of Over-parameterized Deep Equilibrium Models
• Authors: Zenan Ling, Xingyu Xie, Qiuhao Wang, Zongpeng Zhang, Zhouchen Lin
• Abstract summary: A deep equilibrium model (DEQ) is implicitly defined through an equilibrium point of an infinite-depth weight-tied model with an input-injection. Instead of infinite computations, it solves an equilibrium point directly with root-finding and computes gradients with implicit differentiation. We propose a novel probabilistic framework to overcome the technical difficulty in the non-asymptotic analysis of infinite-depth weight-tied models.
• Score: 52.65330015267245
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A deep equilibrium model (DEQ) is implicitly defined through an equilibrium point of an infinite-depth weight-tied model with an input-injection. Instead of infinite computations, it solves an equilibrium point directly with root-finding and computes gradients with implicit differentiation. The training dynamics of over-parameterized DEQs are investigated in this study. By supposing a condition on the initial equilibrium point, we show that the unique equilibrium point always exists during the training process, and the gradient descent is proved to converge to a globally optimal solution at a linear convergence rate for the quadratic loss function. In order to show that the required initial condition is satisfied via mild over-parameterization, we perform a fine-grained analysis on random DEQs. We propose a novel probabilistic framework to overcome the technical difficulty in the non-asymptotic analysis of infinite-depth weight-tied models.

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