Second-Order Neural ODE Optimizer

• URL: http://arxiv.org/abs/2109.14158v1
• Date: Wed, 29 Sep 2021 02:58:18 GMT
• Title: Second-Order Neural ODE Optimizer
• Authors: Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou
• Abstract summary: We show that a specific continuous-time OC methodology, called Differential Programming, can be adopted to derive backward ODEs for higher-order derivatives at the same O(1) memory cost. The resulting method converges much faster than first-order baselines in wall-clock time. Our framework also enables direct architecture optimization, such as the integration time of Neural ODEs, with second-order feedback policies.
• Score: 11.92713188431164
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs). Since their training already involves expensive gradient computation by solving a backward ODE, deriving efficient second-order methods becomes highly nontrivial. Nevertheless, inspired by the recent Optimal Control (OC) interpretation of training deep networks, we show that a specific continuous-time OC methodology, called Differential Programming, can be adopted to derive backward ODEs for higher-order derivatives at the same O(1) memory cost. We further explore a low-rank representation of the second-order derivatives and show that it leads to efficient preconditioned updates with the aid of Kronecker-based factorization. The resulting method converges much faster than first-order baselines in wall-clock time, and the improvement remains consistent across various applications, e.g. image classification, generative flow, and time-series prediction. Our framework also enables direct architecture optimization, such as the integration time of Neural ODEs, with second-order feedback policies, strengthening the OC perspective as a principled tool of analyzing optimization in deep learning.

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