Second-Order Neural ODE Optimizer

Guan-Horng Liu, Tianrong Chen, Evangelos Theodorou

Keywords: Neural ODEs, second-order optimization, continuous-time optimal control

TL;DR: An efficient second-order optimizer for training Neural ODEs that achieves superior convergence against first-order methods in wall-clock time on various applications.

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 – named SNOpt – 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. Our code is available at https://github.com/ghliu/snopt.

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Community Implementations: [ 3 code implementations](https://www.catalyzex.com/paper/arxiv:2109.14158/code)