On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis

• URL: http://arxiv.org/abs/2009.07799v3
• Date: Fri, 30 Aug 2024 14:12:30 GMT
• Title: On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis
• Authors: Zhong Li, Jiequn Han, Weinan E, Qianxiao Li,
• Abstract summary: We consider the simple but representative setting of using continuous-time linear RNNs to learn from data generated by linear relationships. We prove a universal approximation theorem of such linear functionals, and characterize the approximation rate and its relation with memory. A unifying theme uncovered is the non-trivial effect of memory, a notion that can be made precise in our framework.
• Score: 30.75240284934018
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the approximation properties and optimization dynamics of recurrent neural networks (RNNs) when applied to learn input-output relationships in temporal data. We consider the simple but representative setting of using continuous-time linear RNNs to learn from data generated by linear relationships. Mathematically, the latter can be understood as a sequence of linear functionals. We prove a universal approximation theorem of such linear functionals, and characterize the approximation rate and its relation with memory. Moreover, we perform a fine-grained dynamical analysis of training linear RNNs, which further reveal the intricate interactions between memory and learning. A unifying theme uncovered is the non-trivial effect of memory, a notion that can be made precise in our framework, on approximation and optimization: when there is long term memory in the target, it takes a large number of neurons to approximate it. Moreover, the training process will suffer from slow downs. In particular, both of these effects become exponentially more pronounced with memory - a phenomenon we call the "curse of memory". These analyses represent a basic step towards a concrete mathematical understanding of new phenomenon that may arise in learning temporal relationships using recurrent architectures.

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