Gradient Descent Fails to Learn High-frequency Functions and Modular Arithmetic

• URL: http://arxiv.org/abs/2310.12660v2
• Date: Thu, 29 Aug 2024 09:58:47 GMT
• Title: Gradient Descent Fails to Learn High-frequency Functions and Modular Arithmetic
• Authors: Rustem Takhanov, Maxat Tezekbayev, Artur Pak, Arman Bolatov, Zhenisbek Assylbekov,
• Abstract summary: We present a mathematical analysis of limitations and challenges associated with using gradient-based learning techniques. We highlight that the variance of the gradient is negligibly small in both cases when either a frequency or the prime base $p$ is large.
• Score: 8.813846754606898
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Classes of target functions containing a large number of approximately orthogonal elements are known to be hard to learn by the Statistical Query algorithms. Recently this classical fact re-emerged in a theory of gradient-based optimization of neural networks. In the novel framework, the hardness of a class is usually quantified by the variance of the gradient with respect to a random choice of a target function. A set of functions of the form $x\to ax \bmod p$, where $a$ is taken from ${\mathbb Z}_p$, has attracted some attention from deep learning theorists and cryptographers recently. This class can be understood as a subset of $p$-periodic functions on ${\mathbb Z}$ and is tightly connected with a class of high-frequency periodic functions on the real line. We present a mathematical analysis of limitations and challenges associated with using gradient-based learning techniques to train a high-frequency periodic function or modular multiplication from examples. We highlight that the variance of the gradient is negligibly small in both cases when either a frequency or the prime base $p$ is large. This in turn prevents such a learning algorithm from being successful.

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