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arXiv · 2309.15001

Convergence guarantees for forward gradient descent in the linear regression model

Thijs Bos·Johannes Schmidt-Hieber
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Abstract

Renewed interest in the relationship between artificial and biological neural networks motivates the study of gradient-free methods. Considering the linear regression model with random design, we theoretically analyze in this work the biologically motivated (weight-perturbed) forward gradient scheme that is based on random linear combination of the gradient. If d denotes the number of parameters and k the number of samples, we prove that the mean squared error of this method converges for $k\gtrsim d^2\log(d)$ with rate $d^2\log(d)/k.$ Compared to the dimension dependence d for stochastic gradient descent, an additional factor $d\log(d)$ occurs.

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Thijs Bos, Johannes Schmidt-Hieber. 2023-09-26. Convergence guarantees for forward gradient descent in the linear regression model. https://doi.org/10.1016/j.jspi.2024.106174

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