Convergence guarantees for forward gradient descent in the linear regression model

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 $k\gtrsim d^{2}log\left(d\right)$ with rate $d^{2}log\left(d\right)/k$. Compared to the dimension dependence $d$ for stochastic gradient descent, an additional factor $dlog\left(d\right)$ occurs.