Gradient descent, first-order, stochastic

a.k.a. SGD, as seen in deep learning


Stochastic optimization, uses noisy (possibly approximate) 1st-order gradient information to find the argument which minimises

\[ x^*=\operatorname{argmin}_{\mathbf{x}} f(x) \]

for some an objective function \(f:\mathbb{R}^n\to\mathbb{R}\).

That this works with little fuss in very high dimensions is a major pillar of deep learning.

The original version, given in terms of root finding, is (Robbins and Monro 1951) who later generalised analysis in (Robbins and Siegmund 1971), using martingale arguments to analyze convergence. There is some historical context in (Lai 2003) which puts it all in context. That article was written before the current craze for SGD in deep learning; after 2013 or so the problem is rather that there is so much information on the method that the challenge becomes sifting out the AI hype from the useful.

I recommend Francis Bach’s Sum of geometric series trick as an introduction to showing things advanced things about SGD using elementary tools.

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