We review the proof by Robbins and Munro for finding fixed points. Stochastic gradient descent, Q-learning and a bunch of other stochastic algorithms can be seen as variants of this basic algorithm. We review the basic ingredients of the original proof.

Often it is important to find a solution to the equation

by evaluating g at a sequence of points. For instance Newton’s method would perform the updates x_{n+1} = x_n - g(x_n)/g'(x_n). However, Robbins and Munro consider the setting where we cannot directly observe g but we might observe some random variable whose mean is g(x). Thus we observe

and hope solve for g(x) = 0. Notice in this setting, even if we can find g'(x), Newton’s method may not converge. The key idea of Robbins and Munro is to use a schema

where we chose the sequence \{\alpha_n\}_{n=0}^\infty so that

Before proceeding here are a few different use cases:

  • Quartiles. We want to find x such that P(X \leq x ) = p for some fixed p. But we can only sample the random variable X.
  • Regression. We preform regression g(x) = \beta_0 + \beta_1 x, but rather than estimate \beta we want to know where g(x)=0.
  • Optimization. We want to optimize a convex function f(x) whose gradient is g(x). Assume that f(x) = \sum_{k=1}^K f_k(x) for some large K. To find the optimum at g(x)=0 we randomly sample (uniformly) f_k(x) whose gradient, g_k(x), is an bias estimate of g(x).

Robbins-Munro Proof.

We now go through the main steps of Robbins and Munro’s proof. This proof is intentionally not the most general but instead gives the key ideas.

We assume that g(x) is an increasing, smooth, real-valued function from [x_{\min},x_{\max}] to [y_{\min},y_{\max}] and that there is a unique point x^* such that g(x^*)=0. Further assume g'(x^*) \neq 0 and that samples y_n= g(x_n)+\epsilon_n belong to the interval [y_{\min},y_{\max}] and \mathbb E [\epsilon_n] = 0 .

In the steps below we prove the following:

Thrm [Robbins-Munro] If we chose x_n acording to the Robbin-Munro step rule then we have that

where here x^* satisfies g(x^*)=0.

Ex 1. Let b_n = \mathbb E [ (x_n-x^*)^2] . Show that for Robbins-Munro

for terms


Ans 1.

as required.

Ex 2. Argue that d_n \geq \kappa b_n for some constant \kappa > 0.

Ans 2. Since g(x) is increasing, smooth and g'(x^*)>0 it must be that


Ex 3. Argue that if \sum_n \alpha^2_n <\infty then \sum_n \alpha_n d_n < \infty and \lim_{n\rightarrow\infty} b_n exists.

Ans 3. Summing the expression for [[RM:1]] over n gives

Since e_n is bounded and \sum_n a^2_n is finite it must be that \sum_n \alpha_n d_n < \infty and \lim_{n\rightarrow\infty} b_n exists.

Ex 4. Finally argue that if \sum_n a_n = \infty then

Ans 4. We know from [3] and [3] that

Since we choose \{ a_n\} so that \sum_n a_n = \infty . It must be that \liminf_{n\rightarrow \infty} b_n = 0. Further since we know the limit \lim_{n\rightarrow \infty} b_n exists it must be that \lim_{n\rightarrow \infty} b_n = 0, as required.

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