Contraction-based convergence proofs

Every now and then, teaching gives us some really valuable experiences. I’m teaching graduate level AI, and I had a really great experience learning something I had never paid much attention to before. Specifically, I read a section of Russell and Norvig’s book that deals with the convergence of Bellman’s utility updates for Markov decision processes.

When it comes to convergence proofs, the kind I’ve seen usually reason about the high-dimensional curvature of some objective function, or they do nasty things like analyzing the unrolled recursion tree. Yuck! For this reason, when I saw a section on a convergence proof, my immediate instinct was to run away and skip it for my and my students’ sakes.

But the convergence of Bellman’s updates can be proven using the concept of contraction, which says that an operator $f$ on two inputs $x$ and $x'$ makes them “closer,” according to some measure of distance $d$.

$d(x, x') < d(f(x), f(x'))$

This property means that repeating the operator, e.g., $f(f(f(f(x))))$ and $f(f(f(f(x'))))$, makes any two different initializations $x$ and $x'$ get closer and closer, and if the contraction is strong enough, makes them converge to the same value, giving a proof of convergence and uniqueness for the update \$\latex f\$. Beautiful, no?

The work comes in proving the contraction bound. There’s freedom in choosing the distance metric. The stronger the contraction, the faster the convergence. The Bellman updates contract by a constant factor, which I imagine is about as fast a convergence as you can get. And I imagine there can be some traps when things contract, but the amount of contraction becomes infinitesimal; in that case, this argument doesn’t actually show convergence.

I’m now curious what other convergent algorithms can be proven to converge using a contraction argument. Or more importantly, whether I can reach into my bag of algorithms-that-I-can’t-yet-prove-convergence-of and find one that I can finally solve using this approach.

I’m also curious if some type of contraction is a necessary condition for a convergent operator with a unique fixed point. I’m sure I could find that by doing some reading. But first, I have to put out a million tiny fires (this is how I describe my life now as a professor).