Last year I took some time off to study online convex optimization in some detail. The reason for doing that was similar to the reason why at some point I took time off to study spectral graph theory: it was coming up in several papers that I wanted to understand, and I felt that I was missing out by not mastering an important tool. In particular, I wanted to understand:

- The Barak-Hardt-Kale proof of the Impagliazzo hard-core lemma.
- The online convex optimization viewpoint on the Frieze-Kannan weak regularity lemma, on the dense model theorem of (RTTV), and on the abstract weak regularity lemma of (TTV) that were described to me by Madhur Tulsiani a few years ago. Furthermore, I wanted to see if Russel Impagliazzo’s subsequent improvements to the dense model theorem and to the abstract weak regularity lemma could be recovered from this point of view.
- The Arora-Kale algorithms for semidefinite programming, including their nearly linear-time algorithm for approximating the Goemans-Williamson relaxation of Max Cut.
- The meaning of the sentence “multiplicative weights and gradient descent are both special cases of follow-the-regularized-leader, using negative entropy and as regularizer, respectively.”
- The AllenZhu-Liao-Orecchia online optimization proof of the Batson-Spielman-Srivastava sparsification result.

I am happy to say that, except for the “furthermore” part of (2), I achieved my goals. To digest this material a bit better, I came up with the rather ambitious plan of writing a series of posts, in which I would alternate between (i) explaining a notion or theorem from online convex optimization (at a level that someone learning about optimization or machine learning might find useful) and (ii) explaining a complexity-theoretic application. Now that a very intense Spring semester is almost over, I plan to get started on this plan, although it is not clear that I will see it through the end. So stay tuned for the forthcoming first episode, which will be about the good old multiplicative weights algorithm.

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I am looking forward to your exposition. I wanted to ask if there is a paper (or series of papers) on (2)…

Russell has been working on an epic paper on (2) for a while, but I don’t think it’s publicly available yet

Glad to see you are sharing your notes on this! Looking forward.