## Information complexity and applications

I have previously here on the blog posted multiple lectures in my ‘lecture-posts’, or I have combined a lecture with other stuff (e.g. links such as those in the previous ‘random stuff’ post). I think such approaches have made me less likely to post lectures on the blog (if I don’t post a lecture soon after I’ve watched it, my experience tells me that I not infrequently simply never get around to posting it), and combined with this issue is also the issue that I don’t really watch a lot of lectures these days. For these reasons I have decided to start posting single lecture posts here on the blog; when I start thinking about the time expenditure of people reading along here in a way this approach actually also seems justified – although it might take *me* as much time/work to watch and cover, say, 4 lectures as it would take me to read and cover 100 pages of a textbook, the time expenditure required by a reader of the blog would be very different in those two cases (you’ll usually be able to read a post that took me multiple hours to write in a short amount of time, whereas ‘the time advantage’ of the reader is close to negligible (maybe not completely; search costs are not completely irrelevant) in the case of lectures). By posting multiple lectures in the same post I probably decrease the expected value of the time readers spend watching the content I upload, which seems suboptimal.

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Here’s the youtube description of the lecture, which was posted a few days ago on the IAS youtube account:

“Over the past two decades, information theory has reemerged within computational complexity theory as a mathematical tool for obtaining unconditional lower bounds in a number of models, including streaming algorithms, data structures, and communication complexity. Many of these applications can be systematized and extended via the study of information complexity – which treats information revealed or transmitted as the resource to be conserved. In this overview talk we will discuss the two-party information complexity and its properties – and the interactive analogues of classical source coding theorems. We will then discuss applications to exact communication complexity bounds, hardness amplification, and quantum communication complexity.”

He actually decided to skip the quantum communication complexity stuff because of the time constraint. I should note that the lecture was ‘easy enough’ for me to follow most of it, so it is not really *that* difficult, at least not if you know some basic information theory.

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A few links to related stuff (you can take these links as indications of what sort of stuff the lecture is about/discusses, if you’re on the fence about whether or not to watch it):

Computational complexity theory.

Shannon entropy.

Shannon’s source coding theorem.

Communication complexity.

Communications protocol.

Information-based complexity.

Hash function.

From Information to Exact Communication (in the lecture he discusses some aspects covered in this paper).

Unique games conjecture (Two-prover proof systems).

A Counterexample to Strong Parallel Repetition (another paper mentioned/briefly discussed during the lecture).

Pinsker’s inequality.

An interesting aspect I once again noted during this lecture is the sort of loose linkage you sometimes observe between the topics of game theory/microeconomics and computer science. Of course the link is made explicit a few minutes later in the talk when he discusses the unique games conjecture to which I link above, but it’s perhaps worth noting that the link is on display even before that point is reached. Around 38 minutes into the lecture he mentions that one of the relevant proofs ‘involves such things as Lagrange multipliers and optimization’. I was far from surprised, as from a certain point of view the problem he discusses at that point is conceptually very similar to some problems encountered in auction theory, where Lagrange multipliers and optimization problems are frequently encountered… If you are too unfamiliar with that field to realize how the similar problem might appear in an auction theory context, what you have there are instead auction partipants who prefer not to reveal their true willingness to pay; and some auction designs actually work in a very similar manner as does the (pseudo-)protocol described in the lecture, and are thus used to reveal it (for some subset of participants at least)).