# ArXiV notes : July 2012 part 2

Differentially Private Filtering and Differentially Private Kalman Filtering
Jerome Le Ny, George J. Pappas
These papers apply the differential privacy model to classical signal processing and look at the effect of adding noise before filtering and summing versus adding noise after filtering and summing. The key is that the operations have to be causal, I think — we usually think about differential privacy as operating offline or online in very restricted settings, but here the signals are coming in as time series.

Finite sample posterior concentration in high-dimensional regression
Nate Strawn, Artin Armagan, Rayan Saab, Lawrence Carin, David Dunson
They study ultra high-dimensional linear regression (cue guitars and reverb) in the “large p, small n” setting. The goal is to get a Bernstein-von Mises type theorem — e.g. assuming the data comes from a linear model $\beta_0$, then using Bayesian inference the posterior should concentrate around $\beta_0$. They of course need a sparsity assumption, and the prior must assign reasonable mass around the true parameter and assign low probability to non-sparse signals. The methods use some ideas from compressed sensing (the Dantzig selector) and should be of interest to people working in that area.

Identifying Users From Their Rating Patterns
José Bento, Nadia Fawaz, Andrea Montanari, and Stratis Ioannidis
This is a paper about recommender systems as part of the 2011 CAMRa challenge. They look at the problem of re-identification of users in this data and show that looking at the time stamps of movie ratings is much more useful than looking at the rating values. This suggests to me that people should use a masking system like Anant and Parv’s “Incentivizing Anonymous ‘Peer-to-Peer’ Reviews” (P. Venkitasubramaniam and A. Sahai, Allerton 2008) paper.

Michael Kearns, Mallesh M. Pai, Aaron Roth, Jonathan Ullman
This proposes a variant of differential privacy which they call joint differential privacy and look at mechanism designs that satisfy privacy and are incentive compatible. At first glance, these should be incompatible, since the latter implies “revealing the truth.” The model is one in which each agent has a finite set of actions but its own payoff/value is private. This is somewhat out of my area, so I can’t really get the nuances of what’s going on here, but a crucial assumption here is that there are a large number of agents. Joint differential privacy seems to be a form of $(\epsilon,\delta)$ differential privacy on the utility functions of the users.

# Strategies in the Iterated Prisoner’s Dilemma

Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent
William H. Press and Freeman J. Dyson
PNAS 109(26), June 26, 2012

This was an interesting little paper that I learned about that I am still digesting; as I’ve said before, I’m not a game theory guy. They start out with a 2×2 game, the classical Prisoner’s Dilemma with players X and Y — if both cooperate (C), they payoffs are $(R,R)$, if the first (resp. second) defects (D) they are $(T,S)$ (resp. $(S,T)$) and if they both defect then $(P,P)$. The usual conditions are $T > R > P > S$ to keep it interesting.

In an iterated game the players get to play over and over again and try to develop strategies to win more than their opponent. The first interesting (to me) observation in this paper is that strategies which maintain a long memory are dominated by those which maintain a short memory in the sense that if for any long-memory strategy of Y, the payoff for a short-memory strategy of X is the same as a certain short-memory strategy of Y. So if your opponent is playing a short-memory strategy you should also play a similar short-memory strategy.

So with that idea in mind they look at first-order Markov strategies — given the past state (CC, CD, DC, DD) player X randomly chooses whether to cooperate or not in the next round. Thus the problem is specified by 4 conditional probabilities of cooperating conditioned on the previous state, and you can compute a 4×4 Markov matrix of transition probabilities out of the randomized strategies of each player.

Now the game becomes how they payoffs behave under different choices of strategy, which boil down to spectral / structural properties of this Markov matrix. To be honest, I found the discussion in the rest of the paper more confusing than enlightening because the mathematics was a little obscured by the discussion of implications. I realize this is a PNAS paper but it felt a little discursive at times and the mathematical intuition was not entirely clear to me.

Still, it’s a short read and probably worth a peruse.