# yet more not-so-recent hits from ArXiV

Some shorter takes on these papers, some of which I should read in more detail later. I figure I’ll use the blog for some quick notes and to see if any readers have any comments/ideas about these:

Differentially Private Convex Optimization with Piecewise Affine Objectives (Shuo Han, Ufuk Topcu, George J. Pappas) — arXiv:1403.6135 [math.OC]. The idea here is to look at minimizing functions of the form
$f(x) = \max_{i = 1,2, \ldots, m} \{ a_i^{\top} x + b_i \}$
subject to $x$ belonging to some convex polytope $\mathcal{P}$. This is a bit different than the kind of convex programs I’ve been looking at (which are more ERM-like). Such programs occur often in resource allocation problems. Here the private information of users are the offsets $b_i$. They propose a number of methods for generating differentially private approximations to this problem. Analyzing the sensitivity of this optimization is tricky, so they use an upper bound based on the diameter of the feasible set $\mathcal{P}$ to find an appropriate noise variance. The exponential mechanism also gives a feasible mechanism, although the exact dependence of the suboptimality gap on $\epsilon$ is unclear. They also propose a noisy subgradient method where, instead of using SGD, they alter the sampling distribution using the exponential mechanism to choose a gradient step. Some preliminary experiments are also given (although none exploring the dependence on $\epsilon$, which would also be very interesting)!

Assisted Common Information with an Application to Secure Two-Party Sampling (Vinod M. Prabhakaran, Manoj M. Prabhakaran) — arXiv:1206.1282 [cs.IT]. This is the final version of the journal version of a few conference papers that Vinod and Manoj have done on an interesting variant of the Gács-Körner problem. The motivation is from secure multiparty computation — the problem also touches on some work Vinod and I started but is sadly languishing due to the utter overwhelmingness of starting a new job. Hopefully I can get back to it this summer.

Analysis of Distributed Stochastic Dual Coordinate Ascent (Tianbao Yang, Shenghuo Zhu, Rong Jin, Yuanqing Lin) — arXiv:1312.1031 [cs.DC]. The title pretty much sums it up. I’m interested in looking a bit more at the analysis method, since I had a similar algorithm bouncing around my head that I would like to analyze. The main idea is also update the primal variables to achieve a speedup/use a larger step size.

Convergence of Stochastic Proximal Gradient Algorithm (Lorenzo Rosasco, Silvia Villa, Bang Công Vũ) — arXiv:1403.5074 [math.OC]. This is a similar setup as my last post, with a convex objective that has a smooth and non-smooth component. They show convergence in expectation and almost surely. The key here is that they show convergence in an infinite-dimentionsal Hilbert space instead of, say, $\mathbb{R}^d$.