Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size n, there always exists a coloring which achieves discrepancy 6 √ n. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is "truly" constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

We give a simple, multiplicative-weight update algorithm for learning undirected graphical models or Markov random fields (MRFs). The approach is new, and for the well-studied case of Ising models or Boltzmann machines, we obtain an algorithm that uses a nearly optimal number of samples and has running timeÕ(n 2 ) (where n is the dimension), subsuming and improving on all prior work. Additionally, we give the first efficient algorithm for learning Ising models over non-binary alphabets.Our main application is an algorithm for learning the structure of t-wise MRFs with nearlyoptimal sample complexity (up to polynomial losses in necessary terms that depend on the weights) and running time that is n O(t) . In addition, given n O(t) samples, we can also learn the parameters of the model and generate a hypothesis that is close in statistical distance to the true MRF. All prior work runs in time n Ω(d) for graphs of bounded degree d and does not generate a hypothesis close in statistical distance even for t = 3. We observe that our runtime has the correct dependence on n and t assuming the hardness of learning sparse parities with noise.Our algorithm-the Sparsitron-is easy to implement (has only one parameter) and holds in the on-line setting. Its analysis applies a regret bound from Freund and Schapire's classic Hedge algorithm. It also gives the first solution to the problem of learning sparse Generalized Linear Models (GLMs).

We study the natural question of constructing pseudorandom generators (PRGs) for lowdegree polynomial threshold functions (PTFs). We give a PRG with seed-length log n/ǫ O(d) fooling degree d PTFs with error at most ǫ. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error ǫ. For the class of degree 1 threshold functions or halfspaces, previously only PRGs with seedlength O(log n log 2 (1/ǫ)/ǫ 2 ) were known. We improve this dependence on the error parameter and construct PRGs with seedlength O(log n + log 2 (1/ǫ)) that ǫ-fool halfspaces. We also obtain PRGs with similar seed lengths for fooling halfspaces over the n-dimensional unit sphere.The main theme of our constructions and analysis is the use of invariance principles to construct pseudorandom generators. We also introduce the notion of monotone read-once branching programs, which is key to improving the dependence on the error rate ǫ for halfspaces. These techniques may be of independent interest. * A preliminary version of this work appeared in STOC 2010.

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