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A k-lift of an n-vertex base-graph G is a graph H on n × k vertices, where each vertex of G is replaced by k vertices and each edge (u, v) in G is replaced by a matching representing a bijection π uv so that the edges of H are of the form (u, i), (v, π uv (i)). H is a (uniformly) random lift of G if for every edge (u, v) the bijection π uv is chosen… (More)

We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal… (More)

First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to efficient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored due to the high cost of computing the second-order information. In this paper we develop second-order stochastic methods… (More)

- Naman Agarwal, Zeyuan Allen-Zhu, Brian Bullins, Elad Hazan, Tengyu Ma
- 2016

We design a non-convex second-order optimization algorithm that is guaranteed to return an approximate local minimum in time which is linear in the input representation. The previously fastest methods run in time proportional to matrix inversion or worse. The time complexity of our algorithm to find a local minimum is even faster than that of gradient… (More)

We design a non-convex second-order optimization algorithm that is guaranteed to return an approximate local minimum in time which is linear in the input representation. The time complexity of our algorithm to find an approximate local minimum is even faster than that of gradient descent to find a critical point. Our algorithm applies to a general class of… (More)

In this paper, we investigate the validity of the Unique Games Conjecture when the constraint graph is the boolean hypercube. We construct an almost optimal integrality gap instance on the Hypercube for the Goemans-Williamson semidefinite program (SDP) for Max-2-LIN(Z 2). We conjecture that adding triangle inequalities to the SDP provides a polynomial time… (More)

- Naman Agarwal, Karthekeyan Chandrasekaran, Alexandra Kolla, Vivek Madan
- 2015

A k-lift of an n-vertex base graph G is a graph H on n × k vertices, where each vertex v of G is replaced by k vertices v 1 , · · · , v k and each edge (u, v) in G is replaced by a matching representing a bijection π uv so that the edges of H are of the form (u i , v πuv(i)). Lifts have been studied as a means to efficiently construct expanders. In this… (More)

- Naman Agarwal
- 2013

In this survey we consider the problem of quantification of the complexity of secure two-party function evaluation in terms of the number of AND black boxes used by the protocols. We specifically prove explicit lower and upper bounds on the number of ANDs required in case of deterministic and randomized protocols.

In this report, we give a brief overview of Probabilistically Chekable proofs, and present Irit Dinur's proof of the celebrated PCP theorem. We also briefly deal with the importance of the PCP theorem in deciding hardness of approximation of various problems. 1 What are PCPs? Let us recap what we mean by NP. NP is the class of all decision problems which… (More)