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Revision #1 to TR12-064 | 21st March 2013 21:51

Statistical Algorithms and a Lower Bound for Detecting Planted Cliques

Revision #1
Authors: Vitaly Feldman, Elena Grigorescu, Lev Reyzin, Santosh Vempala, Ying Xiao
Accepted on: 21st March 2013 21:51
Keywords:

Abstract:

We introduce a framework for proving lower bounds on computational problems over distributions, based on a class of algorithms called statistical algorithms. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather than directly accessing samples.
Most natural algorithms of interest in theory and in practice, e.g., moments-based methods, local search, standard iterative methods for convex optimization, MCMC and simulated annealing, are statistical algorithms or have statistical counterparts. Our framework is inspired by and generalizes the statistical query model in learning theory [Kearns 1998].

Our main application is a nearly optimal lower bound on the complexity of any statistical algorithm for detecting planted bipartite clique distributions (or planted dense subgraph distributions) when the planted clique has size $O(n^{1/2-\delta})$ for any constant $\delta > 0$. Variants of these problems have been assumed to be hard to prove hardness for other problems and for cryptographic applications. Our lower bounds provide concrete evidence of hardness, thus supporting these assumptions.

Changes to previous version:

This is a major revision of the paper:
1) introduction and overview were substantially revised
2) main lower bounds strengthened
3) new results added
4) a number of minor bugs fixed

Paper:

TR12-064 | 10th May 2012 04:19

Statistical Algorithms and a Lower Bound for Planted Clique

TR12-064
Authors: Vitaly Feldman, Elena Grigorescu, Lev Reyzin, Santosh Vempala, Ying Xiao
Publication: 23rd May 2012 00:50
For specific well-known problems over distributions, we give lower bounds on the complexity of any statistical algorithm. These include an exponential lower bounds for moment maximization in $R^n$, and a nearly optimal lower bound for detecting planted clique distributions when the planted clique has size $O(n^{1/2-\delta})$ for any constant $\delta > 0$. Variants of the latter problem have been assumed to be hard to prove hardness for other problems and for cryptographic applications. Our lower bounds provide concrete evidence supporting these assumptions.