64 research outputs found
Structure learning of antiferromagnetic Ising models
In this paper we investigate the computational complexity of learning the
graph structure underlying a discrete undirected graphical model from i.i.d.
samples. We first observe that the notoriously difficult problem of learning
parities with noise can be captured as a special case of learning graphical
models. This leads to an unconditional computational lower bound of for learning general graphical models on nodes of maximum degree
, for the class of so-called statistical algorithms recently introduced by
Feldman et al (2013). The lower bound suggests that the runtime
required to exhaustively search over neighborhoods cannot be significantly
improved without restricting the class of models.
Aside from structural assumptions on the graph such as it being a tree,
hypertree, tree-like, etc., many recent papers on structure learning assume
that the model has the correlation decay property. Indeed, focusing on
ferromagnetic Ising models, Bento and Montanari (2009) showed that all known
low-complexity algorithms fail to learn simple graphs when the interaction
strength exceeds a number related to the correlation decay threshold. Our
second set of results gives a class of repelling (antiferromagnetic) models
that have the opposite behavior: very strong interaction allows efficient
learning in time . We provide an algorithm whose performance
interpolates between and depending on the strength of the
repulsion.Comment: 15 pages. NIPS 201
Label optimal regret bounds for online local learning
We resolve an open question from (Christiano, 2014b) posed in COLT'14
regarding the optimal dependency of the regret achievable for online local
learning on the size of the label set. In this framework the algorithm is shown
a pair of items at each step, chosen from a set of items. The learner then
predicts a label for each item, from a label set of size and receives a
real valued payoff. This is a natural framework which captures many interesting
scenarios such as collaborative filtering, online gambling, and online max cut
among others. (Christiano, 2014a) designed an efficient online learning
algorithm for this problem achieving a regret of , where
is the number of rounds. Information theoretically, one can achieve a regret of
. One of the main open questions left in this framework
concerns closing the above gap.
In this work, we provide a complete answer to the question above via two main
results. We show, via a tighter analysis, that the semi-definite programming
based algorithm of (Christiano, 2014a), in fact achieves a regret of
. Second, we show a matching computational lower bound. Namely,
we show that a polynomial time algorithm for online local learning with lower
regret would imply a polynomial time algorithm for the planted clique problem
which is widely believed to be hard. We prove a similar hardness result under a
related conjecture concerning planted dense subgraphs that we put forth. Unlike
planted clique, the planted dense subgraph problem does not have any known
quasi-polynomial time algorithms.
Computational lower bounds for online learning are relatively rare, and we
hope that the ideas developed in this work will lead to lower bounds for other
online learning scenarios as well.Comment: 13 pages; Changes from previous version: small changes to proofs of
Theorems 1 & 2, a small rewrite of introduction as well (this version is the
same as camera-ready copy in COLT '15
Fine-grained Search Space Classification for Hard Enumeration Variants of Subset Problems
We propose a simple, powerful, and flexible machine learning framework for
(i) reducing the search space of computationally difficult enumeration variants
of subset problems and (ii) augmenting existing state-of-the-art solvers with
informative cues arising from the input distribution. We instantiate our
framework for the problem of listing all maximum cliques in a graph, a central
problem in network analysis, data mining, and computational biology. We
demonstrate the practicality of our approach on real-world networks with
millions of vertices and edges by not only retaining all optimal solutions, but
also aggressively pruning the input instance size resulting in several fold
speedups of state-of-the-art algorithms. Finally, we explore the limits of
scalability and robustness of our proposed framework, suggesting that
supervised learning is viable for tackling NP-hard problems in practice.Comment: AAAI 201
Average-case Hardness of RIP Certification
The restricted isometry property (RIP) for design matrices gives guarantees
for optimal recovery in sparse linear models. It is of high interest in
compressed sensing and statistical learning. This property is particularly
important for computationally efficient recovery methods. As a consequence,
even though it is in general NP-hard to check that RIP holds, there have been
substantial efforts to find tractable proxies for it. These would allow the
construction of RIP matrices and the polynomial-time verification of RIP given
an arbitrary matrix. We consider the framework of average-case certifiers, that
never wrongly declare that a matrix is RIP, while being often correct for
random instances. While there are such functions which are tractable in a
suboptimal parameter regime, we show that this is a computationally hard task
in any better regime. Our results are based on a new, weaker assumption on the
problem of detecting dense subgraphs
- …