[Paper Review] Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

Ivona Bezáková, Antonio Blanca+3 more|arXiv (Cornell University)|Jan 22, 2019

Machine Learning and Algorithms53 references4 citations

TL;DR

This paper establishes computational lower bounds for identity testing in graphical models, showing that testing identity is as hard as structure learning in antiferromagnetic Ising models and proper colorings. Under the condition that |β|d = ω(log n), identity testing is NP-hard unless RP=NP, and similar hardness holds for colorings under the #BIS assumption, indicating no significant computational advantage over structure learning in these settings.

ABSTRACT

We study the identity testing problem in the context of spin systems or undirected graphical models, where it takes the following form: given the parameter specification of the model $M$ and a sampling oracle for the distribution $μ_{\hat{M}}$ of an unknown model $\hat{M}$, can we efficiently determine if the two models $M$ and $\hat{M}$ are the same? We consider identity testing for both soft-constraint and hard-constraint systems. In particular, we prove hardness results in two prototypical cases, the Ising model and proper colorings, and explore whether identity testing is any easier than structure learning. For the ferromagnetic (attractive) Ising model, Daskalakis et al. (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples. In particular, for $n$-vertex graphs of maximum degree $d$, we prove that if $|β| d = ω(\log{n})$ (where $β$ is the inverse temperature parameter), then there is no polynomial running time identity testing algorithm unless $RP=NP$. We also establish computational lower bounds for a broader set of parameters under the (randomized) exponential time hypothesis. Our proofs utilize insights into the design of gadgets using random graphs in recent works concerning the hardness of approximate counting by Sly (2010). In the hard-constraint setting, we present hardness results for identity testing for proper colorings. Our results are based on the presumed hardness of #BIS, the problem of (approximately) counting independent sets in bipartite graphs. In particular, we prove that identity testing is hard in the same range of parameters where structure learning is known to be hard.

Motivation & Objective

• To investigate whether identity testing in graphical models is computationally easier than structure learning.
• To establish hardness results for identity testing in antiferromagnetic Ising models under parameter regimes where structure learning is known to be hard.
• To extend these hardness results to hard-constraint systems, particularly proper 3-colorings, under the #BIS assumption.
• To determine whether identity testing remains hard even when the true model’s parameters are in a regime where structure learning requires super-polynomial samples.
• To explore the role of the inverse temperature β and graph degree d in determining the complexity of identity testing.

Proposed method

• Prove computational lower bounds for identity testing in antiferromagnetic Ising models using gadgets based on random graphs, inspired by Sly (2010).
• Use the partition function and total variation distance to compare distributions of two models, deriving bounds via configuration counts.
• Construct a family of graphs with phase vectors to analyze the number of valid configurations under different parameter settings.
• Leverage symmetry and known results on the number of proper 3-colorings to derive bounds on |Ω^A| and |Ω^B|, the sets of colorings with specific phase patterns.
• Apply the randomized exponential time hypothesis to extend hardness results beyond the NP-hardness regime.
• Use sampling oracles and phase vector sampling techniques to simulate distributions and test identity in the coloring model.

Experimental results

Research questions

• RQ1 Is identity testing in antiferromagnetic Ising models computationally easier than structure learning?
• RQ2 Under what parameter regimes is identity testing NP-hard for antiferromagnetic Ising models?
• RQ3 Does the hardness of identity testing in proper colorings match the hardness of structure learning under the #BIS assumption?
• RQ4 Can identity testing be solved in polynomial time when |β|d = ω(log n), even if structure learning requires super-polynomial samples?
• RQ5 What is the role of the inverse temperature β and maximum degree d in determining the complexity of identity testing?

Key findings

• Identity testing for antiferromagnetic Ising models is NP-hard unless RP=NP when |β|d = ω(log n), matching the parameter regime where structure learning requires super-polynomial samples.
• For proper 3-colorings, identity testing is hard under the #BIS assumption in the same parameter regime where structure learning is known to be hard.
• The total variation distance between two models’ distributions is bounded by a function of the partition function ratio, enabling hardness proofs via configuration counting.
• When Z₃(H) < 2^{ℓ/k} - 2, the total variation distance is at most 2((Z₃(H)+2)/2^{ℓ/k})^k, which decays exponentially if Z₃(H) is small.
• When Z₃(H) ≥ 2^{ℓ/k} - 2, the total variation distance is bounded below by 1/2(1 - ((Z₃(B)+2)/(Z₃(H)+2))^k), showing a non-trivial separation when Z₃(H) is large.
• The sampling algorithm for the coloring model runs in O(n) time, and the phase vector sampling method enables efficient simulation of the target distribution under the oracle model.