Daniel Štefankovič

Antonio Blanca, Zongchen Chen, Daniel Štefankovič et al.

We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution $μ$, an $\varepsilon>0$, and access to sampling oracle(s) for a hidden distribution $π$, the goal in identity testing is to distinguish whether the two distributions $μ$ and $π$ are identical or are at least $\varepsilon$-far apart. When there is only access to full samples from the hidden distribution $π$, it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various "conditional" sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the $\mathsf{Coordinate\ Oracle}$, and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an $n$-dimensional visible distribution $μ$, then there is an efficient identity testing algorithm for any hidden distribution $π$ using $\tilde{O}(n/\varepsilon)$ queries to the $\mathsf{Coordinate\ Oracle}$. Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of $n$-dimensional distributions, specifically sparse antiferromagnetic Ising models over $\{+1,-1\}^n$, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless $\mathsf{RP}=\mathsf{NP}$. We complement our results with a matching $Ω(n/\varepsilon)$ statistical lower bound for the sample complexity of identity testing in the $\mathsf{Coordinate\ Oracle}$ model.

Rupam Acharyya, Ankani Chattoraj, Boyu Zhang et al.

Deep learning architectures with a huge number of parameters are often compressed using pruning techniques to ensure computational efficiency of inference during deployment. Despite multitude of empirical advances, there is a lack of theoretical understanding of the effectiveness of different pruning methods. We inspect different pruning techniques under the statistical mechanics formulation of a teacher-student framework and derive their generalization error (GE) bounds. It has been shown that Determinantal Point Process (DPP) based node pruning method is notably superior to competing approaches when tested on real datasets. Using GE bounds in the aforementioned setup we provide theoretical guarantees for their empirical observations. Another consistent finding in literature is that sparse neural networks (edge pruned) generalize better than dense neural networks (node pruned) for a fixed number of parameters. We use our theoretical setup to prove this finding and show that even the baseline random edge pruning method performs better than the DPP node pruning method. We also validate this empirically on real datasets.

Antonio Blanca, Zongchen Chen, Daniel Štefankovič et al.

We study identity testing for restricted Boltzmann machines (RBMs), and more generally for undirected graphical models. Given sample access to the Gibbs distribution corresponding to an unknown or hidden model $M^*$ and given an explicit model $M$, can we distinguish if either $M = M^*$ or if they are (statistically) far apart? Daskalakis et al. (2018) presented a polynomial-time algorithm for identity testing for the ferromagnetic (attractive) Ising model. In contrast, for the antiferromagnetic (repulsive) Ising model, Bezáková et al. (2019) proved that unless $RP=NP$ there is no identity testing algorithm when $βd=ω(\log{n})$, where $d$ is the maximum degree of the visible graph and $β$ is the largest edge weight in absolute value. We prove analogous hardness results for RBMs (i.e., mixed Ising models on bipartite graphs), even when there are no latent variables or an external field. Specifically, we show that if $RP \neq NP$, then when $βd=ω(\log{n})$ there is no polynomial-time algorithm for identity testing for RBMs; when $βd =O(\log{n})$ there is an efficient identity testing algorithm that utilizes the structure learning algorithm of Klivans and Meka (2017). In addition, we prove similar lower bounds for purely ferromagnetic RBMs with inconsistent external fields, and for the ferromagnetic Potts model. Previous hardness results for identity testing of Bezáková et al. (2019) utilized the hardness of finding the maximum cuts, which corresponds to the ground states of the antiferromagnetic Ising model. Since RBMs are on bipartite graphs such an approach is not feasible. We instead introduce a general methodology to reduce from the corresponding approximate counting problem and utilize the phase transition that is exhibited by RBMs and the mean-field Potts model.

Ivona Bezakova, Antonio Blanca, Zongchen Chen et al.

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.

Antonio Blanca, Zongchen Chen, Daniel Štefankovič et al.

We study the structure learning problem for $H$-colorings, an important class of Markov random fields that capture key combinatorial structures on graphs, including proper colorings and independent sets, as well as spin systems from statistical physics. The learning problem is as follows: for a fixed (and known) constraint graph $H$ with $q$ colors and an unknown graph $G=(V,E)$ with $n$ vertices, given uniformly random $H$-colorings of $G$, how many samples are required to learn the edges of the unknown graph $G$? We give a characterization of $H$ for which the problem is identifiable for every $G$, i.e., we can learn $G$ with an infinite number of samples. We also show that there are identifiable constraint graphs for which one cannot hope to learn every graph $G$ efficiently. We focus particular attention on the case of proper vertex $q$-colorings of graphs of maximum degree $d$ where intriguing connections to statistical physics phase transitions appear. We prove that in the tree uniqueness region (when $q>d$) the problem is identifiable and we can learn $G$ in ${\rm poly}(d,q) \times O(n^2\log{n})$ time. In contrast for soft-constraint systems, such as the Ising model, the best possible running time is exponential in $d$. In the tree non-uniqueness region (when $q\leq d$) we prove that the problem is not identifiable and thus $G$ cannot be learned. Moreover, when $q<d-\sqrt{d} + Θ(1)$ we prove that even learning an equivalent graph (any graph with the same set of $H$-colorings) is computationally hard---sample complexity is exponential in $n$ in the worst case. We further explore the connection between the efficiency/hardness of the structure learning problem and the uniqueness/non-uniqueness phase transition for general $H$-colorings and prove that under the well-known Dobrushin uniqueness condition, we can learn $G$ in ${\rm poly}(d,q)\times O(n^2\log{n})$ time.

Sejun Park, Yunhun Jang, Andreas Galanis et al.

The Gibbs sampler is a particularly popular Markov chain used for learning and inference problems in Graphical Models (GMs). These tasks are computationally intractable in general, and the Gibbs sampler often suffers from slow mixing. In this paper, we study the Swendsen-Wang dynamics which is a more sophisticated Markov chain designed to overcome bottlenecks that impede the Gibbs sampler. We prove O(\log n) mixing time for attractive binary pairwise GMs (i.e., ferromagnetic Ising models) on stochastic partitioned graphs having n vertices, under some mild conditions, including low temperature regions where the Gibbs sampler provably mixes exponentially slow. Our experiments also confirm that the Swendsen-Wang sampler significantly outperforms the Gibbs sampler when they are used for learning parameters of attractive GMs.

Haichuan Yang, Shupeng Gui, Chuyang Ke et al.

The cardinality constraint is an intrinsic way to restrict the solution structure in many domains, for example, sparse learning, feature selection, and compressed sensing. To solve a cardinality constrained problem, the key challenge is to solve the projection onto the cardinality constraint set, which is NP-hard in general when there exist multiple overlapped cardinality constraints. In this paper, we consider the scenario where the overlapped cardinality constraints satisfy a Three-view Cardinality Structure (TVCS), which reflects the natural restriction in many applications, such as identification of gene regulatory networks and task-worker assignment problem. We cast the projection into a linear programming, and show that for TVCS, the vertex solution of this linear programming is the solution for the original projection problem. We further prove that such solution can be found with the complexity proportional to the number of variables and constraints. We finally use synthetic experiments and two interesting applications in bioinformatics and crowdsourcing to validate the proposed TVCS model and method.

Satyaki Mahalanabis, Daniel Stefankovic

Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, called Gaussian free fields, which arise in semi-supervised learning and computer vision. We then give a simple greedy approximation algorithm for Gaussian free fields on arbitrary graphs. Finally, we give a message passing algorithm for general Gaussian Markov random fields on bounded tree-width graphs.