arXiv subjects

Elchanan Mossel

Publications and source records attributed to Elchanan Mossel.

At least 109 records · Page 6Linked to original sources

arXiv2019

Bayesian Decision Making in Groups is Hard

We study the computations that Bayesian agents undertake when exchanging opinions over a network. The agents act repeatedly on their private information and take myopic actions that maximize their expected utility according to a fully rational posterior belief. We show that such computations are NP-hard for two natural utility functions: one with binary actions, and another where agents reveal their posterior beliefs. In fact, we show that distinguishing between posteriors that are concentrated on different states of the world is NP-hard. Therefore, even approximating the Bayesian posterior beliefs is hard. We also describe a natural search algorithm to compute agents' actions, which we call elimination of impossible signals, and show that if the network is transitive, the algorithm can be modified to run in polynomial time.

math.ST↗

arXiv2019

Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation

The analysis of Belief Propagation and other algorithms for the {\em reconstruction problem} plays a key role in the analysis of community detection in inference on graphs, phylogenetic reconstruction in bioinformatics, and the cavity method in statistical physics. We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) which states that any bounded memory message passing algorithm is statistically much weaker than Belief Propagation for the reconstruction problem. More formally, any recursive algorithm with bounded memory for the reconstruction problem on the trees with the binary symmetric channel has a phase transition strictly below the Belief Propagation threshold, also known as the Kesten-Stigum bound. The proof combines in novel fashion tools from recursive reconstruction, information theory, and optimal transport, and also establishes an asymptotic normality result for BP and other message-passing algorithms near the critical threshold.

cs.IT↗

arXiv2019

How Many Subpopulations is Too Many? Exponential Lower Bounds for Inferring Population Histories

Reconstruction of population histories is a central problem in population genetics. Existing coalescent-based methods, like the seminal work of Li and Durbin (Nature, 2011), attempt to solve this problem using sequence data but have no rigorous guarantees. Determining the amount of data needed to correctly reconstruct population histories is a major challenge. Using a variety of tools from information theory, the theory of extremal polynomials, and approximation theory, we prove new sharp information-theoretic lower bounds on the problem of reconstructing population structure -- the history of multiple subpopulations that merge, split and change sizes over time. Our lower bounds are exponential in the number of subpopulations, even when reconstructing recent histories. We demonstrate the sharpness of our lower bounds by providing algorithms for distinguishing and learning population histories with matching dependence on the number of subpopulations. Along the way and of independent interest, we essentially determine the optimal number of samples needed to learn an exponential mixture distribution information-theoretically, proving the upper bound by analyzing natural (and efficient) algorithms for this problem.

q-bio.PE↗

arXiv2019

Junta correlation is testable

The problem of tolerant junta testing is a natural and challenging problem which asks if the property of a function having some specified correlation with a $k$-Junta is testable. In this paper we give an affirmative answer to this question: We show that given distance parameters $\frac{1}{2} >c_u>c_{\ell} \ge 0$, there is a tester which given oracle access to $f:\{-1,1\}^n \rightarrow \{-1,1\}$, with query complexity $ 2^k \cdot \mathsf{poly}(k,1/|c_u-c_{\ell}|)$ and distinguishes between the following cases: $\mathbf{1.}$ The distance of $f$ from any $k$-junta is at least $c_u$; $\mathbf{2.}$ There is a $k$-junta $g$ which has distance at most $c_\ell$ from $f$. This is the first non-trivial tester (i.e., query complexity is independent of $n$) which works for all $1/2 > c_u > c_\ell \ge 0$. The best previously known results by Blais \emph{et~ al.}, required $c_u \ge 16 c_\ell$. In fact, with the same query complexity, we accomplish the stronger goal of identifying the most correlated $k$-junta, up to permutations of the coordinates. We can further improve the query complexity to $\mathsf{poly}(k, 1/|c_u-c_{\ell}|)$ for the (weaker) task of distinguishing between the following cases: $\mathbf{1.}$ The distance of $f$ from any $k'$-junta is at least $c_u$. $\mathbf{2.}$ There is a $k$-junta $g$ which is at a distance at most $c_\ell$ from $f$. Here $k'=O(k^2/|c_u-c_\ell|)$. Our main tools are Fourier analysis based algorithms that simulate oracle access to influential coordinates of functions.

cs.CC↗

arXiv2019

Harmonicity and invariance on slices of the Boolean cube

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences. We show that any real-valued function on the slice, whose degree when written as a harmonic multi-linear polynomial is $o(\sqrt{n})$, has approximately the same distribution under the slice and cube measure. Our proof is based on a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, ours decomposition is applied in a non-reversible non-stationary setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the facts that 1) one cannot distinguish between the slice and the cube based on functions of $o(n)$ coordinates and 2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is $o(\sqrt{n})$.

math.PR↗

arXiv2018

Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics

There is a common theme to some research questions in additive combinatorics and noise stability. Both study the following basic question: Let $\mathcal{P}$ be a probability distribution over a space $Ω^\ell$ with all $\ell$ marginals equal. Let $\underline{X}^{(1)}, \ldots, \underline{X}^{(\ell)}$ where $\underline{X}^{(j)} = (X_1^{(j)}, \ldots, X_n^{(j)})$ be random vectors such that for every coordinate $i \in [n]$ the tuples $(X_i^{(1)}, \ldots, X_i^{(\ell)})$ are i.i.d. according to $\mathcal{P}$. A central question that is addressed in both areas is: - Does there exist a function $c_{\mathcal{P}}()$ independent of $n$ such that for every $f: Ω^n \to [0, 1]$ with $\mathrm{E}[f(X^{(1)})] = μ> 0$: \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(μ) > 0 \, ? \end{align*} Instances of this question include the finite field model version of Roth's and Szemerédi's theorems as well as Borell's result about the optimality of noise stability of half-spaces. Our goal in this paper is to interpolate between the noise stability theory and the finite field additive combinatorics theory and address the question above in further generality than considered before. In particular, we settle the question for $\ell = 2$ and when $\ell > 2$ and $\mathcal{P}$ has bounded correlation $ρ(\mathcal{P}) < 1$. Under the same conditions we also characterize the _obstructions_ for similar lower bounds in the case of $\ell$ different functions. Part of the novelty in our proof is the combination of analytic arguments from the theories of influences and hyper-contraction with arguments from additive combinatorics.

cs.DM↗

arXiv2018

Learning Restricted Boltzmann Machines via Influence Maximization

Graphical models are a rich language for describing high-dimensional distributions in terms of their dependence structure. While there are algorithms with provable guarantees for learning undirected graphical models in a variety of settings, there has been much less progress in the important scenario when there are latent variables. Here we study Restricted Boltzmann Machines (or RBMs), which are a popular model with wide-ranging applications in dimensionality reduction, collaborative filtering, topic modeling, feature extraction and deep learning. The main message of our paper is a strong dichotomy in the feasibility of learning RBMs, depending on the nature of the interactions between variables: ferromagnetic models can be learned efficiently, while general models cannot. In particular, we give a simple greedy algorithm based on influence maximization to learn ferromagnetic RBMs with bounded degree. In fact, we learn a description of the distribution on the observed variables as a Markov Random Field. Our analysis is based on tools from mathematical physics that were developed to show the concavity of magnetization. Our algorithm extends straighforwardly to general ferromagnetic Ising models with latent variables. Conversely, we show that even for a contant number of latent variables with constant degree, without ferromagneticity the problem is as hard as sparse parity with noise. This hardness result is based on a sharp and surprising characterization of the representational power of bounded degree RBMs: the distribution on their observed variables can simulate any bounded order MRF. This result is of independent interest since RBMs are the building blocks of deep belief networks.

cs.LG↗

arXiv2018

Is your function low-dimensional?

We study the problem of testing if a function depends on a small number of linear directions of its input data. We call a function $f$ a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. In this paper, we study the problem of testing whether a given $n$ variable function $f : \mathbb{R}^n \to \{0,1\}$, is a linear $k$-junta or $ε$-far from all linear $k$-juntas, where the closeness is measured with respect to the Gaussian measure on $\mathbb{R}^n$. Linear $k$-juntas are a common generalization of two fundamental classes from Boolean function analysis (both of which have been studied in property testing) $\textbf{1.}$ $k$- juntas which are functions on the Boolean cube which depend on at most k of the variables and $\textbf{2.}$ intersection of $k$ halfspaces, a fundamental geometric concept class. We show that the class of linear $k$-juntas is not testable, but adding a surface area constraint makes it testable: we give a $\mathsf{poly}(k \cdot s/ε)$-query non-adaptive tester for linear $k$-juntas with surface area at most $s$. We show that the polynomial dependence on $s$ is necessary. Moreover, we show that if the function is a linear $k$-junta with surface area at most $s$, we give a $(s \cdot k)^{O(k)}$-query non-adaptive algorithm to learn the function up to a rotation of the basis. In particular, this implies that we can test the class of intersections of $k$ halfspaces in $\mathbb{R}^n$ with query complexity independent of $n$.

cs.CC↗

arXiv2018

Deep Learning and Hierarchal Generative Models

It is argued that deep learning is efficient for data that is generated from hierarchal generative models. Examples of such generative models include wavelet scattering networks, functions of compositional structure, and deep rendering models. Unfortunately so far, for all such models, it is either not rigorously known that they can be learned efficiently, or it is not known that "deep algorithms" are required in order to learn them. We propose a simple family of "generative hierarchal models" which can be efficiently learned and where "deep" algorithm are necessary for learning. Our definition of "deep" algorithms is based on the empirical observation that deep nets necessarily use correlations between features. More formally, we show that in a semi-supervised setting, given access to low-order moments of the labeled data and all of the unlabeled data, it is information theoretically impossible to perform classification while at the same time there is an efficient algorithm, that given all labelled and unlabeled data, perfectly labels all unlabelled data with high probability. For the proof, we use and strengthen the fact that Belief Propagation does not admit a good approximation in terms of linear functions.

cs.LG↗

arXiv2018

Reasoning in Bayesian Opinion Exchange Networks Is PSPACE-Hard

We study the Bayesian model of opinion exchange of fully rational agents arranged on a network. In this model, the agents receive private signals that are indicative of an unkown state of the world. Then, they repeatedly announce the state of the world they consider most likely to their neighbors, at the same time updating their beliefs based on their neighbors' announcements. This model is extensively studied in economics since the work of Aumann (1976) and Geanakoplos and Polemarchakis (1982). It is known that the agents eventually agree with high probability on any network. It is often argued that the computations needed by agents in this model are difficult, but prior to our results there was no rigorous work showing this hardness. We show that it is PSPACE-hard for the agents to compute their actions in this model. Furthermore, we show that it is equally difficult even to approximate an agent's posterior: It is PSPACE-hard to distinguish between the posterior being almost entirely concentrated on one state of the world or another.

cs.CC↗

arXiv2018

Seeded Graph Matching via Large Neighborhood Statistics

We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erdős-Rényi random graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. For seeded problems, our result provides a significant improvement over previously known results. We show that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices $n$. Moreover, we show the number of seeds needed for exact recovery in polynomial-time can be as low as $n^{3ε}$ in the sparse graph regime (with the average degree smaller than $n^ε$) and $Ω(\log n)$ in the dense graph regime. Our results also shed light on the unseeded problem. In particular, we give sub-exponential time algorithms for sparse models and an $n^{O(\log n)}$ algorithm for dense models for some parameters, including some that are not covered by recent results of Barak et al.

cs.LG↗

arXiv2018

Contextual Stochastic Block Models

We provide the first information theoretic tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in the detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretical necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model.

cs.SI↗

arXiv2018

Broadcasting on Bounded Degree DAGs

We study the following generalization of the well-known model of broadcasting on trees. Consider an infinite directed acyclic graph (DAG) with a unique source node $X$. Let the collection of nodes at distance $k$ from $X$ be called the $k$th layer. At time zero, the source node is given a bit. At time $k\geq 1$, each node in the $(k-1)$th layer inspects its inputs and sends a bit to its descendants in the $k$th layer. Each bit is flipped with a probability of error $δ\in \left(0,\frac{1}{2}\right)$ in the process of transmission. The goal is to be able to recover the original bit with probability of error better than $\frac{1}{2}$ from the values of all nodes at an arbitrarily deep layer $k$. Besides its natural broadcast interpretation, the DAG broadcast is a natural model of noisy computation. Some special cases of the model represent information flow in biological networks, and other cases represent noisy finite automata models. We show that there exist DAGs with bounded degree and layers of size $ω(\log(k))$ that permit recovery provided $δ$ is sufficiently small and find the critical $δ$ for the DAGs constructed. Our result demonstrates a doubly-exponential advantage for storing a bit in bounded degree DAGs compared to trees. On the negative side, we show that if the DAG is a two-dimensional regular grid, then recovery is impossible for any $δ\in \left(0,\frac{1}{2}\right)$ provided all nodes use either AND or XOR for their processing functions.

cs.IT↗

arXiv2018

The Vertex Sample Complexity of Free Energy is Polynomial

We study the following question: given a massive Markov random field on $n$ nodes, can a small sample from it provide a rough approximation to the free energy $\mathcal{F}_n = \log{Z_n}$? Results in graph limit literature by Borgs, Chayes, Lovász, Sós, and Vesztergombi show that for Ising models on $n$ nodes and interactions of strength $Θ(1/n)$, an $ε$ approximation to $\log Z_n / n$ can be achieved by sampling a randomly induced model on $2^{O(1/ε^2)}$ nodes. We show that the sampling complexity of this problem is {\em polynomial in} $1/ε$. We further show a polynomial dependence on $ε$ cannot be avoided. Our results are very general as they apply to higher order Markov random fields. For Markov random fields of order $r$, we obtain an algorithm that achieves $ε$ approximation using a number of samples polynomial in $r$ and $1/ε$ and running time that is $2^{O(1/ε^2)}$ up to polynomial factors in $r$ and $ε$. For ferromagnetic Ising models, the running time is polynomial in $1/ε$. Our results are intimately connected to recent research on the regularity lemma and property testing, where the interest is in finding which properties can tested within $ε$ error in time polynomial in $1/ε$. In particular, our proofs build on results from a recent work by Alon, de la Vega, Kannan and Karpinski, who also introduced the notion of polynomial vertex sample complexity. Another critical ingredient of the proof is an effective bound by the authors of the paper relating the variational free energy and the free energy.

cs.LG↗

arXiv2018

Approximating Partition Functions in Constant Time

We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for approximating the partition function are Markov Chain Monte Carlo and Variational Methods. An impressive body of work in mathematics, physics and theoretical computer science provides conditions under which Markov Chain Monte Carlo methods converge in polynomial time. These methods often lead to polynomial time approximation algorithms for the partition function in cases where the underlying model exhibits correlation decay. There are very few theoretical guarantees for the performance of variational methods. One exception is recent results by Risteski (2016) who considered dense graphical models and showed that using variational methods, it is possible to find an $O(εn)$ additive approximation to the log partition function in time $n^{O(1/ε^2)}$ even in a regime where correlation decay does not hold. We show that under essentially the same conditions, an $O(εn)$ additive approximation of the log partition function can be found in constant time, independent of $n$. In particular, our results cover dense Ising and Potts models as well as dense graphical models with $k$-wise interaction. They also apply for low threshold rank models.

cs.LG↗

arXiv2018

The Mean-Field Approximation: Information Inequalities, Algorithms, and Complexity

The mean field approximation to the Ising model is a canonical variational tool that is used for analysis and inference in Ising models. We provide a simple and optimal bound for the KL error of the mean field approximation for Ising models on general graphs, and extend it to higher order Markov random fields. Our bound improves on previous bounds obtained in work in the graph limit literature by Borgs, Chayes, Lovász, Sós, and Vesztergombi and another recent work by Basak and Mukherjee. Our bound is tight up to lower order terms. Building on the methods used to prove the bound, along with techniques from combinatorics and optimization, we study the algorithmic problem of estimating the (variational) free energy for Ising models and general Markov random fields. For a graph $G$ on $n$ vertices and interaction matrix $J$ with Frobenius norm $\| J \|_F$, we provide algorithms that approximate the free energy within an additive error of $εn \|J\|_F$ in time $\exp(poly(1/ε))$. We also show that approximation within $(n \|J\|_F)^{1-δ}$ is NP-hard for every $δ> 0$. Finally, we provide more efficient approximation algorithms, which find the optimal mean field approximation, for ferromagnetic Ising models and for Ising models satisfying Dobrushin's condition.

cs.LG↗

arXiv2017

Gaussian Bounds for Noise Correlation of Resilient Functions

Gaussian bounds on noise correlation of functions play an important role in hardness of approximation, in quantitative social choice theory and in testing. The author (2008) obtained sharp gaussian bounds for the expected correlation of $\ell$ low influence functions $f^{(1)},\ldots, f^{(\ell)} : Ω^n \to [0,1]$, where the inputs to the functions are correlated via the $n$-fold tensor of distribution $\mathcal{P}$ on $Ω^{\ell}$. It is natural to ask if the condition of low influences can be relaxed to the condition that the function has vanishing Fourier coefficients. Here we answer this question affirmatively. For the case of two functions $f$ and $g$, we further show that if $f,g$ have a noisy inner product that exceeds the gaussian bound, then the Fourier supports of their large coefficients intersect.

math.PR↗

arXiv2017

Distance-based species tree estimation: information-theoretic trade-off between number of loci and sequence length under the coalescent

We consider the reconstruction of a phylogeny from multiple genes under the multispecies coalescent. We establish a connection with the sparse signal detection problem, where one seeks to distinguish between a distribution and a mixture of the distribution and a sparse signal. Using this connection, we derive an information-theoretic trade-off between the number of genes, $m$, needed for an accurate reconstruction and the sequence length, $k$, of the genes. Specifically, we show that to detect a branch of length $f$, one needs $m = Θ(1/[f^{2} \sqrt{k}])$.

math.PR↗