Elchanan Mossel

Elisabetta Cornacchia, Elchanan Mossel · mit

Curriculum learning (CL) - training using samples that are generated and presented in a meaningful order - was introduced in the machine learning context around a decade ago. While CL has been extensively used and analysed empirically, there has been very little mathematical justification for its advantages. We introduce a CL model for learning the class of k-parities on d bits of a binary string with a neural network trained by stochastic gradient descent (SGD). We show that a wise choice of training examples involving two or more product distributions, allows to reduce significantly the computational cost of learning this class of functions, compared to learning under the uniform distribution. Furthermore, we show that for another class of functions - namely the `Hamming mixtures' - CL strategies involving a bounded number of product distributions are not beneficial.

Jason Gaitonde, Elchanan Mossel · mit

We revisit the problem of efficiently learning the underlying parameters of Ising models from data. Current algorithmic approaches achieve essentially optimal sample complexity when given i.i.d. samples from the stationary measure and the underlying model satisfies "width" bounds on the total $\ell_1$ interaction involving each node. We show that a simple existing approach based on node-wise logistic regression provably succeeds at recovering the underlying model in several new settings where these assumptions are violated: (1) Given dynamically generated data from a wide variety of local Markov chains, like block or round-robin dynamics, logistic regression recovers the parameters with optimal sample complexity up to $\log\log n$ factors. This generalizes the specialized algorithm of Bresler, Gamarnik, and Shah [IEEE Trans. Inf. Theory'18] for structure recovery in bounded degree graphs from Glauber dynamics. (2) For the Sherrington-Kirkpatrick model of spin glasses, given $\mathsf{poly}(n)$ independent samples, logistic regression recovers the parameters in most of the known high-temperature regime via a simple reduction to weaker structural properties of the measure. This improves on recent work of Anari, Jain, Koehler, Pham, and Vuong [ArXiv'23] which gives distribution learning at higher temperature. (3) As a simple byproduct of our techniques, logistic regression achieves an exponential improvement in learning from samples in the M-regime of data considered by Dutt, Lokhov, Vuffray, and Misra [ICML'21] as well as novel guarantees for learning from the adversarial Glauber dynamics of Chin, Moitra, Mossel, and Sandon [ArXiv'23]. Our approach thus significantly generalizes the elegant analysis of Wu, Sanghavi, and Dimakis [Neurips'19] without any algorithmic modification.

Jason Gaitonde, Ankur Moitra, Elchanan Mossel · mit

We consider the problem of learning graphical models, also known as Markov random fields (MRFs) from temporally correlated samples. As in many traditional statistical settings, fundamental results in the area all assume independent samples from the distribution. However, these samples generally will not directly correspond to more realistic observations from nature, which instead evolve according to some stochastic process. From the computational lens, even generating a single sample from the true MRF distribution is intractable unless $\mathsf{NP}=\mathsf{RP}$, and moreover, any algorithm to learn from i.i.d. samples requires prohibitive runtime due to hardness reductions to the parity with noise problem. These computational barriers for sampling and learning from the i.i.d. setting severely lessen the utility of these breakthrough results for this important task; however, dropping this assumption typically only introduces further algorithmic and statistical complexities. In this work, we surprisingly demonstrate that the direct trajectory data from a natural evolution of the MRF overcomes the fundamental computational lower bounds to efficient learning. In particular, we show that given a trajectory with $\widetilde{O}_k(n)$ site updates of an order $k$ MRF from the Glauber dynamics, a well-studied, natural stochastic process on graphical models, there is an algorithm that recovers the graph and the parameters in $\widetilde{O}_k(n^2)$ time. By contrast, all prior algorithms for learning order $k$ MRFs inherently suffer from $n^{Θ(k)}$ runtime even in sparse instances due to the reductions to sparse parity with noise. Our results thus surprisingly show that this more realistic, but intuitively less tractable, model for MRFs actually leads to efficiency far beyond what is known and believed to be true in the traditional i.i.d. case.

Rupert Li, Elchanan Mossel · mit

t-SNE has gained popularity as a dimension reduction technique, especially for visualizing data. It is well-known that all dimension reduction techniques may lose important features of the data. We provide a mathematical framework for understanding this loss for t-SNE by establishing a number of results in different scenarios showing how important features of data are lost by using t-SNE.

Han Huang, Pakawut Jiradilok, Elchanan Mossel · mit

We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise distance estimation has been fundamentally constrained by the volumetric barrier, namely the natural sample-spacing scale $n^{-1/d}$ coming from the fact that a generic point of the manifold typically lies at distance of order $n^{-1/d}$ from the nearest sampled point. In this paper, we introduce a novel approach, Orthogonal Ring Distance Estimation Routine (ORDER), which achieves a pointwise distance estimation precision of order $n^{-2/(d+5)}$ up to polylogarithmic factors in $n$ in polynomial time. This strictly beats the volumetric barrier for dimensions $d > 5$. As a consequence of obtaining pointwise precision better than $n^{-1/d}$, we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order $n^{-1/d}$. This matches the Wasserstein convergence rate of empirical measures, demonstrating that our reconstructed graph metric is asymptotically as good as having access to the full pairwise distance matrix of the sampled points. Our results are proven in a very general setting which includes general models of noisy pairwise distances, sparse random geometric graphs, and unknown connection probability functions.

Anna Brandenberger, Cassandra Marcussen, Elchanan Mossel et al. · mit

We study processes of societal knowledge accumulation, where the validity of a new unit of knowledge depends both on the correctness of its derivation and on the validity of the units it depends on. A fundamental question in this setting is: If a constant fraction of the new derivations is wrong, can investing a constant fraction, bounded away from one, of effort ensure that a constant fraction of knowledge in society is valid? Ben-Eliezer, Mikulincer, Mossel, and Sudan (ITCS 2023) introduced a concrete probabilistic model to analyze such questions and showed an affirmative answer to this question. Their study, however, focuses on the simple case where each new unit depends on just one existing unit, and units attach according to a $\textit{preferential attachment rule}$. In this work, we consider much more general families of cumulative knowledge processes, where new units may attach according to varied attachment mechanisms and depend on multiple existing units. We also allow a (random) fraction of insertions of adversarial nodes. We give a robust affirmative answer to the above question by showing that for $\textit{all}$ of these models, as long as many of the units follow simple heuristics for checking a bounded number of units they depend on, all errors will be eventually eliminated. Our results indicate that preserving the quality of large interdependent collections of units of knowledge is feasible, as long as careful but not too costly checks are performed when new units are derived/deposited.

Elchanan Mossel, David Vulakh · mit

In an extant population, how much information do extant individuals provide on the pedigree of their ancestors? Recent work by Kim, Mossel, Ramnarayan and Turner (2020) studied this question under a number of simplifying assumptions, including random mating, fixed length inheritance blocks and sufficiently large founding population. They showed that under these conditions if the average number of offspring is a sufficiently large constant, then it is possible to recover a large fraction of the pedigree structure and genetic content by an algorithm they named REC-GEN. We are interested in studying the performance of REC-GEN on simulated data generated according to the model. As a first step, we improve the running time of the algorithm. However, we observe that even the faster version of the algorithm does not do well in any simulations in recovering the pedigree beyond 2 generations. We claim that this is due to the inbreeding present in any setting where the algorithm can be run, even on simulated data. To support the claim we show that a main step of the algorithm, called ancestral reconstruction, performs accurately in a idealized setting with no inbreeding but performs poorly in random mating populations. To overcome the poor behavior of REC-GEN we introduce a Belief-Propagation based heuristic that accounts for the inbreeding and performs much better in our simulations.

Ilan Doron-Arad, Elchanan Mossel · mit

Theoretical analyses of Empirical Risk Minimization (ERM) are standardly framed within the Real-RAM model of computation. In this setting, training even simple neural networks is known to be $\exists \mathbb{R}$-complete -- a complexity class believed to be harder than NP, that characterizes the difficulty of solving systems of polynomial inequalities over the real numbers. However, this algebraic framework diverges from the reality of digital computation with finite-precision hardware. In this work, we analyze the theoretical complexity of ERM under a realistic bit-level model ($\mathsf{ERM}_{\text{bit}}$), where network parameters and inputs are constrained to be rational numbers with polynomially bounded bit-lengths. Under this model, we reveal a sharp dichotomy in tractability governed by the network's activation function. We prove that for deep networks with {\em any} polynomial activations with rational coefficients and degree at least $2$, the bit-complexity of training is severe: deciding $\mathsf{ERM}_{\text{bit}}$ is $\#P$-Hard, hence believed to be strictly harder than NP-complete problems. Furthermore, we show that determining the sign of a single partial derivative of the empirical loss function is intractable (unlikely in BPP), and deciding a specific bit in the gradient is $\#P$-Hard. This provides a complexity-theoretic perspective for the phenomenon of exploding and vanishing gradients. In contrast, we show that for piecewise-linear activations such as ReLU, the precision requirements remain manageable: $\mathsf{ERM}_{\text{bit}}$ is contained within NP (specifically NP-complete), and standard backpropagation runs in polynomial time. Our results demonstrate that finite-precision constraints are not merely implementation details but fundamental determinants of learnability.

Elchanan Mossel · mit

Recent reports claim that Large Language Models (LLMs) have achieved the ability to derive new science and exhibit human-level general intelligence. We argue that such claims are not rigorous scientific claims, as they do not satisfy Popper's refutability principle (often termed falsifiability), which requires that scientific statements be capable of being disproven. We identify several methodological pitfalls in current AI research on reasoning, including the inability to verify the novelty of findings due to opaque and non-searchable training data, the lack of reproducibility caused by continuous model updates, and the omission of human-interaction transcripts, which obscures the true source of scientific discovery. Additionally, the absence of counterfactuals and data on failed attempts creates a selection bias that may exaggerate LLM capabilities. To address these challenges, we propose guidelines for scientific transparency and reproducibility for research on reasoning by LLMs. Establishing such guidelines is crucial for both scientific integrity and the ongoing societal debates regarding fair data usage.

Ilan Doron-Arad, Idan Mehalel, Elchanan Mossel · mit

Realizable online regression can behave very differently from online classification. Even without any margin or stochastic assumptions, realizability may enforce horizon-free (finite) cumulative loss under metric-like losses, even when the analogous classification problem has an infinite mistake bound. We study realizable online regression in the adversarial model under losses that satisfy an approximate triangle inequality (approximate pseudo-metrics). Recent work of Attias et al. shows that the minimax realizable cumulative loss is characterized by the scaled Littlestone/online dimension $\mathbb{D}_{\mathrm{onl}}$, but this quantity can be difficult to analyze. Our main contribution is a generic potential method that upper bounds $\mathbb{D}_{\mathrm{onl}}$ by a concrete Dudley-type entropy integral that depends only on covering numbers of the hypothesis class under the induced sup pseudo-metric. We define an \emph{entropy potential} $Φ(\mathcal{H})=\int_{0}^{diam(\mathcal{H})} \log N(\mathcal{H},\varepsilon)\,d\varepsilon$, where $N(\mathcal{H},\varepsilon)$ is the $\varepsilon$-covering number of $\mathcal{H}$, and show that for every $c$-approximate pseudo-metric loss, $\mathbb{D}_{\mathrm{onl}}(\mathcal{H})\le O(c)\,Φ(\mathcal{H})$. In particular, polynomial metric entropy implies $Φ(\mathcal{H})<\infty$ and hence a horizon-free realizable cumulative-loss bound with transparent dependence on effective dimension. We illustrate the method on two families. We prove a sharp $q$-vs.-$d$ dichotomy for realizable online learning (finite and efficiently achievable $Θ_{d,q}(L^d)$ total loss for $L$-Lipschitz regression iff $q>d$, otherwise infinite), and for bounded-norm $k$-ReLU networks separate regression (finite loss, even $\widetilde O(k^2)$, and $O(1)$ for one ReLU) from classification (impossible already for $k=2,d=1$).

Renato Ferreira Pinto, Cassandra Marcussen, Elchanan Mossel et al.

We consider the problems of \emph{learning} and \emph{testing} real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and testability have largely been examined only in discrete or restricted settings -- typically with respect to the Hamming distance, which is ill-suited for real-valued functions. In contrast, we study these problems in high dimensions under the standard Gaussian measure, assuming sample access to the function and a mild smoothness condition, namely Lipschitzness. A smoothness assumption is natural and, in fact, necessary even in one dimension: without it, convexity cannot be inferred from finitely many samples. As our main results, we give: - Learning Convex Functions: An agnostic proper learning algorithm for Lipschitz convex functions that achieves error $\varepsilon$ using $n^{O(1/\varepsilon^2)}$ samples, together with a complementary lower bound of $n^{\mathrm{poly}(1/\varepsilon)}$ samples in the \emph{correlational statistical query (CSQ)} model. - Testing Convex Functions: A tolerant (two-sided) tester for convexity of Lipschitz functions with the same sample complexity (as a corollary of our learning result), and a one-sided tester (which never rejects convex functions) using $O(\sqrt{n}/\varepsilon)^n$ samples.

Jason Gaitonde, Frederic Koehler, Elchanan Mossel et al.

We introduce a family of synthetic languages with hierarchical structure -- generated by a broadcast process on trees -- for which the role of context length and reasoning in autoregressive generation can be analyzed precisely. At the heart of our analytic approach is an \emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the \emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the \emph{coloring broadcast process} (a hard-constrained language) in the freezing regime, bounded-context autoregression produces sequences that, with high probability, are inconsistent with \emph{any} valid coloring of the underlying tree. Together these results imply an $Ω(n)$ lower bound on the context length required to faithfully sample length-$n$ sequences. In contrast, we prove that an autoregressive \emph{reasoning} model with only $Θ(\log n)$ working memory can sample exactly from the true language -- an exponential improvement. We confirm both the lower-bound predictions and the reasoning-based upper bound empirically with transformers trained on the synthetic language; the trained models track our asymptotic predictions quantitatively across a wide range of context sizes.

Elisabetta Cornacchia, Dan Mikulincer, Elchanan Mossel

We study how temporal correlations in the data can make certain sparse learning problems efficiently learnable by gradient-based methods. Our focus is on Boolean k-juntas, a canonical sparse learning problem known to pose barriers for gradient-based methods under independent uniform samples. We show that this picture changes when the samples are generated by a lazy random walk on the hypercube. In this setting, the temporal dependencies can be exploited by a two-layer ReLU network trained using stylized-SGD with a temporal-difference loss, which compares target and predicted increments across consecutive samples. For every fixed k, the resulting sample complexity is essentially linear in the ambient dimension d. By contrast, we show that for large-batch gradient methods using standard convex pointwise losses, temporal correlations do not provide the same advantage.

Ilan Doron-Arad, Idan Mehalel, Elchanan Mossel

Autoregressive generation lies at the heart of the mechanism of large language models. It can be viewed as the repeated application of a next-token generator: starting from an input string (prompt), the generator is applied for $M$ steps, and the last generated token is taken as the final output. [Joshi et al., 2025] proposed a PAC model for studying the learnability of the input-output maps arising from this process. We develop an online analogue of this framework, focusing on the mistake bound of learning the final output induced by an unknown next-token generator. We distinguish between two forms of feedback. In the End-to-End model, after each round the learner observes only the final token produced after $M$ autoregressive steps. In the Chain-of-Thought model, the learner is additionally shown the entire $M$-step trajectory. Our goal is to understand how the optimal mistake bound depends on the generation horizon $M$, and to what extent observing intermediate tokens can reduce this dependence. Our main results show that the online theory of autoregressive learning exhibits a qualitative picture analogous to the statistical one found by [Hanneke et al., 2026], but with a different scale of dependence on the generation horizon. In the End-to-End model, we prove a taxonomy of possible mistake-bound growth rates in the generation horizon $M$: essentially any rate between constant and logarithmic can arise. We further show that this logarithmic ceiling is unavoidable. In the Chain-of-Thought model, we show that access to the full generated trajectory eliminates the dependence on $M$ altogether. We also analyze autoregressive linear threshold classes, and prove optimal mistake bounds, as well as a new lower bound for the statistical setting. Along the way, our results resolve several questions left open by [Joshi et al., 2025].

Amit Daniely, Idan Mehalel, Elchanan Mossel · mit

We study online learning of feedforward neural networks with the sign activation function that implement functions from the unit ball in $\mathbb{R}^d$ to a finite label set $\{1, \ldots, Y\}$. First, we characterize a margin condition that is sufficient and in some cases necessary for online learnability of a neural network: Every neuron in the first hidden layer classifies all instances with some margin $γ$ bounded away from zero. Quantitatively, we prove that for any net, the optimal mistake bound is at most approximately $\mathtt{TS}(d,γ)$, which is the $(d,γ)$-totally-separable-packing number, a more restricted variation of the standard $(d,γ)$-packing number. We complement this result by constructing a net on which any learner makes $\mathtt{TS}(d,γ)$ many mistakes. We also give a quantitative lower bound of approximately $\mathtt{TS}(d,γ) \geq \max\{1/(γ\sqrt{d})^d, d\}$ when $γ\geq 1/2$, implying that for some nets and input sequences every learner will err for $\exp(d)$ many times, and that a dimension-free mistake bound is almost always impossible. To remedy this inevitable dependence on $d$, it is natural to seek additional natural restrictions to be placed on the network, so that the dependence on $d$ is removed. We study two such restrictions. The first is the multi-index model, in which the function computed by the net depends only on $k \ll d$ orthonormal directions. We prove a mistake bound of approximately $(1.5/γ)^{k + 2}$ in this model. The second is the extended margin assumption. In this setting, we assume that all neurons (in all layers) in the network classify every ingoing input from previous layer with margin $γ$ bounded away from zero. In this model, we prove a mistake bound of approximately $(\log Y)/ γ^{O(L)}$, where L is the depth of the network.

Elisabetta Cornacchia, Dan Mikulincer, Elchanan Mossel · mit

The problem of learning single index and multi index models has gained significant interest as a fundamental task in high-dimensional statistics. Many recent works have analysed gradient-based methods, particularly in the setting of isotropic data distributions, often in the context of neural network training. Such studies have uncovered precise characterisations of algorithmic sample complexity in terms of certain analytic properties of the target function, such as the leap, information, and generative exponents. These properties establish a quantitative separation between low and high complexity learning tasks. In this work, we show that high complexity cases are rare. Specifically, we prove that introducing a small random perturbation to the data distribution--via a random shift in the first moment--renders any Gaussian single index model as easy to learn as a linear function. We further extend this result to a class of multi index models, namely sparse Boolean functions, also known as Juntas.

Jason Gaitonde, Ankur Moitra, Elchanan Mossel

We study the problem of learning the structure and parameters of the Ising model, a fundamental model of high-dimensional data, when observing the evolution of an associated Markov chain. A recent line of work has studied the natural problem of learning when observing an evolution of the well-known Glauber dynamics [Bresler, Gamarnik, Shah, IEEE Trans. Inf. Theory 2018, Gaitonde, Mossel STOC 2024], which provides an arguably more realistic generative model than the classical i.i.d. setting. However, this prior work crucially assumes that all site update attempts are observed, \emph{even when this attempt does not change the configuration}: this strong observation model is seemingly essential for these approaches. While perhaps possible in restrictive contexts, this precludes applicability to most realistic settings where we can observe \emph{only} the stochastic evolution itself, a minimal and natural assumption for any process we might hope to learn from. However, designing algorithms that succeed in this more realistic setting has remained an open problem [Bresler, Gamarnik, Shah, IEEE Trans. Inf. Theory 2018, Gaitonde, Moitra, Mossel, STOC 2025]. In this work, we give the first algorithms that efficiently learn the Ising model in this much more natural observation model that only observes when the configuration changes. For Ising models with maximum degree $d$, our algorithm recovers the underlying dependency graph in time $\mathsf{poly}(d)\cdot n^2\log n$ and then the actual parameters in additional $\widetilde{O}(2^d n)$ time, which qualitatively matches the state-of-the-art even in the i.i.d. setting in a much weaker observation model. Our analysis holds more generally for a broader class of reversible, single-site Markov chains that also includes the popular Metropolis chain by leveraging more robust properties of reversible Markov chains.

Rupert Li, Elchanan Mossel · mit

Recent works explore deep learning's success by examining functions or data with hierarchical structure. To study the learning complexity of functions with hierarchical structure, we study the noise stability of functions with tree hierarchical structure on independent inputs. We show that if each function in the hierarchy is $\varepsilon$-far from linear, the noise stability is exponentially small in the depth of the hierarchy. Our results have immediate applications for agnostic learning. In the Boolean setting using the results of Dachman-Soled, Feldman, Tan, Wan and Wimmer (2014), our results provide Statistical Query super-polynomial lower bounds for agnostically learning classes that are based on hierarchical functions. We also derive similar SQ lower bounds based on the indicators of crossing events in critical site percolation. These crossing events are not formally hierarchical as we define but still have some hierarchical features as studied in mathematical physics. Using the results of Abbe, Bengio, Cornacchiam, Kleinberg, Lotfi, Raghu and Zhang (2022), our results imply sample complexity lower bounds for learning hierarchical functions with gradient descent on fully connected neural networks. Finally in the Gaussian setting, using the results of Diakonikolas, Kane, Pittas and Zarifis (2021), our results provide super-polynomial lower bounds for agnostic SQ learning.

Jason Gaitonde, Elchanan Mossel · mit

We consider the problem of linear regression with self-selection bias in the unknown-index setting, as introduced in recent work by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [STOC 2023]. In this model, one observes $m$ i.i.d. samples $(\mathbf{x}_{\ell},z_{\ell})_{\ell=1}^m$ where $z_{\ell}=\max_{i\in [k]}\{\mathbf{x}_{\ell}^T\mathbf{w}_i+η_{i,\ell}\}$, but the maximizing index $i_{\ell}$ is unobserved. Here, the $\mathbf{x}_{\ell}$ are assumed to be $\mathcal{N}(0,I_n)$ and the noise distribution $\mathbfη_{\ell}\sim \mathcal{D}$ is centered and independent of $\mathbf{x}_{\ell}$. We provide a novel and near optimally sample-efficient (in terms of $k$) algorithm to recover $\mathbf{w}_1,\ldots,\mathbf{w}_k\in \mathbb{R}^n$ up to additive $\ell_2$-error $\varepsilon$ with polynomial sample complexity $\tilde{O}(n)\cdot \mathsf{poly}(k,1/\varepsilon)$ and significantly improved time complexity $\mathsf{poly}(n,k,1/\varepsilon)+O(\log(k)/\varepsilon)^{O(k)}$. When $k=O(1)$, our algorithm runs in $\mathsf{poly}(n,1/\varepsilon)$ time, generalizing the polynomial guarantee of an explicit moment matching algorithm of Cherapanamjeri, et al. for $k=2$ and when it is known that $\mathcal{D}=\mathcal{N}(0,I_k)$. Our algorithm succeeds under significantly relaxed noise assumptions, and therefore also succeeds in the related setting of max-linear regression where the added noise is taken outside the maximum. For this problem, our algorithm is efficient in a much larger range of $k$ than the state-of-the-art due to Ghosh, Pananjady, Guntuboyina, and Ramchandran [IEEE Trans. Inf. Theory 2022] for not too small $\varepsilon$, and leads to improved algorithms for any $\varepsilon$ by providing a warm start for existing local convergence methods.

Han Huang, Pakawut Jiradilok, Elchanan Mossel · mit

Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work, we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the manifold assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in $\mathbb{R}^N$. Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.

Frederic Koehler, Elchanan Mossel

The study of Markov processes and broadcasting on trees has deep connections to a variety of areas including statistical physics, graphical models, phylogenetic reconstruction, Markov Chain Monte Carlo, and community detection in random graphs. Notably, the celebrated Belief Propagation (BP) algorithm achieves Bayes-optimal performance for the reconstruction problem of predicting the value of the Markov process at the root of the tree from its values at the leaves. Recently, the analysis of low-degree polynomials has emerged as a valuable tool for predicting computational-to-statistical gaps. In this work, we investigate the performance of low-degree polynomials for the reconstruction problem on trees. Perhaps surprisingly, we show that there are simple tree models with $N$ leaves and bounded arity where (1) nontrivial reconstruction of the root value is possible with a simple polynomial time algorithm and with robustness to noise, but not with any polynomial of degree $N^{c}$ for $c > 0$ a constant depending only on the arity, and (2) when the tree is unknown and given multiple samples with correlated root assignments, nontrivial reconstruction of the root value is possible with a simple Statistical Query algorithm but not with any polynomial of degree $N^c$. These results clarify some of the limitations of low-degree polynomials vs. polynomial time algorithms for Bayesian estimation problems. They also complement recent work of Moitra, Mossel, and Sandon who studied the circuit complexity of Belief Propagation. As a consequence of our main result, we show that for some $c' > 0$ depending only on the arity, $\exp(N^{c'})$ many samples are needed for RBF kernel regression to obtain nontrivial correlation with the true regression function (BP). We pose related open questions about low-degree polynomials and the Kesten-Stigum threshold.

Ankur Moitra, Elchanan Mossel, Colin Sandon

In this work, we study the computational complexity of determining whether a machine learning model that perfectly fits the training data will generalizes to unseen data. In particular, we study the power of a malicious agent whose goal is to construct a model g that fits its training data and nothing else, but is indistinguishable from an accurate model f. We say that g strongly spoofs f if no polynomial-time algorithm can tell them apart. If instead we restrict to algorithms that run in $n^c$ time for some fixed $c$, we say that g c-weakly spoofs f. Our main results are 1. Under cryptographic assumptions, strong spoofing is possible and 2. For any c> 0, c-weak spoofing is possible unconditionally While the assumption of a malicious agent is an extreme scenario (hopefully companies training large models are not malicious), we believe that it sheds light on the inherent difficulties of blindly trusting large proprietary models or data.

Ankur Moitra, Elchanan Mossel, Colin Sandon

We study learning Censor Markov Random Fields (abbreviated CMRFs). These are Markov Random Fields where some of the nodes are censored (not observed). We present an algorithm for learning high-temperature CMRFs within o(n) transportation distance. Crucially our algorithm makes no assumption about the structure of the graph or the number or location of the observed nodes. We obtain stronger results for high girth high-temperature CMRFs as well as computational lower bounds indicating that our results can not be qualitatively improved.

Younhun Kim, Elchanan Mossel, Govind Ramnarayan et al.

We introduce a new algorithm called {\sc Rec-Gen} for reconstructing the genealogy or \textit{pedigree} of an extant population purely from its genetic data. We justify our approach by giving a mathematical proof of the effectiveness of {\sc Rec-Gen} when applied to pedigrees from an idealized generative model that replicates some of the features of real-world pedigrees. Our algorithm is iterative and provides an accurate reconstruction of a large fraction of the pedigree while having relatively low \emph{sample complexity}, measured in terms of the length of the genetic sequences of the population. We propose our approach as a prototype for further investigation of the pedigree reconstruction problem toward the goal of applications to real-world examples. As such, our results have some conceptual bearing on the increasingly important issue of genomic privacy.

Anindya De, Elchanan Mossel, Joe Neeman

A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. A recent work of the authors showed that linear $k$-juntas are testable. Thus there exists an algorithm to distinguish between: 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ which is a linear $k$-junta with surface area $s$, 2. $f$ is $ε$-far from any linear $k$-junta with surface area $(1+ε)s$, where the query complexity of the algorithm is independent of the ambient dimension $n$. Following the surge of interest in noise-tolerant property testing, in this paper we prove a noise-tolerant (or robust) version of this result. Namely, we give an algorithm which given any $c>0$, $ε>0$, distinguishes between 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ has correlation at least $c$ with some linear $k$-junta with surface area $s$. 2. $f$ has correlation at most $c-ε$ with any linear $k$-junta with surface area at most $s$. The query complexity of our tester is $k^{\mathsf{poly}(s/ε)}$. Using our techniques, we also obtain a fully noise tolerant tester with the same query complexity for any class $\mathcal{C}$ of linear $k$-juntas with surface area bounded by $s$. As a consequence, we obtain a fully noise tolerant tester with query complexity $k^{O(\mathsf{poly}(\log k/ε))}$ for the class of intersection of $k$-halfspaces (for constant $k$) over the Gaussian space. Our query complexity is independent of the ambient dimension $n$. Previously, no non-trivial noise tolerant testers were known even for a single halfspace.

Vishesh Jain, Frederic Koehler, Jingbo Liu et al.

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.

Elchanan Mossel, Jiaming Xu

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.

Yash Deshpande, Andrea Montanari, Elchanan Mossel et al.

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.

Guy Bresler, Frederic Koehler, Ankur Moitra et al.

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.

Vishesh Jain, Frederic Koehler, Elchanan Mossel

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.

Vishesh Jain, Frederic Koehler, Elchanan Mossel

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.

Vishesh Jain, Frederic Koehler, Elchanan Mossel

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.

Gautam Dasarathy, Elchanan Mossel, Robert Nowak et al.

The reconstruction of a species phylogeny from genomic data faces two significant hurdles: 1) the trees describing the evolution of each individual gene--i.e., the gene trees--may differ from the species phylogeny and 2) the molecular sequences corresponding to each gene often provide limited information about the gene trees themselves. In this paper we consider an approach to species tree reconstruction that addresses both these hurdles. Specifically, we propose an algorithm for phylogeny reconstruction under the multispecies coalescent model with a standard model of site substitution. The multispecies coalescent is commonly used to model gene tree discordance due to incomplete lineage sorting, a well-studied population-genetic effect. In previous work, an information-theoretic trade-off was derived in this context between the number of loci, $m$, needed for an accurate reconstruction and the length of the locus sequences, $k$. It was shown that to reconstruct an internal branch of length $f$, one needs $m$ to be of the order of $1/[f^{2} \sqrt{k}]$. That previous result was obtained under the molecular clock assumption, i.e., under the assumption that mutation rates (as well as population sizes) are constant across the species phylogeny. Here we generalize this result beyond the restrictive molecular clock assumption, and obtain a new reconstruction algorithm that has the same data requirement (up to log factors). Our main contribution is a novel reduction to the molecular clock case under the multispecies coalescent. As a corollary, we also obtain a new identifiability result of independent interest: for any species tree with $n \geq 3$ species, the rooted species tree can be identified from the distribution of its unrooted weighted gene trees even in the absence of a molecular clock.

Jan Hązła, Ali Jadbabaie, Elchanan Mossel et al.

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.

Elchanan Mossel

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.

Elchanan Mossel, Jiaming Xu

There is a recent surge of interest in identifying the sharp recovery thresholds for cluster recovery under the stochastic block model. In this paper, we address the more refined question of how many vertices that will be misclassified on average. We consider the binary form of the stochastic block model, where $n$ vertices are partitioned into two clusters with edge probability $a/n$ within the first cluster, $c/n$ within the second cluster, and $b/n$ across clusters. Suppose that as $n \to \infty$, $a= b+ μ\sqrt{ b} $, $c=b+ ν\sqrt{ b} $ for two fixed constants $μ, ν$, and $b \to \infty$ with $b=n^{o(1)}$. When the cluster sizes are balanced and $μ\neq ν$, we show that the minimum fraction of misclassified vertices on average is given by $Q(\sqrt{v^*})$, where $Q(x)$ is the Q-function for standard normal, $v^*$ is the unique fixed point of $v= \frac{(μ-ν)^2}{16} + \frac{ (μ+ν)^2 }{16} \mathbb{E}[ \tanh(v+ \sqrt{v} Z)],$ and $Z$ is standard normal. Moreover, the minimum misclassified fraction on average is attained by a local algorithm, namely belief propagation, in time linear in the number of edges. Our proof techniques are based on connecting the cluster recovery problem to tree reconstruction problems, and analyzing the density evolution of belief propagation on trees with Gaussian approximations.

Elchanan Mossel, Jiaming Xu

There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs. Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erdős-Rényi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance. In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the $n$ vertices is labeled $+$ or $-$ independently and uniformly at random; each pair of vertices is connected independently with probability $a/n$ if both of them have the same label or $b/n$ otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree $a/b$ is $Θ(1)$ and the average degree $ (a+b) / 2 = n^{o(1)}$, we characterize three different regimes under which a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly. Thus, in contrast to the case of symmetric block models without side information, we show that local algorithms can achieve optimal performance for the block model with side information.

Elchanan Mossel, Sebastien Roch

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 $m = Θ(1/[f^{2} \sqrt{k}])$.

Elchanan Mossel, Mesrob I. Ohannessian

This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete distribution. The probability of this event is referred to as the "missing mass". The impossibility result can then be stated as: the missing mass is not distribution-free PAC-learnable in relative error. The proof is semi-constructive and relies on a coupling argument using a dithered geometric distribution. This result formalizes the folklore that in order to predict rare events, one necessarily needs distributions with "heavy tails".

Varun Kanade, Elchanan Mossel

The theory of learning under the uniform distribution is rich and deep, with connections to cryptography, computational complexity, and the analysis of boolean functions to name a few areas. This theory however is very limited due to the fact that the uniform distribution and the corresponding Fourier basis are rarely encountered as a statistical model. A family of distributions that vastly generalizes the uniform distribution on the Boolean cube is that of distributions represented by Markov Random Fields (MRF). Markov Random Fields are one of the main tools for modeling high dimensional data in many areas of statistics and machine learning. In this paper we initiate the investigation of extending central ideas, methods and algorithms from the theory of learning under the uniform distribution to the setup of learning concepts given examples from MRF distributions. In particular, our results establish a novel connection between properties of MCMC sampling of MRFs and learning under the MRF distribution.

Florent Krzakala, Cristopher Moore, Elchanan Mossel et al.

Spectral algorithms are classic approaches to clustering and community detection in networks. However, for sparse networks the standard versions of these algorithms are suboptimal, in some cases completely failing to detect communities even when other algorithms such as belief propagation can do so. Here we introduce a new class of spectral algorithms based on a non-backtracking walk on the directed edges of the graph. The spectrum of this operator is much better-behaved than that of the adjacency matrix or other commonly used matrices, maintaining a strong separation between the bulk eigenvalues and the eigenvalues relevant to community structure even in the sparse case. We show that our algorithm is optimal for graphs generated by the stochastic block model, detecting communities all the way down to the theoretical limit. We also show the spectrum of the non-backtracking operator for some real-world networks, illustrating its advantages over traditional spectral clustering.