Frederic Koehler

ICML Conference 2024 Conference Paper

Inferring Dynamic Networks from Marginals with Iterative Proportional Fitting

• Serina Chang
• Frederic Koehler
• Zhaonan Qu
• Jure Leskovec
• Johan Ugander

A common network inference problem, arising from real-world data constraints, is how to infer a dynamic network from its time-aggregated adjacency matrix and time-varying marginals (i. e. , row and column sums). Prior approaches to this problem have repurposed the classic iterative proportional fitting (IPF) procedure, also known as Sinkhorn’s algorithm, with promising empirical results. However, the statistical foundation for using IPF has not been well understood: under what settings does IPF provide principled estimation of a dynamic network from its marginals, and how well does it estimate the network? In this work, we establish such a setting, by identifying a generative network model whose maximum likelihood estimates are recovered by IPF. Our model both reveals implicit assumptions on the use of IPF in such settings and enables new analyses, such as structure-dependent error bounds on IPF’s parameter estimates. When IPF fails to converge on sparse network data, we introduce a principled algorithm that guarantees IPF converges under minimal changes to the network structure. Finally, we conduct experiments with synthetic and real-world data, which demonstrate the practical value of our theoretical and algorithmic contributions.

STOC Conference 2024 Conference Paper

Influences in Mixing Measures

• Frederic Koehler
• Noam Lifshitz
• Dor Minzer
• Elchanan Mossel

ICLR Conference 2024 Conference Paper

Sampling Multimodal Distributions with the Vanilla Score: Benefits of Data-Based Initialization

• Frederic Koehler
• Thuy-Duong Vuong

There is a long history, as well as a recent explosion of interest, in statistical and generative modeling approaches based on \emph{score functions} --- derivatives of the log-likelihood of a distribution. In seminal works, Hyv\"arinen proposed vanilla score matching as a way to learn distributions from data by computing an estimate of the score function of the underlying ground truth, and established connections between this method and established techniques like Contrastive Divergence and Pseudolikelihood estimation. It is by now well-known that vanilla score matching has significant difficulties learning multimodal distributions. Although there are various ways to overcome this difficulty, the following question has remained unanswered --- is there a natural way to sample multimodal distributions using just the vanilla score? Inspired by a long line of related experimental works, we prove that the Langevin diffusion with early stopping, initialized at the empirical distribution, and run on a score function estimated from data successfully generates natural multimodal distributions (mixtures of log-concave distributions).

STOC Conference 2024 Conference Paper

Trickle-Down in Localization Schemes and Applications

• Nima Anari
• Frederic Koehler
• Thuy-Duong Vuong

Trickle-down is a phenomenon in high-dimensional expanders with many important applications — for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems — for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include near-linear time sampling algorithms from the antiferromagnetic Ising model and the fixed magnetization (antiferromagnetic or ferromagnetic) Ising model on expanders. For this application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials.

SODA Conference 2024 Conference Paper

Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

• Nima Anari
• Vishesh Jain
• Frederic Koehler
• Huy Tuan Pham
• Thuy-Duong Vuong

NeurIPS Conference 2023 Conference Paper

Feature Adaptation for Sparse Linear Regression

• Jonathan Kelner
• Frederic Koehler
• Raghu Meka
• Dhruv Rohatgi

Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian $N(0, \Sigma)$, and we seek an estimator with small excess risk. If the true signal is $t$-sparse, information-theoretically, it is possible to achieve strong recovery guarantees with only $O(t\log n)$ samples. However, computationally efficient algorithms have sample complexity linear in (some variant of) the *condition number* of $\Sigma$. Classical algorithms such as the Lasso can require significantly more samples than necessary even if there is only a single sparse approximate dependency among the covariates. We provide a polynomial-time algorithm that, given $\Sigma$, automatically adapts the Lasso to tolerate a small number of approximate dependencies. In particular, we achieve near-optimal sample complexity for constant sparsity and if $\Sigma$ has few ``outlier'' eigenvalues. Our algorithm fits into a broader framework of *feature adaptation* for sparse linear regression with ill-conditioned covariates. With this framework, we additionally provide the first polynomial-factor improvement over brute-force search for constant sparsity $t$ and arbitrary covariance $\Sigma$.

ICLR Conference 2023 Conference Paper

Statistical Efficiency of Score Matching: The View from Isoperimetry

• Frederic Koehler
• Alexander Heckett
• Andrej Risteski

Deep generative models parametrized up to a normalizing constant (e.g. energy-based models) are difficult to train by maximizing the likelihood of the data because the likelihood and/or gradients thereof cannot be explicitly or efficiently written down. Score matching is a training method, whereby instead of fitting the likelihood $\log p(x)$ for the training data, we instead fit the score function $\nabla_x \log p(x)$ --- obviating the need to evaluate the partition function. Though this estimator is known to be consistent, its unclear whether (and when) its statistical efficiency is comparable to that of maximum likelihood --- which is known to be (asymptotically) optimal. We initiate this line of inquiry in this paper, and show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated --- i.e. the Poincar\'e, log-Sobolev and isoperimetric constant --- quantities which govern the mixing time of Markov processes like Langevin dynamics. Roughly, we show that the score matching estimator is statistically comparable to the maximum likelihood when the distribution has a small isoperimetric constant. Conversely, if the distribution has a large isoperimetric constant --- even for simple families of distributions like exponential families with rich enough sufficient statistics --- score matching will be substantially less efficient than maximum likelihood. We suitably formalize these results both in the finite sample regime, and in the asymptotic regime. Finally, we identify a direct parallel in the discrete setting, where we connect the statistical properties of pseudolikelihood estimation with approximate tensorization of entropy and the Glauber dynamics.

NeurIPS Conference 2023 Conference Paper

Uniform Convergence with Square-Root Lipschitz Loss

• Lijia Zhou
• Zhen Dai
• Frederic Koehler
• Nati Srebro

We establish generic uniform convergence guarantees for Gaussian data in terms of the Radamacher complexity of the hypothesis class and the Lipschitz constant of the square root of the scalar loss function. We show how these guarantees substantially generalize previous results based on smoothness (Lipschitz constant of the derivative), and allow us to handle the broader class of square-root-Lipschtz losses, which includes also non-smooth loss functions appropriate for studying phase retrieval and ReLU regression, as well as rederive and better understand “optimistic rate” and interpolation learning guarantees.

NeurIPS Conference 2022 Conference Paper

A Non-Asymptotic Moreau Envelope Theory for High-Dimensional Generalized Linear Models

• Lijia Zhou
• Frederic Koehler
• Pragya Sur
• Danica J. Sutherland
• Nati Srebro

We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\ell$ can control the test error under all Moreau envelopes of the loss $\ell$. We use our finite-sample bound to directly recover the “optimistic rate” of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal $\ell_2$-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand’s well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory for generalization that applies outside of proportional scaling regimes, handles model misspecification, and complements existing asymptotic Moreau envelope theories for M-estimation.

STOC Conference 2022 Conference Paper

Entropic independence: optimal mixing of down-up random walks

• Nima Anari
• Vishesh Jain
• Frederic Koehler
• Huy Tuan Pham
• Thuy-Duong Vuong

We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution µ on k -sized subsets of a ground set of elements says that for any (possibly randomly chosen) set S , the relative entropy of a single element of S drawn uniformly at random carries at most O (1/ k ) fraction of the relative entropy of S . Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities, which guarantee entropy contraction not for all distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: µ is entropically independent exactly when a transformed version of the generating polynomial of µ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain (1) tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions, (2) the tight mixing time of O ( n log n ) for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than 1, improving upon the prior quadratic dependence on n , and (3) nearly-linear time O δ ( n ) samplers for the hardcore and Ising models on n -node graphs that have δ-relative gap to the tree-uniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree Δ of the graph, and is therefore optimal even for high-degree graphs, and in fact, is sublinear in the size of the graph for high-degree graphs.

STOC Conference 2022 Conference Paper

Kalman filtering with adversarial corruptions

• Sitan Chen
• Frederic Koehler
• Ankur Moitra
• Morris Yau

Here we revisit the classic problem of linear quadratic estimation, i.e. estimating the trajectory of a linear dynamical system from noisy measurements. The celebrated Kalman filter gives an optimal estimator when the measurement noise is Gaussian, but is widely known to break down when one deviates from this assumption, e.g. when the noise is heavy-tailed. Many ad hoc heuristics have been employed in practice for dealing with outliers. In a pioneering work, Schick and Mitter gave provable guarantees when the measurement noise is a known infinitesimal perturbation of a Gaussian and raised the important question of whether one can get similar guarantees for large and unknown perturbations.

NeurIPS Conference 2022 Conference Paper

Lower Bounds on Randomly Preconditioned Lasso via Robust Sparse Designs

• Jonathan Kelner
• Frederic Koehler
• Raghu Meka
• Dhruv Rohatgi

Sparse linear regression with ill-conditioned Gaussian random covariates is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief. Recent work has shown that, for certain covariance matrices, the broad class of Preconditioned Lasso programs provably cannot succeed on polylogarithmically sparse signals with a sublinear number of samples. However, this lower bound only holds against deterministic preconditioners, and in many contexts randomization is crucial to the success of preconditioners. We prove a stronger lower bound that rules out randomized preconditioners. For an appropriate covariance matrix, we construct a single signal distribution on which any invertibly-preconditioned Lasso program fails with high probability, unless it receives a linear number of samples. Surprisingly, at the heart of our lower bound is a new robustness result in compressed sensing. In particular, we study recovering a sparse signal when a few measurements can be erased adversarially. To our knowledge, this natural question has not been studied before for sparse measurements. We surprisingly show that standard sparse Bernoulli measurements are almost-optimally robust to adversarial erasures: if $b$ measurements are erased, then all but $O(b)$ of the coordinates of the signal are identifiable.

NeurIPS Conference 2022 Conference Paper

Reconstruction on Trees and Low-Degree Polynomials

• 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 are able to prove a result of independent interest regarding the performance of RBF kernel ridge regression for learning to predict the root coloration: for some $c' > 0$ depending only on the arity, $\exp(N^{c'})$ many samples are needed for the 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.

ICLR Conference 2022 Conference Paper

Variational autoencoders in the presence of low-dimensional data: landscape and implicit bias

• Frederic Koehler
• Viraj Mehta
• Chenghui Zhou
• Andrej Risteski

Variational Autoencoders (VAEs) are one of the most commonly used generative models, particularly for image data. A prominent difficulty in training VAEs is data that is supported on a lower dimensional manifold. Recent work by Dai and Wipf (2020) proposes a two-stage training algorithm for VAEs, based on a conjecture that in standard VAE training the generator will converge to a solution with 0 variance which is correctly supported on the ground truth manifold. They gave partial support for this conjecture by showing that some optima of the VAE loss do satisfy this property, but did not analyze the training dynamics. In this paper, we show that for linear encoders/decoders, the conjecture is true—that is the VAE training does recover a generator with support equal to the ground truth manifold—and does so due to an implicit bias of gradient descent rather than merely the VAE loss itself. In the nonlinear case, we show that VAE training frequently learns a higher-dimensional manifold which is a superset of the ground truth manifold.

FOCS Conference 2021 Conference Paper

Chow-Liu++: Optimal Prediction-Centric Learning of Tree Ising Models

• Enric Boix-Adserà
• Guy Bresler
• Frederic Koehler

We consider the problem of learning a tree-structured Ising model from data, such that subsequent predictions computed using the model are accurate. Con-cretely, we aim to learn a model such that posteriors $p$ (Xi| X s) for small sets of variables $S$ are accurate. Since its introduction more than 50 years ago, the Chow-Liu algorithm, which efficiently computes the maximum likelihood tree, has been the benchmark algorithm for learning tree-structured graphical models. A bound on the sample complexity of the Chow-Liu algorithm with respect to the prediction-centric local total variation loss was shown in [7]. While those results demonstrated that it is possible to learn a useful model even when recovering the true underlying graph is impossible, their bound depends on the maximum strength of interactions and thus does not achieve the information-theoretic optimum. In this paper, we introduce a new algorithm that carefully combines elements of the Chow-Liu algorithm with tree metric reconstruction methods to efficiently and optimally learn tree Ising models under a prediction-centric loss. Our algorithm is robust to model misspecification and adver-sarial corruptions. In contrast, we show that the celebrated Chow- Liu algorithm can be arbitrarily suboptimal.

ICML Conference 2021 Conference Paper

Multidimensional Scaling: Approximation and Complexity

• Erik D. Demaine
• Adam Hesterberg
• Frederic Koehler
• Jayson Lynch
• John Urschel

Metric Multidimensional scaling (MDS) is a classical method for generating meaningful (non-linear) low-dimensional embeddings of high-dimensional data. MDS has a long history in the statistics, machine learning, and graph drawing communities. In particular, the Kamada-Kawai force-directed graph drawing method is equivalent to MDS and is one of the most popular ways in practice to embed graphs into low dimensions. Despite its ubiquity, our theoretical understanding of MDS remains limited as its objective function is highly non-convex. In this paper, we prove that minimizing the Kamada-Kawai objective is NP-hard and give a provable approximation algorithm for optimizing it, which in particular is a PTAS on low-diameter graphs. We supplement this result with experiments suggesting possible connections between our greedy approximation algorithm and gradient-based methods.

FOCS Conference 2021 Conference Paper

On the Power of Preconditioning in Sparse Linear Regression

• Jonathan A. Kelner
• Frederic Koehler
• Raghu Meka
• Dhruv Rohatgi

Sparse linear regression is a fundamental problem in high-dimensional statistics, but strikingly little is known about how to efficiently solve it without restrictive conditions on the design matrix. We consider the (correlated) random design setting, where the covariates are independently drawn from a multivariate Gaussian $N(0, \ \Sigma)$, for some $n\times n$ positive semi-definite matrix $\Sigma$, and seek estimators $\hat{w}$ minimizing $(\hat{w}-w^{\ast})^{T}\Sigma(\hat{w}-w^{\ast})$, where $w^{\ast}$ is the k-sparse ground truth. Information theoretically, one can achieve strong error bounds with only $O(k\log n)$ samples for arbitrary $\Sigma$ and $w^{\ast}$; however, no efficient algorithms are known to match these guarantees even with $o(n)$ samples, without further assumptions on $\Sigma$ or $w^{\ast}$. Yet there is little evidence for this gap in the random design setting: computational lower bounds are only known for worst-case design matrices. To date, random-design instances (i. e. specific covariance matrices $\Sigma$ ) have only been proven hard against the Lasso program and variants. More precisely, these “hard” instances can often be solved by Lasso after a simple change-of-basis (i. e. preconditioning). In this work, we give both upper and lower bounds clarifying the power of preconditioning as a tool for solving sparse linear regression problems. On the one hand, we show that the preconditioned Lasso can solve a large class of sparse linear regression problems nearly optimally: it succeeds whenever the dependency structure of the covariates, in the sense of the Markov property, has low treewidth - even if $\Sigma$ is highly ill-conditioned. This upper bound builds on ideas from the wavelet and signal processing literature. As a special case of this result, we give an algorithm for sparse linear regression with covariates from an autoregressive time series model, where we also show that the (usual) Lasso provably fails. On the other hand, we construct (for the first time) random-design instances which are provably hard even for an optimally preconditioned Lasso. In fact, we complete our treewidth classification by proving that for any treewidth-t graph, there exists a Gaussian Markov Random Field on this graph such that the preconditioned Lasso, with any choice of preconditioner, requires $\Omega(t^{1/20})$ samples to recover $O(\log n)$ -sparse signals when covariates are drawn from this model.

FOCS Conference 2021 Conference Paper

Online and Distribution-Free Robustness: Regression and Contextual Bandits with Huber Contamination

• Sitan Chen
• Frederic Koehler
• Ankur Moitra
• Morris Yau

In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for both linear regression and contextual bandits. In fact our algorithms succeed where conventional methods fail. In particular we show strong lower bounds against Huber regression and more generally any convex $M$ -estimator. Our approach is based on a novel alternating minimization scheme that interleaves ordinary least-squares with a simple convex program that finds the optimal reweighting of the distribution under a spectral constraint. Our results obtain essentially optimal dependence on the contamination level η, reach the optimal breakdown point, and naturally apply to infinite dimensional settings where the feature vectors are represented implicitly via a kernel map.

ICML Conference 2021 Conference Paper

Representational aspects of depth and conditioning in normalizing flows

• Frederic Koehler
• Viraj Mehta
• Andrej Risteski

Normalizing flows are among the most popular paradigms in generative modeling, especially for images, primarily because we can efficiently evaluate the likelihood of a data point. This is desirable both for evaluating the fit of a model, and for ease of training, as maximizing the likelihood can be done by gradient descent. However, training normalizing flows comes with difficulties as well: models which produce good samples typically need to be extremely deep – which comes with accompanying vanishing/exploding gradient problems. A very related problem is that they are often poorly \emph{conditioned}: since they are parametrized as invertible maps from $\mathbb{R}^d \to \mathbb{R}^d$, and typical training data like images intuitively is lower-dimensional, the learned maps often have Jacobians that are close to being singular. In our paper, we tackle representational aspects around depth and conditioning of normalizing flows: both for general invertible architectures, and for a particular common architecture, affine couplings. We prove that $\Theta(1)$ affine coupling layers suffice to exactly represent a permutation or $1 \times 1$ convolution, as used in GLOW, showing that representationally the choice of partition is not a bottleneck for depth. We also show that shallow affine coupling networks are universal approximators in Wasserstein distance if ill-conditioning is allowed, and experimentally investigate related phenomena involving padding. Finally, we show a depth lower bound for general flow architectures with few neurons per layer and bounded Lipschitz constant.

NeurIPS Conference 2021 Conference Paper

Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds and Benign Overfitting

• Frederic Koehler
• Lijia Zhou
• Danica J. Sutherland
• Nathan Srebro

We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class’s Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum $\ell_1$-norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings.

NeurIPS Conference 2020 Conference Paper

Classification Under Misspecification: Halfspaces, Generalized Linear Models, and Evolvability

• Sitan Chen
• Frederic Koehler
• Ankur Moitra
• Morris Yau

In this paper, we revisit the problem of distribution-independently learning halfspaces under Massart noise with rate $\eta$. Recent work resolved a long-standing problem in this model of efficiently learning to error $\eta + \epsilon$ for any $\epsilon > 0$, by giving an improper learner that partitions space into $\text{poly}(d, 1/\epsilon)$ regions. Here we give a much simpler algorithm and settle a number of outstanding open questions: (1) We give the first \emph{proper} learner for Massart halfspaces that achieves $\eta + \epsilon$. (2) Based on (1), we develop a blackbox knowledge distillation procedure to convert an arbitrarily complex classifier to an equally good proper classifier. (3) By leveraging a simple but overlooked connection to \emph{evolvability}, we show any SQ algorithm requires super-polynomially many queries to achieve $\mathsf{OPT} + \epsilon$. We then zoom out to study generalized linear models and give an efficient algorithm for learning under a challenging new corruption model generalizing Massart noise. Finally we study our algorithm for learning halfspaces under Massart noise empirically and find that it exhibits some appealing fairness properties as a byproduct of its strong provable robustness guarantees.

NeurIPS Conference 2020 Conference Paper

From Boltzmann Machines to Neural Networks and Back Again

• Surbhi Goel
• Adam Klivans
• Frederic Koehler

Graphical models are powerful tools for modeling high-dimensional data, but learning graphical models in the presence of latent variables is well-known to be difficult. In this work we give new results for learning Restricted Boltzmann Machines, probably the most well-studied class of latent variable models. Our results are based on new connections to learning two-layer neural networks under $\ell_{\infty}$ bounded input; for both problems, we give nearly optimal results under the conjectured hardness of sparse parity with noise. Using the connection between RBMs and feedforward networks, we also initiate the theoretical study of {\em supervised RBMs} \citep{hinton2012practical}, a version of neural-network learning that couples distributional assumptions induced from the underlying graphical model with the architecture of the unknown function class. We then give an algorithm for learning a natural class of supervised RBMs with better runtime than what is possible for its related class of networks without distributional assumptions.

NeurIPS Conference 2020 Conference Paper

Learning Some Popular Gaussian Graphical Models without Condition Number Bounds

• Jonathan Kelner
• Frederic Koehler
• Raghu Meka
• Ankur Moitra

Gaussian Graphical Models (GGMs) have wide-ranging applications in machine learning and the natural and social sciences. In most of the settings in which they are applied, the number of observed samples is much smaller than the dimension and they are assumed to be sparse. While there are a variety of algorithms (e. g. Graphical Lasso, CLIME) that provably recover the graph structure with a logarithmic number of samples, to do so they require various assumptions on the well-conditioning of the precision matrix that are not information-theoretically necessary. Here we give the first fixed polynomial-time algorithms for learning attractive GGMs and walk-summable GGMs with a logarithmic number of samples without any such assumptions. In particular, our algorithms can tolerate strong dependencies among the variables. Our result for structure recovery in walk-summable GGMs is derived from a more general result for efficient sparse linear regression in walk-summable models without any norm dependencies. We complement our results with experiments showing that many existing algorithms fail even in some simple settings where there are long dependency chains. Our algorithms do not.

NeurIPS Conference 2019 Conference Paper

Fast Convergence of Belief Propagation to Global Optima: Beyond Correlation Decay

• Frederic Koehler

Belief propagation is a fundamental message-passing algorithm for probabilistic reasoning and inference in graphical models. While it is known to be exact on trees, in most applications belief propagation is run on graphs with cycles. Understanding the behavior of loopy'' belief propagation has been a major challenge for researchers in machine learning, and several positive convergence results for BP are known under strong assumptions which imply the underlying graphical model exhibits decay of correlations. We show that under a natural initialization, BP converges quickly to the global optimum of the Bethe free energy for Ising models on arbitrary graphs, as long as the Ising model is \emph{ferromagnetic} (i. e. neighbors prefer to be aligned). This holds even though such models can exhibit long range correlations and may have multiple suboptimal BP fixed points. We also show an analogous result for iterating the (naive) mean-field equations; perhaps surprisingly, both results are dimension-free'' in the sense that a constant number of iterations already provides a good estimate to the Bethe/mean-field free energy.

STOC Conference 2019 Conference Paper

Learning restricted Boltzmann machines via influence maximization

• Guy Bresler
• Frederic Koehler
• Ankur Moitra

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.

STOC Conference 2019 Conference Paper

Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective

• Vishesh Jain
• Frederic Koehler
• Andrej Risteski

The free energy is a key quantity of interest in Ising models, but unfortunately, computing it in general is computationally intractable. Two popular (variational) approximation schemes for estimating the free energy of general Ising models (in particular, even in regimes where correlation decay does not hold) are: (i) the mean-field approximation with roots in statistical physics, which estimates the free energy from below, and (ii) hierarchies of convex relaxations with roots in theoretical computer science, which estimate the free energy from above. We show, surprisingly, that the tight regime for both methods to compute the free energy to leading order is identical .

NeurIPS Conference 2017 Conference Paper

Information Theoretic Properties of Markov Random Fields, and their Algorithmic Applications

• Linus Hamilton
• Frederic Koehler
• Ankur Moitra

Markov random fields are a popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for provably learning them relied on exhaustive search, correlation decay or various incoherence assumptions. Bresler gave an algorithm for learning general Ising models on bounded degree graphs. His approach was based on a structural result about mutual information in Ising models. Here we take a more conceptual approach to proving lower bounds on the mutual information. Our proof generalizes well beyond Ising models, to arbitrary Markov random fields with higher order interactions. As an application, we obtain algorithms for learning Markov random fields on bounded degree graphs on $n$ nodes with $r$-order interactions in $n^r$ time and $\log n$ sample complexity. Our algorithms also extend to various partial observation models.

ICML Conference 2016 Conference Paper

Provable Algorithms for Inference in Topic Models

• Sanjeev Arora
• Rong Ge 0001
• Frederic Koehler
• Tengyu Ma 0001
• Ankur Moitra

Recently, there has been considerable progress on designing algorithms with provable guarantees —typically using linear algebraic methods—for parameter learning in latent variable models. Designing provable algorithms for inference has proved more difficult. Here we take a first step towards provable inference in topic models. We leverage a property of topic models that enables us to construct simple linear estimators for the unknown topic proportions that have small variance, and consequently can work with short documents. Our estimators also correspond to finding an estimate around which the posterior is well-concentrated. We show lower bounds that for shorter documents it can be information theoretically impossible to find the hidden topics. Finally, we give empirical results that demonstrate that our algorithm works on realistic topic models. It yields good solutions on synthetic data and runs in time comparable to a single iteration of Gibbs sampling.