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Bryon Aragam

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30 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

Differentiable Structure Learning and Causal Discovery for General Binary Data

Chang Deng
Bryon Aragam

Existing methods for differentiable structure learning in discrete data typically assume that the data are generated from specific structural equation models. However, these assumptions may not align with the true data-generating process, which limits the general applicability of such methods. Furthermore, current approaches often ignore the complex dependence structure inherent in discrete data and consider only linear effects. We propose a differentiable structure learning framework that is capable of capturing arbitrary dependencies among discrete variables. We show that although general discrete models are unidentifiable from purely observational data, it is possible to characterize the complete set of compatible parameters and structures. Additionally, we establish identifiability up to the Markov equivalence class (MEC) under mild assumptions. We formulate the learning problem as a single differentiable optimization task in the most general form, thereby avoiding the unrealistic simplifications adopted by previous methods. Empirical results demonstrate that our approach effectively captures complex relationships in discrete data.

ICML Conference 2025 Conference Paper

Dimension-Independent Rates for Structured Neural Density Estimation

Robert A. Vandermeulen
Wai Ming Tai
Bryon Aragam

We show that deep neural networks can achieve dimension-independent rates of convergence for learning structured densities typical of image, audio, video, and text data. For example, in images, where each pixel becomes independent of the rest of the image when conditioned on pixels at most $t$ steps away, a simple $L^2$-minimizing neural network can attain a rate of $n^{-1/((t+1)^2+4)}$, where $t$ is independent of the ambient dimension $d$, i. e. the total number of pixels. We further provide empirical evidence that, in real-world applications, $t$ is often a small constant, thus effectively circumventing the curse of dimensionality. Moreover, for sequential data (e. g. , audio or text) exhibiting a similar local dependence structure, our analysis shows a rate of $n^{-1/(t+5)}$, offering further evidence of dimension independence in practical scenarios.

NeurIPS Conference 2024 Conference Paper

Breaking the curse of dimensionality in structured density estimation

Robert A. Vandermeulen
Wai Ming Tai
Bryon Aragam

We consider the problem of estimating a structured multivariate density, subject to Markov conditions implied by an undirected graph. In the worst case, without Markovian assumptions, this problem suffers from the curse of dimensionality. Our main result shows how the curse of dimensionality can be avoided or greatly alleviated under the Markov property, and applies to arbitrary graphs. While existing results along these lines focus on sparsity or manifold assumptions, we introduce a new graphical quantity called ``graph resilience'' and show that it dictates the optimal sample complexity. Surprisingly, although one might expect the sample complexity of this problem to scale with local graph parameters such as the degree, this turns out not to be the case. Through explicit examples, we compute uniform deviation bounds and illustrate how the curse of dimensionality in density estimation can thus be circumvented. Notable examples where the rate improves substantially include sequential, hierarchical, and spatial data.

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NeurIPS Conference 2024 Conference Paper

Do LLMs dream of elephants (when told not to)? Latent concept association and associative memory in transformers

Yibo Jiang
Goutham Rajendran
Pradeep Ravikumar
Bryon Aragam

Large Language Models (LLMs) have the capacity to store and recall facts. Through experimentation with open-source models, we observe that this ability to retrieve facts can be easily manipulated by changing contexts, even without altering their factual meanings. These findings highlight that LLMs might behave like an associative memory model where certain tokens in the contexts serve as clues to retrieving facts. We mathematically explore this property by studying how transformers, the building blocks of LLMs, can complete such memory tasks. We study a simple latent concept association problem with a one-layer transformer and we show theoretically and empirically that the transformer gathers information using self-attention and uses the value matrix for associative memory.

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NeurIPS Conference 2024 Conference Paper

From Causal to Concept-Based Representation Learning

Goutham Rajendran
Simon Buchholz
Bryon Aragam
Bernhard Schölkopf
Pradeep Ravikumar

To build intelligent machine learning systems, modern representation learning attempts to recover latent generative factors from data, such as in causal representation learning. A key question in this growing field is to provide rigorous conditions under which latent factors can be identified and thus, potentially learned. Motivated by extensive empirical literature on linear representations and concept learning, we propose to relax causal notions with a geometric notion of concepts. We formally define a notion of concepts and show rigorously that they can be provably recovered from diverse data. Instead of imposing assumptions on the "true" generative latent space, we assume that concepts can be represented linearly in this latent space. The tradeoff is that instead of identifying the "true" generative factors, we identify a subset of desired human-interpretable concepts that are relevant for a given application. Experiments on synthetic data, multimodal CLIP models and large language models supplement our results and show the utility of our approach. In this way, we provide a foundation for moving from causal representations to interpretable, concept-based representations by bringing together ideas from these two neighboring disciplines.

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NeurIPS Conference 2024 Conference Paper

Identifying General Mechanism Shifts in Linear Causal Representations

Tianyu Chen
Kevin Bello
Francesco Locatello
Bryon Aragam
Pradeep Ravikumar

We consider the linear causal representation learning setting where we observe a linear mixing of $d$ unknown latent factors, which follow a linear structural causal model. Recent work has shown that it is possible to recover the latent factors as well as the underlying structural causal model over them, up to permutation and scaling, provided that we have at least $d$ environments, each of which corresponds to perfect interventions on a single latent node (factor). After this powerful result, a key open problem faced by the community has been to relax these conditions: allow for coarser than perfect single-node interventions, and allow for fewer than $d$ of them, since the number of latent factors $d$ could be very large. In this work, we consider precisely such a setting, where we allow a smaller than $d$ number of environments, and also allow for very coarse interventions that can very coarsely \textit{change the entire causal graph over the latent factors}. On the flip side, we relax what we wish to extract to simply the \textit{list of nodes that have shifted between one or more environments}. We provide a surprising identifiability result that it is indeed possible, under some very mild standard assumptions, to identify the set of shifted nodes. Our identifiability proof moreover is a constructive one: we explicitly provide necessary and sufficient conditions for a node to be a shifted node, and show that we can check these conditions given observed data. Our algorithm lends itself very naturally to the sample setting where instead of just interventional distributions, we are provided datasets of samples from each of these distributions. We corroborate our results on both synthetic experiments as well as an interesting psychometric dataset. The code can be found at https: //github. com/TianyuCodings/iLCS.

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NeurIPS Conference 2024 Conference Paper

Markov Equivalence and Consistency in Differentiable Structure Learning

Chang Deng
Kevin Bello
Pradeep Ravikumar
Bryon Aragam

Existing approaches to differentiable structure learning of directed acyclic graphs (DAGs) rely on strong identifiability assumptions in order to guarantee that global minimizers of the acyclicity-constrained optimization problem identifies the true DAG. Moreover, it has been observed empirically that the optimizer may exploit undesirable artifacts in the loss function. We explain and remedy these issues by studying the behavior of differentiable acyclicity-constrained programs under general likelihoods with multiple global minimizers. By carefully regularizing the likelihood, it is possible to identify the sparsest model in the Markov equivalence class, even in the absence of an identifiable parametrization. We first study the Gaussian case in detail, showing how proper regularization of the likelihood defines a score that identifies the sparsest model. Assuming faithfulness, it also recovers the Markov equivalence class. These results are then generalized to general models and likelihoods, where the same claims hold. These theoretical results are validated empirically, showing how this can be done using standard gradient-based optimizers (without resorting to approximations such as Gumbel-Softmax), thus paving the way for differentiable structure learning under general models and losses. Open-source code is available at \url{https: //github. com/duntrain/dagrad}.

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ICML Conference 2024 Conference Paper

On the Origins of Linear Representations in Large Language Models

Yibo Jiang
Goutham Rajendran
Pradeep Ravikumar
Bryon Aragam
Victor Veitch

An array of recent works have argued that high-level semantic concepts are encoded "linearly" in the representation space of large language models. In this work, we study the origins of such linear representations. To that end, we introduce a latent variable model to abstract and formalize the concept dynamics of the next token prediction. We use this formalism to prove that linearity arises as a consequence of the loss function and the implicit bias of gradient descent. The theory is further substantiated empirically via experiments.

NeurIPS Conference 2023 Conference Paper

Assumption violations in causal discovery and the robustness of score matching

Francesco Montagna
Atalanti Mastakouri
Elias Eulig
Nicoletta Noceti
Lorenzo Rosasco
Dominik Janzing
Bryon Aragam
Francesco Locatello

When domain knowledge is limited and experimentation is restricted by ethical, financial, or time constraints, practitioners turn to observational causal discovery methods to recover the causal structure, exploiting the statistical properties of their data. Because causal discovery without further assumptions is an ill-posed problem, each algorithm comes with its own set of usually untestable assumptions, some of which are hard to meet in real datasets. Motivated by these considerations, this paper extensively benchmarks the empirical performance of recent causal discovery methods on observational iid data generated under different background conditions, allowing for violations of the critical assumptions required by each selected approach. Our experimental findings show that score matching-based methods demonstrate surprising performance in the false positive and false negative rate of the inferred graph in these challenging scenarios, and we provide theoretical insights into their performance. This work is also the first effort to benchmark the stability of causal discovery algorithms with respect to the values of their hyperparameters. Finally, we hope this paper will set a new standard for the evaluation of causal discovery methods and can serve as an accessible entry point for practitioners interested in the field, highlighting the empirical implications of different algorithm choices.

NeurIPS Conference 2023 Conference Paper

Global Optimality in Bivariate Gradient-based DAG Learning

Chang Deng
Kevin Bello
Pradeep Ravikumar
Bryon Aragam

Recently, a new class of non-convex optimization problems motivated by the statistical problem of learning an acyclic directed graphical model from data has attracted significant interest. While existing work uses standard first-order optimization schemes to solve this problem, proving the global optimality of such approaches has proven elusive. The difficulty lies in the fact that unlike other non-convex problems in the literature, this problem is not "benign", and possesses multiple spurious solutions that standard approaches can easily get trapped in. In this paper, we prove that a simple path-following optimization scheme globally converges to the global minimum of the population loss in the bivariate setting.

NeurIPS Conference 2023 Conference Paper

iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models

Tianyu Chen
Kevin Bello
Bryon Aragam
Pradeep Ravikumar

Structural causal models (SCMs) are widely used in various disciplines to represent causal relationships among variables in complex systems. Unfortunately, the underlying causal structure is often unknown, and estimating it from data remains a challenging task. In many situations, however, the end goal is to localize the changes (shifts) in the causal mechanisms between related datasets instead of learning the full causal structure of the individual datasets. Some applications include root cause analysis, analyzing gene regulatory network structure changes between healthy and cancerous individuals, or explaining distribution shifts. This paper focuses on identifying the causal mechanism shifts in two or more related datasets over the same set of variables--- without estimating the entire DAG structure of each SCM. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key technical contribution of this work is to show that the Jacobian of the score function for the mixture distribution allows for the identification of shifts under general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach. Code implementing the proposed method is open-source and publicly available at https: //github. com/kevinsbello/iSCAN.

NeurIPS Conference 2023 Conference Paper

Learning Linear Causal Representations from Interventions under General Nonlinear Mixing

Simon Buchholz
Goutham Rajendran
Elan Rosenfeld
Bryon Aragam
Bernhard Schölkopf
Pradeep Ravikumar

We study the problem of learning causal representations from unknown, latent interventions in a general setting, where the latent distribution is Gaussian but the mixing function is completely general. We prove strong identifiability results given unknown single-node interventions, i. e. , without having access to the intervention targets. This generalizes prior works which have focused on weaker classes, such as linear maps or paired counterfactual data. This is also the first instance of identifiability from non-paired interventions for deep neural network embeddings and general causal structures. Our proof relies on carefully uncovering the high-dimensional geometric structure present in the data distribution after a non-linear density transformation, which we capture by analyzing quadratic forms of precision matrices of the latent distributions. Finally, we propose a contrastive algorithm to identify the latent variables in practice and evaluate its performance on various tasks.

ICML Conference 2023 Conference Paper

Learning Mixtures of Gaussians with Censored Data

Wai Ming Tai
Bryon Aragam

We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $ \sum_{i=1}^k w_i \mathcal{N}(\mu_i, \sigma^2), $ i. e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $w_i$ and the means $\mu_i$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $w_i$ and the means $\mu_i$ within $\varepsilon$ error.

NeurIPS Conference 2023 Conference Paper

Learning Nonparametric Latent Causal Graphs with Unknown Interventions

Yibo Jiang
Bryon Aragam

We establish conditions under which latent causal graphs are nonparametrically identifiable and can be reconstructed from unknown interventions in the latent space. Our primary focus is the identification of the latent structure in measurement models without parametric assumptions such as linearity or Gaussianity. Moreover, we do not assume the number of hidden variables is known, and we show that at most one unknown intervention per hidden variable is needed. This extends a recent line of work on learning causal representations from observations and interventions. The proofs are constructive and introduce two new graphical concepts--- imaginary subsets and isolated edges ---that may be useful in their own right. As a matter of independent interest, the proofs also involve a novel characterization of the limits of edge orientations within the equivalence class of DAGs induced by unknown interventions. These are the first results to characterize the conditions under which causal representations are identifiable without making any parametric assumptions in a general setting with unknown interventions and without faithfulness.

ICML Conference 2023 Conference Paper

Optimizing NOTEARS Objectives via Topological Swaps

Chang Deng
Kevin Bello
Bryon Aragam
Pradeep Ravikumar

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimality challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https: //github. com/duntrain/topo.

NeurIPS Conference 2023 Conference Paper

Uncovering Meanings of Embeddings via Partial Orthogonality

Yibo Jiang
Bryon Aragam
Victor Veitch

Machine learning tools often rely on embedding text as vectors of real numbers. In this paper, we study how the semantic structure of language is encoded in the algebraic structure of such embeddings. Specifically, we look at a notion of "semantic independence" capturing the idea that, e. g. , "eggplant" and "tomato" are independent given "vegetable". Although such examples are intuitive, it is difficult to formalize such a notion of semantic independence. The key observation here is that any sensible formalization should obey a set of so-called independence axioms, and thus any algebraic encoding of this structure should also obey these axioms. This leads us naturally to use partial orthogonality as the relevant algebraic structure. We develop theory and methods that allow us to demonstrate that partial orthogonality does indeed capture semantic independence. Complementary to this, we also introduce the concept of independence preserving embeddings where embeddings preserve the conditional independence structures of a distribution, and we prove the existence of such embeddings and approximations to them.

NeurIPS Conference 2022 Conference Paper

DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization

Kevin Bello
Bryon Aragam
Pradeep Ravikumar

The combinatorial problem of learning directed acyclic graphs (DAGs) from data was recently framed as a purely continuous optimization problem by leveraging a differentiable acyclicity characterization of DAGs based on the trace of a matrix exponential function. Existing acyclicity characterizations are based on the idea that powers of an adjacency matrix contain information about walks and cycles. In this work, we propose a new acyclicity characterization based on the log-determinant (log-det) function, which leverages the nilpotency property of DAGs. To deal with the inherent asymmetries of a DAG, we relate the domain of our log-det characterization to the set of $\textit{M-matrices}$, which is a key difference to the classical log-det function defined over the cone of positive definite matrices. Similar to acyclicity functions previously proposed, our characterization is also exact and differentiable. However, when compared to existing characterizations, our log-det function: (1) Is better at detecting large cycles; (2) Has better-behaved gradients; and (3) Its runtime is in practice about an order of magnitude faster. From the optimization side, we drop the typically used augmented Lagrangian scheme and propose DAGMA ($\textit{Directed Acyclic Graphs via M-matrices for Acyclicity}$), a method that resembles the central path for barrier methods. Each point in the central path of DAGMA is a solution to an unconstrained problem regularized by our log-det function, then we show that at the limit of the central path the solution is guaranteed to be a DAG. Finally, we provide extensive experiments for $\textit{linear}$ and $\textit{nonlinear}$ SEMs and show that our approach can reach large speed-ups and smaller structural Hamming distances against state-of-the-art methods. Code implementing the proposed method is open-source and publicly available at https: //github. com/kevinsbello/dagma.

JMLR Journal 2022 Journal Article

Fundamental Limits and Tradeoffs in Invariant Representation Learning

Han Zhao
Chen Dan
Bryon Aragam
Tommi S. Jaakkola
Geoffrey J. Gordon
Pradeep Ravikumar

A wide range of machine learning applications such as privacy-preserving learning, algorithmic fairness, and domain adaptation/generalization among others, involve learning invariant representations of the data that aim to achieve two competing goals: (a) maximize information or accuracy with respect to a target response, and (b) maximize invariance or independence with respect to a set of protected features (e.g.\ for fairness, privacy, etc). Despite their wide applicability, theoretical understanding of the optimal tradeoffs --- with respect to accuracy, and invariance --- achievable by invariant representations is still severely lacking. In this paper, we provide an information theoretic analysis of such tradeoffs under both classification and regression settings. More precisely, we provide a geometric characterization of the accuracy and invariance achievable by any representation of the data; we term this feasible region the information plane. We provide an inner bound for this feasible region for the classification case, and an exact characterization for the regression case, which allows us to either bound or exactly characterize the Pareto optimal frontier between accuracy and invariance. Although our contributions are mainly theoretical, a key practical application of our results is in certifying the potential sub-optimality of any given representation learning algorithm for either classification or regression tasks. Our results shed new light on the fundamental interplay between accuracy and invariance, and may be useful in guiding the design of future representation learning algorithms. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

NeurIPS Conference 2022 Conference Paper

Identifiability of deep generative models without auxiliary information

Bohdan Kivva
Goutham Rajendran
Pradeep Ravikumar
Bryon Aragam

We prove identifiability of a broad class of deep latent variable models that (a) have universal approximation capabilities and (b) are the decoders of variational autoencoders that are commonly used in practice. Unlike existing work, our analysis does not require weak supervision, auxiliary information, or conditioning in the latent space. Specifically, we show that for a broad class of generative (i. e. unsupervised) models with universal approximation capabilities, the side information $u$ is not necessary: We prove identifiability of the entire generative model where we do not observe $u$ and only observe the data $x$. The models we consider match autoencoder architectures used in practice that leverage mixture priors in the latent space and ReLU/leaky-ReLU activations in the encoder, such as VaDE and MFC-VAE. Our main result is an identifiability hierarchy that significantly generalizes previous work and exposes how different assumptions lead to different ``strengths'' of identifiability, and includes certain ``vanilla'' VAEs with isotropic Gaussian priors as a special case. For example, our weakest result establishes (unsupervised) identifiability up to an affine transformation, and thus partially resolves an open problem regarding model identifiability raised in prior work. These theoretical results are augmented with experiments on both simulated and real data.

NeurIPS Conference 2021 Conference Paper

Efficient Bayesian network structure learning via local Markov boundary search

Ming Gao
Bryon Aragam

We analyze the complexity of learning directed acyclic graphical models from observational data in general settings without specific distributional assumptions. Our approach is information-theoretic and uses a local Markov boundary search procedure in order to recursively construct ancestral sets in the underlying graphical model. Perhaps surprisingly, we show that for certain graph ensembles, a simple forward greedy search algorithm (i. e. without a backward pruning phase) suffices to learn the Markov boundary of each node. This substantially improves the sample complexity, which we show is at most polynomial in the number of nodes. This is then applied to learn the entire graph under a novel identifiability condition that generalizes existing conditions from the literature. As a matter of independent interest, we establish finite-sample guarantees for the problem of recovering Markov boundaries from data. Moreover, we apply our results to the special case of polytrees, for which the assumptions simplify, and provide explicit conditions under which polytrees are identifiable and learnable in polynomial time. We further illustrate the performance of the algorithm, which is easy to implement, in a simulation study. Our approach is general, works for discrete or continuous distributions without distributional assumptions, and as such sheds light on the minimal assumptions required to efficiently learn the structure of directed graphical models from data.

NeurIPS Conference 2021 Conference Paper

Learning latent causal graphs via mixture oracles

Bohdan Kivva
Goutham Rajendran
Pradeep Ravikumar
Bryon Aragam

We study the problem of reconstructing a causal graphical model from data in the presence of latent variables. The main problem of interest is recovering the causal structure over the latent variables while allowing for general, potentially nonlinear dependencies. In many practical problems, the dependence between raw observations (e. g. pixels in an image) is much less relevant than the dependence between certain high-level, latent features (e. g. concepts or objects), and this is the setting of interest. We provide conditions under which both the latent representations and the underlying latent causal model are identifiable by a reduction to a mixture oracle. These results highlight an intriguing connection between the well-studied problem of learning the order of a mixture model and the problem of learning the bipartite structure between observables and unobservables. The proof is constructive, and leads to several algorithms for explicitly reconstructing the full graphical model. We discuss efficient algorithms and provide experiments illustrating the algorithms in practice.

NeurIPS Conference 2021 Conference Paper

Structure learning in polynomial time: Greedy algorithms, Bregman information, and exponential families

Goutham Rajendran
Bohdan Kivva
Ming Gao
Bryon Aragam

Greedy algorithms have long been a workhorse for learning graphical models, and more broadly for learning statistical models with sparse structure. In the context of learning directed acyclic graphs, greedy algorithms are popular despite their worst-case exponential runtime. In practice, however, they are very efficient. We provide new insight into this phenomenon by studying a general greedy score-based algorithm for learning DAGs. Unlike edge-greedy algorithms such as the popular GES and hill-climbing algorithms, our approach is vertex-greedy and requires at most a polynomial number of score evaluations. We then show how recent polynomial-time algorithms for learning DAG models are a special case of this algorithm, thereby illustrating how these order-based algorithms can be rigourously interpreted as score-based algorithms. This observation suggests new score functions and optimality conditions based on the duality between Bregman divergences and exponential families, which we explore in detail. Explicit sample and computational complexity bounds are derived. Finally, we provide extensive experiments suggesting that this algorithm indeed optimizes the score in a variety of settings.

NeurIPS Conference 2020 Conference Paper

A polynomial-time algorithm for learning nonparametric causal graphs

Ming Gao
Yi Ding
Bryon Aragam

We establish finite-sample guarantees for a polynomial-time algorithm for learning a nonlinear, nonparametric directed acyclic graphical (DAG) model from data. The analysis is model-free and does not assume linearity, additivity, independent noise, or faithfulness. Instead, we impose a condition on the residual variances that is closely related to previous work on linear models with equal variances. Compared to an optimal algorithm with oracle knowledge of the variable ordering, the additional cost of the algorithm is linear in the dimension $d$ and the number of samples $n$. Finally, we compare the proposed algorithm to existing approaches in a simulation study.

UAI Conference 2020 Conference Paper

Automated Dependence Plots

David I. Inouye
Liu Leqi
Joon Sik Kim
Bryon Aragam
Pradeep Ravikumar

In practical applications of machine learning, it is necessary to look beyond standard metrics such as test accuracy in order to validate various qualitative properties of a model. Partial dependence plots (PDP), including instance-specific PDPs (i. e. , ICE plots), have been widely used as a visual tool to understand or validate a model. Yet, current PDPs suffer from two main drawbacks: (1) a user must manually sort or select interesting plots, and (2) PDPs are usually limited to plots along a single feature. To address these drawbacks, we formalize a method for automating the selection of interesting PDPs and extend PDPs beyond showing single features to show the model response along arbitrary directions, for example in raw feature space or a latent space arising from some generative model. We demonstrate the usefulness of our automated dependence plots (ADP) across multiple use-cases and datasets including model selection, bias detection, understanding out-of-sample behavior, and exploring the latent space of a generative model. The code is available at.

ICML Conference 2019 Conference Paper

Fault Tolerance in Iterative-Convergent Machine Learning

Aurick Qiao
Bryon Aragam
Bingjing Zhang
Eric P. Xing

Machine learning (ML) training algorithms often possess an inherent self-correcting behavior due to their iterative- convergent nature. Recent systems exploit this property to achieve adaptability and efficiency in unreliable computing environments by relaxing the consistency of execution and allowing calculation errors to be self-corrected during training. However, the behavior of such systems are only well understood for specific types of calculation errors, such as those caused by staleness, reduced precision, or asynchronicity, and for specific algorithms, such as stochastic gradient descent. In this paper, we develop a general framework to quantify the effects of calculation errors on iterative-convergent algorithms. We then use this framework to derive a worst-case upper bound on the cost of arbitrary perturbations to model parameters during training and to design new strategies for checkpoint-based fault tolerance. Our system, SCAR, can reduce the cost of partial failures by 78%{–}95% when compared with traditional checkpoint-based fault tolerance across a variety of ML models and training algorithms, providing near-optimal performance in recovering from failures.

NeurIPS Conference 2019 Conference Paper

Globally optimal score-based learning of directed acyclic graphs in high-dimensions

Bryon Aragam
Arash Amini
Qing Zhou

We prove that $\Omega(s\log p)$ samples suffice to learn a sparse Gaussian directed acyclic graph (DAG) from data, where $s$ is the maximum Markov blanket size. This improves upon recent results that require $\Omega(s^{4}\log p)$ samples in the equal variance case. To prove this, we analyze a popular score-based estimator that has been the subject of extensive empirical inquiry in recent years and is known to achieve state-of-the-art results. Furthermore, the approach we study does not require strong assumptions such as faithfulness that existing theory for score-based learning crucially relies on. The resulting estimator is based around a difficult nonconvex optimization problem, and its analysis may be of independent interest given recent interest in nonconvex optimization in machine learning. Our analysis overcomes the drawbacks of existing theoretical analyses, which either fail to guarantee structure consistency in high-dimensions (i. e. learning the correct graph with high probability), or rely on restrictive assumptions. In contrast, we give explicit finite-sample bounds that are valid in the important $p\gg n$ regime.

NeurIPS Conference 2019 Conference Paper

Learning Sample-Specific Models with Low-Rank Personalized Regression

Ben Lengerich
Bryon Aragam
Eric Xing

Modern applications of machine learning (ML) deal with increasingly heterogeneous datasets comprised of data collected from overlapping latent subpopulations. As a result, traditional models trained over large datasets may fail to recognize highly predictive localized effects in favour of weakly predictive global patterns. This is a problem because localized effects are critical to developing individualized policies and treatment plans in applications ranging from precision medicine to advertising. To address this challenge, we propose to estimate sample-specific models that tailor inference and prediction at the individual level. In contrast to classical ML models that estimate a single, complex model (or only a few complex models), our approach produces a model personalized to each sample. These sample-specific models can be studied to understand subgroup dynamics that go beyond coarse-grained class labels. Crucially, our approach does not assume that relationships between samples (e. g. a similarity network) are known a priori. Instead, we use unmodeled covariates to learn a latent distance metric over the samples. We apply this approach to financial, biomedical, and electoral data as well as simulated data and show that sample-specific models provide fine-grained interpretations of complicated phenomena without sacrificing predictive accuracy compared to state-of-the-art models such as deep neural networks.

NeurIPS Conference 2018 Conference Paper

DAGs with NO TEARS: Continuous Optimization for Structure Learning

Xun Zheng
Bryon Aragam
Pradeep Ravikumar
Eric Xing

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: we formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree.

NeurIPS Conference 2018 Conference Paper

The Sample Complexity of Semi-Supervised Learning with Nonparametric Mixture Models

Chen Dan
Liu Leqi
Bryon Aragam
Pradeep Ravikumar
Eric Xing

We study the sample complexity of semi-supervised learning (SSL) and introduce new assumptions based on the mismatch between a mixture model learned from unlabeled data and the true mixture model induced by the (unknown) class conditional distributions. Under these assumptions, we establish an $\Omega(K\log K)$ labeled sample complexity bound without imposing parametric assumptions, where $K$ is the number of classes. Our results suggest that even in nonparametric settings it is possible to learn a near-optimal classifier using only a few labeled samples. Unlike previous theoretical work which focuses on binary classification, we consider general multiclass classification ($K>2$), which requires solving a difficult permutation learning problem. This permutation defines a classifier whose classification error is controlled by the Wasserstein distance between mixing measures, and we provide finite-sample results characterizing the behaviour of the excess risk of this classifier. Finally, we describe three algorithms for computing these estimators based on a connection to bipartite graph matching, and perform experiments to illustrate the superiority of the MLE over the majority vote estimator.

JMLR Journal 2015 Journal Article

Concave Penalized Estimation of Sparse Gaussian Bayesian Networks

Bryon Aragam
Qing Zhou

We develop a penalized likelihood estimation framework to learn the structure of Gaussian Bayesian networks from observational data. In contrast to recent methods which accelerate the learning problem by restricting the search space, our main contribution is a fast algorithm for score-based structure learning which does not restrict the search space in any way and works on high-dimensional data sets with thousands of variables. Our use of concave regularization, as opposed to the more popular $\ell_0$ (e.g. BIC) penalty, is new. Moreover, we provide theoretical guarantees which generalize existing asymptotic results when the underlying distribution is Gaussian. Most notably, our framework does not require the existence of a so-called faithful DAG representation, and as a result, the theory must handle the inherent nonidentifiability of the estimation problem in a novel way. Finally, as a matter of independent interest, we provide a comprehensive comparison of our approach to several standard structure learning methods using open-source packages developed for the R language. Based on these experiments, we show that our algorithm obtains higher sensitivity with comparable false discovery rates for high- dimensional data and scales efficiently as the number of nodes increases. In particular, the total runtime for our method to generate a solution path of 20 estimates for DAGs with 8000 nodes is around one hour. [abs] [ pdf ][ bib ] &copy JMLR 2015. ( edit, beta )