Fair Graph Neural Networks (GNNs) have been extensively studied in graph-based applications. However, most approaches to fair GNNs assume the full availability of demographic information by default, which is often unrealistic due to legal restrictions or privacy concerns, leaving a noticeable gap in methods for addressing bias under such constraints. To this end, we propose a novel method for fair graph learning without demographic information. Our approach leverages a Bayesian variational autoencoder to infer missing demographic information and uses disentangled latent variables to separately capture demographics-related and label-related information, reducing interference when inferring demographic proxies. Additionally, we incorporate a fairness regularizer that enables measuring model fairness without demographics while optimizing the fairness objective. Extensive experiments on three real-world graph datasets demonstrate the proposed method’s effectiveness in improving both fairness and utility.

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.

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}.

Learning the structures of structural equation models (SEMs) as directed acyclic graphs (DAGs) from data is crucial for representing causal relationships in various scientific domains. Instead of estimating individual DAG structures, it is often preferable to directly estimate changes in causal relations between conditions, such as changes in genetic expression between healthy and diseased subjects. This work studies the problem of directly estimating the difference between two linear SEMs, i.e. *without estimating the individual DAG structures*, given two sets of samples drawn from the individual SEMs. We consider general classes of linear SEMs where the noise distributions are allowed to be Gaussian or non-Gaussian and have different noise variances across the variables in the individual SEMs. We rigorously characterize novel conditions related to the topological layering of the structural difference that lead to the *identifiability* of the difference DAG (DDAG). Moreover, we propose an *efficient* algorithm to identify the DDAG via sequential re-estimation of the difference of precision matrices. A surprising implication of our results is that causal changes can be identifiable even between *non-identifiable* models such as Gaussian SEMs with unequal noise variances. Synthetic experiments are presented to validate our theoretical results and to show the scalability of our method.

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.

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.

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.

Performing inference in graphs is a common task within several machine learning problems, e.g., image segmentation, community detection, among others. For a given undirected connected graph, we tackle the statistical problem of exactly recovering an unknown ground-truth binary labeling of the nodes from a single corrupted observation of each edge. Such problem can be formulated as a quadratic combinatorial optimization problem over the boolean hypercube, where it has been shown before that one can (with high probability and in polynomial time) exactly recover the ground-truth labeling of graphs that have an isoperimetric number that grows with respect to the number of nodes (e.g., complete graphs, regular expanders). In this work, we apply a powerful hierarchy of relaxations, known as the sum-of-squares (SoS) hierarchy, to the combinatorial problem. Motivated by empirical evidence on the improvement in exact recoverability, we center our attention on the degree-4 SoS relaxation and set out to understand the origin of such improvement from a graph theoretical perspective. We show that the solution of the dual of the relaxed problem is related to finding edge weights of the Johnson and Kneser graphs, where the weights fulfill the SoS constraints and intuitively allow the input graph to increase its algebraic connectivity. Finally, as byproduct of our analysis, we derive a novel Cheeger-type lower bound for the algebraic connectivity of graphs with signed edge weights.

Inference in structured prediction is naturally modeled with a graph, where the goal is to recover the unknown true label for each node given noisy observations corresponding to nodes and edges. The focus of this paper is on the fundamental limits of exact recovery irrespective of computational efficiency, assuming the generative process proposed by [1]. Analyzing the fundamental limits is crucial for algorithm evaluation and development. In this regard, we establish the information-theoretic limit bounds and show that there exists a gap between the limits and the performance of the existent tractable method [2], implying the need for further development of algorithms for exact inference. The fundamental limit we suggest applies to general connected graphs and involves graphical metrics such as the Cheeger constant and the maximum degree. Finally, we reveal that the sufficient and necessary conditions derived from the limit bounds are tight up to a logarithmic factor for a wide range of graphs.

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.

Robustness of machine learning methods is essential for modern practical applications. Given the arms race between attack and defense mechanisms, it is essential to understand the fundamental limits of any conceivable learning method used in an adversarial setting. In this work, we focus on the problem of learning from noise-injected data, where the existing literature falls short by either assuming a specific adversary model or by over-specifying the learning problem. We shed light on the information-theoretic limits of adversarial learning without assuming a particular adversary. Specifically, we derive a general Le Cam type bound for learning from noise-injected data. Finally, we apply our general bounds to a canonical set of non-trivial learning problems and provide examples of common types of noise-injected data.

Inverse Reinforcement Learning (IRL) is the problem of finding a reward function which describes observed/known expert behavior. The IRL setting is remarkably useful for automated control, in situations where the reward function is difficult to specify manually or as a means to extract agent preference. In this work, we provide a new IRL algorithm for the continuous state space setting with unknown transition dynamics by modeling the system using a basis of orthonormal functions. Moreover, we provide a proof of correctness and formal guarantees on the sample and time complexity of our algorithm. Finally, we present synthetic experiments to corroborate our theoretical guarantees.

Structured prediction can be considered as a generalization of many standard supervised learning tasks, and is usually thought as a simultaneous prediction of multiple labels. One standard approach is to maximize a score function on the space of labels, which usually decomposes as a sum of unary and pairwise potentials, each depending on one or two specific labels, respectively.For this approach, several learning and inference algorithms have been proposed over the years, ranging from exact to approximate methods while balancing the computational complexity.However, in contrast to binary and multiclass classification, results on the necessary number of samples for achieving learning are still limited, even for a specific family of predictors such as factor graphs.In this work, we provide minimax lower bounds for a class of general factor-graph inference models in the context of structured prediction.That is, we characterize the necessary sample complexity for any conceivable algorithm to achieve learning of general factor-graph predictors.

Many inference problems in structured prediction can be modeled as maximizing a score function on a space of labels, where graphs are a natural representation to decompose the total score into a sum of unary (nodes) and pairwise (edges) scores. Given a generative model with an undirected connected graph G and true vector of binary labels $\bar{y}$, it has been previously shown that when G has good expansion properties, such as complete graphs or d-regular expanders, one can exactly recover $\bar{y}$ (with high probability and in polynomial time) from a single noisy observation of each edge and node. We analyze the previously studied generative model by Globerson et al. (2015) under a notion of statistical parity. That is, given a fair binary node labeling, we ask the question whether it is possible to recover the fair assignment, with high probability and in polynomial time, from single edge and node observations. We find that, in contrast to the known trade-offs between fairness and model performance, the addition of the fairness constraint improves the probability of exact recovery. We effectively explain this phenomenon and empirically show how graphs with poor expansion properties, such as grids, are now capable of achieving exact recovery. Finally, as a byproduct of our analysis, we provide a tighter minimum-eigenvalue bound than that which can be derived from Weyl's inequality.

Structured prediction can be thought of as a simultaneous prediction of multiple labels. This is often done by maximizing a score function on the space of labels, which decomposes as a sum of pairwise and unary potentials. The above is naturally modeled with a graph, where edges and vertices are related to pairwise and unary potentials, respectively. We consider the generative process proposed by Globerson et al. (2015) and apply it to general connected graphs. We analyze the structural conditions of the graph that allow for the exact recovery of the labels. Our results show that exact recovery is possible and achievable in polynomial time for a large class of graphs. In particular, we show that graphs that are bad expanders can be exactly recovered by adding small edge perturbations coming from the \Erdos-\Renyi model. Finally, as a byproduct of our analysis, we provide an extension of Cheeger's inequality.

The standard margin-based structured prediction commonly uses a maximum loss over all possible structured outputs. The large-margin formulation including latent variables not only results in a non-convex formulation but also increases the search space by a factor of the size of the latent space. Recent work has proposed the use of the maximum loss over random structured outputs sampled independently from some proposal distribution, with theoretical guarantees. We extend this work by including latent variables. We study a new family of loss functions under Gaussian perturbations and analyze the effect of the latent space on the generalization bounds. We show that the non-convexity of learning with latent variables originates naturally, as it relates to a tight upper bound of the Gibbs decoder distortion with respect to the latent space. Finally, we provide a formulation using random samples and relaxations that produces a tighter upper bound of the Gibbs decoder distortion up to a statistical accuracy, which enables a polynomial time evaluation of the objective function. We illustrate the method with synthetic experiments and a computer vision application.

Causal discovery from empirical data is a fundamental problem in many scientific domains. Observational data allows for identifiability only up to Markov equivalence class. In this paper we first propose a polynomial time algorithm for learning the exact correctly-oriented structure of the transitive reduction of any causal Bayesian network with high probability, by using interventional path queries. Each path query takes as input an origin node and a target node, and answers whether there is a directed path from the origin to the target. This is done by intervening on the origin node and observing samples from the target node. We theoretically show the logarithmic sample complexity for the size of interventional data per path query, for continuous and discrete networks. We then show how to learn the transitive edges using also logarithmic sample complexity (albeit in time exponential in the maximum number of parents for discrete networks), which allows us to learn the full network. We further extend our work by reducing the number of interventional path queries for learning rooted trees. We also provide an analysis of imperfect interventions.