We consider a matrix completion problem that exploits social or item similarity graphs as side information. We develop a universal, parameter-free, and computationally efficient algorithm that starts with hierarchical graph clustering and then iteratively refines estimates both on graph clustering and matrix ratings. Under a hierarchical stochastic block model that well respects practically-relevant social graphs and a low-rank rating matrix model (to be detailed), we demonstrate that our algorithm achieves the information-theoretic limit on the number of observed matrix entries (i.e., optimal sample complexity) that is derived by maximum likelihood estimation together with a lower-bound impossibility result. One consequence of this result is that exploiting the hierarchical structure of social graphs yields a substantial gain in sample complexity relative to the one that simply identifies different groups without resorting to the relational structure across them. We conduct extensive experiments both on synthetic and real-world datasets to corroborate our theoretical results as well as to demonstrate significant performance improvements over other matrix completion algorithms that leverage graph side information.

Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution that best explain the observed data. In the context of text generation, MLE is often used to train generative language models, which can then be used to generate new text. However, we argue that MLE is not always necessary and optimal, especially for closed-ended text generation tasks like machine translation. In these tasks, the goal of model is to generate the most appropriate response, which does not necessarily require it to estimate the entire data distribution with MLE. To this end, we propose a novel class of training objectives based on convex functions, which enables text generation models to focus on highly probable outputs without having to estimate the entire data distribution. We investigate the theoretical properties of the optimal predicted distribution when applying convex functions to the loss, demonstrating that convex functions can sharpen the optimal distribution, thereby enabling the model to better capture outputs with high probabilities. Experiments on various text generation tasks and models show the effectiveness of our approach. It enables autoregressive models to bridge the gap between greedy and beam search, and facilitates the learning of non-autoregressive models with a maximum improvement of 9+ BLEU points. Moreover, our approach also exhibits significant impact on large language models (LLMs), substantially enhancing their generative capability on various tasks.

Tractable probabilistic models such as cutset networks which admit exact linear time posterior marginal inference are often preferred in practice over intractable models such as Bayesian and Markov networks. This is because although tractable models, when learned from data, are slightly inferior to the intractable ones in terms of goodness-of-fit measures such as log-likelihood, they do not use approximate inference at prediction time and as a result exhibit superior predictive performance. In this paper, we consider the problem of improving a tractable model using a large number of local probability estimates, each defined over a small subset of variables that are either available from experts or via an external process. Given a model learned from fully-observed, but small amount of possibly noisy data, the key idea in our approach is to update the parameters of the model via a gradient descent procedure that seeks to minimize a convex combination of two quantities: one that enforces closeness via KL divergence to the local estimates and another that enforces closeness to the given model. We show that although the gradients are NP-hard to compute on arbitrary graphical models, they can be efficiently computed over tractable models. We show via experiments that our approach yields tractable models that are significantly superior to the ones learned from small amount of possibly noisy data, even when the local estimates are inconsistent.

We consider the parameter estimation problem of a probabilistic generative model prescribed using a natural exponential family of distributions. For this problem, the typical maximum likelihood estimator usually overfits under limited training sample size, is sensitive to noise and may perform poorly on downstream predictive tasks. To mitigate these issues, we propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric Kullback-Leibler ball around a parametric nominal distribution. Leveraging the analytical expression of the Kullback-Leibler divergence between two distributions in the same natural exponential family, we show that the min-max estimation problem is tractable in a broad setting, including the robust training of generalized linear models. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.

In this paper, we consider the problem of learning a sparse graph from the Laplacian constrained Gaussian graphical model. This problem can be formulated as a penalized maximum likelihood estimation of the precision matrix under Laplacian structural constraints. Like in the classical graphical lasso problem, recent works made use of the $\ell_1$-norm with the goal of promoting sparsity in the Laplacian constrained precision matrix estimation. However, through empirical evidence, we observe that the $\ell_1$-norm is not effective in imposing a sparse solution in this problem. From a theoretical perspective, we prove that a large regularization parameter will surprisingly lead to a solution representing a fully connected graph instead of a sparse graph. To address this issue, we propose a nonconvex penalized maximum likelihood estimation method, and establish the order of the statistical error. Numerical experiments involving synthetic and real-world data sets demonstrate the effectiveness of the proposed method.

Autoregressive models use chain rule to define a joint probability distribution as a product of conditionals. These conditionals need to be normalized, imposing constraints on the functional families that can be used. To increase flexibility, we propose autoregressive conditional score models (AR-CSM) where we parameterize the joint distribution in terms of the derivatives of univariate log-conditionals (scores), which need not be normalized. To train AR-CSM, we introduce a new divergence between distributions named Composite Score Matching (CSM). For AR-CSM models, this divergence between data and model distributions can be computed and optimized efficiently, requiring no expensive sampling or adversarial training. Compared to previous score matching algorithms, our method is more scalable to high dimensional data and more stable to optimize. We show with extensive experimental results that it can be applied to density estimation on synthetic data, image generation, image denoising, and training latent variable models with implicit encoders.

We give methods for Bayesian inference of directed acyclic graphs, DAGs, and the induced causal effects from passively observed complete data. Our methods build on a recent Markov chain Monte Carlo scheme for learning Bayesian networks, which enables efficient approximate sampling from the graph posterior, provided that each node is assigned a small number K of candidate parents. We present algorithmic techniques to significantly reduce the space and time requirements, which make the use of substantially larger values of K feasible. Furthermore, we investigate the problem of selecting the candidate parents per node so as to maximize the covered posterior mass. Finally, we combine our sampling method with a novel Bayesian approach for estimating causal effects in linear Gaussian DAG models. Numerical experiments demonstrate the performance of our methods in detecting ancestor-descendant relations, and in causal effect estimation our Bayesian method is shown to outperform previous approaches.

Variational Autoencoders (VAEs) are powerful probabilistic models to learn representations of complex data distributions. One important limitation of VAEs is the strong prior assumption that latent representations learned by the model follow a simple uni-modal Gaussian distribution. Further, the variational training procedure poses considerable practical challenges. Recently proposed regularized autoencoders offer a deterministic autoencoding framework, that simplifies the original VAE objective and is significantly easier to train. Since these models only provide weak control over the learned latent distribution, they require an ex-post density estimation step to generate samples comparable to those of VAEs. In this paper, we propose a simple and end-to-end trainable deterministic autoencoding framework, that efficiently shapes the latent space of the model during training and utilizes the capacity of expressive multi-modal latent distributions. The proposed training procedure provides direct evidence if the latent distribution adequately captures complex aspects of the encoded data. We show in experiments the expressiveness and sample quality of our model in various challenging continuous and discrete domains.

We show that the matrix perspective function, which is jointly convex in the Cartesian product of a standard Euclidean vector space and a conformal space of symmetric matrices, has a proximity operator in an almost closed form. The only implicit part is to solve a semismooth, univariate root finding problem. We uncover the connection between our problem of study and the matrix nearness problem. Through this connection, we propose a quadratically convergent Newton algorithm for the root finding problem.Experiments verify that the evaluation of the proximity operator requires at most 8 Newton steps, taking less than 5s for 2000 by 2000 matrices on a standard laptop. Using this routine as a building block, we demonstrate the usefulness of the studied proximity operator in constrained maximum likelihood estimation of Gaussian mean and covariance, peudolikelihood-based graphical model selection, and a matrix variant of the scaled lasso problem.

Analyzing causality in multivariate systems involves establishing how information is generated, distributed and combined, and thus requires tools that capture interactions beyond pairwise relations. Higher-order information theory provides such tools. In particular, Partial Information Decomposition (PID) allows the decomposition of the information that a set of sources provides about a target into redundant, unique, and synergistic components. Yet the mathematical connection between such higher-order information-theoretic measures and causal structure remains undeveloped. Here we establish the first theoretical correspondence between PID components and causal structure in both Bayesian networks and hypergraphs. We first show that in Bayesian networks unique information precisely characterizes direct causal neighbors, while synergy identifies collider relationships. This establishes a localist causal discovery paradigm in which the structure surrounding each variable can be recovered from its immediate informational footprint, eliminating the need for global search over graph space. Extending these results to higher-order systems, we prove that PID signatures in Bayesian hypergraphs differentiate parents, children, co-heads, and co-tails, revealing a higher-order collider effect unique to multi-tail hyperedges. We also present procedures by which our results can be used to characterize systematically the causal structure of Bayesian networks and hypergraphs. Our results position PID as a rigorous, model-agnostic foundation for inferring both pairwise and higher-order causal structure, and introduce a fundamentally local information-theoretic viewpoint on causal discovery.