Trajectory inference seeks to recover the temporal dynamics of a population from snapshots of its (uncoupled) temporal marginals, i.e. where observed particles are \emph{not} tracked over time. Prior works addressed this challenging problem under a stochastic differential equation (SDE) model with a gradient-driven drift in the observed space, introducing a minimum entropy estimator relative to the Wiener measure and a practical grid-free mean-field Langevin (MFL) algorithm using Schr\"odinger bridges. Motivated by the success of observable state space models in the traditional paired trajectory inference problem (e.g. target tracking), we extend the above framework to a class of latent SDEs in the form of \emph{observable state space models}. In this setting, we use partial observations to infer trajectories in the latent space under a specified dynamics model (e.g. the constant velocity/acceleration models from target tracking). We introduce the PO-MFL algorithm to solve this latent trajectory inference problem and provide theoretical guarantees to the partially observed setting. Experiments validate the robustness of our method and the exponential convergence of the MFL dynamics, and demonstrate significant outperformance over the latent-free baseline in key scenarios.

Fine-tuning large language models (LLMs) with low-rank adaptations (LoRAs) has become common practice, often yielding numerous copies of the same LLM differing only in their LoRA updates. This paradigm presents challenges for systems that serve real-time responses to queries that each involve a different LoRA. Prior works optimize the design of such systems but still require continuous loading and offloading of LoRAs, as it is infeasible to store thousands of LoRAs in GPU memory. To mitigate this issue, we investigate the efficacy of compression when serving LoRAs. We propose a method for the joint compression of LoRAs into a shared basis paired with LoRA-specific scaling matrices. We extend our algorithm to learn clusters of LoRAs that are amenable to joint compression, allowing it to scale gracefully to large LoRA collections. Our experiments with up to 1000 LoRAs demonstrate that compressed LoRAs preserve performance while offering major throughput gains in realistic serving scenarios with over a thousand LoRAs, maintaining 80\% of the throughput of serving a single LoRA.

Low-Rank Adaptation (LoRA) has emerged as a highly efficient framework for finetuning the weights of large foundation models, and has become the go-to method for data-driven customization of LLMs. Despite the promise of highly customized behaviors and capabilities, switching between relevant LoRAs in a multiturn setting is inefficient, as the key-value (KV) cache of the entire turn history must be recomputed with the LoRA weights before generation can begin. To address this problem, we propose Activated LoRA (aLoRA), an adapter architecture which modifies the LoRA framework to only adapt weights for the tokens in the sequence after the aLoRA is invoked. This change crucially allows aLoRA to accept the base model's KV cache of the input string, meaning that aLoRA can be instantly activated whenever needed in a chain without recomputing the prior keys and values. This enables building what we call intrinsics, i.e. specialized models invoked to perform well-defined operations on portions of an input chain or conversation that otherwise uses the base model by default. We train a set of aLoRA-based intrinsics models, demonstrating competitive accuracy with standard LoRA while significantly improving inference efficiency. We contributed our Activated LoRA implementation to the Huggingface PEFT library.

Process reward models (PRMs) play a central role in guiding inference-time scaling algorithms for large language models (LLMs). However, we observe that even state-of-the-art PRMs can be poorly calibrated. Specifically, they tend to overestimate the success probability that a partial reasoning step will lead to a correct final answer, particularly when smaller LLMs are used to complete the reasoning trajectory. To address this, we present a calibration approach—performed via quantile regression—that adjusts PRM outputs to better align with true success probabilities. Leveraging these calibrated success estimates and their associated confidence bounds, we introduce an instance-adaptive scaling (IAS) framework that dynamically adjusts the compute budget based on the estimated likelihood that a partial reasoning trajectory will yield a correct final answer. Unlike conventional methods that allocate a fixed number of reasoning trajectories per query, this approach adapts to each instance and reasoning step when using our calibrated PRMs. Experiments on mathematical reasoning benchmarks show that (i) our PRM calibration method achieves small calibration error, outperforming the baseline methods, (ii) calibration is crucial for enabling effective IAS, and (iii) the proposed IAS strategy reduces inference costs while maintaining final answer accuracy, utilizing less compute on more confident problems as desired.

The ability of machine learning (ML) algorithms to generalize well to unseen data has been studied through the lens of information theory, by bounding the generalization error with the input-output mutual information (MI), i.e., the MI between the training data and the learned hypothesis. Yet, these bounds have limited practicality for modern ML applications (e.g., deep learning), due to the difficulty of evaluating MI in high dimensions. Motivated by recent findings on the compressibility of neural networks, we consider algorithms that operate by *slicing* the parameter space, i.e., trained on random lower-dimensional subspaces. We introduce new, tighter information-theoretic generalization bounds tailored for such algorithms, demonstrating that slicing improves generalization. Our bounds offer significant computational and statistical advantages over standard MI bounds, as they rely on scalable alternative measures of dependence, i.e., disintegrated mutual information and $k$-sliced mutual information. Then, we extend our analysis to algorithms whose parameters do not need to exactly lie on random subspaces, by leveraging rate-distortion theory. This strategy yields generalization bounds that incorporate a distortion term measuring model compressibility under slicing, thereby tightening existing bounds without compromising performance or requiring model compression. Building on this, we propose a regularization scheme enabling practitioners to control generalization through compressibility. Finally, we empirically validate our results and achieve the computation of non-vacuous information-theoretic generalization bounds for neural networks, a task that was previously out of reach.

We propose a distributional framework for benchmarking socio-technical risks of foundation models with quantified statistical significance. Our approach hinges on a new statistical relative testing based on first and second order stochastic dominance of real random variables. We show that the second order statistics in this test are linked to mean-risk models commonly used in econometrics and mathematical finance to balance risk and utility when choosing between alternatives. Using this framework, we formally develop a risk-aware approach for foundation model selection given guardrails quantified by specified metrics. Inspired by portfolio optimization and selection theory in mathematical finance, we define a metrics portfolio for each model as a means to aggregate a collection of metrics, and perform model selection based on the stochastic dominance of these portfolios. The statistical significance of our tests is backed theoretically by an asymptotic analysis via central limit theorems instantiated in practice via a bootstrap variance estimate. We use our framework to compare various large language models regarding risks related to drifting from instructions and outputting toxic content.

We consider the issue of calibration in large language models (LLM). Recent studies have found that common interventions such as instruction tuning often result in poorly calibrated LLMs. Although calibration is well-explored in traditional applications, calibrating LLMs is uniquely challenging. These challenges stem as much from the severe computational requirements of LLMs as from their versatility, which allows them to be applied to diverse tasks. Addressing these challenges, we propose THERMOMETER, a calibration approach tailored to LLMs. THERMOMETER learns an auxiliary model, given data from multiple tasks, for calibrating a LLM. It is computationally efficient, preserves the accuracy of the LLM, and produces better-calibrated responses for new tasks. Extensive empirical evaluations across various benchmarks demonstrate the effectiveness of the proposed method.

Parameter-efficient fine-tuning optimizes large, pre-trained foundation models by updating a subset of parameters; in this class, Low-Rank Adaptation (LoRA) is particularly effective. Inspired by an effort to investigate the different roles of LoRA matrices during fine-tuning, this paper characterizes and leverages unexpected asymmetry in the importance of low-rank adapter matrices. Specifically, when updating the parameter matrices of a neural network by adding a product $BA$, we observe that the $B$ and $A$ matrices have distinct functions: $A$ extracts features from the input, while $B$ uses these features to create the desired output. Based on this observation, we demonstrate that fine-tuning $B$ is inherently more effective than fine-tuning $A$, and that a random untrained $A$ should perform nearly as well as a fine-tuned one. Using an information-theoretic lens, we also bound the generalization of low-rank adapters, showing that the parameter savings of exclusively training $B$ improves the bound. We support our conclusions with experiments on RoBERTa, BART-Large, LLaMA-2, and ViTs. The code and data is available at https://github.com/Jiacheng-Zhu-AIML/AsymmetryLoRA

Many modern datasets exhibit dependencies among observations as well as variables. This gives rise to the challenging problem of analyzing high-dimensional matrix-variate data with unknown dependence structures. To address this challenge, Kalaitzis et. al. (2013) proposed the Bigraphical Lasso, an estimator for precision matrices of matrix-normals based on the Cartesian product of graphs. Subsequently, Greenewald, Zhou and Hero (GZH 2019) introduced a multiway tensor generalization of the Bigraphical Lasso, known as the TeraLasso estimator. In this paper, we provide sharper rates of convergence in the Frobenius and operator norm for both Bigraphical and TeraLasso estimators for inverse covariance matrices. This improves upon the rates presented in GZH 2019. In particular, (a) we strengthen the bounds for the relative errors in the operator and Frobenius norm by a factor of approximately$\log p$; (b) Crucially, this improvement allows for finite-sample estimation errors in both norms to be derived for the two-way Kronecker sum model. This closes the gap between the low single-sample error for the two-way model as observed in GZH 2019 and the lack of theoretical guarantee for this particular case. The two-way regime is important because it is the setting that is the most theoretically challenging, and simultaneously the most common in applications. Normality is not needed in our proofs; instead, we consider subgaussian ensembles and derive tight concentration of measure bounds, using tensor unfolding techniques. The proof techniques may be of independent interest.

Training generative models with differential privacy (DP) typically involves injecting noise into gradient updates or adapting the discriminator's training procedure. As a result, such approaches often struggle with hyper-parameter tuning and convergence. We consider the \emph{slicing privacy mechanism} that injects noise into random low-dimensional projections of the private data, and provide strong privacy guarantees for it. These noisy projections are used for training generative models. To enable optimizing generative models using this DP approach, we introduce the \emph{smoothed-sliced $f$-divergence} and show it enjoys statistical consistency. Moreover, we present a kernel-based estimator for this divergence, circumventing the need for adversarial training. Extensive numerical experiments demonstrate that our approach can generate synthetic data of higher quality compared with baselines. Beyond performance improvement, our method, by sidestepping the need for noisy gradients, offers data scientists the flexibility to adjust generator architecture and hyper-parameters, run the optimization over any number of epochs, and even restart the optimization process---all without incurring additional privacy costs.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

Current LLM alignment techniques use pairwise human preferences at a sample level, and as such, they do not imply an alignment on the distributional level. We propose in this paper Alignment via Optimal Transport (AOT), a novel method for distributional preference alignment of LLMs. AOT aligns LLMs on unpaired preference data by making the reward distribution of the positive samples stochastically dominant in the first order on the distribution of negative samples. We introduce a convex relaxation of this first-order stochastic dominance and cast it as an optimal transport problem with a smooth and convex cost. Thanks to the one-dimensional nature of the resulting optimal transport problem and the convexity of the cost, it has a closed-form solution via sorting on empirical measures. We fine-tune LLMs with this AOT objective, which enables alignment by penalizing the violation of the stochastic dominance of the reward distribution of the positive samples on the reward distribution of the negative samples. We analyze the sample complexity of AOT by considering the dual of the OT problem and show that it converges at the parametric rate. Empirically, we show on a diverse set of alignment datasets and LLMs that AOT leads to state-of-the-art models in the 7B family of models when evaluated with Open LLM Benchmarks and AlpacaEval. Code for $\mathsf{AOT}$ is available in the Hugging Face TRL library \url{https://ibm.biz/AOT_TRL}.

Stochastic dominance is an important concept in probability theory, econometrics and social choice theory for robustly modeling agents' preferences between random outcomes. While many works have been dedicated to the univariate case, little has been done in the multivariate scenario, wherein an agent has to decide between different multivariate outcomes. By exploiting a characterization of multivariate first stochastic dominance in terms of couplings, we introduce a statistic that assesses multivariate almost stochastic dominance under the framework of Optimal Transport with a smooth cost. Further, we introduce an entropic regularization of this statistic, and establish a central limit theorem (CLT) and consistency of the bootstrap procedure for the empirical statistic. Armed with this CLT, we propose a hypothesis testing framework as well as an efficient implementation using the Sinkhorn algorithm. We showcase our method in comparing and benchmarking Large Language Models that are evaluated on multiple metrics. Our multivariate stochastic dominance test allows us to capture the dependencies between the metrics in order to make an informed and statistically significant decision on the relative performance of the models.

Given a set of discrete probability distributions, the minimum entropy coupling is the minimum entropy joint distribution that has the input distributions as its marginals. This has immediate relevance to tasks such as entropic causal inference for causal graph discovery and bounding mutual information between variables that we observe separately. Since finding the minimum entropy coupling is NP-Hard, various works have studied approximation algorithms. The work of [Compton, 2022] shows that the greedy coupling algorithm of [Kocaoglu et al., 2017a] is always within $\log_2(e)$ $\approx$ 1.44 bits of the optimal coupling. Moreover, they show that it is impossible to obtain a better approximation guarantee using the majorization lower-bound that all prior works have used: thus establishing a majorization barrier. In this work, we break the majorization barrier by designing a stronger lower-bound that we call the profile method. Using this profile method, we are able to show that the greedy algorithm is always within $\log_2(e)/e$ $\approx$ 0.53 bits of optimal for coupling two distributions (previous best-known bound is within 1 bit), and within $(1 + \log_2(e))/2$ $\approx$ 1.22 bits for coupling any number of distributions (previous best-known bound is within 1.44 bits). We also examine a generalization of the minimum entropy coupling problem: Concave Minimum-Cost Couplings. We are able to obtain similar guarantees for this generalization in terms of the concave cost function. Additionally, we make progress on the open problem of [Kovačević et al., 2015] regarding NP membership of the minimum entropy coupling problem by showing that any hardness of minimum entropy coupling beyond NP comes from the difficulty of computing arithmetic in the complexity class NP. Finally, we present exponential-time algorithms for computing the exactly optimal solution. We experimentally observe that our new profile method lower bound is not only helpful for analyzing the greedy approximation algorithm, but also for improving the speed of our new backtracking-based exact algorithm.

We study the problem of overcoming exponential sample complexity in differential entropy estimation under Gaussian convolutions. Specifically, we consider the estimation of the differential entropy h(X + Z) via n independently and identically distributed samples of X, where X and Z are independent D-dimensional random variables with X sub-Gaussian with bounded second moment and $Z\sim {\mathcal{N}}\left({0,{\sigma ^2}{I_D}}\right)$. Under the absolute-error loss, the above problem has a parametric estimation rate of $\frac{{{c^D}}}{{\sqrt n }}$, which is exponential in data dimension D and often problematic for applications. We overcome this exponential sample complexity by projecting X to a low-dimensional space via principal component analysis (PCA) before the entropy estimation, and show that the asymptotic error overhead vanishes as the unexplained variance of the PCA vanishes. This implies near-optimal performance for inherently low-dimensional structures embedded in high-dimensional spaces, including hidden-layer outputs of deep neural networks (DNN), which can be used to estimate mutual information (MI) in DNNs. We provide numerical results verifying the performance of our PCA approach on Gaussian and spiral data. We also apply our method to analysis of information flow through neural network layers (c.f. information bottleneck), with results measuring mutual information in a noisy fully connected network and a noisy convolutional neural network (CNN) for MNIST classification.12

Quantifying dependence between high-dimensional random variables is central to statistical learning and inference. Two classical methods are canonical correlation analysis (CCA), which identifies maximally correlated projected versions of the original variables, and Shannon's mutual information, which is a universal dependence measure that also captures high-order dependencies. However, CCA only accounts for linear dependence, which may be insufficient for certain applications, while mutual information is often infeasible to compute/estimate in high dimensions. This work proposes a middle ground in the form of a scalable information-theoretic generalization of CCA, termed max-sliced mutual information (mSMI). mSMI equals the maximal mutual information between low-dimensional projections of the high-dimensional variables, which reduces back to CCA in the Gaussian case. It enjoys the best of both worlds: capturing intricate dependencies in the data while being amenable to fast computation and scalable estimation from samples. We show that mSMI retains favorable structural properties of Shannon's mutual information, like variational forms and identification of independence. We then study statistical estimation of mSMI, propose an efficiently computable neural estimator, and couple it with formal non-asymptotic error bounds. We present experiments that demonstrate the utility of mSMI for several tasks, encompassing independence testing, multi-view representation learning, algorithmic fairness, and generative modeling. We observe that mSMI consistently outperforms competing methods with little-to-no computational overhead.

Existing private synthetic data generation algorithms are agnostic to downstream tasks. However, end users may have specific requirements that the synthetic data must satisfy. Failure to meet these requirements could significantly reduce the utility of the data for downstream use. We introduce a post-processing technique that improves the utility of the synthetic data with respect to measures selected by the end user, while preserving strong privacy guarantees and dataset quality. Our technique involves resampling from the synthetic data to filter out samples that do not meet the selected utility measures, using an efficient stochastic first-order algorithm to find optimal resampling weights. Through comprehensive numerical experiments, we demonstrate that our approach consistently improves the utility of synthetic data across multiple benchmark datasets and state-of-the-art synthetic data generation algorithms.

Causal disentanglement aims to uncover a representation of data using latent variables that are interrelated through a causal model. Such a representation is identifiable if the latent model that explains the data is unique. In this paper, we focus on the scenario where unpaired observational and interventional data are available, with each intervention changing the mechanism of a latent variable. When the causal variables are fully observed, statistically consistent algorithms have been developed to identify the causal model under faithfulness assumptions. We here show that identifiability can still be achieved with unobserved causal variables, given a generalized notion of faithfulness. Our results guarantee that we can recover the latent causal model up to an equivalence class and predict the effect of unseen combinations of interventions, in the limit of infinite data. We implement our causal disentanglement framework by developing an autoencoding variational Bayes algorithm and apply it to the problem of predicting combinatorial perturbation effects in genomics.

Entropic causal inference is a recent framework for learning the causal graph between two variables from observational data by finding the information-theoretically simplest structural explanation of the data, i.e., the model with smallest entropy. In our work, we first extend the causal graph identifiability result in the two-variable setting under relaxed assumptions. We then show the first identifiability result using the entropic approach for learning causal graphs with more than two nodes. Our approach utilizes the property that ancestrality between a source node and its descendants can be determined using the bivariate entropic tests. We provide a sound sequential peeling algorithm for general graphs that relies on this property. We also propose a heuristic algorithm for small graphs that shows strong empirical performance. We rigorously evaluate the performance of our algorithms on synthetic data generated from a variety of models, observing improvement over prior work. Finally we test our algorithms on real-world datasets.

The need for efficiently comparing and representing datasets with unknown alignment spans various fields, from model analysis and comparison in machine learning to trend discovery in collections of medical datasets. We use manifold learning to compare the intrinsic geometric structures of different datasets by comparing their diffusion operators, symmetric positive-definite (SPD) matrices that relate to approximations of the continuous Laplace-Beltrami operator from discrete samples. Existing methods typically assume known data alignment and compare such operators in a pointwise manner. Instead, we exploit the Riemannian geometry of SPD matrices to compare these operators and define a new theoretically-motivated distance based on a lower bound of the log-Euclidean metric. Our framework facilitates comparison of data manifolds expressed in datasets with different sizes, numbers of features, and measurement modalities. Our log-Euclidean signature (LES) distance recovers meaningful structural differences, outperforming competing methods in various application domains.

Sliced mutual information (SMI) is defined as an average of mutual information (MI) terms between one-dimensional random projections of the random variables. It serves as a surrogate measure of dependence to classic MI that preserves many of its properties but is more scalable to high dimensions. However, a quantitative characterization of how SMI itself and estimation rates thereof depend on the ambient dimension, which is crucial to the understanding of scalability, remain obscure. This work provides a multifaceted account of the dependence of SMI on dimension, under a broader framework termed $k$-SMI, which considers projections to $k$-dimensional subspaces. Using a new result on the continuity of differential entropy in the 2-Wasserstein metric, we derive sharp bounds on the error of Monte Carlo (MC)-based estimates of $k$-SMI, with explicit dependence on $k$ and the ambient dimension, revealing their interplay with the number of samples. We then combine the MC integrator with the neural estimation framework to provide an end-to-end $k$-SMI estimator, for which optimal convergence rates are established. We also explore asymptotics of the population $k$-SMI as dimension grows, providing Gaussian approximation results with a residual that decays under appropriate moment bounds. All our results trivially apply to SMI by setting $k=1$. Our theory is validated with numerical experiments and is applied to sliced InfoGAN, which altogether provide a comprehensive quantitative account of the scalability question of $k$-SMI, including SMI as a special case when $k=1$.

The estimation of causal treatment effects from observational data is a fundamental problem in causal inference. To avoid bias, the effect estimator must control for all confounders. Hence practitioners often collect data for as many covariates as possible to raise the chances of including the relevant confounders. While this addresses the bias, this has the side effect of significantly increasing the number of data samples required to accurately estimate the effect due to the increased dimensionality. In this work, we consider the setting where out of a large number of covariates $X$ that satisfy strong ignorability, an unknown sparse subset $S$ is sufficient to include to achieve zero bias, i.e. $c$-equivalent to $X$. We propose a common objective function involving outcomes across treatment cohorts with nonconvex joint sparsity regularization that is guaranteed to recover $S$ with high probability under a linear outcome model for $Y$ and subgaussian covariates for each of the treatment cohort. This improves the effect estimation sample complexity so that it scales with the cardinality of the sparse subset $S$ and $\log |X|$, as opposed to the cardinality of the full set $X$. We validate our approach with experiments on treatment effect estimation.

Understanding the generalization of deep neural networks is one of the most important tasks in deep learning. Although much progress has been made, theoretical error bounds still often behave disparately from empirical observations. In this work, we develop margin-based generalization bounds, where the margins are normalized with optimal transport costs between independent random subsets sampled from the training distribution. In particular, the optimal transport cost can be interpreted as a generalization of variance which captures the structural properties of the learned feature space. Our bounds robustly predict the generalization error, given training data and network parameters, on large scale datasets. Theoretically, we demonstrate that the concentration and separation of features play crucial roles in generalization, supporting empirical results in the literature.

Mutual information (MI) is a fundamental measure of statistical dependence, with a myriad of applications to information theory, statistics, and machine learning. While it possesses many desirable structural properties, the estimation of high-dimensional MI from samples suffers from the curse of dimensionality. Motivated by statistical scalability to high dimensions, this paper proposes sliced MI (SMI) as a surrogate measure of dependence. SMI is defined as an average of MI terms between one-dimensional random projections. We show that it preserves many of the structural properties of classic MI, while gaining scalable computation and efficient estimation from samples. Furthermore, and in contrast to classic MI, SMI can grow as a result of deterministic transformations. This enables leveraging SMI for feature extraction by optimizing it over processing functions of raw data to identify useful representations thereof. Our theory is supported by numerical studies of independence testing and feature extraction, which demonstrate the potential gains SMI offers over classic MI for high-dimensional inference.

Optimal transport (OT) is a popular tool in machine learning to compare probability measures geometrically, but it comes with substantial computational burden. Linear programming algorithms for computing OT distances scale cubically in the size of the input, making OT impractical in the large-sample regime. We introduce a practical algorithm, which relies on a quantization step, to estimate OT distances between measures given cheap sample access. We also provide a variant of our algorithm to improve the performance of approximate solvers, focusing on those for entropy-regularized transport. We give theoretical guarantees on the benefits of this quantization step and display experiments showing that it behaves well in practice, providing a practical approximation algorithm that can be used as a drop-in replacement for existing OT estimators.

Optimal transport (OT), and in particular the Wasserstein distance, has seen a surge of interest and applications in machine learning. However, empirical approximation under Wasserstein distances suffers from a severe curse of dimensionality, rendering them impractical in high dimensions. As a result, entropically regularized OT has become a popular workaround. However, while it enjoys fast algorithms and better statistical properties, it looses the metric structure that Wasserstein distances enjoy. This work proposes a novel Gaussian-smoothed OT (GOT) framework, that achieves the best of both worlds: preserving the 1-Wasserstein metric structure while alleviating the empirical approximation curse of dimensionality. Furthermore, as the Gaussian-smoothing parameter shrinks to zero, GOT $\Gamma$-converges towards classic OT (with convergence of optimizers), thus serving as a natural extension. An empirical study that validates the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.

Entropic causal inference is a framework for inferring the causal direction between two categorical variables from observational data. The central assumption is that the amount of unobserved randomness in the system is not too large. This unobserved randomness is measured by the entropy of the exogenous variable in the underlying structural causal model, which governs the causal relation between the observed variables. Kocaoglu et al. conjectured that the causal direction is identifiable when the entropy of the exogenous variable is not too large. In this paper, we prove a variant of their conjecture. Namely, we show that for almost all causal models where the exogenous variable has entropy that does not scale with the number of states of the observed variables, the causal direction is identifiable from observational data. We also consider the minimum entropy coupling-based algorithmic approach presented by Kocaoglu et al., and for the first time demonstrate algorithmic identifiability guarantees using a finite number of samples. We conduct extensive experiments to evaluate the robustness of the method to relaxing some of the assumptions in our theory and demonstrate that both the constant-entropy exogenous variable and the no latent confounder assumptions can be relaxed in practice. We also empirically characterize the number of observational samples needed for causal identification. Finally, we apply the algorithm on Tuebingen cause-effect pairs dataset.

Minimum distance estimation (MDE) gained recent attention as a formulation of (implicit) generative modeling. It considers minimizing, over model parameters, a statistical distance between the empirical data distribution and the model. This formulation lends itself well to theoretical analysis, but typical results are hindered by the curse of dimensionality. To overcome this and devise a scalable finite-sample statistical MDE theory, we adopt the framework of smooth 1-Wasserstein distance (SWD) $\mathsf{W}_1^{(\sigma)}$. The SWD was recently shown to preserve the metric and topological structure of classic Wasserstein distances, while enjoying dimension-free empirical convergence rates. In this work, we conduct a thorough statistical study of the minimum smooth Wasserstein estimators (MSWEs), first proving the estimator's measurability and asymptotic consistency. We then characterize the limit distribution of the optimal model parameters and their associated minimal SWD. These results imply an $O(n^{-1/2})$ generalization bound for generative modeling based on MSWE, which holds in arbitrary dimension. Our main technical tool is a novel high-dimensional limit distribution result for empirical $\mathsf{W}_1^{(\sigma)}$. The characterization of a nondegenerate limit stands in sharp contrast with the classic empirical 1-Wasserstein distance, for which a similar result is known only in the one-dimensional case. The validity of our theory is supported by empirical results, posing the SWD as a potent tool for learning and inference in high dimensions.

A growing body of work has begun to study intervention design for efficient structure learning of causal directed acyclic graphs (DAGs). A typical setting is a \emph{causally sufficient} setting, i.e. a system with no latent confounders, selection bias, or feedback, when the essential graph of the observational equivalence class (EC) is given as an input and interventions are assumed to be noiseless. Most existing works focus on \textit{worst-case} or \textit{average-case} lower bounds for the number of interventions required to orient a DAG. These worst-case lower bounds only establish that the largest clique in the essential graph \textit{could} make it difficult to learn the true DAG. In this work, we develop a \textit{universal} lower bound for single-node interventions that establishes that the largest clique is \textit{always} a fundamental impediment to structure learning. Specifically, we present a decomposition of a DAG into independently orientable components through \emph{directed clique trees} and use it to prove that the number of single-node interventions necessary to orient any DAG in an EC is at least the sum of half the size of the largest cliques in each chain component of the essential graph. Moreover, we present a two-phase intervention design algorithm that, under certain conditions on the chordal skeleton, matches the optimal number of interventions up to a multiplicative logarithmic factor in the number of maximal cliques. We show via synthetic experiments that our algorithm can scale to much larger graphs than most of the related work and achieves better worst-case performance than other scalable approaches. A code base to recreate these results can be found at \url{https://github.com/csquires/dct-policy}.

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating P*Nσ, for Nσ=△N(0, σ2Id), by P̑n*Nσunder different statistical distances, where P̑nis the empirical measure. We examine the convergence in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and χ2-divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance (W1) converges at the rate eO(d)n-1/2in remarkable contrast to a (typical) n-1/drate for unsmoothed W1(and d ≥ 3). Similarly, for the KL divergence, squared 2-Wasserstein distance (W22), and χ2-divergence, the convergence rate is eO(d)n-1, but only if P achieves finite input-output χ2mutual information across the additive white Gaussian noise (AWGN) channel. If the latter condition is not met, the rate changes to ω (n-1) for the KL divergence and W22, while the χ2-divergence becomes infinite - a curious dichotomy. As an application we consider estimating the differential entropy h(S + Z), where S ~ P and Z ~ Nσare independent d-dimensional random variables. The distribution P is unknown and belongs to some nonparametric class, but n independently and identically distributed (i.i.d) samples from it are available. Despite the regularizing effect of noise, we first show that any good estimator (within an additive gap) for this problem must have a sample complexity that is exponential in d. We then leverage the above empirical approximation results to show that the absolute-error risk of the plug-in estimator converges as eO(d)n-1/2, thus attaining the parametric rate in n. This establishes the plug-in estimator as minimax rate-optimal for the considered problem, with sharp dependence of the convergence rate both in n and d. We provide numerical results comparing the performance of the plug-in estimator to that of general-purpose (unstructured) differential entropy estimators (based on kernel density estimation (KDE) or k nearest neighbors (kNN) techniques) applied to samples of S + Z. These results reveal a significant empirical superiority of the plug-in to state-of-the-art KDE and kNN methods. As a motivating utilization of the plug-in approach, we estimate information flows in deep neural networks and discuss Tishby's Information Bottleneck and the compression conjecture, among others.

We study the estimation of the mutual information I(X;T_$\ell$) between the input X to a deep neural network (DNN) and the output vector T_$\ell$ of its $\ell$-th hidden layer (an “internal representation”). Focusing on feedforward networks with fixed weights and noisy internal representations, we develop a rigorous framework for accurate estimation of I(X;T_$\ell$). By relating I(X;T_$\ell$) to information transmission over additive white Gaussian noise channels, we reveal that compression, i.e. reduction in I(X;T_$\ell$) over the course of training, is driven by progressive geometric clustering of the representations of samples from the same class. Experimental results verify this connection. Finally, we shift focus to purely deterministic DNNs, where I(X;T_$\ell$) is provably vacuous, and show that nevertheless, these models also cluster inputs belonging to the same class. The binning-based approximation of I(X;T_$\ell$) employed in past works to measure compression is identified as a measure of clustering, thus clarifying that these experiments were in fact tracking the same clustering phenomenon. Leveraging the clustering perspective, we provide new evidence that compression and generalization may not be causally related and discuss potential future research ideas.

In federated learning problems, data is scattered across different servers and exchanging or pooling it is often impractical or prohibited. We develop a Bayesian nonparametric framework for federated learning with neural networks. Each data server is assumed to provide local neural network weights, which are modeled through our framework. We then develop an inference approach that allows us to synthesize a more expressive global network without additional supervision, data pooling and with as few as a single communication round. We then demonstrate the efficacy of our approach on federated learning problems simulated from two popular image classification datasets.

This paper establishes the optimality of the plugin estimator for the problem of differential entropy estimation under Gaussian convolutions. Specifically, we consider the estimation of the differential entropy h(X + Z), where X and Z are independent d-dimensional random variables with Z ~ N(0, σ2Id). The distribution of X is unknown and belongs to some nonparametric class, but n independently and identically distributed samples from it are available. We first show that despite the regularizing effect of noise, any good estimator (within an additive gap) for this problem must have an exponential in d sample complexity. We then analyze the absolute-error risk of the plug-in estimator and show that it converges as cd√/n, thus attaining the parametric estimation rate. This implies the optimality of the plug-in estimator for the considered problem. We provide numerical results comparing the performance of the plug-in estimator to general-purpose (unstructured) differential entropy estimators (based on kernel density estimation (KDE) or k nearest neighbors (kNN) techniques) applied to samples of X + Z. These results reveal a significant empirical superiority of the plug-in to state-of-the-art KDEand kNN-based methods.

We consider the problem of experimental design for learning causal graphs that have a tree structure. We propose an adaptive framework that determines the next intervention based on a Bayesian prior updated with the outcomes of previous experiments, focusing on the setting where observational data is cheap (assumed infinite) and interventional data is expensive. While information greedy approaches are popular in active learning, we show that in this setting they can be exponentially suboptimal (in the number of interventions required), and instead propose an algorithm that exploits graph structure in the form of a centrality measure. If infinite interventional data is available, we show that the algorithm requires a number of interventions less than or equal to a factor of 2 times the minimum achievable number. We show that the algorithm and the associated theory can be adapted to the setting where each performed intervention yields finitely many samples. Several extensions are also presented, to the case where a specified set of nodes cannot be intervened on, to the case where $K$ interventions are scheduled at once, and to the fully adaptive case where each experiment yields only one sample. In the case of finite interventional data, through simulated experiments we show that our algorithms outperform different adaptive baseline algorithms.

We consider the problem of aggregating models learned from sequestered, possibly heterogeneous datasets. Exploiting tools from Bayesian nonparametrics, we develop a general meta-modeling framework that learns shared global latent structures by identifying correspondences among local model parameterizations. Our proposed framework is model-independent and is applicable to a wide range of model types. After verifying our approach on simulated data, we demonstrate its utility in aggregating Gaussian topic models, hierarchical Dirichlet process based hidden Markov models, and sparse Gaussian processes with applications spanning text summarization, motion capture analysis, and temperature forecasting.

In this work, we present an additive model for space-time data that splits the data into a temporally correlated component and a spatially correlated component. We model the spatially correlated portion using a time-varying Gaussian graphical model. Under assumptions on the smoothness of changes in covariance matrices, we derive strong single sample convergence results, confirming our ability to estimate meaningful graphical structures as they evolve over time. We apply our methodology to the discovery of time-varying spatial structures in human brain fMRI signals.

Contextual bandits have become popular as they offer a middle ground between very simple approaches based on multi-armed bandits and very complex approaches using the full power of reinforcement learning. They have demonstrated success in web applications and have a rich body of associated theoretical guarantees. Linear models are well understood theoretically and preferred by practitioners because they are not only easily interpretable but also simple to implement and debug. Furthermore, if the linear model is true, we get very strong performance guarantees. Unfortunately, in emerging applications in mobile health, the time-invariant linear model assumption is untenable. We provide an extension of the linear model for contextual bandits that has two parts: baseline reward and treatment effect. We allow the former to be complex but keep the latter simple. We argue that this model is plausible for mobile health applications. At the same time, it leads to algorithms with strong performance guarantees as in the linear model setting, while still allowing for complex nonlinear baseline modeling. Our theory is supported by experiments on data gathered in a recently concluded mobile health study.

Recent work has focused on the problem of non-parametric estimation of divergence functionals. Many existing approaches are restrictive in their assumptions on the density support or require difficult calculations at the support boundary which must be known a priori. We derive the MSE convergence rate of a leave-one-out kernel density plug-in divergence functional estimator for general bounded density support sets where knowledge of the support boundary is not required. We generalize the theory of optimally weighted ensemble estimation to derive two estimators that achieve the parametric rate when the densities are sufficiently smooth. The asymptotic distribution of these estimators and tuning parameter selection guidelines are provided. Based on the theory, we propose an empirical estimator of Rényi-α divergence that outperforms the standard kernel density plug-in estimator, especially in higher dimensions.