Gaussian Processes

Deciding on an appropriate intervention requires a causal model of a treatment, the outcome, and potential mediators. Causal mediation analysis lets us distinguish between direct and indirect effects of the intervention, but has mostly been studied in a static setting. In healthcare, data come in the form of complex, irregularly sampled time-series, with dynamic interdependencies between a treatment, outcomes, and mediators across time. Existing approaches to dynamic causal mediation analysis are limited to regular measurement intervals, simple parametric models, and disregard long-range mediator--outcome interactions. To address these limitations, we propose a non-parametric mediator--outcome model where the mediator is assumed to be a temporal point process that interacts with the outcome process. With this model, we estimate the direct and indirect effects of an external intervention on the outcome, showing how each of these affects the whole future trajectory. We demonstrate on semi-synthetic data that our method can accurately estimate direct and indirect effects. On real-world healthcare data, our model infers clinically meaningful direct and indirect effect trajectories for blood glucose after a surgery.

We introduce beta diffusion, a novel generative modeling method that integrates demasking and denoising to generate data within bounded ranges. Using scaled and shifted beta distributions, beta diffusion utilizes multiplicative transitions over time to create both forward and reverse diffusion processes, maintaining beta distributions in both the forward marginals and the reverse conditionals, given the data at any point in time. Unlike traditional diffusion-based generative models relying on additive Gaussian noise and reweighted evidence lower bounds (ELBOs), beta diffusion is multiplicative and optimized with KL-divergence upper bounds (KLUBs) derived from the convexity of the KL divergence. We demonstrate that the proposed KLUBs are more effective for optimizing beta diffusion compared to negative ELBOs, which can also be derived as the KLUBs of the same KL divergence with its two arguments swapped. The loss function of beta diffusion, expressed in terms of Bregman divergence, further supports the efficacy of KLUBs for optimization. Experimental results on both synthetic data and natural images demonstrate the unique capabilities of beta diffusion in generative modeling of range-bounded data and validate the effectiveness of KLUBs in optimizing diffusion models, thereby making them valuable additions to the family of diffusion-based generative models and the optimization techniques used to train them.

Gaussian processes (GPs) can provide a principled approach to uncertainty quantification with easy-to-interpret kernel hyperparameters, such as the lengthscale, which controls the correlation distance of function values.However, selecting an appropriate kernel can be challenging. Deep GPs avoid manual kernel engineering by successively parameterizing kernels with GP layers, allowing them to learn low-dimensional embeddings of the inputs that explain the output data. Following the architecture of deep neural networks, the most common deep GPs warp the input space layer-by-layer but lose all the interpretability of shallow GPs. An alternative construction is to successively parameterize the lengthscale of a kernel, improving the interpretability but ultimately giving away the notion of learning lower-dimensional embeddings. Unfortunately, both methods are susceptible to particular pathologies which may hinder fitting and limit their interpretability. This work proposes a novel synthesis of both previous approaches: {Thin and Deep GP} (TDGP). Each TDGP layer defines locally linear transformations of the original input data maintaining the concept of latent embeddings while also retaining the interpretation of lengthscales of a kernel. Moreover, unlike the prior solutions, TDGP induces non-pathological manifolds that admit learning lower-dimensional representations. We show with theoretical and experimental results that i) TDGP is, unlike previous models, tailored to specifically discover lower-dimensional manifolds in the input data, ii) TDGP behaves well when increasing the number of layers, and iii) TDGP performs well in standard benchmark datasets.

In many scientific applications, we observe a system in different conditions in which its components may change, rather than in isolation. In our work, we are interested in explaining the generating process of such a multi-context system using a finite mixture of causal mechanisms. Recent work shows that this causal model is identifiable from data, but is limited to settings where the sparse mechanism shift hypothesis holds and only a subset of the causal conditionals change. As this assumption is not easily verifiable in practice, we study the more general principle that mechanism shifts are independent, which we formalize using the algorithmic notion of independence. We introduce an approach for causal discovery beyond partially directed graphs using Gaussian Process models, and give conditions under which we provably identify the correct causal model. In our experiments, we show that our method performs well in a range of synthetic settings, on realistic gene expression simulations, as well as on real-world cell signaling data.

Uncertainty estimation is an important research area to make deep neural networks (DNNs) more trustworthy. While extensive research on uncertainty estimation has been conducted with unimodal data, uncertainty estimation for multimodal data remains a challenge. Neural processes (NPs) have been demonstrated to be an effective uncertainty estimation method for unimodal data by providing the reliability of Gaussian processes with efficient and powerful DNNs. While NPs hold significant potential for multimodal uncertainty estimation, the adaptation of NPs for multimodal data has not been carefully studied. To bridge this gap, we propose Multimodal Neural Processes (MNPs) by generalising NPs for multimodal uncertainty estimation. Based on the framework of NPs, MNPs consist of several novel and principled mechanisms tailored to the characteristics of multimodal data. In extensive empirical evaluation, our method achieves state-of-the-art multimodal uncertainty estimation performance, showing its appealing robustness against noisy samples and reliability in out-of-distribution detection with faster computation time compared to the current state-of-the-art multimodal uncertainty estimation method.

Conditional Neural Processes (CNPs) are a class of metalearning models popular for combining the runtime efficiency of amortized inference with reliable uncertainty quantification. Many relevant machine learning tasks, such as in spatio-temporal modeling, Bayesian Optimization and continuous control, inherently contain equivariances – for example to translation – which the model can exploit for maximal performance. However, prior attempts to include equivariances in CNPs do not scale effectively beyond two input dimensions. In this work, we propose Relational Conditional Neural Processes (RCNPs), an effective approach to incorporate equivariances into any neural process model. Our proposed method extends the applicability and impact of equivariant neural processes to higher dimensions. We empirically demonstrate the competitive performance of RCNPs on a large array of tasks naturally containing equivariances.

Distributional shifts pose a significant challenge to achieving robustness in contemporary machine learning. To overcome this challenge, robust satisficing (RS) seeks a robust solution to an unspecified distributional shift while achieving a utility above a desired threshold. This paper focuses on the problem of RS in contextual Bayesian optimization when there is a discrepancy between the true and reference distributions of the context. We propose a novel robust Bayesian satisficing algorithm called RoBOS for noisy black-box optimization. Our algorithm guarantees sublinear lenient regret under certain assumptions on the amount of distribution shift. In addition, we define a weaker notion of regret called robust satisficing regret, in which our algorithm achieves a sublinear upper bound independent of the amount of distribution shift. To demonstrate the effectiveness of our method, we apply it to various learning problems and compare it to other approaches, such as distributionally robust optimization.

We study the problem of collaboratively learning least squares estimates for $m$ agents. Each agent observes a different subset of the features---e.g., containing data collected from sensors of varying resolution. Our goal is to determine how to coordinate the agents in order to produce the best estimator for each agent. We propose a distributed, semi-supervised algorithm Collab, consisting of three steps: local training, aggregation, and distribution. Our procedure does not require communicating the labeled data, making it communication efficient and useful in settings where the labeled data is inaccessible. Despite this handicap, our procedure is nearly asymptotically, local-minimax optimal---even among estimators allowed to communicate the labeled data such as imputation methods. We test our method on US Census data. We also discuss generalizations of our method to non-Gaussian feature settings, non-linear settings, and Federated Learning.

Generative processes that involve solving differential equations, such as diffusion models, frequently necessitate balancing speed and quality. ODE-based samplers are fast but plateau in performance while SDE-based samplers deliver higher sample quality at the cost of increased sampling time. We attribute this difference to sampling errors: ODE-samplers involve smaller discretization errors while stochasticity in SDE contracts accumulated errors. Based on these findings, we propose a novel sampling algorithm called \textit{Restart} in order to better balance discretization errors and contraction. The sampling method alternates between adding substantial noise in additional forward steps and strictly following a backward ODE. Empirically, Restart sampler surpasses previous SDE and ODE samplers in both speed and accuracy. Restart not only outperforms the previous best SDE results, but also accelerates the sampling speed by 10-fold / 2-fold on CIFAR-10 / ImageNet $64{\times} 64$. In addition, it attains significantly better sample quality than ODE samplers within comparable sampling times. Moreover, Restart better balances text-image alignment/visual quality versus diversity than previous samplers in the large-scale text-to-image Stable Diffusion model pre-trained on LAION $512{\times} 512$. Code is available at https://github.com/Newbeeer/diffusion_restart_sampling

While Gaussian processes are a mainstay for various engineering and scientific applications, the uncertainty estimates don't satisfy frequentist guarantees and can be miscalibrated in practice. State-of-the-art approaches for designing calibrated models rely on inflating the Gaussian process posterior variance, which yields confidence intervals that are potentially too coarse. To remedy this, we present a calibration approach that generates predictive quantiles using a computation inspired by the vanilla Gaussian process posterior variance but using a different set of hyperparameters chosen to satisfy an empirical calibration constraint. This results in a calibration approach that is considerably more flexible than existing approaches, which we optimize to yield tight predictive quantiles. Our approach is shown to yield a calibrated model under reasonable assumptions. Furthermore, it outperforms existing approaches in sharpness when employed for calibrated regression.

High training costs of generative models and the need to fine-tune them for specific tasks have created a strong interest in model reuse and composition. A key challenge in composing iterative generative processes, such as GFlowNets and diffusion models, is that to realize the desired target distribution, all steps of the generative process need to be coordinated, and satisfy delicate balance conditions. In this work, we propose Compositional Sculpting: a general approach for defining compositions of iterative generative processes. We then introduce a method for sampling from these compositions built on classifier guidance. We showcase ways to accomplish compositional sculpting in both GFlowNets and diffusion models. We highlight two binary operations $\\unicode{x2014}$ the $\\textit{harmonic mean}\\unicode{x00A0}(p_1 \\otimes p_2$) and the $\\textit{contrast}\\unicode{x00A0}(p_1 \\,\\unicode{x25D1}\\,\\, p_2$) between pairs, and the generalization of these operations to multiple component distributions. We offer empirical results on image and molecular generation tasks. Project codebase: https://github.com/timgaripov/compositional-sculpting.

We introduce GAUCHE, an open-source library for GAUssian processes in CHEmistry. Gaussian processes have long been a cornerstone of probabilistic machine learning, affording particular advantages for uncertainty quantification and Bayesian optimisation. Extending Gaussian processes to molecular representations, however, necessitates kernels defined over structured inputs such as graphs, strings and bit vectors. By providing such kernels in a modular, robust and easy-to-use framework, we seek to enable expert chemists and materials scientists to make use of state-of-the-art black-box optimization techniques. Motivated by scenarios frequently encountered in practice, we showcase applications for GAUCHE in molecular discovery, chemical reaction optimisation and protein design. The codebase is made available at https://github.com/leojklarner/gauche.

Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least squares estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality.

Modern recording techniques now allow us to record from distinct neuronal populations in different brain networks. However, especially as we consider multiple (more than two) populations, new conceptual and statistical frameworks are needed to characterize the multi-dimensional, concurrent flow of signals among these populations. Here, we develop a dimensionality reduction framework that determines (1) the subset of populations described by each latent dimension, (2) the direction of signal flow among those populations, and (3) how those signals evolve over time within and across experimental trials. We illustrate these features in simulation, and further validate the method by applying it to previously studied recordings from neuronal populations in macaque visual areas V1 and V2. Then we study interactions across select laminar compartments of areas V1, V2, and V3d, recorded simultaneously with multiple Neuropixels probes. Our approach uncovered signatures of selective communication across these three areas that related to their retinotopic alignment. This work advances the study of concurrent signaling across multiple neuronal populations.

Autoregressive neural networks within the temporal point process (TPP) framework have become the standard for modeling continuous-time event data. Even though these models can expressively capture event sequences in a one-step-ahead fashion, they are inherently limited for long-term forecasting applications due to the accumulation of errors caused by their sequential nature. To overcome these limitations, we derive ADD-THIN, a principled probabilistic denoising diffusion model for TPPs that operates on entire event sequences. Unlike existing diffusion approaches, ADD-THIN naturally handles data with discrete and continuous components. In experiments on synthetic and real-world datasets, our model matches the state-of-the-art TPP models in density estimation and strongly outperforms them in forecasting.

In this paper, we consider the uncertainty quantification problem for regression models. Specifically, we consider an individual calibration objective for characterizing the quantiles of the prediction model. While such an objective is well-motivated from downstream tasks such as newsvendor cost, the existing methods have been largely heuristic and lack of statistical guarantee in terms of individual calibration. We show via simple examples that the existing methods focusing on population-level calibration guarantees such as average calibration or sharpness can lead to harmful and unexpected results. We propose simple nonparametric calibration methods that are agnostic of the underlying prediction model and enjoy both computational efficiency and statistical consistency. Our approach enables a better understanding of the possibility of individual calibration, and we establish matching upper and lower bounds for the calibration error of our proposed methods. Technically, our analysis combines the nonparametric analysis with a covering number argument for parametric analysis, which advances the existing theoretical analyses in the literature of nonparametric density estimation and quantile bandit problems. Importantly, the nonparametric perspective sheds new theoretical insights into regression calibration in terms of the curse of dimensionality and reconciles the existing results on the impossibility of individual calibration. To our knowledge, we make the first effort to reach both individual calibration and finite-sample guarantee with minimal assumptions in terms of conformal prediction. Numerical experiments show the advantage of such a simple approach under various metrics, and also under covariates shift. We hope our work provides a simple benchmark and a starting point of theoretical ground for future research on regression calibration.

Robots often rely on a repertoire of previously-learned motion policies for performing tasks of diverse complexities. When facing unseen task conditions or when new task requirements arise, robots must adapt their motion policies accordingly. In this context, policy optimization is the \emph{de facto} paradigm to adapt robot policies as a function of task-specific objectives. Most commonly-used motion policies carry particular structures that are often overlooked in policy optimization algorithms. We instead propose to leverage the structure of probabilistic policies by casting the policy optimization as an optimal transport problem. Specifically, we focus on robot motion policies that build on Gaussian mixture models (GMMs) and formulate the policy optimization as a Wassertein gradient flow over the GMMs space. This naturally allows us to constrain the policy updates via the $L^2$-Wasserstein distance between GMMs to enhance the stability of the policy optimization process. Furthermore, we leverage the geometry of the Bures-Wasserstein manifold to optimize the Gaussian distributions of the GMM policy via Riemannian optimization. We evaluate our approach on common robotic settings: Reaching motions, collision-avoidance behaviors, and multi-goal tasks. Our results show that our method outperforms common policy optimization baselines in terms of task success rate and low-variance solutions.

Neural spiking activity is generally variable, non-stationary, and exhibits complex dependencies on covariates, such as sensory input or behavior. These dependencies have been proposed to be signatures of specific computations, and so characterizing them with quantitative rigor is critical for understanding neural computations. Approaches based on point processes provide a principled statistical framework for modeling neural spiking activity. However, currently, they only allow the instantaneous mean, but not the instantaneous variability, of responses to depend on covariates. To resolve this limitation, we propose a scalable Bayesian approach generalizing modulated renewal processes using sparse variational Gaussian processes. We leverage pathwise conditioning for computing nonparametric priors over conditional interspike interval distributions and rely on automatic relevance determination to detect lagging interspike interval dependencies beyond renewal order. After systematically validating our method on synthetic data, we apply it to two foundational datasets of animal navigation: head direction cells in freely moving mice and hippocampal place cells in rats running along a linear track. Our model exhibits competitive or better predictive power compared to state-of-the-art baselines, and outperforms them in terms of capturing interspike interval statistics. These results confirm the importance of modeling covariate-dependent spiking variability, and further analyses of our fitted models reveal rich patterns of variability modulation beyond the temporal resolution of flexible count-based approaches.

Modern reinforcement learning (RL) has shown empirical success in various real world settings with complex models and large state-action spaces. The existing analytical results, however, typically focus on settings with a small number of state-actions or simple models such as linearly modeled state-action value functions. To derive RL policies that efficiently handle large state-action spaces with more general value functions, some recent works have considered nonlinear function approximation using kernel ridge regression. We propose $\pi$-KRVI, an optimistic modification of least-squares value iteration, when the action-value function is represented by an RKHS. We prove the first order-optimal regret guarantees under a general setting. Our results show a significant polynomial in the number of episodes improvement over the state of the art. In particular, with highly non-smooth kernels (such as Neural Tangent kernel or some Matérn kernels) the existing results lead to trivial (superlinear in the number of episodes) regret bounds. We show a sublinear regret bound that is order optimal in the cases where a lower bound on regret is known (which includes the kernels mentioned above).

Kullback-Leibler (KL) divergence is one of the most important measures to calculate the difference between probability distributions. In this paper, we theoretically study several properties of KL divergence between multivariate Gaussian distributions. Firstly, for any two $n$-dimensional Gaussian distributions $\mathcal{N}_1$ and $\mathcal{N}_2$, we prove that when $KL(\mathcal{N}_2||\mathcal{N}_1)\leq \varepsilon\ (\varepsilon>0)$ the supremum of $KL(\mathcal{N}_1||\mathcal{N}_2)$ is $(1/2)\left((-W_{0}(-e^{-(1+2\varepsilon)}))^{-1}+\log(-W_{0}(-e^{-(1+2\varepsilon)})) -1 \right)$, where $W_0$ is the principal branch of Lambert $W$ function. For small $\varepsilon$, the supremum is $\varepsilon + 2\varepsilon^{1.5} + O(\varepsilon^2)$. This quantifies the approximate symmetry of small KL divergence between Gaussian distributions. We further derive the infimum of $KL(\mathcal{N}_1||\mathcal{N}_2)$ when $KL(\mathcal{N}_2||\mathcal{N}_1)\geq M\ (M>0)$. We give the conditions when the supremum and infimum can be attained. Secondly, for any three $n$-dimensional Gaussian distributions $\mathcal{N}_1$, $\mathcal{N}_2$, and $\mathcal{N}_3$, we theoretically show that an upper bound of $KL(\mathcal{N}_1||\mathcal{N}_3)$ is $3\varepsilon_1+3\varepsilon_2+2\sqrt{\varepsilon_1\varepsilon_2}+o(\varepsilon_1)+o(\varepsilon_2)$ when $KL(\mathcal{N}_1||\mathcal{N}_2)\leq \varepsilon_1$ and $KL(\mathcal{N}_2||\mathcal{N}_3)\leq \varepsilon_2$ ($\varepsilon_1,\varepsilon_2\ge 0$). This reveals that KL divergence between Gaussian distributions follows a relaxed triangle inequality. Note that, all these bounds in the theorems presented in this work are independent of the dimension $n$. Finally, we discuss several applications of our theories in deep learning, reinforcement learning, and sample complexity research.