Thompson sampling (TS) is widely used in sequential decision making due to its ease of use and appealing empirical performance. However, many existing analytical and empirical results for TS rely on restrictive assumptions on reward distributions, such as belonging to conjugate families, which limits their applicability in realistic scenarios. Moreover, sequential decision making problems are often carried out in a batched manner, either due to the inherent nature of the problem or to serve the purpose of reducing communication and computation costs. In this work, we jointly study these problems in two popular settings, namely, stochastic multi-armed bandits (MABs) and infinite-horizon reinforcement learning (RL), where TS is used to learn the unknown reward distributions and transition dynamics, respectively. We propose batched $\textit{Langevin Thompson Sampling}$ algorithms that leverage MCMC methods to sample from approximate posteriors with only logarithmic communication costs in terms of batches. Our algorithms are computationally efficient and maintain the same order-optimal regret guarantees of $\mathcal{O}(\log T)$ for stochastic MABs, and $\mathcal{O}(\sqrt{T})$ for RL. We complement our theoretical findings with experimental results.

Shepard's universal law of generalization is a remarkable hypothesis about how intelligent organisms should perceive similarity. In its broadest form, the universal law states that the level of perceived similarity between a pair of stimuli should decay as a concave function of their distance when embedded in an appropriate psychological space. While extensively studied, evidence in support of the universal law has relied on low-dimensional stimuli and small stimulus sets that are very different from their real-world counterparts. This is largely because pairwise comparisons - as required for similarity judgments - scale quadratically in the number of stimuli. We provide direct evidence for the universal law in a naturalistic high-dimensional regime by analyzing an existing dataset of 214, 200 human similarity judgments and a newly collected dataset of 390, 819 human generalization judgments (N = 2406 US participants) across three sets of natural images. The Universal Law of Generalization Holds for Naturalistic Stimuli Statement of Relevance Humans constantly form generalizations, whether when trying to identify the color of an object or reasoning about which action to take based on past experiences.

Graph neural networks have been extensively studied for learning with inter-connected data. Despite this, recent evidence has revealed GNNs' deficiencies related to over-squashing, heterophily, handling long-range dependencies, edge incompleteness and particularly, the absence of graphs altogether. While a plausible solution is to learn new adaptive topology for message passing, issues concerning quadratic complexity hinder simultaneous guarantees for scalability and precision in large networks. In this paper, we introduce a novel all-pair message passing scheme for efficiently propagating node signals between arbitrary nodes, as an important building block for a pioneering Transformer-style network for node classification on large graphs, dubbed as \textsc{NodeFormer}. Specifically, the efficient computation is enabled by a kernerlized Gumbel-Softmax operator that reduces the algorithmic complexity to linearity w.r.t. node numbers for learning latent graph structures from large, potentially fully-connected graphs in a differentiable manner. We also provide accompanying theory as justification for our design. Extensive experiments demonstrate the promising efficacy of the method in various tasks including node classification on graphs (with up to 2M nodes) and graph-enhanced applications (e.g., image classification) where input graphs are missing.

The Gaussian process latent variable model (GPLVM) is a popular probabilistic method used for nonlinear dimension reduction, matrix factorization, and state-space modeling. Inference for GPLVMs is computationally tractable only when the data likelihood is Gaussian. Moreover, inference for GPLVMs has typically been restricted to obtaining maximum a posteriori point estimates, which can lead to overfitting, or variational approximations, which mischaracterize the posterior uncertainty. Here, we present a method to perform Markov chain Monte Carlo (MCMC) inference for generalized Bayesian nonlinear latent variable modeling. The crucial insight necessary to generalize GPLVMs to arbitrary observation models is that we approximate the kernel function in the Gaussian process mappings with random Fourier features; this allows us to compute the gradient of the posterior in closed form with respect to the latent variables. We show that we can generalize GPLVMs to non-Gaussian observations, such as Poisson, negative binomial, and multinomial distributions, using our random feature latent variable model (RFLVM). Our generalized RFLVMs perform on par with state-of-the-art latent variable models on a wide range of applications, including motion capture, images, and text data for the purpose of estimating the latent structure and imputing the missing data of these complex data sets. Keywords: Latent variable modeling, Gaussian processes, probabilistic modeling.

Many Artificial Intelligence (AI) algorithms are inspired by physics and employ stochastic fluctuations. We connect these physics-inspired AI algorithms by unifying them under a single mathematical framework that we call Thermodynamic AI. Seemingly disparate algorithmic classes can be described by this framework, for example, (1) Generative diffusion models, (2) Bayesian neural networks, (3) Monte Carlo sampling and (4) Simulated annealing. Such Thermodynamic AI algorithms are currently run on digital hardware, ultimately limiting their scalability and overall potential. Stochastic fluctuations naturally occur in physical thermodynamic systems, and such fluctuations can be viewed as a computational resource. Hence, we propose a novel computing paradigm, where software and hardware become inseparable. Our algorithmic unification allows us to identify a single full-stack paradigm, involving Thermodynamic AI hardware, that could accelerate such algorithms. We contrast Thermodynamic AI hardware with quantum computing where noise is a roadblock rather than a resource. Thermodynamic AI hardware can be viewed as a novel form of computing, since it uses a novel fundamental building block. We identify stochastic bits (s-bits) and stochastic modes (s-modes) as the respective building blocks for discrete and continuous Thermodynamic AI hardware. In addition to these stochastic units, Thermodynamic AI hardware employs a Maxwell's demon device that guides the system to produce non-trivial states. We provide a few simple physical architectures for building these devices and we develop a formalism for programming the hardware via gate sequences. We hope to stimulate discussion around this new computing paradigm. Beyond acceleration, we believe it will impact the design of both hardware and algorithms, while also deepening our understanding of the connection between physics and intelligence.

A common challenge in applying graph machine learning methods is that the underlying graph Learning the graph structure underlying a set of smooth signals of a system is often unknown. Although different is a classical problem in GSP. Well-established methods graph inference methods have been proposed optimize a graph representation, usually the graph adjacency for continuous graph signals, inferring the matrix or the graph Laplacian, so that the total variation of graph structure underlying other types of data, given signals will be minimal on the learned graph (Dong such as discrete counts, is under-explored. In et al., 2016; Kalofolias, 2016; Egilmez et al., 2017; Kumar this paper, we generalize a graph signal processing et al., 2020). However, smooth graph signals are rarely (GSP) framework for learning a graph from encountered in the real world and one is often required to smooth graph signals to the exponential family deal with noisy signals.

In several practical applications of federated learning (FL), the clients are highly heterogeneous in terms of both their data and compute resources, and therefore enforcing the same model architecture for each client is very limiting. Moreover, the need for uncertainty quantification and data privacy constraints are often particularly amplified for clients that have limited local data. This paper presents a unified FL framework to simultaneously address all these constraints and concerns, based on training customized local Bayesian models that learn well even in the absence of large local datasets. A Bayesian framework provides a natural way of incorporating supervision in the form of prior distributions. We use priors in the functional (output) space of the networks to facilitate collaboration across heterogeneous clients. Moreover, formal differential privacy guarantees are provided for this framework. Experiments on standard FL datasets demonstrate that our approach outperforms strong baselines in both homogeneous and heterogeneous settings and under strict privacy constraints, while also providing characterizations of model uncertainties.

Causal structure learning, a prominent technique for encoding cause and effect relationships among variables, through Bayesian Networks (BNs). Merely recovering causal structures from real-world observed data lacks precision, while the development of Large Language Models (LLM) is opening a new frontier of causality. LLM presents strong capability in discovering causal relationships between variables with the "text" inputs defining the investigated variables, leading to a potential new hierarchy and new ladder of causality. We aim an critical issue in the emerging topic of LLM based causal structure learning, to tackle erroneous prior causal statements from LLM, which is seldom considered in the current context of expert dominating prior resources. As a pioneer attempt, we propose a BN learning strategy resilient to prior errors without need of human intervention. Focusing on the edge-level prior, we classify the possible prior errors into three types: order-consistent, order-reversed, and irrelevant, and provide their theoretical impact on the Structural Hamming Distance (SHD) under the presumption of sufficient data. Intriguingly, we discover and prove that only the order-reversed error contributes to an increase in a unique acyclic closed structure, defined as a "quasi-circle". Leveraging this insight, a post-hoc strategy is employed to identify the order-reversed prior error by its impact on the increment of "quasi-circles". Through empirical evaluation on both real and synthetic datasets, we demonstrate our strategy's robustness against prior errors. Specifically, we highlight its substantial ability to resist order-reversed errors while maintaining the majority of correct prior knowledge.

Bayesian neural networks often approximate the weight-posterior with a Gaussian distribution. However, practical posteriors are often, even locally, highly non-Gaussian, and empirical performance deteriorates. We propose a simple parametric approximate posterior that adapts to the shape of the true posterior through a Riemannian metric that is determined by the log-posterior gradient. We develop a Riemannian Laplace approximation where samples naturally fall into weight-regions with low negative log-posterior. We show that these samples can be drawn by solving a system of ordinary differential equations, which can be done efficiently by leveraging the structure of the Riemannian metric and automatic differentiation. Empirically, we demonstrate that our approach consistently improves over the conventional Laplace approximation across tasks. We further show that, unlike the conventional Laplace approximation, our method is not overly sensitive to the choice of prior, which alleviates a practical pitfall of current approaches.

In the field of decision trees, most previous studies have difficulty ensuring the statistical optimality of a prediction of new data and suffer from overfitting because trees are usually used only to represent prediction functions to be constructed from given data. In contrast, some studies, including this paper, used the trees to represent stochastic data observation processes behind given data. Moreover, they derived the statistically optimal prediction, which is robust against overfitting, based on the Bayesian decision theory by assuming a prior distribution for the trees. However, these studies still have a problem in computing this Bayes optimal prediction because it involves an infeasible summation for all division patterns of a feature space, which is represented by the trees and some parameters. In particular, an open problem is a summation with respect to combinations of division axes, i.e., the assignment of features to inner nodes of the tree. We solve this by a Markov chain Monte Carlo method, whose step size is adaptively tuned according to a posterior distribution for the trees.