Past decades have witnessed a great interest in the distinction and connection between neural network learning and kernel learning. Recent advancements have made theoretical progress in connecting infinite-wide neural networks and Gaussian processes. Two predominant approaches have emerged: the Neural Network Gaussian Process (NNGP) and the Neural Tangent Kernel (NTK). The former, rooted in Bayesian inference, represents a zero-order kernel, while the latter, grounded in the tangent space of gradient descents, is a first-order kernel. In this paper, we present the Unified Neural Kernel (UNK), which characterizes the learning dynamics of neural networks with gradient descents and parameter initialization. The proposed UNK kernel maintains the limiting properties of both NNGP and NTK, exhibiting behaviors akin to NTK with a finite learning step and converging to NNGP as the learning step approaches infinity. Besides, we also theoretically characterize the uniform tightness and learning convergence of the UNK kernel, providing comprehensive insights into this unified kernel. Experimental results underscore the effectiveness of our proposed method.
We present SIM-FSVGD for learning robot dynamics from data. As opposed to traditional methods, SIM-FSVGD leverages low-fidelity physical priors, e.g., in the form of simulators, to regularize the training of neural network models. While learning accurate dynamics already in the low data regime, SIM-FSVGD scales and excels also when more data is available. We empirically show that learning with implicit physical priors results in accurate mean model estimation as well as precise uncertainty quantification. We demonstrate the effectiveness of SIM-FSVGD in bridging the sim-to-real gap on a high-performance RC racecar system. Using model-based RL, we demonstrate a highly dynamic parking maneuver with drifting, using less than half the data compared to the state of the art.
The advent of machine learning has led to transformative changes across various domains, but the sensitive nature of data raises concerns about privacy and security. While Local Differential Privacy (LDP) has been a cornerstone in addressing these concerns, recent research has proposed privacy concepts aligned with the Bayesian inference perspective of an adversary, such as Average Bayesian Privacy (ABP) and Maximum Bayesian Privacy (MBP). This paper explores the intricate relationships between LDP, ABP, and MBP, and their implications for algorithmic robustness. We establish theoretical connections between these privacy notions, proving that LDP implies MBP and vice versa under certain conditions, and deriving bounds connecting MBP and ABP. We also investigate the relationship between PAC robust learning and privacy preservation, demonstrating how to derive PAC robustness from privacy-preserving algorithms and construct privacy-preserving algorithms from PAC robust ones. Our findings provide valuable insights for constructing privacy-preserving and robust machine learning algorithms.
Intelligent tutoring systems optimize the selection and timing of learning materials to enhance understanding and long-term retention. This requires estimates of both the learner's progress (''knowledge tracing''; KT), and the prerequisite structure of the learning domain (''knowledge mapping''). While recent deep learning models achieve high KT accuracy, they do so at the expense of the interpretability of psychologically-inspired models. In this work, we present a solution to this trade-off. PSI-KT is a hierarchical generative approach that explicitly models how both individual cognitive traits and the prerequisite structure of knowledge influence learning dynamics, thus achieving interpretability by design. Moreover, by using scalable Bayesian inference, PSI-KT targets the real-world need for efficient personalization even with a growing body of learners and learning histories. Evaluated on three datasets from online learning platforms, PSI-KT achieves superior multi-step predictive accuracy and scalable inference in continual-learning settings, all while providing interpretable representations of learner-specific traits and the prerequisite structure of knowledge that causally supports learning. In sum, predictive, scalable and interpretable knowledge tracing with solid knowledge mapping lays a key foundation for effective personalized learning to make education accessible to a broad, global audience.
Rankings are a type of preference elicitation that arise in experiments where assessors arrange items, for example, in decreasing order of utility. Orderings of n items labelled {1,...,n} denoted are permutations that reflect strict preferences. For a number of reasons, strict preferences can be unrealistic assumptions for real data. For example, when items share common traits it may be reasonable to attribute them equal ranks. Also, there can be different importance attributions to decisions that form the ranking. In a situation with, for example, a large number of items, an assessor may wish to rank at top a certain number items; to rank other items at the bottom and to express indifference to all others. In addition, when aggregating opinions, a judging body might be decisive about some parts of the rank but ambiguous for others. In this paper we extend the well-known Mallows (Mallows, 1957) model (MM) to accommodate item indifference, a phenomenon that can be in place for a variety of reasons, such as those above mentioned.The underlying grouping of similar items motivates the proposed Clustered Mallows Model (CMM). The CMM can be interpreted as a Mallows distribution for tied ranks where ties are learned from the data. The CMM provides the flexibility to combine strict and indifferent relations, achieving a simpler and robust representation of rank collections in the form of ordered clusters. Bayesian inference for the CMM is in the class of doubly-intractable problems since the model's normalisation constant is not available in closed form. We overcome this challenge by sampling from the posterior with a version of the exchange algorithm \citep{murray2006}. Real data analysis of food preferences and results of Formula 1 races are presented, illustrating the CMM in practical situations.
Machine learning models often perform poorly under subpopulation shifts in the data distribution. Developing methods that allow machine learning models to better generalize to such shifts is crucial for safe deployment in real-world settings. In this paper, we develop a family of group-aware prior (GAP) distributions over neural network parameters that explicitly favor models that generalize well under subpopulation shifts. We design a simple group-aware prior that only requires access to a small set of data with group information and demonstrate that training with this prior yields state-of-the-art performance -- even when only retraining the final layer of a previously trained non-robust model. Group aware-priors are conceptually simple, complementary to existing approaches, such as attribute pseudo labeling and data reweighting, and open up promising new avenues for harnessing Bayesian inference to enable robustness to subpopulation shifts.
Bayesian inference for complex models with an intractable likelihood can be tackled using algorithms performing many calls to computer simulators. These approaches are collectively known as "simulation-based inference" (SBI). Recent SBI methods have made use of neural networks (NN) to provide approximate, yet expressive constructs for the unavailable likelihood function and the posterior distribution. However, the trade-off between accuracy and computational demand leaves much space for improvement. In this work, we propose an alternative that provides both approximations to the likelihood and the posterior distribution, using structured mixtures of probability distributions. Our approach produces accurate posterior inference when compared to state-of-the-art NN-based SBI methods, even for multimodal posteriors, while exhibiting a much smaller computational footprint. We illustrate our results on several benchmark models from the SBI literature and on a biological model of the translation kinetics after mRNA transfection.
Spatiotemporal datasets, which consist of spatially-referenced time series, are ubiquitous in diverse applications, such as air pollution monitoring, disease tracking, and cloud-demand forecasting. As the scale of modern datasets increases, there is a growing need for statistical methods that are flexible enough to capture complex spatiotemporal dynamics and scalable enough to handle many observations. This article introduces the Bayesian Neural Field (BayesNF), a domain-general statistical model that infers rich spatiotemporal probability distributions for data-analysis tasks including forecasting, interpolation, and variography. BayesNF integrates a deep neural network architecture for high-capacity function estimation with hierarchical Bayesian inference for robust predictive uncertainty quantification. Evaluations against prominent baselines show that BayesNF delivers improvements on prediction problems from climate and public health data containing tens to hundreds of thousands of measurements. Accompanying the paper is an open-source software package (https://github.com/google/bayesnf) that runs on GPU and TPU accelerators through the JAX machine learning platform.
We present Bayesian Diffusion Models (BDM), a prediction algorithm that performs effective Bayesian inference by tightly coupling the top-down (prior) information with the bottom-up (data-driven) procedure via joint diffusion processes. We show the effectiveness of BDM on the 3D shape reconstruction task. Compared to prototypical deep learning data-driven approaches trained on paired (supervised) data-labels (e.g. image-point clouds) datasets, our BDM brings in rich prior information from standalone labels (e.g. point clouds) to improve the bottom-up 3D reconstruction. As opposed to the standard Bayesian frameworks where explicit prior and likelihood are required for the inference, BDM performs seamless information fusion via coupled diffusion processes with learned gradient computation networks. The specialty of our BDM lies in its capability to engage the active and effective information exchange and fusion of the top-down and bottom-up processes where each itself is a diffusion process. We demonstrate state-of-the-art results on both synthetic and real-world benchmarks for 3D shape reconstruction.
In-context learning is a promising approach for online policy learning of offline reinforcement learning (RL) methods, which can be achieved at inference time without gradient optimization. However, this method is hindered by significant computational costs resulting from the gathering of large training trajectory sets and the need to train large Transformer models. We address this challenge by introducing an In-context Exploration-Exploitation (ICEE) algorithm, designed to optimize the efficiency of in-context policy learning. Unlike existing models, ICEE performs an exploration-exploitation trade-off at inference time within a Transformer model, without the need for explicit Bayesian inference. Consequently, ICEE can solve Bayesian optimization problems as efficiently as Gaussian process biased methods do, but in significantly less time. Through experiments in grid world environments, we demonstrate that ICEE can learn to solve new RL tasks using only tens of episodes, marking a substantial improvement over the hundreds of episodes needed by the previous in-context learning method.
Regression methods are fundamental for scientific and technological applications. However, fitted models can be highly unreliable outside of their training domain, and hence the quantification of their uncertainty is crucial in many of their applications. Based on the solution of a constrained optimization problem, we propose "prediction rigidities" as a method to obtain uncertainties of arbitrary pre-trained regressors. We establish a strong connection between our framework and Bayesian inference, and we develop a last-layer approximation that allows the new method to be applied to neural networks. This extension affords cheap uncertainties without any modification to the neural network itself or its training procedure. We show the effectiveness of our method on a wide range of regression tasks, ranging from simple toy models to applications in chemistry and meteorology.
Accurate representations of unknown and sub-grid physical processes through parameterizations (or closure) in numerical simulations with quantified uncertainty are critical for resolving the coarse-grained partial differential equations that govern many problems ranging from weather and climate prediction to turbulence simulations. Recent advances have seen machine learning (ML) increasingly applied to model these subgrid processes, resulting in the development of hybrid physics-ML models through the integration with numerical solvers. In this work, we introduce a novel framework for the joint estimation of physical parameters and machine learning parameterizations with uncertainty quantification. Our framework incorporates online training and efficient Bayesian inference within a high-dimensional parameter space, facilitated by differentiable programming. This proof of concept underscores the substantial potential of differentiable programming in synergistically combining machine learning with differential equations, thereby enhancing the capabilities of hybrid physics-ML modeling.
Recovering dynamical equations from observed noisy data is the central challenge of system identification. We develop a statistical mechanics approach to analyze sparse equation discovery algorithms, which typically balance data fit and parsimony via hyperparameter tuning. In this framework, statistical mechanics offers tools to analyze the interplay between complexity and fitness similarly to that of entropy and energy in physical systems. To establish this analogy, we define the hyperparameter optimization procedure as a two-level Bayesian inference problem that separates variable selection from coefficient inference and enables the computation of the posterior parameter distribution in closed form. Our approach provides uncertainty quantification, crucial in the low-data limit that is frequently encountered in real-world applications. A key advantage of employing statistical mechanical concepts, such as free energy and the partition function, is to connect the large data limit to thermodynamic limit and characterize the sparsity- and noise-induced phase transitions that delineate correct from incorrect identification. We thus provide a method for closed-loop inference, estimating the noise in a given model and checking if the model is tolerant to that noise amount. This perspective of sparse equation discovery is versatile and can be adapted to various other equation discovery algorithms.
In recent years, denoising problems have become intertwined with the development of deep generative models. In particular, diffusion models are trained like denoisers, and the distribution they model coincide with denoising priors in the Bayesian picture. However, denoising through diffusion-based posterior sampling requires the noise level and covariance to be known, preventing blind denoising. We overcome this limitation by introducing Gibbs Diffusion (GDiff), a general methodology addressing posterior sampling of both the signal and the noise parameters. Assuming arbitrary parametric Gaussian noise, we develop a Gibbs algorithm that alternates sampling steps from a conditional diffusion model trained to map the signal prior to the family of noise distributions, and a Monte Carlo sampler to infer the noise parameters. Our theoretical analysis highlights potential pitfalls, guides diagnostic usage, and quantifies errors in the Gibbs stationary distribution caused by the diffusion model. We showcase our method for 1) blind denoising of natural images involving colored noises with unknown amplitude and spectral index, and 2) a cosmology problem, namely the analysis of cosmic microwave background data, where Bayesian inference of "noise" parameters means constraining models of the evolution of the Universe.
Transport-based density estimation methods are receiving growing interest because of their ability to efficiently generate samples from the approximated density. We further invertigate the sequential transport maps framework proposed from arXiv:2106.04170 arXiv:2303.02554, which builds on a sequence of composed Knothe-Rosenblatt (KR) maps. Each of those maps are built by first estimating an intermediate density of moderate complexity, and then by computing the exact KR map from a reference density to the precomputed approximate density. In our work, we explore the use of Sum-of-Squares (SoS) densities and $\alpha$-divergences for approximating the intermediate densities. Combining SoS densities with $\alpha$-divergence interestingly yields convex optimization problems which can be efficiently solved using semidefinite programming. The main advantage of $\alpha$-divergences is to enable working with unnormalized densities, which provides benefits both numerically and theoretically. In particular, we provide a new convergence analyses of the sequential transport maps based on information geometric properties of $\alpha$-divergences. The choice of intermediate densities is also crucial for the efficiency of the method. While tempered (or annealed) densities are the state-of-the-art, we introduce diffusion-based intermediate densities which permits to approximate densities known from samples only. Such intermediate densities are well-established in machine learning for generative modeling. Finally we propose low-dimensional maps (or lazy maps) for dealing with high-dimensional problems and numerically demonstrate our methods on Bayesian inference problems and unsupervised learning tasks.
Extracting information efficiently from quantum systems is a major component of quantum information processing tasks. Randomized measurements, or classical shadows, enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Prepending a shallow random quantum circuit before measurements maintains this experimental friendliness, but also has favorable sample complexities for observables beyond low-weight Paulis, including high-weight Paulis and global low-rank properties such as fidelity. However, in realistic scenarios, quantum noise accumulated with each additional layer of the shallow circuit biases the results. To address these challenges, we propose the \emph{robust shallow shadows protocol}. Our protocol uses Bayesian inference to learn the experimentally relevant noise model and mitigate it in postprocessing. This mitigation introduces a bias-variance trade-off: correcting for noise-induced bias comes at the cost of a larger estimator variance. Despite this increased variance, as we demonstrate on a superconducting quantum processor, our protocol correctly recovers state properties such as expectation values, fidelity, and entanglement entropy, while maintaining a lower sample complexity compared to the random single qubit measurement scheme. We also theoretically analyze the effects of noise on sample complexity and show how the optimal choice of the shallow shadow depth varies with noise strength. This combined theoretical and experimental analysis positions the robust shallow shadow protocol as a scalable, robust, and sample-efficient protocol for characterizing quantum states on current quantum computing platforms.
The expectation maximization (EM) algorithm is a widespread method for empirical Bayesian inference, but its expectation step (E-step) is often intractable. Employing a stochastic approximation scheme with Markov chain Monte Carlo (MCMC) can circumvent this issue, resulting in an algorithm known as MCMC-SAEM. While theoretical guarantees for MCMC-SAEM have previously been established, these results are restricted to the case where asymptotically unbiased MCMC algorithms are used. In practice, MCMC-SAEM is often run with asymptotically biased MCMC, for which the consequences are theoretically less understood. In this work, we fill this gap by analyzing the asymptotics and non-asymptotics of SAEM with biased MCMC steps, particularly the effect of bias. We also provide numerical experiments comparing the Metropolis-adjusted Langevin algorithm (MALA), which is asymptotically unbiased, and the unadjusted Langevin algorithm (ULA), which is asymptotically biased, on synthetic and real datasets. Experimental results show that ULA is more stable with respect to the choice of Langevin stepsize and can sometimes result in faster convergence.
People often give instructions whose meaning is ambiguous without further context, expecting that their actions or goals will disambiguate their intentions. How can we build assistive agents that follow such instructions in a flexible, context-sensitive manner? This paper introduces cooperative language-guided inverse plan search (CLIPS), a Bayesian agent architecture for pragmatic instruction following and goal assistance. Our agent assists a human by modeling them as a cooperative planner who communicates joint plans to the assistant, then performs multimodal Bayesian inference over the human's goal from actions and language, using large language models (LLMs) to evaluate the likelihood of an instruction given a hypothesized plan. Given this posterior, our assistant acts to minimize expected goal achievement cost, enabling it to pragmatically follow ambiguous instructions and provide effective assistance even when uncertain about the goal. We evaluate these capabilities in two cooperative planning domains (Doors, Keys & Gems and VirtualHome), finding that CLIPS significantly outperforms GPT-4V, LLM-based literal instruction following and unimodal inverse planning in both accuracy and helpfulness, while closely matching the inferences and assistive judgments provided by human raters.
This paper introduces a quasi-Bayesian method that integrates frequentist nonparametric estimation with Bayesian inference in a two-stage process. Applied to an endogenous discrete choice model, the approach first uses kernel or sieve estimators to estimate the control function nonparametrically, followed by Bayesian methods to estimate the structural parameters. This combination leverages the advantages of both frequentist tractability for nonparametric estimation and Bayesian computational efficiency for complicated structural models. We analyze the asymptotic properties of the resulting quasi-posterior distribution, finding that its mean provides a consistent estimator for the parameters of interest, although its quantiles do not yield valid confidence intervals. However, bootstrapping the quasi-posterior mean accounts for the estimation uncertainty from the first stage, thereby producing asymptotically valid confidence intervals.
We propose a novel deep learning framework, named SYMHnet, which employs a graph neural network and a bidirectional long short-term memory network to cooperatively learn patterns from solar wind and interplanetary magnetic field parameters for short-term forecasts of the SYM-H index based on 1-minute and 5-minute resolution data. SYMHnet takes, as input, the time series of the parameters' values provided by NASA's Space Science Data Coordinated Archive and predicts, as output, the SYM-H index value at time point t + w hours for a given time point t where w is 1 or 2. By incorporating Bayesian inference into the learning framework, SYMHnet can quantify both aleatoric (data) uncertainty and epistemic (model) uncertainty when predicting future SYM-H indices. Experimental results show that SYMHnet works well at quiet time and storm time, for both 1-minute and 5-minute resolution data. The results also show that SYMHnet generally performs better than related machine learning methods. For example, SYMHnet achieves a forecast skill score (FSS) of 0.343 compared to the FSS of 0.074 of a recent gradient boosting machine (GBM) method when predicting SYM-H indices (1 hour in advance) in a large storm (SYM-H = -393 nT) using 5-minute resolution data. When predicting the SYM-H indices (2 hours in advance) in the large storm, SYMHnet achieves an FSS of 0.553 compared to the FSS of 0.087 of the GBM method. In addition, SYMHnet can provide results for both data and model uncertainty quantification, whereas the related methods cannot.