In physics, density $\rho(\cdot)$ is a fundamentally important scalar function to model, since it describes a scalar field or a probability density function that governs a physical process. Modeling $\rho(\cdot)$ typically scales poorly with parameter space, however, and quickly becomes prohibitively difficult and computationally expensive. One promising avenue to bypass this is to leverage the capabilities of denoising diffusion models often used in high-fidelity image generation to parameterize $\rho(\cdot)$ from existing scientific data, from which new samples can be trivially sampled from. In this paper, we propose $\rho$-Diffusion, an implementation of denoising diffusion probabilistic models for multidimensional density estimation in physics, which is currently in active development and, from our results, performs well on physically motivated 2D and 3D density functions. Moreover, we propose a novel hashing technique that allows $\rho$-Diffusion to be conditioned by arbitrary amounts of physical parameters of interest.

Model explanations are very valuable for interpreting and debugging prediction models. We study a specific kind of global explanations called Concept Explanations, where the goal is to interpret a model using human-understandable concepts. Recent advances in multi-modal learning rekindled interest in concept explanations and led to several label-efficient proposals for estimation. However, existing estimation methods are unstable to the choice of concepts or dataset that is used for computing explanations. We observe that instability in explanations is due to high variance in point estimation of importance scores. We propose an uncertainty aware Bayesian estimation method, which readily improved reliability of the concept explanations. We demonstrate with theoretical analysis and empirical evaluation that explanations computed by our method are more reliable while also being label-efficient and faithful.

Variational autoencoder (VAE) and generative adversarial networks (GAN) have found widespread applications in clustering and have achieved significant success. However, the potential of these approaches may be limited due to VAE's mediocre generation capability or GAN's well-known instability during adversarial training. In contrast, denoising diffusion probabilistic models (DDPMs) represent a new and promising class of generative models that may unlock fresh dimensions in clustering. In this study, we introduce an innovative expectation-maximization (EM) framework for clustering using DDPMs. In the E-step, we aim to derive a mixture of Gaussian priors for the subsequent M-step. In the M-step, our focus lies in learning clustering-friendly latent representations for the data by employing the conditional DDPM and matching the distribution of latent representations to the mixture of Gaussian priors. We present a rigorous theoretical analysis of the optimization process in the M-step, proving that the optimizations are equivalent to maximizing the lower bound of the Q function within the vanilla EM framework under certain constraints. Comprehensive experiments validate the advantages of the proposed framework, showcasing superior performance in clustering, unsupervised conditional generation and latent representation learning.

We consider the problem of quantifying uncertainty over expected cumulative rewards in model-based reinforcement learning. In particular, we focus on characterizing the variance over values induced by a distribution over MDPs. Previous work upper bounds the posterior variance over values by solving a so-called uncertainty Bellman equation (UBE), but the over-approximation may result in inefficient exploration. We propose a new UBE whose solution converges to the true posterior variance over values and leads to lower regret in tabular exploration problems. We identify challenges to apply the UBE theory beyond tabular problems and propose a suitable approximation. Based on this approximation, we introduce a general-purpose policy optimization algorithm, Q-Uncertainty Soft Actor-Critic (QU-SAC), that can be applied for either risk-seeking or risk-averse policy optimization with minimal changes. Experiments in both online and offline RL demonstrate improved performance compared to other uncertainty estimation methods.

Generative models have gained popularity for their potential applications in imaging science, such as image reconstruction, posterior sampling and data sharing. Flow-based generative models are particularly attractive due to their ability to tractably provide exact density estimates along with fast, inexpensive and diverse samples. Training such models, however, requires a large, high quality dataset of objects. In applications such as computed imaging, it is often difficult to acquire such data due to requirements such as long acquisition time or high radiation dose, while acquiring noisy or partially observed measurements of these objects is more feasible. In this work, we propose AmbientFlow, a framework for learning flow-based generative models directly from noisy and incomplete data. Using variational Bayesian methods, a novel framework for establishing flow-based generative models from noisy, incomplete data is proposed. Extensive numerical studies demonstrate the effectiveness of AmbientFlow in learning the object distribution. The utility of AmbientFlow in a downstream inference task of image reconstruction is demonstrated.

Gradient-based sampling algorithms have demonstrated their effectiveness in text generation, especially in the context of controlled text generation. However, there exists a lack of theoretically grounded and principled approaches for this task. In this paper, we take an important step toward building a principled approach for sampling from language models with gradient-based methods. We use discrete distributions given by language models to define densities and develop an algorithm based on Hamiltonian Monte Carlo to sample from them. We name our gradient-based technique Structured Voronoi Sampling (SVS). In an experimental setup where the reference distribution is known, we show that the empirical distribution of SVS samples is closer to the reference distribution compared to alternative sampling schemes. Furthermore, in a controlled generation task, SVS is able to generate fluent and diverse samples while following the control targets significantly better than other methods.

In the face of dataset shift, model calibration plays a pivotal role in ensuring the reliability of machine learning systems. Calibration error (CE) is an indicator of the alignment between the predicted probabilities and the classifier accuracy. While prior works have delved into the implications of dataset shift on calibration, existing CE estimators assume access to labels from the target domain, which are often unavailable in practice, i.e., when the model is deployed and used. This work addresses such challenging scenario, and proposes a novel CE estimator under label shift, which is characterized by changes in the marginal label distribution $p(Y)$, while keeping the conditional $p(X|Y)$ constant between the source and target distributions. Our contribution is an approach, which, by leveraging importance re-weighting of the labeled source distribution, provides consistent and asymptotically unbiased CE estimation with respect to the shifted target distribution. Empirical results across diverse real-world datasets, under various conditions and label-shift intensities, demonstrate the effectiveness and reliability of the proposed estimator.

Modern reinforcement learning (RL) can be categorized into online and offline variants. As a pivotal aspect of both online and offline RL, current research on the Bellman equation revolves primarily around optimization techniques and performance enhancement rather than exploring the inherent structural properties of the Bellman error, such as its distribution characteristics. This study investigates the distribution of the Bellman approximation error through iterative exploration of the Bellman equation with the observation that the Bellman error approximately follows the Logistic distribution. Based on this, we proposed the utilization of the Logistic maximum likelihood function (LLoss) as an alternative to the commonly used mean squared error (MSELoss) that assumes a Normal distribution for Bellman errors. We validated the hypotheses through extensive numerical experiments across diverse online and offline environments. In particular, we applied the Logistic correction to loss functions in various RL baseline methods and observed that the results with LLoss consistently outperformed the MSE counterparts. We also conducted the Kolmogorov-Smirnov tests to confirm the reliability of the Logistic distribution. Moreover, our theory connects the Bellman error to the proportional reward scaling phenomenon by providing a distribution-based analysis. Furthermore, we applied the bias-variance decomposition for sampling from the Logistic distribution. The theoretical and empirical insights of this study lay a valuable foundation for future investigations and enhancements centered on the distribution of Bellman error.

We study experiment design for unique identification of the causal graph of a simple SCM, where the graph may contain cycles. The presence of cycles in the structure introduces major challenges for experiment design as, unlike acyclic graphs, learning the skeleton of causal graphs with cycles may not be possible from merely the observational distribution. Furthermore, intervening on a variable in such graphs does not necessarily lead to orienting all the edges incident to it. In this paper, we propose an experiment design approach that can learn both cyclic and acyclic graphs and hence, unifies the task of experiment design for both types of graphs. We provide a lower bound on the number of experiments required to guarantee the unique identification of the causal graph in the worst case, showing that the proposed approach is order-optimal in terms of the number of experiments up to an additive logarithmic term. Moreover, we extend our result to the setting where the size of each experiment is bounded by a constant. For this case, we show that our approach is optimal in terms of the size of the largest experiment required for uniquely identifying the causal graph in the worst case.

Proper scoring rules evaluate the quality of probabilistic predictions, playing an essential role in the pursuit of accurate and well-calibrated models. Every proper score decomposes into two fundamental components -- proper calibration error and refinement -- utilizing a Bregman divergence. While uncertainty calibration has gained significant attention, current literature lacks a general estimator for these quantities with known statistical properties. To address this gap, we propose a method that allows consistent, and asymptotically unbiased estimation of all proper calibration errors and refinement terms. In particular, we introduce Kullback--Leibler calibration error, induced by the commonly used cross-entropy loss. As part of our results, we prove the relation between refinement and f-divergences, which implies information monotonicity in neural networks, regardless of which proper scoring rule is optimized. Our experiments validate empirically the claimed properties of the proposed estimator and suggest that the selection of a post-hoc calibration method should be determined by the particular calibration error of interest.