We study the problem of representation transfer in offline Reinforcement Learning (RL), where a learner has access to episodic data from a number of source tasks collected a priori, and aims to learn a shared representation to be used in finding a good policy for a target task. Unlike in online RL where the agent interacts with the environment while learning a policy, in the offline setting there cannot be such interactions in either the source tasks or the target task; thus multi-task offline RL can suffer from incomplete coverage. We propose an algorithm to compute pointwise uncertainty measures for the learnt representation, and establish a data-dependent upper bound for the suboptimality of the learnt policy for the target task. Our algorithm leverages the collective exploration done by source tasks to mitigate poor coverage at some points by a few tasks, thus overcoming the limitation of needing uniformly good coverage for a meaningful transfer by existing offline algorithms. We complement our theoretical results with empirical evaluation on a rich-observation MDP which requires many samples for complete coverage. Our findings illustrate the benefits of penalizing and quantifying the uncertainty in the learnt representation.

We study automated security response for an IT infrastructure and formulate the interaction between an attacker and a defender as a partially observed, non-stationary game. We relax the standard assumption that the game model is correctly specified and consider that each player has a probabilistic conjecture about the model, which may be misspecified in the sense that the true model has probability 0. This formulation allows us to capture uncertainty about the infrastructure and the intents of the players. To learn effective game strategies online, we design a novel method where a player iteratively adapts its conjecture using Bayesian learning and updates its strategy through rollout. We prove that the conjectures converge to best fits, and we provide a bound on the performance improvement that rollout enables with a conjectured model. To characterize the steady state of the game, we propose a variant of the Berk-Nash equilibrium. We present our method through an advanced persistent threat use case. Simulation studies based on testbed measurements show that our method produces effective security strategies that adapt to a changing environment. We also find that our method enables faster convergence than current reinforcement learning techniques.

This paper introduces a Bayesian framework for image inversion by deriving a probabilistic counterpart to the regularization-by-denoising (RED) paradigm. It additionally implements a Monte Carlo algorithm specifically tailored for sampling from the resulting posterior distribution, based on an asymptotically exact data augmentation (AXDA). The proposed algorithm is an approximate instance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo step. The proposed method is applied to common imaging tasks such as deblurring, inpainting and super-resolution, demonstrating its efficacy through extensive numerical experiments. These contributions advance Bayesian inference in imaging by leveraging data-driven regularization strategies within a probabilistic framework.

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. In contrast to the prevalent paradigm of solving this problem by minimizing a loss functional, we have given a direct one-shot construction together with optimal error bounds under the manifold assumption; i.e., one assumes that the data is sampled from an unknown sub-manifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace-Beltrami operator or coordinate charts, and using this information for function approximation. This two step approach implies some extra errors in the approximation stemming from basic quantities of the data in addition to the errors inherent in function approximation. In Neural Networks, 132:253268, 2020, we have proposed a one-shot direct method to achieve function approximation without requiring the extraction of any information about the manifold other than its dimension. However, one cannot pin down the class of approximants used in that paper. In this paper, we view the unknown manifold as a sub-manifold of an ambient hypersphere and study the question of constructing a one-shot approximation using the spherical polynomials based on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively "rough" functions.

Fine-tuning large language models can improve task specific performance, although a general understanding of what the fine-tuned model has learned, forgotten and how to trust its predictions is still missing. We derive principled uncertainty quantification for fine-tuned LLMs with posterior approximations using computationally efficient low-rank adaptation ensembles. We analyze three common multiple-choice datasets using low-rank adaptation ensembles based on Mistral-7b, and draw quantitative and qualitative conclusions on their perceived complexity and model efficacy on the different target domains during and after fine-tuning. In particular, backed by the numerical experiments, we hypothesise about signals from entropic uncertainty measures for data domains that are inherently difficult for a given architecture to learn.

The second inequality can be proved by resorting to the first one.

A persistent challenge in conditional image synthesis has been to generate diverse output images from the same input image despite only one output image being observed per input image. GAN-based methods are prone to mode collapse, which leads to low diversity. To get around this, we leverage Implicit Maximum Likelihood Estimation (IMLE) which can overcome mode collapse fundamentally. IMLE uses the same generator as GANs but trains it with a different, non-adversarial objective which ensures each observed image has a generated sample nearby. Unfortunately, to generate high-fidelity images, prior IMLE-based methods require a large number of samples, which is expensive. In this paper, we propose a new method to get around this limitation, which we dub Conditional Hierarchical IMLE (CHIMLE), which can generate high-fidelity images without requiring many samples. We show CHIMLE significantly outperforms the prior best IMLE, GAN and diffusion-based methods in terms of image fidelity and mode coverage across four tasks, namely night-to-day, 16 single image super-resolution, image colourization and image decompression. Quantitatively, our method improves Fréchet Inception Distance (FID) by 36.9% on average compared to the prior best IMLE-based method, and by 27.5% on average compared to the best non-IMLE-based generalpurpose methods. More results and code are available on the project website at https://niopeng.github.io/CHIMLE/.

Any inference procedure that is too computationally expensive to be run on the full posterior can instead be run inexpensively on the coreset, with results that approximate those on the full data. However, current approaches are limited by either a significant run-time or the need for the user to specify a low-cost approximation to the full posterior. We propose a Bayesian coreset construction algorithm that first selects a uniformly random subset of data, and then optimizes the weights using a novel quasi-Newton method. Our algorithm is a simple to implement, black-box method, that does not require the user to specify a low-cost posterior approximation. It is the first to come with a general high-probability bound on the KL divergence of the output coreset posterior. Experiments demonstrate that our method provides significant improvements in coreset quality against alternatives with comparable construction times, with far less storage cost and user input required.

Such structure can be expressed in the pattern of interactions between explanatory latent variables captured through a probabilistic graphical model.

Kernel herding belongs to a family of deterministic quadratures that seek to minimize the worst-case integration error over a reproducing kernel Hilbert space (RKHS). In spite of strong experimental support, it has revealed difficult to prove that this worst-case error decreases at a faster rate than the standard square root of the number of quadrature nodes, at least in the usual case where the RKHS is infinite-dimensional. In this theoretical paper, we study a joint probability distribution over quadrature nodes, whose support tends to minimize the same worst-case error as kernel herding. We prove that it does outperform i.i.d. Monte Carlo, in the sense of coming with a tighter concentration inequality on the worst-case integration error. While not improving the rate yet, this demonstrates that the mathematical tools of the study of Gibbs measures can help understand to what extent kernel herding and its variants improve on computationally cheaper methods. Moreover, we provide early experimental evidence that a faster rate of convergence, though not worst-case, is likely.