We study the phase transition phenomenon inherent in the shuffled (permuted) regression problem, which has found numerous applications in databases, privacy, data analysis, etc. In this study, we aim to precisely identify the locations of the phase transition points by leveraging techniques from message passing (MP). In our analysis, we first transform the permutation recovery problem into a probabilistic graphical model. We then leverage the analytical tools rooted in the message passing (MP) algorithm and derive an equation to track the convergence of the MP algorithm. By linking this equation to the branching random walk process, we are able to characterize the impact of the signal-to-noise-ratio ($\snr$) on the permutation recovery. Depending on whether the signal is given or not, we separately investigate the oracle case and the non-oracle case. The bottleneck in identifying the phase transition regimes lies in deriving closed-form formulas for the corresponding critical points, but only in rare scenarios can one obtain such precise expressions. To tackle this technical challenge, this study proposes the Gaussian approximation method, which allows us to obtain the closed-form formulas in almost all scenarios. In the oracle case, our method can fairly accurately predict the phase transition $\snr$. In the non-oracle case, our algorithm can predict the maximum allowed number of permuted rows and uncover its dependency on the sample number.

Bayesian Generalized Linear Models (GLMs) define a flexible probabilistic framework to model categorical, ordinal and continuous data, and are widely used in practice. However, exact inference in GLMs is prohibitively expensive for large datasets, thus requiring approximations in practice. The resulting approximation error adversely impacts the reliability of the model and is not accounted for in the uncertainty of the prediction. In this work, we introduce a family of iterative methods that explicitly model this error. They are uniquely suited to parallel modern computing hardware, efficiently recycle computations, and compress information to reduce both the time and memory requirements for GLMs. As we demonstrate on a realistically large classification problem, our method significantly accelerates training by explicitly trading off reduced computation for increased uncertainty.

Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable invertible transformations. Therefore, once the manifold is properly aligned via a reconstruction loss, the probability density is tractable on the manifold and maximum likelihood can be used to optimize the network parameters. Naturally, the lower-dimensional representation of the data requires an injective-mapping. Recent approaches were able to enforce that the density aligns with the modelled manifold, while efficiently calculating the density volume-change term when embedding to the higher-dimensional space. However, unless the injective-mapping is analytically predefined, the learned manifold is not necessarily an efficient representation of the data. Namely, the latent dimensions of such models frequently learn an entangled intrinsic basis, with degenerate information being stored in each dimension. Alternatively, if a locally orthogonal and/or sparse basis is to be learned, here coined canonical intrinsic basis, it can serve in learning a more compact latent space representation. Toward this end, we propose a canonical manifold learning flow method, where a novel optimization objective enforces the transformation matrix to have few prominent and non-degenerate basis functions. We demonstrate that by minimizing the off-diagonal manifold metric elements $\ell_1$-norm, we can achieve such a basis, which is simultaneously sparse and/or orthogonal. Canonical manifold flow yields a more efficient use of the latent space, automatically generating fewer prominent and distinct dimensions to represent data, and a better approximation of target distributions than other manifold flow methods in most experiments we conducted, resulting in lower FID scores.

To make reinforcement learning more sample efficient, we need better credit assignment methods that measure an action's influence on future rewards. Building upon Hindsight Credit Assignment (HCA), we introduce Counterfactual Contribution Analysis (COCOA), a new family of model-based credit assignment algorithms. Our algorithms achieve precise credit assignment by measuring the contribution of actions upon obtaining subsequent rewards, by quantifying a counterfactual query: 'Would the agent still have reached this reward if it had taken another action?'. We show that measuring contributions w.r.t. rewarding states, as is done in HCA, results in spurious estimates of contributions, causing HCA to degrade towards the high-variance REINFORCE estimator in many relevant environments. Instead, we measure contributions w.r.t. rewards or learned representations of the rewarding objects, resulting in gradient estimates with lower variance. We run experiments on a suite of problems specifically designed to evaluate long-term credit assignment capabilities. By using dynamic programming, we measure ground-truth policy gradients and show that the improved performance of our new model-based credit assignment methods is due to lower bias and variance compared to HCA and common baselines. Our results demonstrate how modeling action contributions towards rewarding outcomes can be leveraged for credit assignment, opening a new path towards sample-efficient reinforcement learning.

Researchers in explainable artificial intelligence have developed numerous methods for helping users understand the predictions of complex supervised learning models. By contrast, explaining the $\textit{uncertainty}$ of model outputs has received relatively little attention. We adapt the popular Shapley value framework to explain various types of predictive uncertainty, quantifying each feature's contribution to the conditional entropy of individual model outputs. We consider games with modified characteristic functions and find deep connections between the resulting Shapley values and fundamental quantities from information theory and conditional independence testing. We outline inference procedures for finite sample error rate control with provable guarantees, and implement efficient algorithms that perform well in a range of experiments on real and simulated data. Our method has applications to covariate shift detection, active learning, feature selection, and active feature-value acquisition.

Diffusion processes are a class of stochastic differential equations (SDEs) providing a rich family of expressive models that arise naturally in dynamic modelling tasks. Probabilistic inference and learning under generative models with latent processes endowed with a non-linear diffusion process prior are intractable problems. We build upon work within variational inference, approximating the posterior process as a linear diffusion process, and point out pathologies in the approach. We propose an alternative parameterization of the Gaussian variational process using a site-based exponential family description. This allows us to trade a slow inference algorithm with fixed-point iterations for a fast algorithm for convex optimization akin to natural gradient descent, which also provides a better objective for learning model parameters.

Gibbs samplers [12] are a family of Markov Chain Monte Carlo (MCMC) algorithms [10] commonly used in various scientific fields. In the context of Bayesian Statistics, they are routinely employed to draw samples from posterior distributions of unknown parameters conditional to the observed data [28, 37]. Like most MCMC methods, they are guaranteed to converge to the correct posterior distribution as the number of iterations tends to infinity under mild assumptions [54]. However, understanding how quickly this convergence occurs, for example by quantifying the so-called mixing time of the Markov chain generated by the algorithm, is in general a hard task. In this paper we address this question for Gibbs samplers targeting certain classes of high-dimensional Bayesian hierarchical models. Analysing convergence properties, such as mixing times, is the key technical step needed to rigorously quantify the computational cost of MCMC algorithms.

Computational models and simulations are not just appealing because of their intrinsic characteristics across spatiotemporal scales, scalability, and predictive power, but also because the set of problems in cancer biomedicine that can be addressed computationally exceeds the set of those amenable to analytical solutions. Agent-based models and simulations are especially interesting candidates among computational modelling strategies in cancer research due to their capabilities to replicate realistic local and global interaction dynamics at a convenient and relevant scale. Yet, the absence of methods to validate the consistency of the results across scales can hinder adoption by turning fine-tuned models into black boxes. This review compiles relevant literature to explore strategies to leverage high-fidelity simulations of multi-scale, or multi-level, cancer models with a focus on validation approached as simulation calibration. We argue that simulation calibration goes beyond parameter optimization by embedding informative priors to generate plausible parameter configurations across multiple dimensions.

Optimal topology synthesis is generally resource-intensive and time-consuming to be done in real time for large ad-hoc robot networks. One should only perform topology re-computations if the probability of topology recoverability after the occurrence of any fault surpasses that of its irrecoverability. We formulate this problem as a binary classification problem. Then, we develop a two-pathway data-driven model based on Bayesian Gaussian mixture models that predicts the solution to a typical problem by two different pre-fault and post-fault prediction pathways. The results, obtained by the integration of the predictions of those pathways, clearly indicate the success of our model in solving the topology (ir)recoverability prediction problem compared to the best of current strategies found in the literature.

Experimental design is a fundamental problem in many science and engineering fields. In this problem, sample efficiency is crucial due to the time, money, and safety costs of real-world design evaluations. Existing approaches either rely on active data collection or access to large, labeled datasets of past experiments, making them impractical in many real-world scenarios. In this work, we address the more challenging yet realistic setting of few-shot experimental design, where only a few labeled data points of input designs and their corresponding values are available. We approach this problem as a conditional generation task, where a model conditions on a few labeled examples and the desired output to generate an optimal input design. To this end, we introduce Experiment Pretrained Transformers (ExPT), a foundation model for few-shot experimental design that employs a novel combination of synthetic pretraining with in-context learning. In ExPT, we only assume knowledge of a finite collection of unlabelled data points from the input domain and pretrain a transformer neural network to optimize diverse synthetic functions defined over this domain. Unsupervised pretraining allows ExPT to adapt to any design task at test time in an in-context fashion by conditioning on a few labeled data points from the target task and generating the candidate optima. We evaluate ExPT on few-shot experimental design in challenging domains and demonstrate its superior generality and performance compared to existing methods. The source code is available at https://github.com/tung-nd/ExPT.git.