We consider a teacher-student model of supervised learning with a fully-trained 2-layer neural network whose width $k$ and input dimension $d$ are large and proportional. We compute the Bayes-optimal generalisation error of the network for any activation function in the regime where the number of training data $n$ scales quadratically with the input dimension, i.e., around the interpolation threshold where the number of trainable parameters $kd+k$ and of data points $n$ are comparable. Our analysis tackles generic weight distributions. Focusing on binary weights, we uncover a discontinuous phase transition separating a "universal" phase from a "specialisation" phase. In the first, the generalisation error is independent of the weight distribution and decays slowly with the sampling rate $n/d^2$, with the student learning only some non-linear combinations of the teacher weights. In the latter, the error is weight distribution-dependent and decays faster due to the alignment of the student towards the teacher network. We thus unveil the existence of a highly predictive solution near interpolation, which is however potentially hard to find.

This paper considers the problem of estimating chirp parameters from a noisy mixture of chirps. While a rich body of work exists in this area, challenges remain when extending these techniques to chirps of higher order polynomials. We formulate this as a non-convex optimization problem and propose a modified Langevin Monte Carlo (LMC) sampler that exploits the average curvature of the objective function to reliably find the minimizer. Results show that our Curvature-guided LMC (CG-LMC) algorithm is robust and succeeds even in low SNR regimes, making it viable for practical applications.

The Job Shop Scheduling Problem (JSSP) is a well-known optimization problem in manufacturing, where the goal is to determine the optimal sequence of jobs across different machines to minimize a given objective. In this work, we focus on minimising the weighted sum of job completion times. We explore the potential of Monte Carlo Tree Search (MCTS), a heuristic-based reinforcement learning technique, to solve large-scale JSSPs, especially those with recirculation. We propose several Markov Decision Process (MDP) formulations to model the JSSP for the MCTS algorithm. In addition, we introduce a new synthetic benchmark derived from real manufacturing data, which captures the complexity of large, non-rectangular instances often encountered in practice. Our experimental results show that MCTS effectively produces good-quality solutions for large-scale JSSP instances, outperforming our constraint programming approach.

Existing actor-critic algorithms, which are popular for continuous control reinforcement learning (RL) tasks, suffer from poor sample efficiency due to lack of principled exploration mechanism within them. Motivated by the success of Thompson sampling for efficient exploration in RL, we propose a novel model-free RL algorithm, Langevin Soft Actor Critic (LSAC), which prioritizes enhancing critic learning through uncertainty estimation over policy optimization. LSAC employs three key innovations: approximate Thompson sampling through distributional Langevin Monte Carlo (LMC) based $Q$ updates, parallel tempering for exploring multiple modes of the posterior of the $Q$ function, and diffusion synthesized state-action samples regularized with $Q$ action gradients. Our extensive experiments demonstrate that LSAC outperforms or matches the performance of mainstream model-free RL algorithms for continuous control tasks. Notably, LSAC marks the first successful application of an LMC based Thompson sampling in continuous control tasks with continuous action spaces.

We would like a robot to navigate to a goal location while minimizing state uncertainty. To aid the robot in this endeavor, maps provide a prior belief over the location of objects and regions of interest. To localize itself within the map, a robot identifies mapped landmarks using its sensors. However, as the time between map creation and robot deployment increases, portions of the map can become stale, and landmarks, once believed to be permanent, may disappear. We refer to the propensity of a landmark to disappear as landmark evanescence. Reasoning about landmark evanescence during path planning, and the associated impact on localization accuracy, requires analyzing the presence or absence of each landmark, leading to an exponential number of possible outcomes of a given motion plan. To address this complexity, we develop BRULE, an extension of the Belief Roadmap. During planning, we replace the belief over future robot poses with a Gaussian mixture which is able to capture the effects of landmark evanescence. Furthermore, we show that belief updates can be made efficient, and that maintaining a random subset of mixture components is sufficient to find high quality solutions. We demonstrate performance in simulated and real-world experiments. Software is available at https://bit.ly/BRULE.

Time-sensitive machine learning benefits from Sequential Probability Ratio Test (SPRT), which provides an optimal stopping time for early classification of time series. However, in finite horizon scenarios, where input lengths are finite, determining the optimal stopping rule becomes computationally intensive due to the need for backward induction, limiting practical applicability. We thus introduce FIRMBOUND, an SPRT-based framework that efficiently estimates the solution to backward induction from training data, bridging the gap between optimal stopping theory and real-world deployment. It employs density ratio estimation and convex function learning to provide statistically consistent estimators for sufficient statistic and conditional expectation, both essential for solving backward induction; consequently, FIRMBOUND minimizes Bayes risk to reach optimality. Additionally, we present a faster alternative using Gaussian process regression, which significantly reduces training time while retaining low deployment overhead, albeit with potential compromise in statistical consistency. Experiments across independent and identically distributed (i.i.d.), non-i.i.d., binary, multiclass, synthetic, and real-world datasets show that FIRMBOUND achieves optimalities in the sense of Bayes risk and speed-accuracy tradeoff. Furthermore, it advances the tradeoff boundary toward optimality when possible and reduces decision-time variance, ensuring reliable decision-making. Code is publicly available at https://github.com/Akinori-F-Ebihara/FIRMBOUND

Quantifying uncertainty in neural networks is a highly relevant problem which is essential to many applications. The two predominant paradigms to tackle this task are Bayesian neural networks (BNNs) and deep ensembles. Despite some similarities between these two approaches, they are typically surmised to lack a formal connection and are thus understood as fundamentally different. BNNs are often touted as more principled due to their reliance on the Bayesian paradigm, whereas ensembles are perceived as more ad-hoc; yet, deep ensembles tend to empirically outperform BNNs, with no satisfying explanation as to why this is the case. In this work we bridge this gap by showing that deep ensembles perform exact Bayesian averaging with a posterior obtained with an implicitly learned data-dependent prior. In other words deep ensembles are Bayesian, or more specifically, they implement an empirical Bayes procedure wherein the prior is learned from the data. This perspective offers two main benefits: (i) it theoretically justifies deep ensembles and thus provides an explanation for their strong empirical performance; and (ii) inspection of the learned prior reveals it is given by a mixture of point masses -- the use of such a strong prior helps elucidate observed phenomena about ensembles. Overall, our work delivers a newfound understanding of deep ensembles which is not only of interest in it of itself, but which is also likely to generate future insights that drive empirical improvements for these models.

We propose a novel deep learning method which combines classical regularization with data augmentation for estimating myelin water fraction (MWF) in the brain via biexponential analysis. Our aim is to design an accurate deep learning technique for analysis of signals arising in magnetic resonance relaxometry. In particular, we study the biexponential model, one of the signal models used for MWF estimation. We greatly extend our previous work on \emph{input layer regularization (ILR)} in several ways. We now incorporate optimal regularization parameter selection via a dedicated neural network or generalized cross validation (GCV) on a signal-by-signal, or pixel-by-pixel, basis to form the augmented input signal, and now incorporate estimation of MWF, rather than just exponential time constants, into the analysis. On synthetically generated data, our proposed deep learning architecture outperformed both classical methods and a conventional multi-layer perceptron. On in vivo brain data, our architecture again outperformed other comparison methods, with GCV proving to be somewhat superior to a NN for regularization parameter selection. Thus, ILR improves estimation of MWF within the biexponential model. In addition, classical methods such as GCV may be combined with deep learning to optimize MWF imaging in the human brain.

Safe reinforcement learning has traditionally relied on predefined constraint functions to ensure safety in complex real-world tasks, such as autonomous driving. However, defining these functions accurately for varied tasks is a persistent challenge. Recent research highlights the potential of leveraging pre-acquired task-agnostic knowledge to enhance both safety and sample efficiency in related tasks. Building on this insight, we propose a novel method to learn shared constraint distributions across multiple tasks. Our approach identifies the shared constraints through imitation learning and then adapts to new tasks by adjusting risk levels within these learned distributions. This adaptability addresses variations in risk sensitivity stemming from expert-specific biases, ensuring consistent adherence to general safety principles even with imperfect demonstrations. Our method can be applied to control and navigation domains, including multi-task and meta-task scenarios, accommodating constraints such as maintaining safe distances or adhering to speed limits. Experimental results validate the efficacy of our approach, demonstrating superior safety performance and success rates compared to baselines, all without requiring task-specific constraint definitions. These findings underscore the versatility and practicality of our method across a wide range of real-world tasks.

Recent works in Learning-Based Model Predictive Control of dynamical systems show impressive sample complexity performances using criteria from Information Theory to accelerate the learning procedure. However, the sequential exploration opportunities are limited by the system local state, restraining the amount of information of the observations from the current exploration trajectory. This article resolves this limitation by introducing temporal abstraction through the framework of Semi-Markov Decision Processes. The framework increases the total information of the gathered data for a fixed sampling budget, thus reducing the sample complexity.