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A computational framework that leverages data from self-consistent field theory simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented. This is a substantial two-dimensional extension of the framework introduced in [1]. Several innovations and improvements are proposed. (1) A Sobolev space-trained, convolutional neural network (CNN) is employed to handle the exponential dimension increase of the discretized, local average monomer density fields and to strongly enforce both spatial translation and rotation invariance of the predicted, field-theoretic intensive Hamiltonian. (2) A generative adversarial network (GAN) is introduced to efficiently and accurately predict saddle point, local average monomer density fields without resorting to gradient descent methods that employ the training set. This GAN approach yields important savings of both memory and computational cost. (3) The proposed machine learning framework is successfully applied to 2D cell size optimization as a clear illustration of its broad potential to accelerate the exploration of parameter space for discovering polymer nanostructures. Extensions to three-dimensional phase discovery appear to be feasible.

The Makespan Scheduling problem is an extensively studied NP-hard problem, and its simplest version looks for an allocation approach for a set of jobs with deterministic processing times to two identical machines such that the makespan is minimized. However, in real life scenarios, the actual processing time of each job may be stochastic around the expected value with a variance, under the influence of external factors, and the actual processing times of these jobs may be correlated with covariances. Thus within this paper, we propose a chance-constrained version of the Makespan Scheduling problem and investigate the theoretical performance of the classical Randomized Local Search and (1+1) EA for it. More specifically, we first study two variants of the Chance-constrained Makespan Scheduling problem and their computational complexities, then separately analyze the expected runtime of the two algorithms to obtain an optimal solution or almost optimal solution to the instances of the two variants. In addition, we investigate the experimental performance of the two algorithms for the two variants.

In data-driven drug discovery, designing molecular descriptors is a very important task. Deep generative models such as variational autoencoders (VAEs) offer a potential solution by designing descriptors as probabilistic latent vectors derived from molecular structures. These models can be trained on large datasets, which have only molecular structures, and applied to transfer learning. Nevertheless, the approximate posterior distribution of the latent vectors of the usual VAE assumes a simple multivariate Gaussian distribution with zero covariance, which may limit the performance of representing the latent features. To overcome this limitation, we propose a novel molecular deep generative model that incorporates a hierarchical structure into the probabilistic latent vectors. We achieve this by a denoising diffusion probabilistic model (DDPM). We demonstrate that our model can design effective molecular latent vectors for molecular property prediction from some experiments by small datasets on physical properties and activity. The results highlight the superior prediction performance and robustness of our model compared to existing approaches.

Particle Markov Chain Monte Carlo (PMCMC) is a general computational approach to Bayesian inference for general state space models. Our article scales up PMCMC in terms of the number of observations and parameters by generating the parameters that are highly correlated with the states \lq integrated out\rq{} in a pseudo marginal step; the rest of the parameters are generated conditional on the states. The novel contribution of our article is to make the pseudo-marginal step much more efficient by positively correlating the numerator and denominator in the Metropolis-Hastings acceptance probability. This is done in a novel way by expressing the target density of the PMCMC in terms of the basic uniform or normal random numbers used in the sequential Monte Carlo algorithm instead of the standard way in terms of state particles. We also show that the new sampler combines and generalizes two separate particle MCMC approaches: particle Gibbs and the correlated pseudo marginal Metropolis-Hastings. We investigate the performance of the hybrid sampler empirically by applying it to univariate and multivariate stochastic volatility models having both a large number of parameters and a large number of latent states and show that it is much more efficient than competing PMCMC methods.

We propose a new generative hierarchical clustering model that learns a flexible tree-based posterior distribution over latent variables. The proposed Tree Variational Autoencoder (TreeVAE) hierarchically divides samples according to their intrinsic characteristics, shedding light on hidden structure in the data. It adapts its architecture to discover the optimal tree for encoding dependencies between latent variables. The proposed tree-based generative architecture permits lightweight conditional inference and improves generative performance by utilizing specialized leaf decoders. We show that TreeVAE uncovers underlying clusters in the data and finds meaningful hierarchical relations between the different groups on a variety of datasets, including real-world imaging data. We present empirically that TreeVAE provides a more competitive log-likelihood lower bound than the sequential counterparts. Finally, due to its generative nature, TreeVAE is able to generate new samples from the discovered clusters via conditional sampling.

Consistent weighted least square estimators are proposed for a wide class of nonparametric regression models with random regression function, where this real-valued random function of $k$ arguments is assumed to be continuous with probability 1. We obtain explicit upper bounds for the rate of uniform convergence in probability of the new estimators to the unobservable random regression function for both fixed or random designs. In contrast to the predecessors' results, the bounds for the convergence are insensitive to the correlation structure of the $k$-variate design points. As an application, we study the problem of estimating the mean and covariance functions of random fields with additive noise under dense data conditions. The theoretical results of the study are illustrated by simulation examples which show that the new estimators are more accurate in some cases than the Nadaraya--Watson ones. An example of processing real data on earthquakes in Japan in 2012--2021 is included.

Longitudinal studies with binary or ordinal responses are widely encountered in various disciplines, where the primary focus is on the temporal evolution of the probability of each response category. Traditional approaches build from the generalized mixed effects modeling framework. Even amplified with nonparametric priors placed on the fixed or random effects, such models are restrictive due to the implied assumptions on the marginal expectation and covariance structure of the responses. We tackle the problem from a functional data analysis perspective, treating the observations for each subject as realizations from subject-specific stochastic processes at the measured times. We develop the methodology focusing initially on binary responses, for which we assume the stochastic processes have Binomial marginal distributions. Leveraging the logits representation, we model the discrete space processes through sequences of continuous space processes. We utilize a hierarchical framework to model the mean and covariance kernel of the continuous space processes nonparametrically and simultaneously through a Gaussian process prior and an Inverse-Wishart process prior, respectively. The prior structure results in flexible inference for the evolution and correlation of binary responses, while allowing for borrowing of strength across all subjects. The modeling approach can be naturally extended to ordinal responses. Here, the continuation-ratio logits factorization of the multinomial distribution is key for efficient modeling and inference, including a practical way of dealing with unbalanced longitudinal data. The methodology is illustrated with synthetic data examples and an analysis of college students' mental health status data.

Computing the marginal likelihood (also called the Bayesian model evidence) is an important task in Bayesian model selection, providing a principled quantitative way to compare models. The learned harmonic mean estimator solves the exploding variance problem of the original harmonic mean estimation of the marginal likelihood. The learned harmonic mean estimator learns an importance sampling target distribution that approximates the optimal distribution. While the approximation need not be highly accurate, it is critical that the probability mass of the learned distribution is contained within the posterior in order to avoid the exploding variance problem. In previous work a bespoke optimization problem is introduced when training models in order to ensure this property is satisfied. In the current article we introduce the use of normalizing flows to represent the importance sampling target distribution. A flow-based model is trained on samples from the posterior by maximum likelihood estimation. Then, the probability density of the flow is concentrated by lowering the variance of the base distribution, i.e. by lowering its "temperature", ensuring its probability mass is contained within the posterior. This approach avoids the need for a bespoke optimisation problem and careful fine tuning of parameters, resulting in a more robust method. Moreover, the use of normalizing flows has the potential to scale to high dimensional settings. We present preliminary experiments demonstrating the effectiveness of the use of flows for the learned harmonic mean estimator. The harmonic code implementing the learned harmonic mean, which is publicly available, has been updated to now support normalizing flows.

Due to its empirical success in few-shot classification and reinforcement learning, meta-learning has recently received significant interest. Meta-learning methods leverage data from previous tasks to learn a new task in a sample-efficient manner. In particular, model-agnostic methods look for initialisation points from which gradient descent quickly adapts to any new task. Although it has been empirically suggested that such methods perform well by learning shared representations during pretraining, there is limited theoretical evidence of such behavior. More importantly, it has not been rigorously shown that these methods still learn a shared structure, despite architectural misspecifications. In this direction, this work shows, in the limit of an infinite number of tasks, that first-order ANIL with a linear two-layer network architecture successfully learns linear shared representations. This result even holds with a misspecified network parameterisation; having a width larger than the dimension of the shared representations results in an asymptotically low-rank solution. The learnt solution then yields a good adaptation performance on any new task after a single gradient step. Overall this illustrates how well model-agnostic methods such as first-order ANIL can learn shared representations.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.