The Schr\"odinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schr\"odinger bridge remain an active area of research. We prove an equivalence between the SBP and maximum likelihood estimation enabling direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

This paper proposes methods for unsupervised lexical acquisition for relative spatial concepts using spoken user utterances. A robot with a flexible spoken dialog system must be able to acquire linguistic representation and its meaning specific to an environment through interactions with humans as children do. Specifically, relative spatial concepts (e.g., front and right) are widely used in our daily lives, however, it is not obvious which object is a reference object when a robot learns relative spatial concepts. Therefore, we propose methods by which a robot without prior knowledge of words can learn relative spatial concepts. The methods are formulated using a probabilistic model to estimate the proper reference objects and distributions representing concepts simultaneously. The experimental results show that relative spatial concepts and a phoneme sequence representing each concept can be learned under the condition that the robot does not know which located object is the reference object. Additionally, we show that two processes in the proposed method improve the estimation accuracy of the concepts: generating candidate word sequences by class n-gram and selecting word sequences using location information. Furthermore, we show that clues to reference objects improve accuracy even though the number of candidate reference objects increases.

We study learning algorithms for the classical Markovian bandit problem with discount. We explain how to adapt PSRL [24] and UCRL2 [2] to exploit the problem structure. These variants are called MB-PSRL and MB-UCRL2. While the regret bound and runtime of vanilla implementations of PSRL and UCRL2 are exponential in the number of bandits, we show that the episodic regret of MB-PSRL and MB-UCRL2 is $\tilde O(S\sqrt{nK})$ where $K$ is the number of episodes, n is the number of bandits and S is the number of states of each bandit (the exact bound in $S$, $n$ and $K$ is given in the paper). Up to a factor $\sqrt S$, this matches the lower bound of $\Omega(\sqrt{SnK}$) that we also derive in the paper. MB-PSRL is also computationally efficient: its runtime is linear in the number of bandits. We further show that this linear runtime cannot be achieved by adapting classical non-Bayesian algorithms such as UCRL2 or UCBVI to Markovian bandit problems. Finally, we perform numerical experiments that confirm that MB-PSRL outperforms other existing algorithms in practice, both in terms of regret and of computation time.

"For how many days during the past 30 days was your mental health not good?" The responses to this question measure self-reported mental health and can be linked to important covariates in the National Health and Nutrition Examination Survey (NHANES). However, these count variables present major distributional challenges: the data are overdispersed, zero-inflated, bounded by 30, and heaped in five- and seven-day increments. To meet these challenges, we design a semiparametric estimation and inference framework for count data regression. The data-generating process is defined by simultaneously transforming and rounding (STAR) a latent Gaussian regression model. The transformation is estimated nonparametrically and the rounding operator ensures the correct support for the discrete and bounded data. Maximum likelihood estimators are computed using an EM algorithm that is compatible with any continuous data model estimable by least squares. STAR regression includes asymptotic hypothesis testing and confidence intervals, variable selection via information criteria, and customized diagnostics. Simulation studies validate the utility of this framework. STAR is deployed to study the factors associated with self-reported mental health and demonstrates substantial improvements in goodness-of-fit compared to existing count data regression models.

Recent years have witnessed an upsurge of interest in employing flexible machine learning models for instrumental variable (IV) regression, but the development of uncertainty quantification methodology is still lacking. In this work we present a scalable quasi-Bayesian procedure for IV regression, building upon the recently developed kernelized IV models. Contrary to Bayesian modeling for IV, our approach does not require additional assumptions on the data generating process, and leads to a scalable approximate inference algorithm with time cost comparable to the corresponding point estimation methods. Our algorithm can be further extended to work with neural network models. We analyze the theoretical properties of the proposed quasi-posterior, and demonstrate through empirical evaluation the competitive performance of our method.

Imitation learning enables high-fidelity, vision-based learning of policies within rich, photorealistic environments. However, such techniques often rely on traditional discrete-time neural models and face difficulties in generalizing to domain shifts by failing to account for the causal relationships between the agent and the environment. In this paper, we propose a theoretical and experimental framework for learning causal representations using continuous-time neural networks, specifically over their discrete-time counterparts. We evaluate our method in the context of visual-control learning of drones over a series of complex tasks, ranging from short- and long-term navigation, to chasing static and dynamic objects through photorealistic environments. Our results demonstrate that causal continuous-time deep models can perform robust navigation tasks, where advanced recurrent models fail. These models learn complex causal control representations directly from raw visual inputs and scale to solve a variety of tasks using imitation learning.

Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quantity of interest as a random variable, and to constrain this variable using data generated during the course of a traditional numerical method. However, data may be nonlinearly related to the quantity of interest, rendering the proper conditioning of random variables difficult and limiting the range of numerical tasks that can be addressed. Instead, this paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method. A convergent sequence of approximations to the quantity of interest constitute a dataset, from which the limiting quantity of interest can be extrapolated, in a probabilistic analogue of Richardson's deferred approach to the limit. This black box approach (1) massively expands the range of tasks to which probabilistic numerics can be applied, (2) inherits the features and performance of state-of-the-art numerical methods, and (3) enables provably higher orders of convergence to be achieved. Applications are presented for nonlinear ordinary and partial differential equations, as well as for eigenvalue problems-a setting for which no probabilistic numerical methods have yet been developed.

Kernel selection plays a central role in determining the performance of Gaussian Process (GP) models, as the chosen kernel determines both the inductive biases and prior support of functions under the GP prior. This work addresses the challenge of constructing custom kernel functions for high-dimensional GP regression models. Drawing inspiration from recent progress in deep learning, we introduce a novel approach named KITT: Kernel Identification Through Transformers. KITT exploits a transformer-based architecture to generate kernel recommendations in under 0.1 seconds, which is several orders of magnitude faster than conventional kernel search algorithms. We train our model using synthetic data generated from priors over a vocabulary of known kernels. By exploiting the nature of the self-attention mechanism, KITT is able to process datasets with inputs of arbitrary dimension. We demonstrate that kernels chosen by KITT yield strong performance over a diverse collection of regression benchmarks.

Semantic representation and inference is essential for Natural Language Processing (NLP). The state of the art for semantic representation and inference is deep learning, and particularly Recurrent Neural Networks (RNNs), Convolutional Neural Networks (CNNs), and transformer Self-Attention models. This thesis investigates the use of deep learning for novel semantic representation and inference, and makes contributions in the following three areas: creating training data, improving semantic representations and extending inference learning. In terms of creating training data, we contribute the largest publicly available dataset of real-life factual claims for the purpose of automatic claim verification (MultiFC), and we present a novel inference model composed of multi-scale CNNs with different kernel sizes that learn from external sources to infer fact checking labels. In terms of improving semantic representations, we contribute a novel model that captures non-compositional semantic indicators. By definition, the meaning of a non-compositional phrase cannot be inferred from the individual meanings of its composing words (e.g., hot dog). Motivated by this, we operationalize the compositionality of a phrase contextually by enriching the phrase representation with external word embeddings and knowledge graphs. Finally, in terms of inference learning, we propose a series of novel deep learning architectures that improve inference by using syntactic dependencies, by ensembling role guided attention heads, incorporating gating layers, and concatenating multiple heads in novel and effective ways. This thesis consists of seven publications (five published and two under review).

A central goal of machine learning is to learn robust representations that capture the causal relationship between inputs features and output labels. However, minimizing empirical risk over finite or biased datasets often results in models latching on to spurious correlations between the training input/output pairs that are not fundamental to the problem at hand. In this paper, we define and analyze robust and spurious representations using the information-theoretic concept of minimal sufficient statistics. We prove that even when there is only bias of the input distribution (i.e. covariate shift), models can still pick up spurious features from their training data. Group distributionally robust optimization (DRO) provides an effective tool to alleviate covariate shift by minimizing the worst-case training loss over a set of pre-defined groups. Inspired by our analysis, we demonstrate that group DRO can fail when groups do not directly account for various spurious correlations that occur in the data. To address this, we further propose to minimize the worst-case losses over a more flexible set of distributions that are defined on the joint distribution of groups and instances, instead of treating each group as a whole at optimization time. Through extensive experiments on one image and two language tasks, we show that our model is significantly more robust than comparable baselines under various partitions. Our code is available at https://github.com/violet-zct/group-conditional-DRO.