Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.

Contemporary robots have become exceptionally skilled at achieving specific tasks in structured environments. However, they often fail when faced with the limitless permutations of real-world unstructured environments. This motivates robotics methods which learn from experience, rather than follow a pre-defined set of rules. In this thesis, we present a range of learning-based methods aimed at enabling robots, operating in dynamic and unstructured environments, to better understand their surroundings, anticipate the actions of others, and take informed actions accordingly. In the first part of the thesis, we investigate methods which leverage learning to represent the structure and motion in a robot's operating environment, in a continuous manner.

Score-based methods have demonstrated their effectiveness in discovering causal relationships by scoring different causal structures based on their goodness of fit to the data. Recently, Huang et al. proposed a generalized score function that can handle general data distributions and causal relationships by modeling the relations in reproducing kernel Hilbert space (RKHS). The selection of an appropriate kernel within this score function is crucial for accurately characterizing causal relationships and ensuring precise causal discovery. However, the current method involves manual heuristic selection of kernel parameters, making the process tedious and less likely to ensure optimality. In this paper, we propose a kernel selection method within the generalized score function that automatically selects the optimal kernel that best fits the data. Specifically, we model the generative process of the variables involved in each step of the causal graph search procedure as a mixture of independent noise variables. Based on this model, we derive an automatic kernel selection method by maximizing the marginal likelihood of the variables involved in each search step. We conduct experiments on both synthetic data and real-world benchmarks, and the results demonstrate that our proposed method outperforms heuristic kernel selection methods.

Existing methods for graph out-of-distribution (OOD) generalization primarily rely on empirical studies on synthetic datasets. Such approaches tend to overemphasize the causal relationships between invariant sub-graphs and labels, thereby neglecting the non-negligible role of environment in real-world scenarios. In contrast to previous studies that impose rigid independence assumptions on environments and invariant sub-graphs, this paper presents the theorems of environment-label dependency and mutable rationale invariance, where the former characterizes the usefulness of environments in determining graph labels while the latter refers to the mutable importance of graph rationales. Based on analytic investigations, a novel variational inference based method named ``Probability Dependency on Environments and Rationales for OOD Graphs on Real-world Data'' (DEROG) is introduced. To alleviate the adverse effect of unknown prior knowledge on environments and rationales, DEROG utilizes generalized Bayesian inference. Further, DEROG employs an EM-based algorithm for optimization. Finally, extensive experiments on real-world datasets under different distribution shifts are conducted to show the superiority of DEROG. Our code is publicly available at https://anonymous.4open.science/r/DEROG-536B.

We describe a new method for estimating the direction of sound in a reverberant environment from basic principles of sound propagation. The method utilizes SNR-adaptive features from time-delay and energy of the directional components after acoustic wave decomposition of the observed sound field to estimate the line-of-sight direction under noisy and reverberant conditions. The effectiveness of the approach is established with measured data of different microphone array configurations under various usage scenarios.

The ensemble deep random vector functional link (edRVFL) neural network has demonstrated the ability to address the limitations of conventional artificial neural networks. However, since edRVFL generates features for its hidden layers through random projection, it can potentially lose intricate features or fail to capture certain non-linear features in its base models (hidden layers). To enhance the feature learning capabilities of edRVFL, we propose a novel edRVFL based on fuzzy inference system (edRVFL-FIS). The proposed edRVFL-FIS leverages the capabilities of two emerging domains, namely deep learning and ensemble approaches, with the intrinsic IF-THEN properties of fuzzy inference system (FIS) and produces rich feature representation to train the ensemble model. Each base model of the proposed edRVFL-FIS encompasses two key feature augmentation components: a) unsupervised fuzzy layer features and b) supervised defuzzified features. The edRVFL-FIS model incorporates diverse clustering methods (R-means, K-means, Fuzzy C-means) to establish fuzzy layer rules, resulting in three model variations (edRVFL-FIS-R, edRVFL-FIS-K, edRVFL-FIS-C) with distinct fuzzified features and defuzzified features. Within the framework of edRVFL-FIS, each base model utilizes the original, hidden layer and defuzzified features to make predictions. Experimental results, statistical tests, discussions and analyses conducted across UCI and NDC datasets consistently demonstrate the superior performance of all variations of the proposed edRVFL-FIS model over baseline models. The source codes of the proposed models are available at https://github.com/mtanveer1/edRVFL-FIS.

We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. Subsequently, such differences are integrated, weighted by their associated uncertainties, using a maximum-likelihood formulation. This procedure can be seen as an extension to a multidimensional setting of the thermodynamic integration, a technique developed in statistical physics. The method leverages the manifold hypothesis, estimating quantities within the intrinsic data manifold without defining an explicit coordinate map. It does not rely on any binning or space partitioning, but rather on the construction of a neighbourhood graph based on an adaptive bandwidth selection procedure. BMTI mitigates the limitations commonly associated with traditional nonparametric density estimators, effectively reconstructing smooth profiles even in high-dimensional embedding spaces. The method is tested on a variety of complex synthetic high-dimensional datasets, where it is shown to outperform traditional estimators, and is benchmarked on realistic datasets from the chemical physics literature.

We introduce a new observational setting for Positive Unlabeled (PU) data where the observations at prediction time are also labeled. This occurs commonly in practice -- we argue that the additional information is important for prediction, and call this task "augmented PU prediction". We allow for labeling to be feature dependent. In such scenario, Bayes classifier and its risk is established and compared with a risk of a classifier which for unlabeled data is based only on predictors. We introduce several variants of the empirical Bayes rule in such scenario and investigate their performance. We emphasise dangers (and ease) of applying classical classification rule in the augmented PU scenario -- due to no preexisting studies, an unaware researcher is prone to skewing the obtained predictions. We conclude that the variant based on recently proposed variational autoencoder designed for PU scenario works on par or better than other considered variants and yields advantage over feature-only based methods in terms of accuracy for unlabeled samples.

Emotion recognition is the technology-driven process of identifying and categorizing human emotions from various data sources, such as facial expressions, voice patterns, body motion, and physiological signals, such as EEG. These physiological indicators, though rich in data, present challenges due to their complexity and variability, necessitating sophisticated feature selection and extraction methods. NGN, an unsupervised learning algorithm, effectively adapts to input spaces without predefined grid structures, improving feature extraction from physiological data. Furthermore, the incorporation of fuzzy logic enables the handling of fuzzy data by introducing reasoning that mimics human decision-making. The combination of PSO with XGBoost aids in optimizing model performance through efficient hyperparameter tuning and decision process optimization. This study explores the integration of Neural-Gas Network (NGN), XGBoost, Particle Swarm Optimization (PSO), and fuzzy logic to enhance emotion recognition using physiological signals. Our research addresses three critical questions concerning the improvement of XGBoost with PSO and fuzzy logic, NGN's effectiveness in feature selection, and the performance comparison of the PSO-fuzzy XGBoost classifier with standard benchmarks. Acquired results indicate that our methodologies enhance the accuracy of emotion recognition systems and outperform other feature selection techniques using the majority of classifiers, offering significant implications for both theoretical advancement and practical application in emotion recognition technology.

We investigate the role of uncertainty in decision-making problems with natural language as input. For such tasks, using Large Language Models as agents has become the norm. However, none of the recent approaches employ any additional phase for estimating the uncertainty the agent has about the world during the decision-making task. We focus on a fundamental decision-making framework with natural language as input, which is the one of contextual bandits, where the context information consists of text. As a representative of the approaches with no uncertainty estimation, we consider an LLM agent with a greedy policy, which picks the action corresponding to the largest predicted reward. We compare this baseline to LLM agents that make active use of uncertainty estimation by integrating the uncertainty in a Thompson Sampling policy. We employ different techniques for uncertainty estimation, such as Laplace Approximation, Dropout, and Epinets. We empirically show on real-world data that the greedy policy performs worse than the Thompson Sampling policies. These findings suggest that, while overlooked in the LLM literature, uncertainty improves performance on bandit tasks with LLM agents.