Machine learning (ML) is the study of computer algorithms that improve automatically through experience. It is seen as a subset of artificial intelligence. Machine learning algorithms build a mathematical model based on sample data, known as "training data", in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as email filtering and computer vision, where it is difficult or infeasible to develop conventional algorithms to perform the needed tasks. Machine learning is closely related to computational statistics, which focuses on making predictions using computers.

Constructing good representations is critical for learning complex tasks in a sample efficient manner. In the context of meta-learning, representations can be constructed from common patterns of previously seen tasks so that a future task can be learned quickly. While recent works show the benefit of subspace-based representations, such results are limited to linear-regression tasks. This work explores a more general class of nonlinear tasks with applications ranging from binary classification, generalized linear models and neural nets. We prove that subspace-based representations can be learned in a sample-efficient manner and provably benefit future tasks in terms of sample complexity. Numerical results verify the theoretical predictions in classification and neural-network regression tasks.

High-dimensional black-box optimisation remains an important yet notoriously challenging problem. Despite the success of Bayesian optimisation methods on continuous domains, domains that are categorical, or that mix continuous and categorical variables, remain challenging. We propose a novel solution -- we combine local optimisation with a tailored kernel design, effectively handling high-dimensional categorical and mixed search spaces, whilst retaining sample efficiency. We further derive convergence guarantee for the proposed approach. Finally, we demonstrate empirically that our method outperforms the current baselines on a variety of synthetic and real-world tasks in terms of performance, computational costs, or both.

A fundamental tenet of learning theory is that a trade-off exists between the complexity of a prediction rule and its ability to generalize. The double-decent phenomenon shows that modern machine learning models do not obey this paradigm: beyond the interpolation limit, the test error declines as model complexity increases. We investigate over-parameterization in linear regression using the recently proposed predictive normalized maximum likelihood (pNML) learner which is the min-max regret solution for individual data. We derive an upper bound of its regret and show that if the test sample lies mostly in a subspace spanned by the eigenvectors associated with the large eigenvalues of the empirical correlation matrix of the training data, the model generalizes despite its over-parameterized nature. We demonstrate the use of the pNML regret as a point-wise learnability measure on synthetic data and that it can successfully predict the double-decent phenomenon using the UCI dataset.

Gaussian processes (GPs) are nonparametric Bayesian models that have been applied to regression and classification problems. One of the approaches to alleviate their cubic training cost is the use of local GP experts trained on subsets of the data. In particular, product-of-expert models combine the predictive distributions of local experts through a tractable product operation. While these expert models allow for massively distributed computation, their predictions typically suffer from erratic behaviour of the mean or uncalibrated uncertainty quantification. By calibrating predictions via a tempered softmax weighting, we provide a solution to these problems for multiple product-of-expert models, including the generalised product of experts and the robust Bayesian committee machine. Furthermore, we leverage the optimal transport literature and propose a new product-of-expert model that combines predictions of local experts by computing their Wasserstein barycenter, which can be applied to both regression and classification.

We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces the long-standing curse of dimensionality. We overcome this challenge by exploiting the intrinsic low-rank structure in the difference between the posterior and prior distributions. The parameters are projected into a low-dimensional subspace to alleviate the approximation error of KDE in high dimensions. We formulate a projected Wasserstein gradient flow and analyze its convergence property under mild assumptions. Several numerical experiments illustrate the accuracy, convergence, and complexity scalability of pWGD with respect to parameter dimension, sample size, and processor cores.

Cyber Physical Systems (CPS) are characterized by their ability to integrate the physical and information or cyber worlds. Their deployment in critical infrastructure have demonstrated a potential to transform the world. However, harnessing this potential is limited by their critical nature and the far reaching effects of cyber attacks on human, infrastructure and the environment. An attraction for cyber concerns in CPS rises from the process of sending information from sensors to actuators over the wireless communication medium, thereby widening the attack surface. Traditionally, CPS security has been investigated from the perspective of preventing intruders from gaining access to the system using cryptography and other access control techniques. Most research work have therefore focused on the detection of attacks in CPS. However, in a world of increasing adversaries, it is becoming more difficult to totally prevent CPS from adversarial attacks, hence the need to focus on making CPS resilient. Resilient CPS are designed to withstand disruptions and remain functional despite the operation of adversaries. One of the dominant methodologies explored for building resilient CPS is dependent on machine learning (ML) algorithms. However, rising from recent research in adversarial ML, we posit that ML algorithms for securing CPS must themselves be resilient. This paper is therefore aimed at comprehensively surveying the interactions between resilient CPS using ML and resilient ML when applied in CPS. The paper concludes with a number of research trends and promising future research directions. Furthermore, with this paper, readers can have a thorough understanding of recent advances on ML-based security and securing ML for CPS and countermeasures, as well as research trends in this active research area.

Meta learning with multiple objectives can be formulated as a Multi-Objective Bi-Level optimization Problem (MOBLP) where the upper-level subproblem is to solve several possible conflicting targets for the meta learner. However, existing studies either apply an inefficient evolutionary algorithm or linearly combine multiple objectives as a single-objective problem with the need to tune combination weights. In this paper, we propose a unified gradient-based Multi-Objective Meta Learning (MOML) framework and devise the first gradient-based optimization algorithm to solve the MOBLP by alternatively solving the lower-level and upper-level subproblems via the gradient descent method and the gradient-based multi-objective optimization method, respectively. Theoretically, we prove the convergence properties of the proposed gradient-based optimization algorithm. Empirically, we show the effectiveness of the proposed MOML framework in several meta learning problems, including few-shot learning, neural architecture search, domain adaptation, and multi-task learning.

Gaussian noise injections (GNIs) are a family of simple and widely-used regularisation methods for training neural networks, where one injects additive or multiplicative Gaussian noise to the network activations at every iteration of the optimisation algorithm, which is typically chosen as stochastic gradient descent (SGD). In this paper we focus on the so-called `implicit effect' of GNIs, which is the effect of the injected noise on the dynamics of SGD. We show that this effect induces an asymmetric heavy-tailed noise on SGD gradient updates. In order to model this modified dynamics, we first develop a Langevin-like stochastic differential equation that is driven by a general family of asymmetric heavy-tailed noise. Using this model we then formally prove that GNIs induce an `implicit bias', which varies depending on the heaviness of the tails and the level of asymmetry. Our empirical results confirm that different types of neural networks trained with GNIs are well-modelled by the proposed dynamics and that the implicit effect of these injections induces a bias that degrades the performance of networks.

SGD with Momentum (SGDM) is widely used for large scale optimization of machine learning problems. Yet, the theoretical understanding of this algorithm is not complete. In fact, even the most recent results require changes to the algorithm like an averaging scheme and a projection onto a bounded domain, which are never used in practice. Also, no lower bound is known for SGDM. In this paper, we prove for the first time that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate of SGDM suffers from an error $\Omega(\frac{\log T}{\sqrt{T}})$ after $T$ steps. Based on this fact, we study a new class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based SGDM algorithms with \emph{increasing momentum} and \emph{shrinking updates}. For these algorithms, we show that the last iterate has optimal convergence $O (\frac{1}{\sqrt{T}})$ for unconstrained convex optimization problems. Further, we show that in the interpolation setting with convex and smooth functions, our new SGDM algorithm automatically converges at a rate of $O(\frac{\log T}{T})$. Empirical results are shown as well.