Machine learning has become one of the main components for task automation in many application domains. Despite the advancements and impressive achievements of machine learning, it has been shown that learning algorithms can be compromised by attackers both at training and test time. Machine learning systems are especially vulnerable to adversarial examples where small perturbations added to the original data points can produce incorrect or unexpected outputs in the learning algorithms at test time. Mitigation of these attacks is hard as adversarial examples are difficult to detect. Existing related work states that the security of machine learning systems against adversarial examples can be weakened when feature selection is applied to reduce the systems' complexity. In this paper, we empirically disprove this idea, showing that the relative distortion that the attacker has to introduce to succeed in the attack is greater when the target is using a reduced set of features. We also show that the minimal adversarial examples differ statistically more strongly from genuine examples with a lower number of features. However, reducing the feature count can negatively impact the system's performance. We illustrate the trade-off between security and accuracy with specific examples. We propose a design methodology to evaluate the security of machine learning classifiers with embedded feature selection against adversarial examples crafted using different attack strategies.

Stochastic gradient descent (SGD) (Robbins and Monro, 1951) and its variants have been widely used in training deep neural networks. Among those variants, adaptive gradient methods (AdaGrad) (Duchi et al., 2011; McMahan and Streeter, 2010), which scale each coordinate of the gradient by a function of past gradients, can achieve better performance than vanilla SGD in practice when the gradients are sparse. An intuitive explanation for the success of AdaGrad is that it automatically adjusts the learning rate for each feature based on the partial gradient, which accelerates the convergence. However, AdaGrad was later found to demonstrate degraded performance especially in cases where the loss function is nonconvex or the gradient is dense, due to rapid decay of learning rate.

Generalising well in supervised learning tasks relies on correctly extrapolating the training data to a large region of the input space. One way to achieve this is to constrain the predictions to be invariant to transformations on the input that are known to be irrelevant (e.g. translation). Commonly, this is done through data augmentation, where the training set is enlarged by applying hand-crafted transformations to the inputs. We argue that invariances should instead be incorporated in the model structure, and learned using the marginal likelihood, which correctly rewards the reduced complexity of invariant models. We demonstrate this for Gaussian process models, due to the ease with which their marginal likelihood can be estimated. Our main contribution is a variational inference scheme for Gaussian processes containing invariances described by a sampling procedure. We learn the sampling procedure by back-propagating through it to maximise the marginal likelihood.

Despite the fact that nonlinear subspace learning techniques (e.g. manifold learning) have successfully applied to data representation, there is still room for improvement in explainability (explicit mapping), generalization (out-of-samples), and cost-effectiveness (linearization). To this end, a novel linearized subspace learning technique is developed in a joint and progressive way, called \textbf{j}oint and \textbf{p}rogressive \textbf{l}earning str\textbf{a}teg\textbf{y} (J-Play), with its application to multi-label classification. The J-Play learns high-level and semantically meaningful feature representation from high-dimensional data by 1) jointly performing multiple subspace learning and classification to find a latent subspace where samples are expected to be better classified; 2) progressively learning multi-coupled projections to linearly approach the optimal mapping bridging the original space with the most discriminative subspace; 3) locally embedding manifold structure in each learnable latent subspace. Extensive experiments are performed to demonstrate the superiority and effectiveness of the proposed method in comparison with previous state-of-the-art methods.

Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis. Since its formulation in 2006, NGCA has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable $X$ in $n$-dimensional Euclidean space. There is an unknown subspace $U$ of the $n$-dimensional Euclidean space such that the orthogonal projection of $X$ onto $U$ is standard multidimensional Gaussian and the orthogonal projection of $X$ onto $V$, the orthogonal complement of $U$, is non-Gaussian, in the sense that all its one-dimensional marginals are different from the Gaussian in a certain metric defined in terms of moments. The NGCA problem is to approximate the non-Gaussian subspace $V$ given samples of $X$. Vectors in $V$ corresponds to "interesting" directions, whereas vectors in $U$ correspond to the directions where data is very noisy. The most interesting applications of the NGCA model is for the case when the magnitude of the noise is comparable to that of the true signal, a setting in which traditional noise reduction techniques such as PCA don't apply directly. NGCA is also related to dimensionality reduction and to other data analysis problems such as ICA. NGCA-like problems have been studied in statistics for a long time using techniques such as projection pursuit. We give an algorithm that takes polynomial time in the dimension $n$ and has an inverse polynomial dependence on the error parameter measuring the angle distance between the non-Gaussian subspace and the subspace output by the algorithm. Our algorithm is based on relative entropy as the contrast function and fits under the projection pursuit framework. The techniques we develop for analyzing our algorithm maybe of use for other related problems.

From self-driving vehicles and back-flipping robots to virtual assistants who book our next appointment at the hair salon or at that restaurant for dinner - machine learning systems are becoming increasingly ubiquitous. The main reason for this is that these methods boast remarkable predictive capabilities. However, most of these models remain black boxes, meaning that it is very challenging for humans to follow and understand their intricate inner workings. Consequently, interpretability has suffered under this ever-increasing complexity of machine learning models. Especially with regards to new regulations, such as the General Data Protection Regulation (GDPR), the necessity for plausibility and verifiability of predictions made by these black boxes is indispensable. Driven by the needs of industry and practice, the research community has recognised this interpretability problem and focussed on developing a growing number of so-called explanation methods over the past few years. These methods explain individual predictions made by black box machine learning models and help to recover some of the lost interpretability. With the proliferation of these explanation methods, it is, however, often unclear, which explanation method offers a higher explanation quality, or is generally better-suited for the situation at hand. In this thesis, we thus propose an axiomatic framework, which allows comparing the quality of different explanation methods amongst each other. Through experimental validation, we find that the developed framework is useful to assess the explanation quality of different explanation methods and reach conclusions that are consistent with independent research.

We introduce a few variants on Frank-Wolfe style algorithms suitable for large scale optimization. We show how to modify the standard Frank-Wolfe algorithm using stochastic gradients, approximate subproblem solutions, and sketched decision variables in order to scale to enormous problems while preserving (up to constants) the optimal convergence rate $\mathcal{O}(\frac{1}{k})$.

In the last decade, a considerable research effort has been devoted to developing adaptive algorithms based on kernel functions. One of the main features of these algorithms is that they form a family of universal approximation techniques, solving problems with nonlinearities elegantly. In this paper, we present data-selective adaptive kernel normalized least-mean square (KNLMS) algorithms that can increase their learning rate and reduce their computational complexity. In fact, these methods deal with kernel expansions, creating a growing structure also known as the dictionary, whose size depends on the number of observations and their innovation. The algorithms described herein use an adaptive step-size to accelerate the learning and can offer an excellent tradeoff between convergence speed and steady state, which allows them to solve nonlinear filtering and estimation problems with a large number of parameters without requiring a large computational cost. The data-selective update scheme also limits the number of operations performed and the size of the dictionary created by the kernel expansion, saving computational resources and dealing with one of the major problems of kernel adaptive algorithms. A statistical analysis is carried out along with a computational complexity analysis of the proposed algorithms. Simulations show that the proposed KNLMS algorithms outperform existing algorithms in examples of nonlinear system identification and prediction of a time series originating from a nonlinear difference equation.

In recent years, we've seen a resurgence in AI, or artificial intelligence, and machine learning. Machine learning has led to some amazing results, like being able to analyze medical images and predict diseases on-par with human experts. Google's AlphaGo program was able to beat a world champion in the strategy game go using deep reinforcement learning. Machine learning is even being used to program self driving cars, which is going to change the automotive industry forever. Imagine a world with drastically reduced car accidents, simply by removing the element of human error.

Mathematics is the bedrock of any contemporary discipline of science. It is no surprise then that, almost all the techniques of modern data science (including all of the machine learning) have some deep mathematical underpinning or the other. Sometimes, as a data scientist (or even as a junior analyst on the team), you have to learn those foundational mathematics by heart to use or apply the techniques properly, other times you can just get by using an API or the out-of-box algorithm. However, having a solid understanding of the math behind the cool algorithm you are using to create meaningful product recommendation for your users, will never hurt you. More often than not, it should give you an edge among your peers and make you more confident.