Continual learning aims to learn new tasks without forgetting previously learned ones. This is especially challenging when one cannot access data from previous tasks and when the model has a fixed capacity. Current regularization-based continual learning algorithms need an external representation and extra computation to measure the parameters' importance. In contrast, we propose Uncertainty-guided Continual Bayesian Neural Networks (UCB), where the learning rate adapts according to the uncertainty defined in the probability distribution of the weights in networks. Uncertainty is a natural way to identify what to remember and what to change as we continually learn, allowing to mitigate catastrophic forgetting. We also show a variant of our model, which uses uncertainty for weight pruning and retains task performance after pruning by saving binary masks per tasks. We evaluate our UCB approach extensively on diverse object classification datasets with short and long sequences of tasks and report superior or on-par performance compared to existing approaches. Additionally, we show that our model does not necessarily need task information at test time, i.e. it does not presume knowledge of which task a sample belongs to.

Unlike supervised learning, unsupervised learning not working with labeled data, it is not showing the machine the correct answer. Instead, it is using different algorithms to let the machine create connections by studying and observing the data. Learning and improving by trial and error is the key to unsupervised learning. However, the Knowledge Discovery process is the field of data mining is concerned with the development of methods, techniques and algorithm which can make sense of the available data. It is useful in finding trends, patterns, correlations and anomalies in the databases which is helpful to make accurate decisions for the future.

We propose a novel feature coding method that exploits invariance. We consider the setting where the transformations that preserve the image contents compose a finite group of orthogonal matrices. This is the case in many image transformations such as image rotations and image flipping. We prove that the group-invariant feature vector contains sufficient discriminative information when we learn a linear classifier using convex loss minimization. From this result, we propose a novel feature modeling for principal component analysis, and k-means clustering, which are used for most feature coding methods, and global feature functions that explicitly consider the group action. Although the global feature functions are complex nonlinear functions in general, we can calculate the group action on this space easily by constructing the functions as the tensor product representations of basic representations, resulting in the explicit form of invariant feature functions. We demonstrate the effectiveness of our methods on several image datasets.

In this thesis, we draw inspiration from both classical system identification and modern machine learning in order to solve estimation problems for real-world, physical systems. The main approach to estimation and learning adopted is optimization based. Concepts such as regularization will be utilized for encoding of prior knowledge and basis-function expansions will be used to add nonlinear modeling power while keeping data requirements practical. The thesis covers a wide range of applications, many inspired by applications within robotics, but also extending outside this already wide field. Usage of the proposed methods and algorithms are in many cases illustrated in the real-world applications that motivated the research. Topics covered include dynamics modeling and estimation, model-based reinforcement learning, spectral estimation, friction modeling and state estimation and calibration in robotic machining. In the work on modeling and identification of dynamics, we develop regularization strategies that allow us to incorporate prior domain knowledge into flexible, overparameterized models. We make use of classical control theory to gain insight into training and regularization while using flexible tools from modern deep learning. A particular focus of the work is to allow use of modern methods in scenarios where gathering data is associated with a high cost. In the robotics-inspired parts of the thesis, we develop methods that are practically motivated and ensure that they are implementable also outside the research setting. We demonstrate this by performing experiments in realistic settings and providing open-source implementations of all proposed methods and algorithms.

The main theme of this work is a unifying algorithm, abbreviated as L2S, that can deal with (strongly) convex and nonconvex empirical risk minimization (ERM) problems. It broadens a recently developed variance reduction method known as SARAH. L2S enjoys a linear convergence rate for strongly convex problems, which also implies the last iteration of SARAH's inner loop converges linearly. For convex problems, different from SARAH, L2S can afford step and mini-batch sizes not dependent on the data size $n$, and the complexity needed to guarantee $\mathbb{E}[\|\nabla F(\mathbf{x}) \|^2] \leq \epsilon$ is ${\cal O}(n+ n/\epsilon)$. For nonconvex problems on the other hand, the complexity is ${\cal O}(n+ \sqrt{n}/\epsilon)$. Parallel to L2S there are a few side results. Leveraging an aggressive step size, D2S is proposed, which provides a more efficient alternative to L2S and SARAH-like algorithms. Specifically, D2S requires a reduced IFO complexity of ${\cal O}\big( (n+ \bar{\kappa}) \ln (1/\epsilon) \big)$ for strongly convex problems. Moreover, to avoid the tedious selection of the optimal step size, an automatic tuning scheme is developed, which obtains comparable empirical performance with SARAH using judiciously tuned step size.

Identifying statistical dependence between the features and the label is a fundamental problem in supervised learning. This paper presents a framework for estimating dependence between numerical features and a categorical label using generalized Gini distance, an energy distance in reproducing kernel Hilbert spaces (RKHS). Two Gini distance based dependence measures are explored: Gini distance covariance and Gini distance correlation. Unlike Pearson covariance and correlation, which do not characterize independence, the above Gini distance based measures define dependence as well as independence of random variables. The test statistics are simple to calculate and do not require probability density estimation. Uniform convergence bounds and asymptotic bounds are derived for the test statistics. Comparisons with distance covariance statistics are provided. It is shown that Gini distance statistics converge faster than distance covariance statistics in the uniform convergence bounds, hence tighter upper bounds on both Type I and Type II errors. Moreover, the probability of Gini distance covariance statistic under-performing the distance covariance statistic in Type II error decreases to 0 exponentially with the increase of the sample size. Extensive experimental results are presented to demonstrate the performance of the proposed method.

We consider the classic supervised learning problem, where a continuous non-negative random label $Y$ (i.e. a random duration) is to be predicted based upon observing a random vector $X$ valued in $\mathbb{R}^d$ with $d\geq 1$ by means of a regression rule with minimum least square error. In various applications, ranging from industrial quality control to public health through credit risk analysis for instance, training observations can be right censored, meaning that, rather than on independent copies of $(X,Y)$, statistical learning relies on a collection of $n\geq 1$ independent realizations of the triplet $(X, \; \min\{Y,\; C\},\; \delta)$, where $C$ is a nonnegative r.v. with unknown distribution, modeling censorship and $\delta=\mathbb{I}\{Y\leq C\}$ indicates whether the duration is right censored or not. As ignoring censorship in the risk computation may clearly lead to a severe underestimation of the target duration and jeopardize prediction, we propose to consider a plug-in estimate of the true risk based on a Kaplan-Meier estimator of the conditional survival function of the censorship $C$ given $X$, referred to as Kaplan-Meier risk, in order to perform empirical risk minimization. It is established, under mild conditions, that the learning rate of minimizers of this biased/weighted empirical risk functional is of order $O_{\mathbb{P}}(\sqrt{\log(n)/n})$ when ignoring model bias issues inherent to plug-in estimation, as can be attained in absence of censorship. Beyond theoretical results, numerical experiments are presented in order to illustrate the relevance of the approach developed.

One reason for the emergence of bias in AI systems is biased data -- datasets that may not be true representations of the underlying distributions -- and may over or under-represent groups with respect to protected attributes such as gender or race. We consider the problem of correcting such biases and learning distributions that are "fair", with respect to measures such as proportional representation and statistical parity, from the given samples. Our approach is based on a novel formulation of the problem of learning a fair distribution as a maximum entropy optimization problem with a given expectation vector and a prior distribution. Technically, our main contributions are: (1) a new second-order method to compute the (dual of the) maximum entropy distribution over an exponentially-sized discrete domain that turns out to be faster than previous methods, and (2) methods to construct prior distributions and expectation vectors that provably guarantee that the learned distributions satisfy a wide class of fairness criteria. Our results also come with quantitative bounds on the total variation distance between the empirical distribution obtained from the samples and the learned fair distribution. Our experimental results include testing our approach on the COMPAS dataset and showing that the fair distributions not only improve disparate impact values but when used to train classifiers only incur a small loss of accuracy.

Metric-based few-shot learning methods try to overcome the difficulty due to the lack of training examples by learning embedding to make comparison easy. We propose a novel algorithm to generate class representatives for few-shot classification tasks. As a probabilistic model for learned features of inputs, we consider a mixture of von Mises-Fisher distributions which is known to be more expressive than Gaussian in a high dimensional space. Then, from a discriminative classifier perspective, we get a better class representative considering inter-class correlation which has not been addressed by conventional few-shot learning algorithms. We apply our method to \emph{mini}ImageNet and \emph{tiered}ImageNet datasets, and show that the proposed approach outperforms other comparable methods in few-shot classification tasks.

Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that certain Gaussian posterior approximations for Bayesian DNNs are equivalent to GP posteriors. As a result, we can obtain a GP kernel and a nonlinear feature map simply by training the DNN. Surprisingly, the resulting kernel is the neural tangent kernel which has desirable theoretical properties for infinitely-wide DNNs. We show feature maps obtained on real datasets and demonstrate the use of the GP marginal likelihood to tune hyperparameters of DNNs. Our work aims to facilitate further research on combining DNNs and GPs in practical settings.