Unsatisfying accuracy of learning methods is mostly caused by omitting the influence of important parameters such as membership assignments, type of data objects, and distance or similarity functions. The proposed method, called Bounded Fuzzy Possibilistic Method (BFPM) addresses different issues that previous clustering or classification methods have not sufficiently considered in their membership assignments. In fuzzy methods, the object's memberships should sum to 1. Hence, any data object may obtain full membership in at most one cluster or class. Possibilistic methods relax this condition, but the method can be satisfied with the results even if just an arbitrary object obtains the membership from just one cluster, which prevents the objects' movement analysis. Whereas, BFPM differs from previous fuzzy and possibilistic approaches by removing these restrictions. Furthermore, BFPM provides the flexible search space for objects' movement analysis. Data objects are also considered as fundamental keys in learning methods, and knowing the exact type of objects results in providing a suitable environment for learning algorithms. The Thesis introduces a new type of object, called critical, as well as categorizing data objects into two different categories: structural-based and behavioural-based. Critical objects are considered as causes of miss-classification and miss-assignment in learning procedures. The Thesis also proposes new methodologies to study the behaviour of critical objects with the aim of evaluating objects' movements (mutation) from one cluster or class to another. The Thesis also introduces a new type of feature, called dominant, that is considered as one of the causes of miss-classification and miss-assignments. Then the Thesis proposes new sets of similarity functions, called Weighted Feature Distance (WFD) and Prioritized Weighted Feature Distance (PWFD).

Collective learning methods exploit relations among data points to enhance classification performance. However, such relations, represented as edges in the underlying graphical model, expose an extra attack surface to the adversaries. We study adversarial robustness of an important class of such graphical models, Associative Markov Networks (AMN), to structural attacks, where an attacker can modify the graph structure at test time. We formulate the task of learning a robust AMN classifier as a bi-level program, where the inner problem is a challenging non-linear integer program that computes optimal structural changes to the AMN. To address this technical challenge, we first relax the attacker problem, and then use duality to obtain a convex quadratic upper bound for the robust AMN problem. We then prove a bound on the quality of the resulting approximately optimal solutions, and experimentally demonstrate the efficacy of our approach. Finally, we apply our approach in a transductive learning setting, and show that robust AMN is much more robust than state-of-the-art deep learning methods, while sacrificing little in accuracy on non-adversarial data.

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithm's robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.

Intrinsically motivated spontaneous exploration is a key enabler of autonomous lifelong learning in human children. It enables the discovery and acquisition of large repertoires of skills through self-generation, self-selection, self-ordering and self-experimentation of learning goals. We present an algorithmic approach called Intrinsically Motivated Goal Exploration Processes (IMGEP) to enable similar properties of autonomous or self-supervised learning in machines. The IMGEP algorithmic architecture relies on several principles: 1) self-generation of goals, generalized as fitness functions; 2) selection of goals based on intrinsic rewards; 3) exploration with incremental goal-parameterized policy search and exploitation of the gathered data with a batch learning algorithm; 4) systematic reuse of information acquired when targeting a goal for improving towards other goals. We present a particularly efficient form of IMGEP, called Modular Population-Based IMGEP, that uses a population-based policy and an object-centered modularity in goals and mutations. We provide several implementations of this architecture and demonstrate their ability to automatically generate a learning curriculum within several experimental setups including a real humanoid robot that can explore multiple spaces of goals with several hundred continuous dimensions. While no particular target goal is provided to the system, this curriculum allows the discovery of skills that act as stepping stone for learning more complex skills, e.g. nested tool use. We show that learning diverse spaces of goals with intrinsic motivations is more efficient for learning complex skills than only trying to directly learn these complex skills.

Control schemes that learn using measurement data collected online are increasingly promising for the control of complex and uncertain systems. However, in most approaches of this kind, learning is viewed as a side effect that passively improves control performance, e.g., by updating a model of the system dynamics. Determining how improvements in control performance due to learning can be actively exploited in the control synthesis is still an open research question. In this paper, we present AntLer, a design algorithm for learning-based control laws that anticipates learning, i.e., that takes the impact of future learning in uncertain dynamic settings explicitly into account. AntLer expresses system uncertainty using a non-parametric probabilistic model. Given a cost function that measures control performance, AntLer chooses the control parameters such that the expected cost of the closed-loop system is minimized approximately. We show that AntLer approximates an optimal solution arbitrarily accurately with probability one. Furthermore, we apply AntLer to a nonlinear system, which yields better results compared to the case where learning is not anticipated.

Positive semidefinite matrix factorization (PSDMF) expresses each entry of a nonnegative matrix as the inner product of two positive semidefinite (psd) matrices. When all these psd matrices are constrained to be diagonal, this model is equivalent to nonnegative matrix factorization. Applications include combinatorial optimization, quantum-based statistical models, and recommender systems, among others. However, despite the increasing interest in PSDMF, only a few PSDMF algorithms were proposed in the literature. In this paper, we show that PSDMF algorithms can be designed based on phase retrieval (PR) and affine rank minimization (ARM) algorithms. This procedure allows a significant shortcut in designing new PSDMF algorithms, as it allows to leverage some of the useful numerical properties of existing PR and ARM methods to the PSDMF framework. Motivated by this idea, we introduce a new family of PSDMF algorithms based on singular value projection (SVP) and iterative hard thresholding (IHT). This family subsumes previously-proposed projected gradient PSDMF methods; additionally, we show a new connection between SVP-based methods and majorization-minimization. Numerical experiments show that our proposed methods outperform state-of-the-art coordinate descent algorithms in terms of convergence speed and computational complexity, in certain scenarios. In certain cases, our proposed normalized-IHT-based method is the only algorithm able to find a solution. These results support our claim that the PSDMF framework can inherit desired numerical properties from PR and ARM algorithms, leading to more efficient PSDMF algorithms, and motivate further study of the links between these models.

The recently proposed distribution correction estimation (DICE) family of estimators has advanced the state of the art in off-policy evaluation from behavior-agnostic data. While these estimators all perform some form of stationary distribution correction, they arise from different derivations and objective functions. In this paper, we unify these estimators as regularized Lagrangians of the same linear program. The unification allows us to expand the space of DICE estimators to new alternatives that demonstrate improved performance. More importantly, by analyzing the expanded space of estimators both mathematically and empirically we find that dual solutions offer greater flexibility in navigating the tradeoff between optimization stability and estimation bias, and generally provide superior estimates in practice.

Optimization is an essential component for solving problems in wide-ranging fields. Ideally, the objective function should be designed such that the solution is unique and the optimization problem can be solved stably. However, the objective function used in a practical application is usually non-convex, and sometimes it even has an infinite set of solutions. To address this issue, we propose to learn the solution manifold in optimization. We train a model conditioned on the latent variable such that the model represents an infinite set of solutions. In our framework, we reduce this problem to density estimation by using importance sampling, and the latent representation of the solutions is learned by maximizing the variational lower bound. We apply the proposed algorithm to motion-planning problems, which involve the optimization of high-dimensional parameters. The experimental results indicate that the solution manifold can be learned with the proposed algorithm, and the trained model represents an infinite set of homotopic solutions for motion-planning problems.

The theory of discrete-time online learning has been successfully applied in many problems that involve sequential decision-making under uncertainty. However, in many applications including contractual hiring in online freelancing platforms and server allocation in cloud computing systems, the outcome of each action is observed only after a random and action-dependent time. Furthermore, as a consequence of certain ethical and economic concerns, the controller may impose deadlines on the completion of each task, and require fairness across different groups in the allocation of total time budget $B$. In order to address these applications, we consider continuous-time online learning problem with fairness considerations, and present a novel framework based on continuous-time utility maximization. We show that this formulation recovers reward-maximizing, max-min fair and proportionally fair allocation rules across different groups as special cases. We characterize the optimal offline policy, which allocates the total time between different actions in an optimally fair way (as defined by the utility function), and impose deadlines to maximize time-efficiency. In the absence of any statistical knowledge, we propose a novel online learning algorithm based on dual ascent optimization for time averages, and prove that it achieves $\tilde{O}(B^{-1/2})$ regret bound.

Control of wire-borne underactuated brachiating robots requires a robust feedback control design that can deal with dynamic uncertainties, actuator constraints and unmeasurable states. In this paper, we develop a robust feedback control for brachiating on flexible cables, building on previous work on optimal trajectory generation and time-varying LQR controller design. We propose a novel simplified model for approximation of the flexible cable dynamics, which enables inclusion of parametric model uncertainties in the system. We then use semidefinite programming (SDP) and sum-of-squares (SOS) optimization to synthesize a time-varying feedback control with formal robustness guarantees to account for model uncertainties and unmeasurable states in the system. Through simulation, hardware experiments and comparison with a time-varying LQR controller, it is shown that the proposed robust controller results in relatively large robust backward reachable sets and is able to reliably track a pre-generated optimal trajectory and achieve the desired brachiating motion in the presence of parametric model uncertainties, actuator limits, and unobservable states.