"Have you ever ordered 43 Chicken McNuggets at McDonald's?" When I first heard this story it completely blew my mind, but it is actually true that the Chicken McNuggets have a mathematical story, and it is pretty interesting. Originally, you only have boxes of 6,9 and 20 Chicken McNuggets. While he was eating with his son, the mathematician Henri Picciotto started to think about the actual numbers that he could order with a combination of these three values. This number is known as McNuggets Number. We will start by giving an in depth detail of the discrete domain that we are considering and we will end up solving an optimization problem about it.

This paper demonstrates how to recover causal graphs from the score of the data distribution in non-linear additive (Gaussian) noise models. Using score matching algorithms as a building block, we show how to design a new generation of scalable causal discovery methods. To showcase our approach, we also propose a new efficient method for approximating the score's Jacobian, enabling to recover the causal graph. Empirically, we find that the new algorithm, called SCORE, is competitive with state-of-the-art causal discovery methods while being significantly faster.

A critical change is happening in today's Internet of Things (IoT). The computational power of edge devices is steadily increasing. AI chips are rapidly infiltrating the market, smart phones nowadays have compute power comparable to everyday use laptops (Samsung 2019), Tesla just boasted that its autopilot system has computing power of more than 3000 MacBook pros (CleanTechnica 2021) and small local computers such as Raspberry Pis have become common place in many applications especially manufacturing (Al-Ali et al. 2018). This opens a new paradigm for data analytics in IoT; one that exploits local computing power to process more of the user's data where it is created. This future of IoT has been recently termed as the "The Internet of Federated Things (IoFT)" (Kontar et al. 2021) where the term federated, refers to some autonomy for IoT devices and is inspired by the explosive recent interest in federated data science.

We argue that results produced by a heuristic optimisation algorithm cannot be considered reproducible unless the algorithm fully specifies what should be done with solutions generated outside the domain, even in the case of simple box constraints. Currently, in the field of heuristic optimisation, such specification is rarely mentioned or investigated due to the assumed triviality or insignificance of this question. Here, we demonstrate that, at least in algorithms based on Differential Evolution, this choice induces notably different behaviours - in terms of performance, disruptiveness and population diversity. This is shown theoretically (where possible) for standard Differential Evolution in the absence of selection pressure and experimentally for the standard and state-of-the-art Differential Evolution variants on special test function $f_0$ and BBOB benchmarking suite, respectively. Moreover, we demonstrate that the importance of this choice quickly grows with problem's dimensionality. Different Evolution is not at all special in this regard - there is no reason to presume that other heuristic optimisers are not equally affected by the aforementioned algorithmic choice. Thus, we urge the field of heuristic optimisation to formalise and adopt the idea of a new algorithmic component in heuristic optimisers, which we call here a strategy of dealing with infeasible solutions. This component needs to be consistently (a) specified in algorithmic descriptions to guarantee reproducibility of results, (b) studied to better understand its impact on algorithm's performance in a wider sense and (c) included in the (automatic) algorithmic design. All of these should be done even for problems with box constraints.

Within machine learning, NNLS problems arise whenever having negative labels is not meaningful, for example, when labels represent quantities like prices, age, pixel intensities, chemical concentrations, or frequency counts. NNLS is also widely used as a subroutine in nonnegative matrix factorization to extract sparse features in applications like image processing, computational biology, clustering, collaborative filtering, and community detection [Gil14]. From a statistical perspective, NNLS problems can be shown to possess a regularization property that enforces sparsity similar to LASSO [Tib96], while being comparatively simpler, without the need to tune a regularization parameter or perform cross-validation [SH14, BEZ08, FK14]. From an algorithmic standpoint, the nonnegativity constraint in NNLS problems is typically viewed as an obstacle: most NNLS algorithms need to perform additional work to handle it, and the problem is considered harder than unconstrained least squares. However, in many applications that use NNLS, the data is also nonnegative. This is true, for example, in problems arising in image processing, computational genomics, functional MRI, and in applications traditionally addressed using nonnegative matrix factorization. We argue in this paper that when the data for NNLS is nonnegative, it is in fact possible to obtain stronger guarantees than for traditional least squares problems.

We analyse the convergence of the proximal gradient algorithm for convex composite problems in the presence of gradient and proximal computational inaccuracies. We derive new tighter deterministic and probabilistic bounds that we use to verify a simulated (MPC) and a synthetic (LASSO) optimization problems solved on a reduced-precision machine in combination with an inaccurate proximal operator. We also show how the probabilistic bounds are more robust for algorithm verification and more accurate for application performance guarantees. Under some statistical assumptions, we also prove that some cumulative error terms follow a martingale property. And conforming to observations, e.g., in \cite{schmidt2011convergence}, we also show how the acceleration of the algorithm amplifies the gradient and proximal computational errors.

This thesis rigorously studies fundamental reinforcement learning (RL) methods in modern practical considerations, including robust RL, distributional RL, and offline RL with neural function approximation. The thesis first prepares the readers with an overall overview of RL and key technical background in statistics and optimization. In each of the settings, the thesis motivates the problems to be studied, reviews the current literature, provides computationally efficient algorithms with provable efficiency guarantees, and concludes with future research directions. The thesis makes fundamental contributions to the three settings above, both algorithmically, theoretically, and empirically, while staying relevant to practical considerations.

The traveling salesman problem is a fundamental combinatorial optimization problem with strong exact algorithms. However, as problems scale up, these exact algorithms fail to provide a solution in a reasonable time. To resolve this, current works look at utilizing deep learning to construct reasonable solutions. Such efforts have been very successful, but tend to be slow and compute intensive. This paper exemplifies the integration of entropic regularized optimal transport techniques as a layer in a deep reinforcement learning network. We show that we can construct a model capable of learning without supervision and inferences significantly faster than current autoregressive approaches. We also empirically evaluate the benefits of including optimal transport algorithms within deep learning models to enforce assignment constraints during end-to-end training.

A flying trapeze act can be a challenging task for a robotics system since some act requires the performer to catch another trapeze or catcher at the end after being airborne. The objective of this paper is to design and validate a motion planning algorithm for a two-link free-flying acrobatic robot that can accurately land on another trapeze after free-flying in the air. First, the proposed algorithm plan the robot trajectory with the non-linear constrained optimization method. Then, a feedback controller is implemented to stabilize the posture. However, since the spatial position of the center-of-mass of the robot cannot be controlled, this paper proposes a trajectory correction scheme that manipulates the robot's posture such that the robot is still able to land on the target. Lastly, the whole algorithm is validated in the simulation that mimics real-world circumstances.

World Endurance Championship (WEC) racing events are characterised by a relevant performance gap among competitors. The fastest vehicles category, consisting in hybrid vehicles, has to respect energy usage constraints set by the technical regulation. Considering absence of competitors, i.e. traffic conditions, the optimal energy usage strategy for lap time minimisation is typically computed through a constrained optimisation problem. To the best of our knowledge, the majority of state-of-the-art works neglects competitors. This leads to a mismatch with the real world, where traffic generates considerable time losses. To bridge this gap, we propose a new framework to offline compute optimal strategies for the powertrain energy management considering competitors. Through analysis of the available data from previous events, statistics on the sector times and overtaking probabilities are extracted to encode the competitors' behaviour. Adopting a multi-agent model, the statistics are then used to generate realistic Monte Carlo (MC) simulation of their position along the track. The simulator is then adopted to identify the optimal strategy as follows. We develop a longitudinal vehicle model for the ego-vehicle and implement an optimisation problem for lap time minimisation in absence of traffic, based on Genetic Algorithms. Solving the optimisation problem for a variety of constraints generates a set of candidate optimal strategies. Stochastic Dynamic Programming is finally implemented to choose the best strategy considering competitors, whose motion is generated by the MC simulator. Our approach, validated on data from a real stint of race, allows to significantly reduce the lap time.