Towards trustworthy machine learning with kernels

Machine Learning has become an indispensable aspect of various safety-critical industries like healthcare, law, and automotive. Hence, it is crucial to ensure that our machine learning models function appropriately and instil trust among their users. This thesis focuses on improving the safety and transparency of Machine Learning by advocating for more principled uncertainty quantification and more effective explainability tools. Specifically, the use of Kernel Mean Embeddings (KME) and Gaussian Processes (GP) is prevalent in this work since they can represent probability distribution with minimal distributional assumptions and capture uncertainty well, respectively. I dedicate Chapter 2 to introduce these two methodologies. Chapter 3 demonstrates an effective use of these methods in conjunction with each other to tackle a statistical downscaling problem, in which a Deconditional Gaussian process is proposed. Chapter 4 considers a causal data fusion problem, where multiple causal graphs are combined for inference. I introduce BayesIMP, an algorithm built using KME and GPs, to draw causal conclusion while accounting for the uncertainty in the data and model. In Chapter 5, I present RKHS-SHAP to model explainability for kernel methods that utilizes Shapley values. Specifically, I propose to estimate the value function in the cooperative game using KMEs, circumventing the need for any parametric density estimations. A Shapley regulariser is also proposed to regulate the amount of contributions certain features can have to the model. Chapter 6 presents a generalised preferential Gaussian processes for modelling preference with non-rankable structure, which sets the scene for Chapter 7, where I built upon my research and propose Pref-SHAP to explain preference models

A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games

Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $\mathcal{O}(d/\epsilon^2)$ iterations to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $\mathcal{O}(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games.Comment: 53 pages, 7 figures, QTML 2023 (Accepted (Long Talk)

Smart Senja electrical network expansion modeling

The addition of variable renewable energy sources into the electrical energy systems of the world has been increasing in recent years. This form of distributed energy production with high production volatility can introduce massive challenges in operating a lower voltage distribution network. One of these affected networks is on the island of Senja in northern Norway, with an eldering radial electrical network with a single connection to the national transmission grid. In this study, prescriptive analysis of the network through mathematical optimization is implemented to investigate if there are more effective solutions to this problem other than building more electrical lines. In selected parts of the island, the electrical network experiences electrical faults of different magnitude and concern affecting 1500 hours a year. In this thesis, the model GenX is presented which prescribes solutions reducing these faults to zero while also cutting costs compared to the baseline scenario of today’s system. Results from the model indicate that simple installments of distributed power generation in conjunction with electrical energy storage drastically improve network capacity and industrial expansion opportunities. Also investigated is the feasibility of operating the electrical network on the island without any connection to the external grid. Meant as a proof of concept for the application of mathematical optimization on electrical grids in other more remote parts of the world. The model proves that investments in local electricity production positively impact the system at a fraction of the cost of building new regional distribution infrastructure. Finally, some drawbacks of the chosen analytical tool used to construct the mathematical optimization model are presented alongside selected methods applicable to apprehend or circumvent these limitations

Implication of Regret on Mutual Fund Managers' Risk-Shifting Decision

We investigate whether regret can explain mutual fund managers' risk-shifting behavior. We propose a theoretical framework by introducing a modi_ed utility function for mutual fund managers who are both risk averse and regret averse. The empirical tests of the proposed framework imply that mutual fund managers who perform worse than their peers (i.e., who exhibit return-regret) tend to have a positive risk-shifting, whereas those who have a higher portfolio volatility (i.e., who exhibit variance-regret) tend to have a negative risk-shifting behavior over the next period. Furthermore, we document that the e_ect of variance regret is more signi_cant for institutional funds than for retail funds. Finally, when considering fund ows, the return-regret e_ect is more signifcant than the variance-regret e_ect, conrming that investors' outows are mainly due fund managers' bad performance relative to their peers. The results are robust to using alternative measures of regret based on funds' potential benchmarks

Implication of Regret on Mutual Fund Managers' Risk-Shifting Decision

We investigate whether regret can explain mutual fund managers' risk-shifting behavior. We propose a theoretical framework by introducing a modi_ed utility function for mutual fund managers who are both risk averse and regret averse. The empirical tests of the proposed framework imply that mutual fund managers who perform worse than their peers (i.e., who exhibit return-regret) tend to have a positive risk-shifting, whereas those who have a higher portfolio volatility (i.e., who exhibit variance-regret) tend to have a negative risk-shifting behavior over the next period. Furthermore, we document that the e_ect of variance regret is more signi_cant for institutional funds than for retail funds. Finally, when considering fund ows, the return-regret e_ect is more signifcant than the variance-regret e_ect, conrming that investors' outows are mainly due fund managers' bad performance relative to their peers. The results are robust to using alternative measures of regret based on funds' potential benchmarks

Implication of Regret on Mutual Funds Managers Risk-Shifting Decision

We investigate whether regret can explain mutual fund managers’ risk-shifting behav-ior. We propose a theoretical framework by introducing a modified utility functionfor mutual fund managers who are both risk averse and regret averse. The empiricaltests of the proposed framework imply that mutual fund managers who perform worsethan their peers (i.e., who exhibit return-regret) tend to have a positive risk-shifting,whereas those who have a higher portfolio volatility (i.e., who exhibit variance-regret)tend to have a negative risk-shifting behavior over the next period. Furthermore, wedocument that the effect of variance regret is more significant for institutional fundsthan for retail funds. Finally, when considering fund flows, the return-regret effect ismore significant than the variance-regret effect, confirming that investors’ outflows aremainly due fund managers’ bad performance relative to their peers. The results arerobust to using alternative measures of regret based on funds’ potential benchmarks

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more