As machine learning technology gets applied to actual products and solutions, new challenges have emerged. Models unexpectedly fail to generalize to small changes in the distribution, tend to be confident on novel data they have never seen, or cannot communicate the rationale behind their decisions effectively with the end users. Collectively, we face a trustworthiness issue with the current machine learning technology. This textbook on Trustworthy Machine Learning (TML) covers a theoretical and technical background of four key topics in TML: Out-of-Distribution Generalization, Explainability, Uncertainty Quantification, and Evaluation of Trustworthiness. We discuss important classical and contemporary research papers of the aforementioned fields and uncover and connect their underlying intuitions. The book evolved from the homonymous course at the University of T\"ubingen, first offered in the Winter Semester of 2022/23. It is meant to be a stand-alone product accompanied by code snippets and various pointers to further sources on topics of TML. The dedicated website of the book is https://trustworthyml.io/.

Differentially private gradient descent (DP-GD) has been extremely effective both theoretically, and in practice, for solving private empirical risk minimization (ERM) problems. In this paper, we focus on understanding the impact of the clipping norm, a critical component of DP-GD, on its convergence. We provide the first formal convergence analysis of clipped DP-GD. More generally, we show that the value which one sets for clipping really matters: done wrong, it can dramatically affect the resulting quality; done properly, it can eliminate the dependence of convergence on the model dimensionality. We do this by showing a dichotomous behavior of the clipping norm. First, we show that if the clipping norm is set smaller than the optimal, even by a constant factor, the excess empirical risk for convex ERMs can increase from $O(1/n)$ to $\Omega(1)$, where $n$ is the number of data samples. Next, we show that, regardless of the value of the clipping norm, clipped DP-GD minimizes a well-defined convex objective over an unconstrained space, as long as the underlying ERM is a generalized linear problem. Furthermore, if the clipping norm is set within at most a constant factor higher than the optimal, then one can obtain an excess empirical risk guarantee that is independent of the dimensionality of the model space. Finally, we extend our result to non-convex generalized linear problems by showing that DP-GD reaches a first-order stationary point as long as the loss is smooth, and the convergence is independent of the dimensionality of the model space.