Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of L2-Boosting in a high-dimensional setting under early stopping. We close a gap in the literature and provide the rate of convergence of L2-Boosting in a high-dimensional setting under approximate sparsity and without beta-min condition. We also show that the rate of convergence of the classical L2-Boosting depends on the design matrix described by a sparse eigenvalue condition. To show the latter results, we derive new, improved approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of L2-Boosting. These results might be of independent interest. Moreover, we introduce so-called "restricted" L2-Boosting. The restricted L2-Boosting algorithm sticks to the set of the previously chosen variables, exploits the information contained in these variables first and then only occasionally allows to add new variables to this set. We derive the rate of convergence for restricted L2-Boosting under early stopping which is close to the convergence rate of Lasso in an approximate sparse, high-dimensional setting without beta-min condition. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Finally, we present simulation studies to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, L2-Boosting clearly outperforms Lasso. An empirical illustration and the proofs are contained in the Appendix.

DoubleML is an open-source Python library implementing the double machine learning framework of Chernozhukov et al. (2018) for a variety of causal models. It contains functionalities for valid statistical inference on causal parameters when the estimation of nuisance parameters is based on machine learning methods. The object-oriented implementation of DoubleML provides a high flexibility in terms of model specifications and makes it easily extendable. The package is distributed under the MIT license and relies on core libraries from the scientific Python ecosystem: scikit-learn, numpy, pandas, scipy, statsmodels and joblib. Source code, documentation and an extensive user guide can be found at https://github.com/DoubleML/doubleml-for-py and https://docs.doubleml.org.

The multi-touch-based pinch to zoom, drag and flick to pan metaphor has gained wide popularity on mobile displays, where it is the paradigm of choice for navigating 2D documents. But is finger-based navigation really the gold standard' In this paper, we present a comprehensive user study with 40 participants, in which we systematically compared the Pinch-Drag-Flick approach with a technique that relies on spatial manipulation, such as lifting a display up/down to zoom. While we solely considered known techniques, we put considerable effort in implementing both input strategies on popular consumer hardware (iPhone, iPad). Our results show that spatial manipulation can significantly outperform traditional Pinch-Drag-Flick. Given the carefully optimized prototypes, we are confident to have found strong arguments that future generations of mobile devices could rely much more on spatial interaction principles.

Lightweight spatially aware displays (Tangible Magic Lenses) are an effective approach for exploring complex information spaces within a tabletop environment. One way of using the 3D space above a horizontal surface is to divide it into discrete parallel layers stacked upon each other. Horizontal and vertical lens movements are essential tasks for the style of multi-layer interaction associated with it. We conducted a comprehensive user study with 18 participants investigating fundamental issues such as optimal number of layers and their thickness, movement and holding accuracies, and physical boundaries of the interaction volume. Findings include a rather limited overall interaction height (44 cm), a different minimal layer thickness for vertical and horizontal search tasks (1 cm/4 cm), a reasonable maximum number of layers depending on the primary task, and a convenience zone in the middle for horizontal search. Derived from that, design guidelines are also presented.

Today's window systems present the information in a graphical and thereby a spatial manner making the text-only access of a standard Braille device insufficient to enable blind users an equivalent exploration of the data. In this paper we present the planar Braille Window System (BWS) designed for a tactile display consisting of a pin-matrix of 120 columns and 60 rows. The system is composed of six separate regions enabling the user to receive different types of information simultaneously. The content of the main region containing Braille windows can be shown in various manners (text-or graphics-based) through four different views. The interaction within our Braille Window System is implemented not only by keyboard shortcuts but also by the use of multitouch gestures. Therefore the user is able to interact directly on the touch-sensitive display. A study conducted with eight blind users has confirmed the concept of Braille windows, regions and views. Especially the gestural input for exploring details of the content offers new possibilities in interacting within a GUI.