Karlson Pfannschmidt

Michael Dellnitz, Eyke Hüllermeier, Marvin Lücke et al.

Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it either results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge--Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave sub-optimal when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalises better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.

Karlson Pfannschmidt, Eyke Hüllermeier

We consider the problem of learning to choose from a given set of objects, where each object is represented by a feature vector. Traditional approaches in choice modelling are mainly based on learning a latent, real-valued utility function, thereby inducing a linear order on choice alternatives. While this approach is suitable for discrete (top-1) choices, it is not straightforward how to use it for subset choices. Instead of mapping choice alternatives to the real number line, we propose to embed them into a higher-dimensional utility space, in which we identify choice sets with Pareto-optimal points. To this end, we propose a learning algorithm that minimizes a differentiable loss function suitable for this task. We demonstrate the feasibility of learning a Pareto-embedding on a suite of benchmark datasets.

Karlson Pfannschmidt, Pritha Gupta, Björn Haddenhorst et al.

Choice functions accept a set of alternatives as input and produce a preferred subset of these alternatives as output. We study the problem of learning such functions under conditions of context-dependence of preferences, which means that the preference in favor of a certain choice alternative may depend on what other options are also available. In spite of its practical relevance, this kind of context-dependence has received little attention in preference learning so far. We propose a suitable model based on context-dependent (latent) utility functions, thereby reducing the problem to the task of learning such utility functions. Practically, this comes with a number of challenges. For example, the set of alternatives provided as input to a choice function can be of any size, and the output of the function should not depend on the order in which the alternatives are presented. To meet these requirements, we propose two general approaches based on two representations of context-dependent utility functions, as well as instantiations in the form of appropriate end-to-end trainable neural network architectures. Moreover, to demonstrate the performance of both networks, we present extensive empirical evaluations on both synthetic and real-world datasets.

Karlson Pfannschmidt, Pritha Gupta, Eyke Hüllermeier

Object ranking is an important problem in the realm of preference learning. On the basis of training data in the form of a set of rankings of objects, which are typically represented as feature vectors, the goal is to learn a ranking function that predicts a linear order of any new set of objects. Current approaches commonly focus on ranking by scoring, i.e., on learning an underlying latent utility function that seeks to capture the inherent utility of each object. These approaches, however, are not able to take possible effects of context-dependence into account, where context-dependence means that the utility or usefulness of an object may also depend on what other objects are available as alternatives. In this paper, we formalize the problem of context-dependent ranking and present two general approaches based on two natural representations of context-dependent ranking functions. Both approaches are instantiated by means of appropriate neural network architectures, which are evaluated on suitable benchmark task.