Konstantin Sonntag

Augustina C. Amakor, Konstantin Sonntag, Sebastian Peitz

Sparsity is a highly desired feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability of models (due to the smaller number of relevant features), and robustness. For linear models, it is well known that there exists a \emph{regularization path} connecting the sparsest solution in terms of the $\ell^1$ norm, i.e., zero weights and the non-regularized solution. Very recently, there was a first attempt to extend the concept of regularization paths to DNNs by means of treating the empirical loss and sparsity ($\ell^1$ norm) as two conflicting criteria and solving the resulting multiobjective optimization problem for low-dimensional DNN. However, due to the non-smoothness of the $\ell^1$ norm and the high number of parameters, this approach is not very efficient from a computational perspective for high-dimensional DNNs. To overcome this limitation, we present an algorithm that allows for the approximation of the entire Pareto front for the above-mentioned objectives in a very efficient manner for high-dimensional DNNs with millions of parameters. We present numerical examples using both deterministic and stochastic gradients. We furthermore demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization. To the best of our knowledge, this is the first algorithm to compute the regularization path for non-convex multiobjective optimization problems (MOPs) with millions of degrees of freedom.

Junaid Akhter, Paul David Fährmann, Konstantin Sonntag et al.

Recently, there has been an increasing interest in the application of multiobjective optimization (MOO) in machine learning (ML). This interest is driven by the numerous real-life situations where multiple objectives must be optimized simultaneously. A key aspect of MOO is the existence of a Pareto set, rather than a single optimal solution, which represents the optimal trade-offs between different objectives. Despite its potential, there is a noticeable lack of satisfactory literature serving as an entry-level guide for ML practitioners aiming to apply MOO effectively. In this paper, our goal is to provide such a resource and highlight pitfalls to avoid. We begin by establishing the groundwork for MOO, focusing on well-known approaches such as the weighted sum (WS) method, alongside more advanced techniques like the multiobjective gradient descent algorithm (MGDA). We critically review existing studies across various ML fields where MOO has been applied and identify challenges that can lead to incorrect interpretations. One of these fields is physics informed neural networks (PINNs), which we use as a guiding example to carefully construct experiments illustrating these pitfalls. By comparing WS and MGDA with one of the most common evolutionary algorithms, NSGA-II, we demonstrate that difficulties can arise regardless of the specific MOO method used. We emphasize the importance of understanding the specific problem, the objective space, and the selected MOO method, while also noting that neglecting factors such as convergence criteria can result in misleading experiments.