Real-world problems are often multi-objective, with decision-makers unable to specify a priori which trade-off between the conflicting objectives is preferable. Intuitively, building machine learning solutions in such cases would entail providing multiple predictions that span and uniformly cover the Pareto front of all optimal trade-off solutions. We propose a novel approach for multi-objective training of neural networks to approximate the Pareto front during inference. In our approach, we train the neural networks multi-objectively using a dynamic loss function, wherein each network’s losses (corresponding to multiple objectives) are weighted by their hypervolume maximizing gradients. Experiments on different multi-objective problems show that our approach returns well-spread outputs across different trade-offs on the approximated Pareto front without requiring the trade-off vectors to be specified a priori. Further, results of comparisons with the state-of-the-art approaches highlight the added value of our proposed approach, especially in cases where the Pareto front is asymmetric.

Evolutionary algorithms (EAs) are well-known to be well suited for multi-objective (MO) optimization. However, especially in the case of real-valued variables, classic domination-based approaches are known to lose selection pressure when approaching the Pareto set. Indicator-based approaches, such as optimizing the uncrowded hypervolume (UHV), can overcome this issue and ensure that individual solutions converge to the Pareto set. Recently, a gradient-based UHV algorithm, known as UHV-ADAM, was shown to be more efficient than (UHV-based) EAs if few local optima are present. Combining the two techniques could exploit synergies, i.e., the EA could be leveraged to avoid local optima while the efficiency of gradient algorithms could speed up convergence to the Pareto set. It is a priori however not clear what would be the best way to make such a combination. In this work, therefore, we study the use of a dynamic resource allocation scheme to create hybrid UHV-based algorithms. On several bi-objective benchmarks, we find that the hybrid algorithms produce similar or better results than the EA or gradient-based algorithm alone, even when finite differences are used to approximate gradients. The implementation of the hybrid algorithm is available at https://github.com/damyha/uncrowded-hypervolume.

Anatomical landmark correspondences in medical images can provide additional guidance information for the alignment of two images, which, in turn, is crucial for many medical applications. However, manual landmark annotation is labor-intensive. Therefore, we propose an end-to-end deep learning approach to automatically detect landmark correspondences in pairs of two-dimensional (2D) images. Our approach consists of a Siamese neural network, which is trained to identify salient locations in images as landmarks and predict matching probabilities for landmark pairs from two different images. We trained our approach on 2D transverse slices from 168 lower abdominal Computed Tomography (CT) scans. We tested the approach on 22,206 pairs of 2D slices with varying levels of intensity, affine, and elastic transformations. The proposed approach finds an average of 639, 466, and 370 landmark matches per image pair for intensity, affine, and elastic transformations, respectively, with spatial matching errors of at most 1 mm. Further, more than 99% of the landmark pairs are within a spatial matching error of 2 mm, 4 mm, and 8 mm for image pairs with intensity, affine, and elastic transformations, respectively. To investigate the utility of our developed approach in a clinical setting, we also tested our approach on pairs of transverse slices selected from follow-up CT scans of three patients. Visual inspection of the results revealed landmark matches in both bony anatomical regions as well as in soft tissues lacking prominent intensity gradients.