Benchmark suites are crucial for assessing the performance of evolutionary algorithms, but the constituent problems are often too complex to provide clear intuition about an algorithm's strengths and weaknesses. To address this gap, we introduce DOSSIER ("Diagnostic Overview of Selection Schemes In Evolutionary Runs"), a diagnostic suite initially composed of eight handcrafted metrics. These metrics are designed to empirically measure specific capacities for exploitation, exploration, and their interactions. We consider exploitation both with and without constraints, and we divide exploration into two aspects: diversity exploration (the ability to simultaneously explore multiple pathways) and valley-crossing exploration (the ability to cross wider and wider fitness valleys). We apply DOSSIER to six popular selection schemes: truncation, tournament, fitness sharing, lexicase, nondominated sorting, and novelty search. Our results confirm that simple schemes (e.g., tournament and truncation) emphasized exploitation. For more sophisticated schemes, however, our diagnostics revealed interesting dynamics. Lexicase selection performed moderately well across all diagnostics that did not incorporate valley crossing, but faltered dramatically whenever valleys were present, performing worse than even random search. Fitness sharing was the only scheme to effectively contend with valley crossing but it struggled with the other diagnostics. Our study highlights the utility of using diagnostics to gain nuanced insights into selection scheme characteristics, which can inform the design of new selection methods.
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In several branches of the social sciences and humanities, surveys based on standardized questionnaires are a prominent research tool. While there are a variety of ways to analyze the data, some standard procedures have become established. When those surveys want to analyze differences in the answer patterns of different groups (e.g., countries, gender, age, ...), these procedures can only be carried out in a meaningful way if there is measurement invariance, i.e., the measured construct has psychometric equivalence across groups. As recently raised as an open problem by Sauerwein et al. (2021), new evaluation methods that work in the absence of measurement invariance are needed. This paper promotes an unsupervised learning-based approach to such research data by proposing a procedure that works in three phases: data preparation, clustering of questionnaires, and measuring similarity based on the obtained clustering and the properties of each group. We generate synthetic data in three data sets, which allows us to compare our approach with the PCA approach under measurement invariance and under violated measurement invariance. As a main result, we obtain that the approach provides a natural comparison between groups and a natural description of the response patterns of the groups. Moreover, it can be safely applied to a wide variety of data sets, even in the absence of measurement invariance. Finally, this approach allows us to translate (violations of) measurement invariance into a meaningful measure of similarity.
Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.
This study elaborates a text-based metric to quantify the unique position of stylized scientific research, characterized by its innovative integration of diverse knowledge components and potential to pivot established scientific paradigms. Our analysis reveals a concerning decline in stylized research, highlighted by its comparative undervaluation in terms of citation counts and protracted peer-review duration. Despite facing these challenges, the disruptive potential of stylized research remains robust, consistently introducing groundbreaking questions and theories. This paper posits that substantive reforms are necessary to incentivize and recognize the value of stylized research, including optimizations to the peer-review process and the criteria for evaluating scientific impact. Embracing these changes may be imperative to halt the downturn in stylized research and ensure enduring scholarly exploration in endless frontiers.
We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.
We present a novel clustering algorithm, visClust, that is based on lower dimensional data representations and visual interpretation. Thereto, we design a transformation that allows the data to be represented by a binary integer array enabling the use of image processing methods to select a partition. Qualitative and quantitative analyses measured in accuracy and an adjusted Rand-Index show that the algorithm performs well while requiring low runtime and RAM. We compare the results to 6 state-of-the-art algorithms with available code, confirming the quality of visClust by superior performance in most experiments. Moreover, the algorithm asks for just one obligatory input parameter while allowing optimization via optional parameters. The code is made available on GitHub and straightforward to use.
We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally, we show some numerical experiments confirming the theoretical results, and a couple of applications in view of bifurcation analysis.
We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.
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