The two-timescale gradient descent-ascent (GDA) is a canonical gradient algorithm designed to find Nash equilibria in min-max games. We analyze the two-timescale GDA by investigating the effects of learning rate ratios on convergence behavior in both finite-dimensional and mean-field settings. In particular, for finite-dimensional quadratic min-max games, we obtain long-time convergence in near quasi-static regimes through the hypocoercivity method. For mean-field GDA dynamics, we investigate convergence under a finite-scale ratio using a mixed synchronous-reflection coupling technique.

Interpreting black-box machine learning models is challenging due to their strong dependence on data and inherently non-parametric nature. This paper reintroduces the concept of importance through "Marginal Variable Importance Metric" (MVIM), a model-agnostic measure of predictor importance based on the true conditional expectation function. MVIM evaluates predictors' influence on continuous or discrete outcomes. A permutation-based estimation approach, inspired by \citet{breiman2001random} and \citet{fisher2019all}, is proposed to estimate MVIM. MVIM estimator is biased when predictors are highly correlated, as black-box models struggle to extrapolate in low-probability regions. To address this, we investigated the bias-variance decomposition of MVIM to understand the source and pattern of the bias under high correlation. A Conditional Variable Importance Metric (CVIM), adapted from \citet{strobl2008conditional}, is introduced to reduce this bias. Both MVIM and CVIM exhibit a quadratic relationship with the conditional average treatment effect (CATE).

Many scientific problems involve data that is embedded in a space with periodic boundary conditions. This can for instance be related to an inherent cyclic or rotational symmetry in the data or a spatially extended periodicity. When analyzing such data, well-tailored methods are needed to obtain efficient approaches that obey the periodic boundary conditions of the problem. In this work, we present a method for applying a clustering algorithm to data embedded in a periodic domain based on the DBSCAN algorithm, a widely used unsupervised machine learning method that identifies clusters in data. The proposed method internally leverages the conventional DBSCAN algorithm for domains with open boundaries, such that it remains compatible with all optimized implementations for neighborhood searches in open domains. In this way, it retains the same optimized runtime complexity of $O(N\log N)$. We demonstrate the workings of the proposed method using synthetic data in one, two and three dimensions and also apply it to a real-world example involving the clustering of bubbles in a turbulent flow. The proposed approach is implemented in a ready-to-use Python package that we make publicly available.

Gradient-based Discrete Samplers (GDSs) are effective for sampling discrete energy landscapes. However, they often stagnate in complex, non-convex settings. To improve exploration, we introduce the Discrete Replica EXchangE Langevin (DREXEL) sampler and its variant with Adjusted Metropolis (DREAM). These samplers use two GDSs at different temperatures and step sizes: one focuses on local exploitation, while the other explores broader energy landscapes. When energy differences are significant, sample swaps occur, which are determined by a mechanism tailored for discrete sampling to ensure detailed balance. Theoretically, we prove both DREXEL and DREAM converge asymptotically to the target energy and exhibit faster mixing than a single GDS. Experiments further confirm their efficiency in exploring non-convex discrete energy landscapes.

Learning curves are a concept from social sciences that has been adopted in the context of machine learning to assess the performance of a learning algorithm with respect to a certain resource, e.g., the number of training examples or the number of training iterations. Learning curves have important applications in several machine learning contexts, most notably in data acquisition, early stopping of model training, and model selection. For instance, learning curves can be used to model the performance of the combination of an algorithm and its hyperparameter configuration, providing insights into their potential suitability at an early stage and often expediting the algorithm selection process. Various learning curve models have been proposed to use learning curves for decision making. Some of these models answer the binary decision question of whether a given algorithm at a certain budget will outperform a certain reference performance, whereas more complex models predict the entire learning curve of an algorithm. We contribute a framework that categorises learning curve approaches using three criteria: the decision-making situation they address, the intrinsic learning curve question they answer and the type of resources they use. We survey papers from the literature and classify them into this framework.

Excited-state nonadiabatic simulations with quantum mechanics/molecular mechanics (QM/MM) are essential to understand photoinduced processes in explicit environments. However, the high computational cost of the underlying quantum chemical calculations limits its application in combination with trajectory surface hopping methods. Here, we use FieldSchNet, a machine-learned interatomic potential capable of incorporating electric field effects into the electronic states, to replace traditional QM/MM electrostatic embedding with its ML/MM counterpart for nonadiabatic excited state trajectories. The developed method is applied to furan in water, including five coupled singlet states. Our results demonstrate that with sufficiently curated training data, the ML/MM model reproduces the electronic kinetics and structural rearrangements of QM/MM surface hopping reference simulations. Furthermore, we identify performance metrics that provide robust and interpretable validation of model accuracy.

Rainfall prediction remains a persistent challenge due to the highly nonlinear and complex nature of meteorological data. Existing approaches lack systematic utilization of grid search for optimal hyperparameter tuning, relying instead on heuristic or manual selection, frequently resulting in sub-optimal results. Additionally, these methods rarely incorporate newly constructed meteorological features such as differences between temperature and humidity to capture critical weather dynamics. Furthermore, there is a lack of systematic evaluation of ensemble learning techniques and limited exploration of diverse advanced models introduced in the past one or two years. To address these limitations, we propose a robust ensemble learning grid search-tuned framework (RAINER) for rainfall prediction. RAINER incorporates a comprehensive feature engineering pipeline, including outlier removal, imputation of missing values, feature reconstruction, and dimensionality reduction via Principal Component Analysis (PCA). The framework integrates novel meteorological features to capture dynamic weather patterns and systematically evaluates non-learning mathematical-based methods and a variety of machine learning models, from weak classifiers to advanced neural networks such as Kolmogorov-Arnold Networks (KAN). By leveraging grid search for hyperparameter tuning and ensemble voting techniques, RAINER achieves promising results within real-world datasets.

Power transformers are subjected to electrical currents and temperature fluctuations that, if not properly controlled, can lead to major deterioration of their insulation system. Therefore, monitoring the temperature of a power transformer is fundamental to ensure a long-term operational life. Models presented in the IEC 60076-7 and IEEE standards, for example, monitor the temperature by calculating the top-oil and the hot-spot temperatures. However, these models are not very accurate and rely on the power transformers' properties. This paper focuses on finding an alternative method to predict the top-oil temperatures given previous measurements. Given the large quantities of data available, machine learning methods for time series forecasting are analyzed and compared to the real measurements and the corresponding prediction of the IEC standard. The methods tested are Artificial Neural Networks (ANNs), Time-series Dense Encoder (TiDE), and Temporal Convolutional Networks (TCN) using different combinations of historical measurements. Each of these methods outperformed the IEC 60076-7 model and they are extended to estimate the temperature rise over ambient. To enhance prediction reliability, we explore the application of quantile regression to construct prediction intervals for the expected top-oil temperature ranges. The best-performing model successfully estimates conditional quantiles that provide sufficient coverage.

Conditional mean independence (CMI) testing is crucial for statistical tasks including model determination and variable importance evaluation. In this work, we introduce a novel population CMI measure and a bootstrap-based testing procedure that utilizes deep generative neural networks to estimate the conditional mean functions involved in the population measure. The test statistic is thoughtfully constructed to ensure that even slowly decaying nonparametric estimation errors do not affect the asymptotic accuracy of the test. Our approach demonstrates strong empirical performance in scenarios with high-dimensional covariates and response variable, can handle multivariate responses, and maintains nontrivial power against local alternatives outside an $n^{-1/2}$ neighborhood of the null hypothesis. We also use numerical simulations and real-world imaging data applications to highlight the efficacy and versatility of our testing procedure.

The state-of-the-art online learning models generally conduct a single online gradient descent when a new sample arrives and thus suffer from suboptimal model weights. To this end, we introduce an online broad learning system framework with closed-form solutions for each online update. Different from employing existing incremental broad learning algorithms for online learning tasks, which tend to incur degraded accuracy and expensive online update overhead, we design an effective weight estimation algorithm and an efficient online updating strategy to remedy the above two deficiencies, respectively. Specifically, an effective weight estimation algorithm is first developed by replacing notorious matrix inverse operations with Cholesky decomposition and forward-backward substitution to improve model accuracy. Second, we devise an efficient online updating strategy that dramatically reduces online update time. Theoretical analysis exhibits the splendid error bound and low time complexity of our model. The most popular test-then-training evaluation experiments on various real-world datasets prove its superiority and efficiency. Furthermore, our framework is naturally extended to data stream scenarios with concept drift and exceeds state-of-the-art baselines.