We introduce the Integrated Tsallis Combination (ITC), a hybrid impurity measure for decision tree learning that combines normalized Tsallis entropy with an exponential polarization component. While many existing measures sacrifice theoretical soundness for computational efficiency or vice versa, ITC provides a mathematically principled framework that balances both aspects. The core innovation lies in the complementarity between Tsallis entropy's information-theoretic foundations and the polarization component's sensitivity to distributional asymmetry. We establish key theoretical properties-concavity under explicit parameter conditions, proper boundary conditions, and connections to classical measures-and provide a rigorous justification for the hybridization strategy. Through an extensive comparative evaluation on seven benchmark datasets comparing 23 impurity measures with five-fold repetition, we show that simple parametric measures (Tsallis $α=0.5$) achieve the highest average accuracy ($91.17\%$), while ITC variants yield competitive results ($88.38-89.16\%$) with strong theoretical guarantees. Statistical analysis (Friedman test: $χ^2=3.89$, $p=0.692$) reveals no significant global differences among top performers, indicating practical equivalence for many applications. ITC's value resides in its solid theoretical grounding-proven concavity under suitable conditions, flexible parameterization ($α$, $β$, $γ$), and computational efficiency $O(K)$-making it a rigorous, generalizable alternative when theoretical guarantees are paramount. We provide guidelines for measure selection based on application priorities and release an open-source implementation to foster reproducibility and further research.

Scorio.jl is a Julia package for evaluating and ranking systems from repeated responses to shared tasks. It provides a common tensor-based interface for direct score-based, pairwise, psychometric, voting, graph, and listwise methods, so the same benchmark can be analyzed under multiple ranking assumptions. We describe the package design, position it relative to existing Julia tools, and report pilot experiments on synthetic rank recovery, stability under limited trials, and runtime scaling.

This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a $\mathcal{O}(1/k)$ best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the $n$-finite-sum and expectation settings are $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, respectively, when employing loopless-SVRG or SAGA, where $ε$ is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ for the $n$-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.

Most machine learning methods assume fixed probability distributions, limiting their applicability in nonstationary real-world scenarios. While continual learning methods address this issue, current approaches often rely on black-box models or require extensive user intervention for interpretability. We propose SyMPLER (Systems Modeling through Piecewise Linear Evolving Regression), an explainable model for time series forecasting in nonstationary environments based on dynamic piecewise-linear approximations. Unlike other locally linear models, SyMPLER uses generalization bounds from Statistical Learning Theory to automatically determine when to add new local models based on prediction errors, eliminating the need for explicit clustering of the data. Experiments show that SyMPLER can achieve comparable performance to both black-box and existing explainable models while maintaining a human-interpretable structure that reveals insights about the system's behavior. In this sense, our approach conciliates accuracy and interpretability, offering a transparent and adaptive solution for forecasting nonstationary time series.

Staged tree models enhance Bayesian networks by incorporating context-specific dependencies through a stage-based structure. In this study, we present a new framework for estimating staged trees using hierarchical clustering on the probability simplex, utilizing simplex basesd divergences. We conduct a thorough evaluation of several distance and divergence metrics including Total Variation, Hellinger, Fisher, and Kaniadakis; alongside various linkage methods such as Ward.D2, average, complete, and McQuitty. We conducted the simulation experiments that reveals Total Variation, especially when combined with Ward.D2 linkage, consistently produces staged trees with better model fit, structure recovery, and computational efficiency. We assess performance by utilizing relative Bayesian Information Criterion (BIC), and Hamming distance. Our findings indicate that although Backward Hill Climbing (BHC) delivers competitive outcomes, it incurs a significantly higher computational cost. On the other, Total Variation divergence with Ward.D2 linkage, achieves similar performance while providing significantly better computational efficiency, making it a more viable option for large-scale or time sensitive tasks.

Recovering the random graph model from an observed collection of networks is known to present significant challenges in the setting, where the networks do not share a common node set and have different sizes. More specifically, the goal is the estimation of the graphon function that parametrizes the nonparametric exchangeable random graph model. Existing methods typically suffer from either limited accuracy or high computational complexity. We introduce a new histogram-based estimator with low algorithmic complexity that achieves high accuracy by jointly aligning the nodes of all graphs, in contrast to most conventional methods that order nodes graph by graph. Consistency results of the proposed graphon estimator are established. A numerical study shows that the proposed estimator outperforms existing methods in terms of accuracy, especially when the dataset comprises only small and variable-size networks. Moreover, the computing time of the new method is considerably shorter than that of other consistent methodologies. Additionally, when applied to a graph neural network classification task, the proposed estimator enables more effective data augmentation, yielding improved performance across diverse real-world datasets.

Model-based approaches for (bio)process systems often suffer from incomplete knowledge of the underlying physical, chemical, or biological laws. Universal differential equations, which embed neural networks within differential equations, have emerged as powerful tools to learn this missing physics from experimental data. However, neural networks are inherently opaque, motivating their post-processing via symbolic regression to obtain interpretable mathematical expressions. Genetic algorithm-based symbolic regression is a popular approach for this post-processing step, but provides only point estimates and cannot quantify the confidence we should place in a discovered equation. We address this limitation by applying Bayesian symbolic regression, which uses Reversible Jump Markov Chain Monte Carlo to sample from the posterior distribution over symbolic expression trees. This approach naturally quantifies uncertainty in the recovered model structure. We demonstrate the methodology on a Lotka-Volterra predator-prey system and then show how a well-designed experiment leads to lower uncertainty in a fed-batch bioreactor case study.

In a spiking neural network, is it enough for each neuron to spike at most once? In recent work, approximation bounds for spiking neural networks have been derived, quantifying how well they can fit target functions. However, these results are only valid for neurons that spike at most once, which is commonly thought to be a strong limitation. Here, we show that the opposite is true for a large class of spiking neuron models, including the commonly used leaky integrate-and-fire model with subtractive reset: for every approximation bound that is valid for a set of multi-spike neural networks, there is an equivalent set of single-spike neural networks with only linearly more neurons (in the maximum number of spikes) for which the bound holds. The same is true for the reverse direction too, showing that regarding their approximation capabilities in general machine learning tasks, single-spike and multi-spike neural networks are equivalent. Consequently, many approximation results in the literature for single-spike neural networks also hold for the multi-spike case.

Large language models (LLMs) have achieved remarkable performance on diverse benchmarks, yet existing evaluation practices largely rely on coarse summary metrics that obscure underlying reasoning abilities. In this work, we propose novel methodologies to adapt cognitive diagnosis models (CDMs) in psychometrics to LLM evaluation, enabling fine-grained diagnosis via multidimensional discrete capability profiles and interpretable characterizations of LLM strengths and weaknesses. First, to enable CDM-based evaluation at benchmark scale (more than 1000 items), we propose a scalable method that jointly estimates LLM mastery profiles and the item-attribute Q-matrix, addressing key challenges posed by high-dimensional latent attributes (K > 20), large item pools, and the prohibitive computational cost of existing marginal maximum likelihood-based estimation. Second, we incorporate item-level textual information to construct AI-embedding-informed priors for the Q-matrix, stabilizing high-dimensional estimation while reducing reliance on costly human specification. We develop an efficient stochastic-approximation algorithm to jointly estimate LLM mastery profiles and the Q-matrix that balances data fit with text-embedding-informed priors. Simulation studies demonstrate accurate parameter recovery. An application to the MATH Level 5 benchmark illustrates the practical utility of our method for LLM evaluation and uncovers useful insights into LLMs' fine-grained capabilities.

In this paper, we analyze the two time-scale stochastic approximation (TTSSA) algorithm introduced in Borkar (1997) using a martingale approach. This approach leads to simple sufficient conditions for the iterations to be bounded almost surely, as well as estimates on the rate of convergence of the mean-squared error of the TTSSA algorithm to zero. Our theory is applicable to nonlinear equations, in contrast to many papers in the TTSSA literature which assume that the equations are linear. The convergence of TTSSA is proved in the "almost sure" sense, in contrast to earlier papers on TTSSA that establish convergence in distribution, convergence in the mean, and the like. Moreover, in this paper we establish different rates of convergence for the fast and the slow subsystems, perhaps for the first time. Finally, all of the above results to continue to hold in the case where the two measurement errors have nonzero conditional mean, and/or have conditional variances that grow without bound as the iterations proceed. This is in contrast to previous papers which assumed that the errors form a martingale difference sequence with uniformly bounded conditional variance. It is shown that when the measurement errors have zero conditional mean and the conditional variance remains bounded, the mean-squared error of the iterations converges to zero at a rate of $o(t^{-η})$ for all $η\in (0,1)$. This improves upon the rate of $O(t^{-2/3})$ proved in Doan (2023) (which is the best bound available to date). Our bound is virtually the same as the rate of $O(t^{-1})$ proved in Doan (2024), but for a Polyak-Ruppert averaged version of TTSSA, and not directly. Rates of convergence are also established for the case where the errors have nonzero conditional mean and/or unbounded conditional variance.