In this paper, we propose an optimally structured gradient coding scheme to mitigate the straggler problem in distributed learning. Conventional gradient coding methods often assume homogeneous straggler models or rely on excessive data replication, limiting performance in real-world heterogeneous systems. To address these limitations, we formulate an optimization problem minimizing residual error while ensuring unbiased gradient estimation by explicitly considering individual straggler probabilities. We derive closed-form solutions for optimal encoding and decoding coefficients via Lagrangian duality and convex optimization, and propose data allocation strategies that reduce both redundancy and computation load. We also analyze convergence behavior for $λ$-strongly convex and $μ$-smooth loss functions. Numerical results show that our approach significantly reduces the impact of stragglers and accelerates convergence compared to existing methods.

Vertex hunting (VH) is the task of estimating a simplex from noisy data points and has many applications in areas such as network and text analysis. We introduce a new variant, semi-supervised vertex hunting (SSVH), in which partial information is available in the form of barycentric coordinates for some data points, known only up to an unknown transformation. To address this problem, we develop a method that leverages properties of orthogonal projection matrices, drawing on novel insights from linear algebra. We establish theoretical error bounds for our method and demonstrate that it achieves a faster convergence rate than existing unsupervised VH algorithms. Finally, we apply SSVH to two practical settings, semi-supervised network mixed membership estimation and semi-supervised topic modeling, resulting in efficient and scalable algorithms.

Constrained clustering leverages limited domain knowledge to improve clustering performance and interpretability, but incorporating pairwise must-link and cannot-link constraints is an NP-hard challenge, making global optimization intractable. Existing mixed-integer optimization methods are confined to small-scale datasets, limiting their utility. We propose Sample-Driven Constrained Group-Based Branch-and-Bound (SDC-GBB), a decomposable branch-and-bound (BB) framework that collapses must-linked samples into centroid-based pseudo-samples and prunes cannot-link through geometric rules, while preserving convergence and guaranteeing global optimality. By integrating grouped-sample Lagrangian decomposition and geometric elimination rules for efficient lower and upper bounds, the algorithm attains highly scalable pairwise k-Means constrained clustering via parallelism. Experimental results show that our approach handles datasets with 200,000 samples with cannot-link constraints and 1,500,000 samples with must-link constraints, which is 200 - 1500 times larger than the current state-of-the-art under comparable constraint settings, while reaching an optimality gap of less than 3%. In providing deterministic global guarantees, our method also avoids the search failures that off-the-shelf heuristics often encounter on large datasets.

Knowledge Distillation (KD) aims to train a lightweight student model by transferring knowledge from a large, high-capacity teacher. Recent studies have shown that leveraging diverse teacher perspectives can significantly improve distillation performance; however, achieving such diversity typically requires multiple teacher networks, leading to high computational costs. In this work, we propose a novel cost-efficient knowledge augmentation method for KD that generates diverse multi-views by attaching multiple branches to a single teacher. To ensure meaningful semantic variation across multi-views, we introduce two angular diversity objectives: 1) constrained inter-angle diversify loss, which maximizes angles between augmented views while preserving proximity to the original teacher output, and 2) intra-angle diversify loss, which encourages an even distribution of views around the original output. The ensembled knowledge from these angularly diverse views, along with the original teacher, is distilled into the student. We further theoretically demonstrate that our objectives increase the diversity among ensemble members and thereby reduce the upper bound of the ensemble's expected loss, leading to more effective distillation. Experimental results show that our method surpasses an existing knowledge augmentation method across diverse configurations. Moreover, the proposed method is compatible with other KD frameworks in a plug-and-play fashion, providing consistent improvements in generalization performance.

Graph Neural Networks (GNNs) have revolutionized the field of graph learning by learning expressive graph representations from massive graph data. As a common pattern to train powerful GNNs, the "pre-training, adaptation" scheme first pre-trains GNNs over unlabeled graph data and subsequently adapts them to specific downstream tasks. In the adaptation phase, graph prompting is an effective strategy that modifies input graph data with learnable prompts while keeping pre-trained GNN models frozen. Typically, existing graph prompting studies mainly focus on *feature-oriented* methods that apply graph prompts to node features or hidden representations. However, these studies often achieve suboptimal performance, as they consistently overlook the potential of *topology-oriented* prompting, which adapts pre-trained GNNs by modifying the graph topology. In this study, we conduct a pioneering investigation of graph prompting in terms of graph topology. We propose the first **Graph** **T**opology-**O**riented **P**rompting (GraphTOP) framework to effectively adapt pre-trained GNN models for downstream tasks. More specifically, we reformulate topology-oriented prompting as an edge rewiring problem within multi-hop local subgraphs and relax it into the continuous probability space through reparameterization while ensuring tight relaxation and preserving graph sparsity. Extensive experiments on five graph datasets under four pre-training strategies demonstrate that our proposed GraphTOP outshines six baselines on multiple node classification datasets. Our code is available at https://github.com/xbfu/GraphTOP.

Controlling high-dimensional stochastic systems, critical in robotics, autonomous vehicles, and hyperchaotic systems, faces the curse of dimensionality, lacks temporal abstraction, and often fails to ensure stochastic stability. To overcome these limitations, this study introduces the Multi-Timescale Lyapunov-Constrained Hierarchical Reinforcement Learning (MTLHRL) framework. MTLHRL integrates a hierarchical policy within a semi-Markov Decision Process (SMDP), featuring a high-level policy for strategic planning and a low-level policy for reactive control, which effectively manages complex, multi-timescale decision-making and reduces dimensionality overhead. Stability is rigorously enforced using a neural Lyapunov function optimized via Lagrangian relaxation and multi-timescale actor-critic updates, ensuring mean-square boundedness or asymptotic stability in the face of stochastic dynamics. The framework promotes efficient and reliable learning through trust-region constraints and decoupled optimization. Extensive simulations on an 8D hyperchaotic system and a 5-DOF robotic manipulator demonstrate MTLHRL's empirical superiority. It significantly outperforms baseline methods in both stability and performance, recording the lowest error indices (e.g., Integral Absolute Error (IAE): 3.912 in hyperchaotic control and IAE: 1.623 in robotics), achieving faster convergence, and exhibiting superior disturbance rejection. MTLHRL offers a theoretically grounded and practically viable solution for robust control of complex stochastic systems.

Forecasting on widely used benchmark time series data (e.g., ETT, Electricity, Taxi, and Exchange Rate, etc.) has favored smooth, seasonal series, but network telemetry time series -- traffic measurements at service, IP, or subnet granularity -- are instead highly bursty and intermittent, with heavy-tailed bursts and highly variable inactive periods. These properties place the latter in the statistical regimes made famous and popularized more than 20 years ago by B.~Mandelbrot. Yet forecasting such time series with modern-day AI architectures remains underexplored. We introduce NetBurst, an event-centric framework that reformulates forecasting as predicting when bursts occur and how large they are, using quantile-based codebooks and dual autoregressors. Across large-scale sets of production network telemetry time series and compared to strong baselines, such as Chronos, NetBurst reduces Mean Average Scaled Error (MASE) by 13--605x on service-level time series while preserving burstiness and producing embeddings that cluster 5x more cleanly than Chronos. In effect, our work highlights the benefits that modern AI can reap from leveraging Mandelbrot's pioneering studies for forecasting in bursty, intermittent, and heavy-tailed regimes, where its operational value for high-stakes decision making is of paramount interest.

We present memory-based language modeling as an efficient, eco-friendly alternative to deep neural network-based language modeling. It offers log-linearly scalable next-token prediction performance and strong memorization capabilities. Implementing fast approximations of k-nearest neighbor classification, memory-based language modeling leaves a relatively small ecological footprint both in training and in inference mode, as it relies fully on CPUs and attains low token latencies. Its internal workings are simple and fully transparent. We compare our implementation of memory-based language modeling, OLIFANT, with GPT-2 and GPT-Neo on next-token prediction accuracy, estimated emissions and speeds, and offer some deeper analyses of the model.

Homophily, the tendency of nodes from the same class to connect, is a fundamental property of real-world graphs, underpinning structural and semantic patterns in domains such as citation networks and social networks. Existing methods exploit homophily through designing homophily-aware GNN architectures or graph structure learning strategies, yet they primarily focus on GNN learning with training graphs. However, in real-world scenarios, test graphs often suffer from data quality issues and distribution shifts, such as domain shifts across users from different regions in social networks and temporal evolution shifts in citation network graphs collected over varying time periods. These factors significantly compromise the pre-trained model's robustness, resulting in degraded test-time performance. With empirical observations and theoretical analysis, we reveal that transforming the test graph structure by increasing homophily in homophilic graphs or decreasing it in heterophilic graphs can significantly improve the robustness and performance of pre-trained GNNs on node classifications, without requiring model training or update. Motivated by these insights, a novel test-time graph structural transformation method grounded in homophily, named GrapHoST, is proposed. Specifically, a homophily predictor is developed to discriminate test edges, facilitating adaptive test-time graph structural transformation by the confidence of predicted homophily scores. Extensive experiments on nine benchmark datasets under a range of test-time data quality issues demonstrate that GrapHoST consistently achieves state-of-the-art performance, with improvements of up to 10.92%. Our code has been released at https://github.com/YanJiangJerry/GrapHoST.

International Journal of Applied Mathematics Volume 38 No. 5 s, 2025 ISSN: 1311 - 1728 (printed version); ISSN: 1314 - 8060 (on - line version) Received: August 0 7, 2025 550 Abstract The growing instability of both global and domestic economic environments has increased the risk of financial distress at the household level. However, traditional econometric models often rely on delayed and aggregated data, limiting their effectiveness. This study introduces a machine learning - based early warning system that utilizes real - time digital and macroeconomic signals to identify financial distress in near real - time. Using a panel dataset of 750 households tracked over three monitoring rounds spa nning 13 months, the framework combines socioeconomic attributes, macroeconomic indicators (such as GDP growth, inflation, and foreign exchange fluctuations), and digital economy measures (including ICT demand and market volatility). Through data preproces sing and feature engineering, we introduce lagged variables, volatility measures, and interaction terms to capture both gradual and sudden changes in financial stability. We benchmark baseline classifiers, such as logistic regression and decision trees, ag ainst advanced ensemble models including random forests, XGBoost, and LightGBM. Our results indicate that the engineered features from the digital economy significantly enhance predictive accuracy. The system performs reliably for both binary distress dete ction and multi - class severity classification, with SHAP - based explanations identifying inflation volatility and ICT demand as key predictors. Crucially, the framework is International Journal of Applied Mathematics Volume 38 No. 5 s, 2025 ISSN: 1311 - 1728 (printed version); ISSN: 1314 - 8060 (on - line version) Received: August 0 7, 2025 551 By implementing machine learning in a transparent and interpretable manner, this study demonstrates the feasibility and impact of providing near - real - time early warnings of financial distress. This offers actionable insights that can strengthen household resilience and guide preemptive intervention strategies. Keywords: Financial Distress, Early Warning Systems, Machine Learning, Digital Economy, Temporal Classification, Explainable AI 1. Introduction 1.1 Background and Motivation The prediction of financial distress has long been recognized as a critical element for ensuring economic resilience and mitigating systemic risk across households, firms, and national economies.