Recently, many efficient algorithms for minimax problems have been proposed, but there are relatively few methods for solving nonsmooth constrained minimax problems. This article focuses on developing a distributed algorithm to address this underexplored class of constrained nonsmooth minimax optimization challenges. To be specific, the distributed nonsmooth convex-concave minimax problem with inequality constraints for multiagent systems is considered, where the two subsystems have opposite objectives, minimization and maximization, respectively. Individual agents cooperate with their neighbors in their own subsystem and compete with agents in the other subsystem, and agents have only partial knowledge of the other subsystem. We propose a distributed continuous-time penalty-based algorithm that adaptively determines appropriate penalty gains. In particular, the proposed algorithm is an adaptive strategy that eliminates Lagrangian multipliervariables and avoids explicit estimation of exact penalty parameters. Furthermore, we prove that the state solution of our algorithm achieves group consensus and converges to the saddle point of the minimax problem. Finally, numerical simulations demonstrate the effectiveness and superiority of the algorithm.
We present an algorithm for efficient evaluation of Boys functions F0,…,Fkmax tailored to modern computing architectures, in particular graphical processing units, where maximum throughput is high and data movement is costly. The method combines rational minimax approximations with upward and downward recurrence relations. The non-negative real axis is partitioned into three regions, [0, ∞⟩ = A ∪ B ∪ C, where regions A and B are treated using rational minimax approximations and region C by an asymptotic approximation. This formulation avoids lookup tables and irregular memory access, making it well-suited for hardware with high maximum throughput and low latency. The rational minimax coefficients are generated using the rational Remez algorithm. For a target maximum absolute error of ɛtol = 5 × 10-14, the corresponding approximation regions and coefficients for Boys functions F0, …, F32 are provided in Appendix D.
This paper considers minimax and adaptive transfer learning for nonparametric classification under the posterior drift model with distributed differential privacy constraints. Our study is conducted within a heterogeneous framework, encompassing diverse sample sizes, varying privacy parameters, and data heterogeneity across different servers. We first establish the minimax misclassification rate, precisely characterizing the effects of privacy constraints, source samples, and target samples on classification accuracy. The results reveal interesting phase transition phenomena and highlight the intricate trade-offs between preserving privacy and achieving classification accuracy. We then develop a data-driven adaptive classifier that achieves the optimal rate within a logarithmic factor across a large collection of parameter spaces while satisfying the same set of differential privacy constraints. Simulation studies and real-world data applications further elucidate the theoretical analysis with numerical results.
The primary challenge in earthquake emergency response is the effective dispatch of large-scale rescue teams to disaster areas after an earthquake disaster occurs, especially as this often involves time constraints and road network interruptions. To solve this, we propose a minimax difference submatrix (MDS) algorithm combined with k-means clustering. First, the k-means method is utilized to cluster the rescue locations with similar features, thereby transforming the non-standard assignment problem (rescue team ≠ rescue location) into a solvable linear assignment problem (LAP). After obtaining the cost matrix, the principle of minimax difference is established; that is, the columns with the largest differences between the maximum and minimum values are selected in sequence for priority assignment. By comparing the results of numerical experiments, it can be seen that the MDS method reduces the computational load effectively compared with the Hungarian algorithm, and can obtain a high-quality approximate optimal solution for the linear assignment problem. Finally, through the case study of simulating the IX-X earthquake intensity scenario in City T, it is demonstrated that the MDS method is an efficient and stable approach for solving the problem of large-scale real-time rescue in earthquake emergencies.
In this paper, we present a novel non-convex tensor completion model specifically tailored for multidimensional data. Our approach introduces a three-directional non-convex tensor rank surrogate regularized by the Minimax Concave Penalty (MCP) function. Crucially, the method processes data by simultaneously exploiting low-rank structures across its three modal directions, with the MCP function effectively mitigating the over-penalization of large singular values-a common drawback in convex nuclear norm minimization. To address the inherent challenges of this non-convex optimization, we develop an innovative approximate convex model that accurately captures the original formulation's essence. We then develop a robust convex Alternating Direction Method of Multipliers (ADMM)-based algorithm, supported by a rigorous convergence guarantee, ensuring both theoretical soundness and practical reliability. Extensive experiments on a variety of real-world datasets demonstrate the superior performance and robustness of the proposed method compared to state-of-the-art approaches.
When constructing models of the world, we aim for optimal compressions: models that include as few details as possible while remaining as accurate as possible. But which details-or features measured in data-should we choose to include in a model? Here, using the minimum description length principle, we show that the optimal features are the ones that produce the maximum entropy model with minimum entropy, thus yielding a minimax entropy principle. We review applications, which range from machine learning to optimal models of biological networks. Naive implementations, however, are limited to systems with small numbers of states and features. We therefore require new theoretical insights and computational techniques to construct optimal compressions of high-dimensional datasets arising in large-scale experiments.
Minimax optimization is gaining increasing attention in modern machine learning applications. Driven by large-scale models and massive volumes of data collected from edge devices, as well as the concern to preserve client privacy, distributed minimax optimization algorithms become popular, such as Local Stochastic Gradient Descent Ascent (Local-SGDA), and Local Decentralized SGDA (Local-DSGDA). While most existing research on distributed minimax algorithms focuses on convergence rates and communication efficiency, their generalization performance remains largely unexplored, whereas generalization ability is a pivotal indicator for evaluating the holistic performance of a model when fed with unknown data. In this paper, we propose the stability-based generalization analytical framework for Distributed-SGDA, which unifies two popular distributed minimax algorithms including Local-SGDA and Local-DSGDA, and conduct a comprehensive analysis of stability error, generalization gap, and population risk across different metrics under various settings, e.g., (S)C-(S)C, PL-SC, and NC-NC cases. Our theoretical results reveal the trade-off between the generalization gap and optimization error and suggest hyperparameters choice to obtain the optimal population risk. Numerical experiments for Local-SGDA and Local-DSGDA validate the theoretical results.
Large language models (LLMs) are increasingly used by patients to obtain health information. Postoperative rehabilitation after anterior cruciate ligament reconstruction (ACLR) has distinct phase boundaries and safety considerations. Therefore, responses should be not only clear and understandable, but also medically accurate, safe, and stage-fit. This study compared the performance of three publicly accessible LLMs in standardized post-ACLR rehabilitation question answering. This was a standardized, blinded, expert-rated comparative evaluation study. On a single prespecified data collection day in March 2026, 30 English-language rehabilitation questions were submitted separately to GPT-5.4, Doubao, and MiniMax-M2.7. The questions covered five postoperative rehabilitation phases. Responses were anonymized and randomly reordered before blinded rating by five orthopaedic clinicians across five domains: Accuracy, Safety, Stage-fit, Completeness, and Understandability. Paired non-parametric tests, effect size analyses, intraclass correlation coefficients, and linear mixed-effects modelling were used for statistical analysis. A total of 90 model-generated responses and 450 expert rating records were included. Overall scores differed significantly among the three models (Friedman χ² = 46.067, P < 0.001; Kendall's W = 0.768). GPT-5.4 achieved the highest overall score (4.61 ± 0.13), followed by MiniMax-M2.7 (4.53 ± 0.19), whereas Doubao had the lowest score (3.86 ± 0.29). GPT-5.4 performed best in Accuracy, Safety, and Stage-fit; MiniMax-M2.7 achieved the highest score for Completeness; and Doubao achieved the highest mean score for Understandability. Inter-rater agreement was good [ICC(3,k) = 0.893], and sensitivity analysis supported the primary findings. The three models showed distinct rating profiles in standardized single-turn post-ACLR rehabilitation question answering. Evaluation of patient-facing rehabilitation information should not rely solely on linguistic fluency, but should prioritize medical accuracy, safety, and Stage-fit. These findings provide preliminary benchmark evidence in a phase-sensitive rehabilitation setting, but they should not be interpreted as evidence supporting clinical implementation, clinician substitution, or patient benefit.
As a variant of the Area Under the ROC Curve (AUC), the partial AUC (PAUC) focuses on a specific range of false positive rate (FPR) and/or true positive rate (TPR) in the ROC curve. It is a pivotal evaluation metric in real-world scenarios with both class imbalance and decision constraints. However, selecting instances within these constrained intervals during its calculation is NP-hard, and thus typically requires approximation techniques for practical resolution. Despite the progress made in PAUC optimization over the last few years, most existing methods still suffer from uncontrollable approximation errors or a limited scalability when optimizing the approximate PAUC objectives. In this paper, we close the approximation gap of PAUC optimization by presenting two simple instance-wise minimax reformulations: one with an asymptotically vanishing gap, the other with the unbiasedness at the cost of more variables. Our key idea is to first establish an equivalent instance-wise problem to lower the time complexity, simplify the complicated sample selection procedure by threshold learning, and then apply different smoothing techniques. Equipped with an efficient solver, the resulting algorithms enjoy a linear per-iteration computational complexity w.r.t. the sample size and a convergence rate of $O(\epsilon ^{-1/3})$O(ε-1/3) for typical one-way and two-way PAUCs. Moreover, we provide a tight generalization bound of our minimax reformulations. The result explicitly demonstrates the impact of the TPR/FPR constraints $\alpha$α/$\beta$β on the generalization and exhibits a sharp order of $\tilde{O}(\alpha ^{-1}n_+^{-1} + \beta ^{-1}n_-^{-1})$O˜(α-1n+-1+β-1n--1). Finally, extensive experiments on several benchmark datasets validate the strength of our proposed methods.
The Information Maximizing Generative Adversarial Network (InfoGAN) can be formulated as a minimax problem involving a generator and a discriminator, augmented by a mutual information regularization term. Despite strong empirical performance, rigorous generalization guarantees for InfoGAN-type objectives remain limited, particularly when additional structural components are introduced. In this paper, we study an InfoGAN-inspired adversarial framework obtained by removing the latent code component and introducing an explicit regularization term on the generator, yielding an analytically tractable generator-regularized adversarial objective. We establish generalization error bounds by analyzing the gap between empirical and population objective functions using Rademacher complexity arguments for the discriminator, the generator, and their composition. The resulting bounds reveal explicit n -1/2 and m -1/2 decay rates with respect to the discriminator and generator sample sizes and clarify the role of the generator regularization parameter. The theory is further specialized to two-layer neural networks with Lipschitz continuous and non-decreasing activation functions, where explicit entropy-based complexity bounds are derived. Experiments on the CIFAR-10 dataset validate the predicted scaling behavior and demonstrate that the generalization gap decreases systematically as sample size increases, highlighting the stabilizing effect of generator regularization. Overall, this work provides one of the first rigorous generalization analyses for an InfoGAN-inspired adversarial objective with explicit generator regularization.
Classifiers trained on labeled source data may yield misleading results when applied to unlabeled target data drawn from a different distribution. Transfer learning can rectify this by transferring knowledge from source to target data, but its effectiveness frequently relies on stringent assumptions, such as label shift or strong separation conditions. We introduce a novel general conditional shift assumption, which encompasses label shift as a special case and facilitates the identifiability of both the target distribution and the shift function without requiring a separation condition. Our classifier is constructed by integrating deep neural networks (DNNs) and a pseudo-maximum likelihood approach. We establish asymptotic error bounds for our DNN-based classifier and estimators of the conditional probabilities ${ \boldsymbol{\eta }_{P}}$ for source data and the target label distribution $\boldsymbol{\pi }_{Q}$, in terms of the intrinsic dimension of ${ \boldsymbol{\eta }_{P}}$. Notably, the excess risk of the proposed classifier achieves the optimal minimax rate, up to a logarithmic factor. Our method not only eliminates the need to estimate the shift function, but also alleviates the curse of dimensionality when ${ \boldsymbol{\eta }_{P}}$ exhibits a low-dimensional structure. Numerical simulations, along with an analysis of an Alzheimer's disease dataset, underscore its exceptional performance.
Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw observations cannot be shared with a trusted central curator, and privacy must be enforced at the data source, which prevents the computation of exact CUSUM statistics. We therefore introduce a local differentially private (DP) variant called LDP-CUSUM, which first applies a local DP mechanism to transform the raw data into privatized observations and then applies a CUSUM procedure to detect the change. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the tradeoff among privacy level, false alarm rate, and detection efficiency. Numerical simulations and a real-data case study were conducted to demonstrate the detection efficiency of our proposed LDP-CUSUM across various scenarios.
Dataset distillation (DD) aims to synthesize a more compact dataset than the original one and models trained on it are expected to have the same generalization capabilities as on the original dataset. Previous work via a generative model (GM) faces several limitations. First, GM struggles to generate representative samples due to a lack of constraints. Second, it overlooks the relationships between generated samples, limiting its effectiveness. In this paper, a new noise-unconstrained GM-based DD framework is proposed. In the distillation stage, an adaptive matching coefficient is introduced to align generated images with representative class elements and the MiniMax loss function is extended to reduce the optimization difficulty. In the deployment stage, features among each generative image are ensembled by gradient-matching based DD. Theoretical analysis based on McDiarmid's inequality demonstrates that the proposed components can reduce the generalization error of the original baseline method. We also provide insights into the potential of generated images as an effective proxy dataset for DD. For example, on the ImageWoof dataset with 50 distilled images per class using a 6-layer ConvNet for evaluation, generated images outperform 25%, 50%, and 75% original images by 8.4%, 6.3%, and 8.3% in distillation performance. Our method effectively handles both low- and high-resolution datasets, with experiments on 11 benchmarks demonstrating its efficacy.
Radiomics, the extraction of features from medical images, has shown promise in cancer prognostics. However, its high-dimensional nature poses challenges for feature selection and model stability. Traditional least absolute shrinkage and selection operator (Lasso) tends to select only one feature from correlated groups, leading to unstable feature selection. This study aimed to compare group penalty models with Lasso in selecting radiomic features for cancer prognosis. We analyzed 590 lung adenocarcinoma lesions for predicting early-stage spread through air spaces (STAS) and 194 meningioma cases for tumor grade prediction. Various group penalty models, including group Lasso, group minimax concave penalty (MCP), group smoothly clipped absolute deviation (SCAD), and adaptive penalization regression with external covariates using variational Bayes (graper), were employed alongside Lasso. Features were organized into natural groups based on segmentation regions and mathematical properties. Model performance was assessed using 10-fold cross-validation, evaluating area under the curve (AUC) and feature selection stability through Jaccard Index. The best-performing models were validated on independent test sets. For lung adenocarcinomas, group SCAD achieved the highest cross-validation AUC of 0.804 [standard deviation (SD) =0.056] with Jaccard Index of 0.613, compared to Lasso's AUC of 0.776 (SD =0.059) and Jaccard Index of 0.503. In the test set, both models showed comparable performance (group SCAD: AUC =0.874; Lasso: AUC =0.877; P=0.896). For meningiomas, group Lasso achieved the highest cross-validation AUC of 0.816 (SD =0.143) with Jaccard Index of 0.7, versus Lasso's AUC of 0.743 (SD =0.223) and Jaccard Index of 0.41. Test set validation showed no significant difference (group Lasso: AUC =0.877; Lasso: AUC =0.835; P=0.391). Group penalty models demonstrated superior feature selection stability while maintaining comparable predictive performance to Lasso. By selecting biologically meaningful feature groups rather than individual features, these models enhance interpretability and align better with clinical reasoning, offering a robust framework for radiomics-based cancer prognostics.
Federated learning (FL) is a promising framework that models distributed machine learning while protecting the privacy of clients. However, FL suffers performance degradation from heterogeneous and limited data. To alleviate the degradation, we present a novel personalized Bayesian FL approach named pFedBayes. By using the trained global distribution from the server as the prior distribution of each client, each client adjusts its own distribution by minimizing the sum of the reconstruction error over its personalized data and the KL divergence with the downloaded global distribution. Then, we propose a sparse personalized Bayesian FL approach named sFedBayes to enhance the inference efficiency. To overcome the extreme heterogeneity in non-i.i.d. data, we propose a clustered Bayesian FL model named cFedbayes by learning different prior distributions for different clients. Theoretical analysis gives the generalization error bound of three approaches and shows that the generalization error rates of the proposed approaches achieve minimax optimality up to a logarithmic factor. Moreover, cFedBayes achieves a cluster-level generalization error bound, rather than a single uniform bound in pFedBayes. Numerous experiments demonstrate that the proposed approaches have better performance than other advanced personalized methods on private models in the presence of heterogeneous and limited data.
We propose an optimal phase II design for double-arm two-stage clinical trials using the restricted mean survival time (RMST) to measure the between-arm difference. In the superiority test of the proposed design, the null hypothesis is rejected when the between-arm RMST difference R ^ E - R ^ C > m $$ {\hat{R}}_E-{\hat{R}}_C>m $$ , and the RMST in the experimental arm R ^ E > q $$ {\hat{R}}_E>q $$ , where the critical values m $$ m $$ and q $$ q $$ are determined by an adaptive probability cutoff function with accumulated sample size at each stage and the asymptotic normality of RMST values. Compared to rejection rule based on R ^ E - R ^ C > m $$ {\hat{R}}_E-{\hat{R}}_C>m $$ only, the sculpted critical region requires smaller sample size based on the same type I error and power level. In the two-stage Minimax and Optimal design, simulations demonstrate that the Sculpted RMST approach offers lower total sample size, earlier interim time (smaller interim sample size under constant accrual rate), and smaller expected sample sizes compared to the log-rank test and simple RMST difference test. Besides, the method can be easily extended to multi-stage sequential design based on the adaptive probability cutoff function. The global robustness of type I error deviation when the survival parameter drift from the assumption is discussed. Implementation of the proposed design on real world trial data is also provided. The R package ScuRMST for implementing the design is now available on Github.
Distributed statistical modeling is a powerful tool for dealing with large-scale datasets while maintaining data privacy. In this study, we propose a data-driven weighted aggregation procedure that leverages model prediction performance and is adaptable to heterogeneous distributed environments. The proposed procedures utilize the squared prediction error matrix as the main transmitted quantity, with its dimension being the square of the number of workers, ensuring communication efficiency. We show that the proposed estimates have asymptotical optimal weights in terms of quadratic loss and corresponding risk. The limits of data-driven weights are also derived. We also study the minimax property of the proposed nonparametric function estimates. To examine the finite sample performance of the proposed procedure, we conduct Monte Carlo simulation studies. Furthermore, we illustrate the proposed methodology via an empirical analysis of a real-world dataset on heart rate prediction.
Arm D of the AcSé-ESMART proof-of-concept phase I/II platform trial aimed to define the recommended phase II dose (RP2D), pharmacokinetics, activity, and biomarkers of the PARP inhibitor olaparib with irinotecan in pediatric patients with recurrent/refractory malignancies. Olaparib was administered orally twice daily on days 1 to 10 and irinotecan intravenously on days 4 to 8 of a 21-day cycle. Dose escalation followed the continuous reassessment method; activity was assessed in diverse tumor types (cohort 1) and Ewing sarcoma (cohort 2) according to a minimax Simon 2-stage design. Cohorts were enriched for alterations in homologous recombination repair (HRR) pathways. Seventy patients (median age, 14.9 years; range, 5.0-23.8) were included, 34 with diverse tumor types (25 with HRR gene alterations) and 36 with Ewing sarcoma. Sixty-six patients received 348 treatment cycles (median, 2; range, 1-51) over four dose levels. Main toxicities were gastrointestinal and myelosuppression; the RP2D was olaparib 90 mg/m2 twice daily and irinotecan 20 mg/m2/day. Olaparib exposure in children was equivalent to that in adults. The overall response rate was 9.1% (cohort 1, 11.8%; cohort 2, 6.3%). Four patients with osteosarcoma, pineoblastoma, choroid plexus carcinoma, and neuroblastoma experienced a partial response and were treated for nine to 51 cycles. Two patients with Ewing sarcoma experienced a complete and a partial response for 10 and 42 cycles, respectively. Genetic analyses suggest a high aneuploidy score possibly associated with objective response and prolonged stable disease. Olaparib combined with irinotecan demonstrated activity in pediatric tumors, which was enriched among tumors that exhibited aneuploidy.
Deep learning (DL) supervised techniques have been extensively employed in magnetic resonance imaging (MRI) reconstruction, delivering notable performance enhancements over traditional non-DL methods. Nonetheless, these models have vulnerabilities during testing such as their susceptibility to worst-case or noise-based measurement perturbations, variations in training/testing settings like acceleration factors, contrast, $k$ -space sampling locations, and distribution shifts stemming from unseen lesions and different anatomies. This article addresses these robustness challenges by leveraging diffusion models (DMs). In particular, we present a robustification strategy that improves the resilience of DL-based MRI reconstruction methods by utilizing pretrained DMs as purifiers. We dub our method as robust DL-based MRI with diffusion purification (RODIO). In contrast to conventional robustification methods for DL-based MRI reconstruction, such as adversarial training (AT), our proposed approach eliminates the need to tackle a minimax optimization problem. It only necessitates efficient fine-tuning on purified examples. Our experimental results underscore the effectiveness of our approach in addressing the mentioned instabilities, outperforming standalone diffusion-based MRI reconstructors and leading robustification methods for deep supervised MRI reconstruction, including AT and randomized smoothing (RS). Our experiments demonstrate: 1) the adaptability of our approach across multiple DL-based supervised MRI reconstruction models; 2) compatibility with accelerated diffusion-based samplers; 3) robustness to data with unseen lesions; and 4) effectiveness when applied to unsupervised single-shot generative reconstructors.
Second-line treatment for metastatic colorectal cancer (mCRC) typically involves oxaliplatin- or irinotecan-based doublet chemotherapy with or without anti-angiogenic antibodies. Triplet regimens such as FOLFOXIRI have demonstrated synergy and improved efficacy as first-line therapy. Surufatinib, an oral multi-kinase inhibitor targeting VEGFR1-3, FGFR1, and CSF-1R, may enhance chemotherapy efficacy. We evaluated surufatinib combined with doublet (FOLFOX/FOLFIRI) versus triplet (FOLFOXIRI) chemotherapy as second-line treatment for mCRC. This multicentre, open-label, randomized phase-II trial used Simon's minimax two-stage design. Eligible patients had mCRC progressing on or within 6 months after first-line doublet chemotherapy. Patients were randomized 1:1 to surufatinib 250 mg once daily plus either mFOLFOX6/FOLFIRI (doublet cohort, selected based on prior regimen) or FOLFOXIRI (triplet cohort). The primary endpoint was objective response rate (ORR). From September 2021 to November 2023, 57 patients were randomized (28 per cohort after one withdrawal). In the doublet cohort, ORR was 35.7% (95% CI: 18.6-55.9), median progression-free survival (PFS) was 5.4 months (95% CI: 3.8-7.0), and median overall survival (OS) was 19.0 months (95% CI: 9.2-28.8). In the triplet cohort, ORR was 39.3% (95% CI: 21.5-59.4), median PFS was 5.8 months (95% CI: 3.3-8.2), and median OS was 10.9 months (95% CI: 6.0-15.8). Grade ≥3 treatment-emergent adverse events occurred more frequently in the triplet (71.4%) versus doublet (57.1%) cohort, with higher rates of treatment delays (89.3% versus 72.0%) and discontinuations (25.0% versus 14.3%). Surufatinib plus doublet chemotherapy showed encouraging antitumor activity and acceptable tolerability in second-line mCRC, warranting further evaluation in a larger randomized trial. In contrast, surufatinib plus triplet chemotherapy was associated with increased toxicity, more frequent treatment delays or discontinuations, and shorter overall survival; this combination is not recommended for further investigation in this setting.ClinicalTrials.gov: NCT04734249Date of registration: January 31, 2021. Surufatinib plus doublet chemotherapy achieved promising efficacy and tolerability in second-line metastatic colorectal cancer, with ORR 35.7%, median PFS 5.4 months, and median OS 19.0 months.Surufatinib plus triplet chemotherapy (FOLFOXIRI) offered no clear advantage over the doublet regimen and was limited by greater toxicity and inferior overall survival.