Adaptive Sampling

Training neural networks to satisfy universal constraints over continuous domains poses unique challenges. Common examples include Lyapunov Neural Networks (Lyapunov NNs) and Physics-Informed Neural Networks (PINNs), where analytical solutions are generally either unavailable or overly restrictive. Sample-based methods are therefore commonly used to enforce these constraints, and the choice of samples has a substantial impact on convergence speed, stability, and solution quality. Most existing methods rely on fixed heuristics or handcrafted rules, and are suboptimal in practice. In this paper, we aim to improve upon them by learning, from data and experience, how to dynamically and iteratively adjust the samples in response to the model's evolving learning performance. Trained by reinforcement learning, the learned policy improves empirical constraint satisfaction on test problems while significantly improving efficiency. We validate the approach on both Lyapunov NNs and PINNs, and demonstrate its broader applicability to domains where adaptive input selection is essential for effective training.

Batch selection is crucial for improving both training efficiency and predictive performance in deep multi-label classification (MLC). Existing batch selection methods typically rely on a single metric to assess instance importance and use static label weights to distinguish label significance, neglecting the dynamic evolution of metric utility and label significance during training. In addition, the method that explicitly exploits label correlations is largely affected by abundant irrelevant labels and insensitive to local label distributions. To address these issues, we propose D2ACE, a novel multi-label batch selection method guided by Dual Dynamics and Adaptive Correlation Enhancement. D2ACE explicitly captures metric and label-level training dynamics by combining stage-wise Bernoulli mixture sampling, which balances uncertainty and noise-resistant hardness, with dynamic label weighting to recalibrate label priorities at each epoch based on current metric statistics. Furthermore, D2ACE introduces a local context-aware correlation enhancement to focus on relevant labels with instance-adaptive dependencies. Extensive experiments on tabular and image benchmarks demonstrate that D2ACE outperforms existing batch selection approaches across various deep MLC models, achieving stronger predictive performance and more efficient correlation modeling.

Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\rightarrow$ 16M $\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$\times$ and the corresponding MSE by up to 13$\times$. These results highlight data distribution as a key factor in operator learning and position M$^3$ as a scalable, data-efficient approach for physically consistent modeling. Code is available at https://github.com/PhysDataRefine/M3.

As an extension of the classical multi-armed bandit problem, multi-fidelity multi-armed bandits (MF-MAB) enable individual arms to be evaluated using diverse feedback sources that vary in both cost and accuracy. Prior stochastic models typically assume fixed low-to-high fidelity discrepancies, whereas modern proxy sources, such as learning-based simulators and Large Language Models (LLMs), can be improved using additional calibration. We investigate adaptive MF-MAB with improving proxy sources, and focus on the canonical two-fidelity case in which the low-fidelity source becomes more informative with repeated use. To capture this dynamic, we introduce a selected-average mismatch bound that converts dynamic low-fidelity observations into improvement-aware confidence bounds for the high-fidelity target. We propose the Threshold-Based Adaptive Continuation Companion (TACC), an optimistic algorithm that uses a bounded continuation rule to decide when low-fidelity sampling remains cost-effective and when to escalate. We prove an instance-dependent regret bound showing that, for detected intermediate arms, adaptive continuation replaces logarithmic high-fidelity confirmation with bounded low-fidelity continuation. Experiments on synthetic bandits and an LLM-as-a-judge policy-evaluation task examine when continuation improves cost-weighted regret.

Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential operators, where non-normality can lead to significant transient amplification and sensitivity to perturbations that are not captured by eigenvalue analysis alone. At large scales, commonly used numerical approaches for pseudospectra computation can become computationally demanding, as they require repeated auxiliary computations to identify spectrally sensitive regions in the complex plane. We present a neural network-based approach that predicts sensitive regions directly from matrix features, thereby avoiding exhaustive pseudospectra evaluation across the entire complex plane. We calibrate the prediction threshold on validation data to ensure reliable coverage of sensitive regions. The trained neural network guides the selection of grid points requiring full computation, enabling focused computation only where necessary. The approach provides a practical preprocessing strategy for efficient pseudospectra computation. Numerical experiments on non-normal banded matrices demonstrate substantial speedup compared to full grid-based numerical evaluation while maintaining high accuracy in identifying sensitive regions.

Standard diffusion models for graph generation typically rely on uniform time-stepping, an approach that overlooks the non-homogeneous dynamics of distributional evolution on complex manifolds. In this paper, we present an information-geometric framework that reinterprets the diffusion sampling trajectory as a parametric curve on a Riemannian manifold. Our key observation is that the Fisher-Rao metric provides a principled measure of the intrinsic distance. By analyzing this metric, we derive the Drift Variation Score (DVS), a geometry-aware indicator that quantifies the instantaneous rate of distributional change. Unlike prior heuristic-based adaptive samplers, our DVS solver enforces a constant informational speed on the statistical manifold, automatically maintaining a uniform rate of distributional change along the sampling trajectory. This equal arc-length strategy ensures that each discretization step contributes equally to the information speed. Theoretical analysis verifies that DVS characterizes the local stiffness of the sampling dynamics in the Fisher-Rao sense. Experimental results on molecule and social network generation show that DVS significantly improves structural fidelity and sampling efficiency. Code is at https://github.com/kunzhan/DVS

We introduce surrogate functionals: machine-learned energy functionals for orbital-free density functional theory (OF-DFT) which are defined not by universal fidelity to a physical reference, but merely by the requirement that density optimization with a fixed procedure yields the true ground-state density. Helpfully, training surrogate functionals requires only ground-state densities, no energies or gradients away from the ground state. We here propose a gradient-descent-improvement loss that guarantees exponential convergence of the density to the ground state, and combine it with an adaptive sampling scheme that concentrates learning around the optimization trajectories actually visited during inference. On the QM9 and QMugs benchmarks, surrogate functionals achieve density errors competitive with or improving upon the state of the art for fully supervised machine-learned OF-DFT, while eliminating the need for the $O(N^3)$ orthononormalization step required by prior work, yielding improved runtime scaling for larger systems.

Safety filters provide a practical approach for enforcing safety constraints in autonomous systems. While learning-based tools scale to high-dimensional systems, their performance depends on informative data that includes states likely to lead to constraint violation, which can be difficult to efficiently sample in complex, high-dimensional systems. In this work, we characterize trajectories that barely avoid safety violations using the Pontryagin Maximum Principle. These boundary trajectories are used to guide data collection for learned Hamilton-Jacobi Reachability, concentrating learning efforts near safety-critical states to improve efficiency. The learned Control Barrier Value Function is then used directly for safety filtering. Simulations and experimental validation on a shared-control automotive racing application demonstrate PMP sampling improves learning efficiency, yielding faster convergence, reduced failure rates, and improved safe set reconstruction, with wall times around 3ms.

Deep neural networks are typically trained by uniformly sampling large datasets across epochs, despite evidence that not all samples contribute equally throughout learning. Recent work shows that progressively reducing the amount of training data can improve efficiency and generalization, but existing methods rely on fixed schedules that do not adapt during training. In this work, we propose Adaptive Data Dropout, a simple framework that dynamically adjusts the subset of training data based on performance feedback. Inspired by self-regulated learning, our approach treats data selection as an adaptive process, increasing or decreasing data exposure in response to changes in training accuracy. We introduce a lightweight stochastic update mechanism that modulates the dropout schedule online, allowing the model to balance exploration and consolidation over time. Experiments on standard image classification benchmarks show that our method reduces effective training steps while maintaining competitive accuracy compared to static data dropout strategies. These results highlight adaptive data selection as a promising direction for efficient and robust training. Code will be released.

Task-adapted compressed sensing magnetic resonance imaging (CS-MRI) is emerging to address the specific demands of downstream clinical tasks with significantly fewer k-space measurements than required by Nyquist sampling. However, existing task-adapted CS-MRI methods suffer from the uncertainty problem for medical diagnosis and cannot achieve adaptive sampling in end-to-end optimization with reconstruction or clinical tasks. To address these limitations, we propose the first task-adapted CS-MRI from the information-theoretic perspective to simultaneously achieve probabilistic inference for uncertainty prediction and adapt to arbitrary sampling ratios and versatile clinical applications. Specifically, we formalize the task-adapted CS-MRI optimization problem by maximizing the mutual information between undersampled k-space measurements and clinical tasks to enable probabilistic inference for addressing the uncertainty problem. We leverage amortized optimization and construct tractable variational bounds for mutual information to jointly optimize sampling, reconstruction, and task-inference models, which enables flexible sampling ratio control using a single end-to-end trained model. Furthermore, the proposed framework addresses two kinds of distinct clinical scenarios within a unified approach, i.e., i) joint task and reconstruction, where reconstruction serves as an auxiliary process to enhance task performance; and ii) task implementation with suppressed reconstruction, applicable for privacy protection. Extensive experiments on large-scale MRI datasets demonstrate that the proposed framework achieves highly competitive performance on standard metrics like Dice compared to deterministic counterpart but provides better distribution matching to the ground-truth posterior distribution as measured by the generalized energy distance (GED).

Standard resampling ratios (e.g., $α\approx 0.632$) are widely used as default baselines in ensemble learning for three decades. However, how these ratios interact with a base learner's intrinsic functional complexity in finite samples lacks a exact mathematical characterization. We leverage the Hoeffding-ANOVA decomposition to derive the first exact, finite-sample variance decomposition for subagging, applicable to any symmetric base learner without requiring asymptotic limits or smoothness assumptions. We establish that subagging operates as a deterministic low-pass spectral filter: it preserves low-order structural signals while attenuating $c$-th order interaction variance by a geometric factor approaching $α^c$. This decoupling reveals why default baselines often under-regularize high-capacity interpolators, which instead require smaller $α$ to exponentially suppress spurious high-order noise. To operationalize these insights, we propose a complexity-guided adaptive subsampling algorithm, empirically demonstrating that dynamically calibrating $α$ to the learner's complexity spectrum consistently improves generalization over static baselines.

Edge-based multimodal medical monitoring requires models that balance diagnostic accuracy with severe energy constraints. Continuous acquisition of ECG, PPG, EMG, and IMU streams rapidly drains wearable batteries, often limiting operation to under 10 hours, while existing systems overlook the high temporal redundancy present in physiological signals. We introduce Adaptive Multimodal Intelligence (AMI), an end-to-end framework that jointly learns when to sense and how to infer. AMI integrates three components: (1) a lightweight Agentic Modality Controller that uses differentiable Gumbel-Sigmoid gating to dynamically select active sensors based on model confidence and task relevance; (2) a Learned Sigma-Delta Sensing module that applies patch-wise Delta-Sigma operations with learnable thresholds to skip temporally redundant samples; and (3) a Foundation-backed Multimodal Prediction Model built on unimodal foundation encoders and a cross-modal transformer with temporal context, enabling robust fusion even under gated or missing inputs. These components are trained jointly via a multi-objective loss combining classification accuracy, sparsity regularization, cross-modal alignment, and predictive coding. AMI is hardware-aware, supporting dynamic computation graphs and masked operations, leading to real energy and latency savings. Across MHEALTH, HMC Sleep, and WESAD datasets, it reduces sensor usage by 48.8% while improving state-of-the-art accuracy by 1.9% on average.

In Bayesian optimization, Thompson sampling selects the evaluation point by sampling from the posterior distribution over the objective function maximizer. Because this sampling problem is intractable for Gaussian process (GP) surrogates, the posterior distribution is typically restricted to fixed discretizations (i.e., candidate points) that become exponentially sparse as dimensionality increases. While previous works aim to increase candidate point density through scalable GP approximations, our orthogonal approach increases density by adaptively reducing the search space during sampling. Specifically, we introduce Adaptive Candidate Thompson Sampling (ACTS), which generates candidate points in subspaces guided by the gradient of a surrogate model sample. ACTS is a simple drop-in replacement for existing TS methods -- including those that use trust regions or other local approximations -- producing better samples of maxima and improved optimization across synthetic and real-world benchmarks.

The Shapley value, and its broader family of semi-values, has received much attention in various attribution problems. A fundamental and long-standing challenge is their efficient approximation, since exact computation generally requires an exponential number of utility queries in the number of players $n$. To meet the challenges of large-scale applications, we explore the limits of efficiently approximating semi-values under a $Θ(n)$ space constraint. Building upon a vector concentration inequality, we establish a theoretical framework that enables sharper query complexities for existing unbiased randomized algorithms. Within this framework, we systematically develop a linear-space algorithm that requires $O(\frac{n}{ε^{2}}\log\frac{1}δ)$ utility queries to ensure $P(\|\hat{\boldsymbolφ}-\boldsymbolφ\|_{2}\geqε)\leq δ$ for all commonly used semi-values. In particular, our framework naturally bridges OFA, unbiased kernelSHAP, SHAP-IQ and the regression-adjusted approach, and definitively characterizes when paired sampling is beneficial. Moreover, our algorithm allows explicit minimization of the mean squared error $\mathbb{E}[\|\hat{\boldsymbolφ}-\boldsymbolφ\|_{2}^{2}]$ for each specific utility function. Accordingly, we introduce the first adaptive, linear-time, linear-space randomized algorithm, Adalina, that theoretically achieves improved mean squared error. All of our theoretical findings are experimentally validated. Our code is available at https://github.com/watml/adalina.

In this paper, we study the problem of mean estimation under 1-bit communication constraints. We propose a novel adaptive mean estimator based solely on randomized threshold queries, where each 1-bit outcome indicates whether a given sample exceeds a sequentially chosen threshold. Our estimator is $(ε, δ)$-PAC for any distribution with a bounded mean $μ\in [-λ, λ]$ and a bounded $k$-th central moment $\mathbb{E}[|X-μ|^k] \le σ^k$ for any fixed $k > 1$. Moreover, our sample complexity is order-optimal in all such tail regimes, i.e., for every such $k$ value. For $k \neq 2$, our estimator's sample complexity matches the unquantized minimax lower bounds plus an unavoidable $O(\log(λ/σ))$ localization cost. For the finite-variance case ($k=2$), our estimator's sample complexity has an extra multiplicative $O(\log(σ/ε))$ penalty, and we establish a novel information-theoretic lower bound showing that this penalty is a fundamental limit of 1-bit quantization. We also establish a significant adaptivity gap: for both threshold queries and more general interval queries, the sample complexity of any non-adaptive estimator must scale linearly with the search space parameter $λ/σ$, rendering it vastly less sample efficient than our adaptive approach. Finally, we present algorithmic variants that (i) handle an unknown sampling budget, (ii) adapt to an unknown scale parameter $σ$ given (possibly loose) bounds, (iii) require only two stages of adaptivity to achieve order-optimal sample complexity at the expense of more general 1-bit queries, and (iv) leverage multiple local samples per 1-bit query to proportionally reduce communication costs.

High-dimensional hyperspectral imaging (HSI) enables the visualization of ultrafast molecular dynamics and complex, heterogeneous spectra. However, applying this capability to resolve spatially varying vibrational couplings in two-dimensional infrared (2DIR) spectroscopy, a type of coherent multidimensional spectroscopy (CMDS), necessitates prohibitively long data acquisition, driven by dense Nyquist sampling requirements and the need for extensive signal accumulation. To address this challenge, we introduce a physics-informed neural representation approach that efficiently reconstructs dense spatially-resolved 2DIR hyperspectral images from sparse experimental measurements. In particular, we used a multilayer perceptron (MLP) to model the relationship between the sub-sampled 4D coordinates and their corresponding spectral intensities, and recover densely sampled 4D spectra from limited observations. The reconstruction results demonstrate that our method, using a fraction of the samples, faithfully recovers both oscillatory and non-oscillatory spectral dynamics in experimental measurement. Moreover, we develop a loss-aware adaptive sampling method to progressively introduce potentially informative samples for iterative data collection while conducting experiments. Experimental results show that the proposed approach achieves high-fidelity spectral recovery using only $1/32$ of the sampling budget, as opposed to exhaustive sampling, effectively reducing total experiment time by up to 32-fold. This framework offers a scalable solution for accelerating any experiments with hypercube data, including multidimensional spectroscopy and hyperspectral imaging, paving the way for rapid chemical imaging of transient biological and material systems.

Multi-channel sensor networks in industrial IoT often exceed available bandwidth. We propose PCA-Triage, a streaming algorithm that converts incremental PCA loadings into proportional per-channel sampling rates under a bandwidth budget. PCA-Triage runs in O(wdk) time with zero trainable parameters (0.67 ms per decision). We evaluate on 7 benchmarks (8--82 channels) against 9 baselines. PCA-Triage is the best unsupervised method on 3 of 6 datasets at 50% bandwidth, winning 5 of 6 against every baseline with large effect sizes (r = 0.71--0.91). On TEP, it achieves F1 = 0.961 +/- 0.001 -- within 0.1% of full-data performance -- while maintaining F1 > 0.90 at 30% budget. Targeted extensions push F1 to 0.970. The algorithm is robust to packet loss and sensor noise (3.7--4.8% degradation under combined worst-case).

With the rapid advancements in deep learning techniques, wearable sensor-aided animal activity recognition (AAR) has demonstrated promising performance, thereby improving livestock management efficiency as well as animal health and welfare monitoring. However, existing research often prioritizes overall performance, overlooking the fact that classification accuracies for specific animal behavioral categories may remain unsatisfactory. This issue typically stems from suboptimal sampling rates or class imbalance problems. To address these challenges and achieve high classification accuracy across all individual behaviors in farm animals, we propose a novel Individual-Behavior-Aware Network (IBA-Net). This network enhances the recognition of each specific behavior by simultaneously customizing features and calibrating the classifier. Specifically, considering that different behaviors require varying sampling rates to achieve optimal performance, we design a Mixture-of-Experts (MoE)-based Feature Customization (MFC) module. This module adaptively fuses data from multiple sampling rates, capturing customized features tailored to various animal behaviors. Additionally, to mitigate classifier bias toward majority classes caused by class imbalance, we develop a Neural Collapse-driven Classifier Calibration (NC3) module. This module introduces a fixed equiangular tight frame (ETF) classifier during the classification stage, maximizing the angles between pair-wise classifier vectors and thereby improving the classification performance for minority classes. To validate the effectiveness of IBA-Net, we conducted experiments on three public datasets covering goat, cattle, and horse activity recognition. The results demonstrate that our method consistently outperforms existing approaches across all datasets.

The increasing capabilities of machine learning models, such as vision-language and multimodal language models, are placing growing demands on data in automotive systems engineering, making the quality and relevance of collected data enablers for the development and validation of such systems. Traditional Big Data approaches focus on large-scale data collection and offline processing, while Smart Data approaches improve data selection strategies but still rely on centralized and offline post-processing. This paper introduces the concept of Fast Data for automotive systems engineering. The approach shifts data selection and recording onto the vehicle as the data source. By enabling real-time, context-aware decisions on whether and which data should be recorded, data collection can be directly aligned with data quality objectives and collection strategies within a closed-loop. This results in datasets with higher relevance, improved coverage of critical scenarios, and increased information density, while at the same time reducing irrelevant data and associated costs. The proposed approach provides a structured foundation for designing data collection strategies that are aligned with the needs of modern machine learning algorithms. It supports efficient data acquisition and contributes to scalable and cost-effective ML development processes in automotive systems engineering.

Nonconvex multi-well energies in cell-induced phase transitions give rise to fine-scale microstructures, low-regularity transition layers and sharp interfaces, all of which pose numerical challenges for physics-informed learning. Here we introduce biomimetic physics-informed neural networks (Bio-PINNs), which implement a near-to-far curriculum by progressively revealing the computational domain away from the cell boundary and combining this schedule with a deformation-uncertainty proxy that concentrates collocation points near evolving transition layers and tether-forming regions. Across single-cell and multicellular benchmarks, Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual-driven adaptive baselines.

Large language models (LLMs) have achieved remarkable success in various natural language processing tasks, yet they remain prone to generating factually incorrect outputs known as hallucinations. While recent approaches have shown promise for hallucination detection by repeatedly sampling from LLMs and quantifying the semantic inconsistency among the generated responses, they rely on fixed sampling budgets that fail to adapt to query complexity, resulting in computational inefficiency. We propose an Adaptive Bayesian Estimation framework for Semantic Entropy with Guided Semantic Exploration, which dynamically adjusts sampling requirements based on observed uncertainty. Our approach employs a hierarchical Bayesian framework to model the semantic distribution, enabling dynamic control of sampling iterations through variance-based thresholds that terminate generation once sufficient certainty is achieved. We also develop a perturbation-based importance sampling strategy to systematically explore the semantic space. Extensive experiments on four QA datasets demonstrate that our method achieves superior hallucination detection performance with significant efficiency gains. In low-budget scenarios, our approach requires about 50% fewer samples to achieve comparable detection performance to existing methods, while delivers an average AUROC improvement of 12.6% under the same sampling budget.

Current network data telemetry pipelines consist of massive streams of fine-grained Key Performance Indicators (KPIs) from multiple distributed sources towards central aggregators, making data storage, transmission, and real-time analysis increasingly unsustainable. This work presents a generative AI (GenAI)-driven sampling and hybrid compression framework that redesigns network telemetry from a goal-oriented perspective. Unlike conventional approaches that passively compress fully observed data, our approach jointly optimizes what to observe and how to encode it, guided by the relevance of information to downstream tasks. The framework integrates adaptive sampling policies, using adaptive masking techniques, with generative modeling to identify patterns and preserve critical features across temporal and spatial dimensions. The selectively acquired data are further processed through a hybrid compression scheme that combines traditional lossless coding with GenAI-driven, lossy compression. Experimental results on real network datasets demonstrate over 50$\%$ reductions in sampling and data transfer costs, while maintaining comparable reconstruction accuracy and goal-oriented analytical fidelity in downstream tasks.

Researchers train neural simulators on uniformly sampled numerical simulation data. But under the same budget, does systematically sampled data provide the most effective information? A fundamental yet unformalized problem is how to sample training data for neural simulators so as to maximize rollout accuracy. Existing data sampling methods either tend to collapse into locally high-information-density regions, or preserve diversity but remain insufficiently model-specific, often leading to performance that is no better than uniform sampling. To address this, we propose a data sampling method tailored to neural simulators, Gradient-Informed Temporal Sampling (GITS). GITS jointly optimizes pilot-model local gradients and set-level temporal coverage, thereby effectively balancing model specificity and dynamical information. Compared with multiple sampling baselines, the data selected by GITS achieves lower rollout error across multiple PDE systems, model backbones and sample ratios. Furthermore, ablation studies demonstrate the necessity and complementarity of the two optimization objectives in GITS. In addition, we analyze the successful sampling patterns of GITS as well as the typical PDE systems and model backbones on which GITS fails.

Database Management Systems (DBMS) are crucial for efficient data management and access control, but their administration remains challenging for Database Administrators (DBAs). Tuning, in particular, is known to be difficult. Modern systems have many tuning parameters, but only a subset significantly impacts performance. Focusing on these influential parameters reduces the search space and optimizes performance. Current methods rely on costly warm-up phases and human expertise to identify important tuning parameters. In this paper, we present DOT, a dynamic knob selection and online sampling DBMS tuning algorithm. DOT uses Recursive Feature Elimination with Cross-Validation (RFECV) to prune low-importance tuning parameters and a Likelihood Ratio Test (LRT) strategy to balance exploration and exploitation. For parameter search, DOT uses a Bayesian Optimization (BO) algorithm to optimize configurations on-the-fly, eliminating the need for warm-up phases or prior knowledge (although existing knowledge can be incorporated). Experiments show that DOT achieves matching or outperforming performance compared to state-of-the-art tuners while substantially reducing tuning overhead.

Probabilistic language generators are theoretically modeled as discrete stochastic processes, yet standard decoding strategies (Top-k, Top-p) impose static truncation rules that fail to accommodate the dynamic information density of natural language. This misalignment often forces a suboptimal trade-off: static bounds are either too restrictive for high-entropy creative generation or too permissive for low-entropy logical reasoning. In this work, we formalize the generation process as a trajectory through a relative probability manifold. We introduce Top-b (Adaptive Relative Band Sampling), a decoding strategy that regulates the candidate set via a dynamic bandwidth coefficient coupled strictly to the instantaneous Shannon entropy of the model's distribution. We provide a theoretical framework demonstrating that Top-b acts as a variance-minimizing operator on the tail distribution. Empirical validation on GPQA and GSM8K benchmarks indicates that Top-b significantly reduces generation entropy and inter-decoding variance while maintaining competitive reasoning accuracy, effectively approximating a self-regulating control system for autoregressive generation.

Surrogate models provide compact relations between user-defined input parameters and output quantities of interest, enabling the efficient evaluation of complex parametric systems in many-query settings. Such capabilities are essential in a wide range of applications, including optimisation, control, data assimilation, uncertainty quantification, and emerging digital twin technologies in various fields such as manufacturing, personalised healthcare, smart cities, and sustainability. This article reviews established methodologies for constructing surrogate models exploiting either knowledge of the governing laws and the dynamical structure of the system (physics-based) or experimental observations (data-driven), as well as hybrid approaches combining these two paradigms. By revisiting the design of a surrogate model as a functional approximation problem, existing methodologies are reviewed in terms of the choice of (i) a reduced basis and (ii) a suitable approximation criterion. The paper reviews methodologies pertaining to the field of Scientific Machine Learning, and it aims at synthesising established knowledge, recent advances, and new perspectives on: dimensionality reduction, physics-based, and data-driven surrogate modelling based on proper orthogonal decomposition, proper generalised decomposition, and artificial neural networks; multi-fidelity methods to exploit information from sources with different fidelities; adaptive sampling, enrichment, and data augmentation techniques to enhance the quality of surrogate models.

In Reinforcement Learning with Verifiable Rewards (RLVR), constructing a robust advantage baseline is critical for policy gradients, effectively guiding the policy model to reinforce desired behaviors. Recent research has introduced Generalist Value Models (such as $V_0$), which achieve pre-trained value estimation by explicitly encoding model capabilities in-context, eliminating the need to synchronously update the value model alongside the policy model. In this paper, we propose $V_{0.5}$, which adaptively fuses the baseline predicted by such value model (acting as a prior) with the empirical mean derived from sparse rollouts. This constructs a robust baseline that balances computational efficiency with extremely low variance. Specifically, we introduce a real-time statistical testing and dynamic budget allocation. This balances the high variance caused by sparse sampling against the systematic bias (or hallucinations) inherent in the value model's prior. By constructing a hypothesis test to evaluate the prior's reliability in real-time, the system dynamically allocates additional rollout budget on demand. This mechanism minimizes the baseline estimator's Mean Squared Error (MSE), guaranteeing stable policy gradients, even under extreme sparsity with a group size of 4. Extensive evaluations across six mathematical reasoning benchmarks demonstrate that $V_{0.5}$ significantly outperforms GRPO and DAPO, achieving faster convergence and over some 10% performance improvement.

Active learning for regression reduces labeling costs by selecting the most informative samples. Improved Greedy Sampling is a prominent method that balances feature-space diversity and output-space uncertainty using a static, multiplicative rule. We propose Weighted improved Greedy Sampling (WiGS), which replaces this framework with a dynamic, additive criterion. We formulate weight selection as a reinforcement learning problem, enabling an agent to adapt the exploration-investigation balance throughout learning. Experiments on 18 benchmark datasets and a synthetic environment show WiGS outperforms iGS and other baseline methods in both accuracy and labeling efficiency, particularly in domains with irregular data density where the baseline's multiplicative rule ignores high-error samples in dense regions.

Statistical inference with bandit data presents fundamental challenges owing to adaptive sampling, which violates the independence assumptions underlying classical asymptotic theory. Recent work has identified stability~\citep{laiwei82} as a sufficient condition for valid inference under adaptivity. This paper first provides a refined stability condition, stated in terms of the iterates of an online algorithm, and shows that a large class of regularized stochastic-mirror-descent-style algorithms satisfy it. This refined condition allows us to strengthen the asymptotic results of~\citet{laiwei82} in several ways. First, we derive a non-asymptotic Berry--Esseen bound for the empirical reward estimates under adaptive sampling. Second, we derive matching non-asymptotic upper and lower bounds on the regret of the proposed algorithm, yielding a precise characterization of its regret. Third, we show that these regularized algorithms preserve asymptotic normality and valid inference under a prescribed level of adversarial corruption. Finally, we show that regularization is necessary rather than incidental: Lai--Wei stability is incompatible with the optimal $O(\sqrt{T})$ regret rate -- the rate attained by unregularized algorithms such as EXP3 -- so that a controlled, polylogarithmic inflation in regret is the price of valid inference.

Large language models (LLMs) can achieve strong reasoning performance through chain-of-thought (CoT) reasoning, yet they often generate unnecessarily long reasoning paths that incur high inference cost. Self-consistency-based approaches push accuracy higher still, but they require sampling and aggregating multiple reasoning trajectories, leading to substantial computational overhead. In this paper, we introduce a confidence-aware selective sampling framework that, at inference time, analyzes a single reasoning trajectory to adaptively determine whether to rely on that trajectory alone or trigger multi-path sampling. The framework uses trajectory-level numeric features and sentence-level linguistic features extracted from reasoning states to guide selective multi-path reasoning. We train it on MedQA and evaluate it in-domain on MedQA and under calibration-only transfer on MathQA, MedMCQA, and MMLU, without further fine-tuning. Experimental results show that the proposed framework maintains comparable performance to full and efficient multi-path reasoning baselines, with accuracy changes of $-0.41 \pm 0.58$ and $-0.31 \pm 0.58$ percentage points, respectively, while reducing token usage by $71.7 \pm 5.0$ and $36.6 \pm 9.1$. These findings demonstrate that reasoning trajectories contain rich signals for uncertainty estimation, enabling a simple, transferable mechanism to balance accuracy and efficiency in LLM reasoning.