Early Stopping

We study compute reduction in neural networks through a unified partial versus full computation view, captured by one-shot magnitude pruning in the static regime and early exit in the adaptive regime. In an asymptotic single-neuron model, we prove a concentration theorem for one-shot magnitude pruning with explicit rates. We also introduce the conditional perceptron for early exit and show that its excess generalization error decays as a power of the compute gap, with an exponent that grows to infinity as the alignment between partial and full computations tends to one. We then extend the analysis to deep networks, characterizing how pruning-induced distortions accumulate with depth and deriving a corresponding compute-accuracy tradeoff for frozen-backbone early exit under a neural network Gaussian process model. Numerical simulations corroborate the predicted scaling laws.

A natural way to cut reasoning-model inference cost is to repeatedly probe a single partial trajectory for its current answer and stop once probes agree -- self-consensus. We ask whether any such rule is both safe and token-saving, and whether one can be selected once and reused. A preregistered sweep of 3,520 consensus rules, replayed on frozen trajectories from two models and three benchmarks, clears none of three acceptance gates fixed in advance; the frontier reproduces on a held-out split and on two unseen models -- while a boundary-confidence control (DEER) swept through the same pipeline clears all three. The reason lies in the signal: agreement establishes that the current answer persists under a fixed probing procedure, not that the reasoning has terminated -- a consensus-termination gap. Stopping on it commits non-terminal answers. At a rule still saving 32% of the tokens, one stop in nine fires on an answer the trajectory itself later abandons, and most of those stops cut off a correction it would otherwise have made. Widening the agreement window does not remove them: the share levels off near 7%, and by then the saving has fallen to 8%. Probe re-wording and a hand-labelled error taxonomy show the agreed answer is often a placeholder the model had not settled on. Used on its own as the stop signal, agreement fails not because it is insufficiently strict, but because it repeatedly measures the wrong object.

In overparameterised classification, training data can be linearly separable even when the underlying distribution is not. In this setting, gradient descent (GD) on the logistic loss diverges in norm while converging in direction to a max-margin interpolating classifier, whose implicit bias can be statistically suboptimal. In this work, we show that early stopping can overcome this suboptimality: in a Gaussian mixture model with label-flipping noise, GD stopped at an appropriate oracle time achieves minimax-optimal excess zero-one risk for covariance spectra with fast and continuous decay, including polynomial and exponential spectral decays. Our analysis combines a sharp upper bound for the early-stopped iterate with a matching statistical lower bound over arbitrary classifiers, yielding optimal rates that are validated by experiments. A central technical contribution is a new calibration result that converts excess logistic risk into excess zero-one risk; it handles the model misspecification induced by the label-flipping noise, and removes the square-root rate in standard bounds. We also establish a lower bound for linear interpolators, showing that interpolation can require exponentially more samples than early stopping to achieve the same excess risk.

Binary neural networks are very attractive for constrained deployment, enabling small footprint and low-power inference. For binary activations, the dot products become sign-controlled additions or subtractions, but the number of operations is unchanged. Indeed, every neuron or output channel still accumulates all of its input, even though only the sign will be retained, which is often wasteful. As the accumulation progresses, the running partial sum frequently drifts so far from zero that its final sign becomes highly predictable long before the last term is reached; every contribution evaluated after that point changes the value of the sum but not the final output activation. This paper turns this observation into a post-training early-stopping mechanism. We characterize the behavior of the running accumulations on the training dataset and use this information to predict the final sign as soon as possible. No model parameter is retrained. We count the number of operations under an idealized ordering of weights. On VGG11 applied to the CIFAR-10 dataset, the method removes $86.6\%$ of the accumulation terms of the deepest convolution for a $0.37$-point accuracy drop, and $25\%$ of the full-network arithmetic when used on the three deepest convolutions simultaneously, for a $1.36$-point drop.

Diffusion policies are a powerful policy class for continuous control, but their iterative denoising process creates a substantial computational bottleneck. Reducing this cost requires adapting the number of denoising steps to the difficulty of each action while preserving task performance. We introduce Prefix-Optimal Generative Policies (POGP), a framework that learns a prefix value function at every intermediate denoising step through a Bellman-style recursion over the denoising chain. The prefix value function serves two purposes: it provides an auxiliary training objective that encourages intermediate outputs to become high-quality actions, and it enables a test-time stopping rule that terminates denoising when additional steps are unlikely to produce meaningful improvement. Across four MuJoCo environments and comparisons with 12 baselines, POGP reduces the required number of denoising iterations by approximately 2.7-fold while retaining near-full task performance. Compared with state-of-the-art dynamic diffusion baselines, prefix training also improves final task performance by approximately 3.5%. These results indicate that supervising intermediate denoising steps is useful not only for adaptive early stopping, but also as an auxiliary objective that improves the learned policy.

Many high-resolution imaging systems face the same fundamental question: when have enough measurements been collected to reconstruct an image accurately? We develop Conformalized Rate-Adaptive Sensing (CoRAS), a method that adaptively chooses an acquisition or compression rate for each image while keeping the reconstruction error below a target level with high probability. As measurements are collected, an image reconstruction model gradually recovers the true image, producing a reconstruction path over acquisition rates. CoRAS uses this path up to an early decision time to estimate the target stopping time, defined as the first time at which the reconstruction error falls below the target level. It then calibrates this estimate using images with similar early reconstruction behavior, producing an upper bound on the stopping time with marginal and approximate conditional coverage guarantees. Experiments on image datasets show that CoRAS attains the target stopping-time coverage, uses fewer measurements on average than fixed-rate stopping rules, and assigns more measurements to images that are harder to reconstruct.

Verify-repair loops are a standard means for large language model (LLM) agents to correct faulty plans in code generation, mathematical reasoning, and tool use. When both the verifier and the repairer are noisy, repair can damage already-correct plans, and reported acceptance keeps rising while true validity falls, so existing methods lack a principled basis for deciding when repair should stop. We propose VRR-Stop, a robust stopping framework for noisy verify-repair-repeat (VRR) loops. A four-parameter noise model separates verifier false acceptance and false rejection from the repair and damage behavior of the repairer. Belief filtering turns repeated verification votes into an estimate of committed validity, and the loop commits or repairs according to the sign of the true marginal gain, which requires only sign identifiability rather than accurate recovery of all parameters. When verifier discrimination approaches zero, calibration itself fails and estimation error can flip the stopping sign, so we pair VRR-Stop with VRR-Guard, an estimation-free fallback that replaces the incumbent candidate only under a sufficient verification margin. On a GSM8K stress setting, VRR-Stop improves final true validity by 60.6 percentage points over fixed five-round repair at an average cost of 0.72 repair rounds. Across settings, stopping reliability is governed jointly by verifier discrimination and the decision margin rather than by the absolute size of estimation error.

Existing methods for testing deep neural networks (DNNs) primarily prioritize test inputs likely to reveal model faults under a fixed labeling budget. In practice, choosing that budget is difficult: too little testing misses failures, while too much incurs unnecessary labeling costs. This work studies the stopping problem in DNN testing. We formulate testing as a cost--benefit decision process in which labeling an input incurs cost $c$ and discovering a fault yields value $v$. Based on this formulation, we introduce \textit{AdaStop}, a framework that estimates the marginal fault discovery rate during testing and stops labeling when the estimated rate falls below the threshold $τ= c/v$. Experiments across multiple datasets, architectures, and selection strategies show that $65$--$84\%$ of faults can be discovered using only $9$--$31\%$ of the labeling budget.

Reasoning models spend test-time compute unevenly across instances, and a growing family of early-exit rules -- confidence thresholds, entropy monitors, answer-stability checks, and learned stoppers -- promises to reclaim the waste. These rules, however, are evaluated under heterogeneous protocols that leave the deployment question unanswered: at a fixed tolerance for losing correct answers, which policy saves more compute, and does the saving survive probe overhead? We answer this question with a controlled study across 18 task-model settings spanning GSM8K, MATH-500, MMLU-Pro, AIME-90, and GPQA on Qwen3 and DeepSeek-R1-distilled models, using LearnStop, a hidden-state-free logistic stopper over prefix-observable features, as the learned policy instrument. Under matched lost-correct risk at $α$ = 0.15, with the scalar competitor selected on calibration data from confidence, entropy, confidence-leap, and run-stability exits, the answer forms three regimes. Learned stopping wins on all four primary Qwen3 free-form math settings (+3.2 to +21.2 pp additional total-token saving); calibrated scalar exits win on multiple-choice MMLU-Pro; and small hard benchmarks (AIME-90, GPQA) admit no certifiable aggressive policy at all. A trajectory decomposition predicts the regime: learning pays where answers oscillate and correctness evidence is spread across complementary signals, while a single confidence threshold suffices where most instances are already solved at the first checkpoint. Cost accounting sharpens the picture further -- the same policy that saves 32% of tokens under KV-cache forking costs 121% extra under black-box repeated prefilling. Together, these results replace the single-method race with a decision procedure for choosing a stopping rule from the trajectory structure and serving regime of the target workload.

Looped Transformers, which repeatedly apply a shared transformer block, are an architecturally natural fit for variable-length algorithmic tasks. Although they can exhibit strong length generalization beyond the length of training sequences, this behavior is brittle, yielding high out-of-distribution (OOD) variance, even across well-performing in-distribution solutions. We trace this variance to the spurious correlation in simple algorithmic tasks between sequence length and number of loops. Introducing stochasticity into the number of loops during training sharply reduces OOD variance and stabilizes predictions across inference-time loop counts. To improve upon heuristic randomization schemes, we further analyze RL-Halting as a learned stochastic schedule and find that it generally improves the accuracy-stability trade-off. Across binary addition, Dyck-1, Unique Set, and Copy, learned stochastic stopping often improves this trade-off but can also stabilize a suboptimal computation. Our work suggests that "when to stop" should be treated as a training-time design choice, not merely an inference-time computation-allocation rule.

We introduce a pre-registered screening rule that decides, before any implementation, whether an evolutionary / population / lifecycle outer loop over neural-network parameters or structure is worth building. Such outer loops cost 10^2-10^3x their gradient inner loop, yet whether they beat a cheap single-shot alternative is usually discovered only after the expense is paid. Our rule computes, at a Phase-0 gate, a single number: the recovery R = s/G, the best single-shot gradient/curvature statistic's gain s divided by the best gain G of any cheap method evaluated, and prescribes skipping the outer loop when R >= 90%. We validate the rule on a within-lab series of pre-registered outer-loop bets (two analyzed cases plus a disclosed file drawer): in both analyzed cases a static or single-shot computation captured the effect on the project's own metric, the gate fired (R approximately 1.0 in both cases; approximately 0.95 under a stricter metric on one), and the outer loop was abandoned, including one case where a companion factorial decomposition localizes the apparent win to a static substrate change with the evolutionary lifecycle contributing no detectable gain. On one project the gate cost about 50-70 GPU-hours and screened out an estimated 400+ GPU-hours (first cell only) plus weeks of implementation, a 6-8x saving. The rule is prospectively falsifiable: a task with R < 90% where the outer loop still fails to beat single-shot would refute it.

Randomized Smoothing (RS) provides rigorous robustness guarantees for neural networks without architectural constraints, yet its adoption is limited by extreme computational costs. Standard RS requires tens of thousands of model evaluations per input and forces practitioners to commit to fixed sample sizes a priori. In this work, we present a novel meta-learning framework for anytime-valid certified robustness that adaptively deploys computational resources. By using a lightweight meta-learner to predict image-specific priors for a sequential E-process, we achieve a 20-fold reduction in sample complexity compared to traditional methods while maintaining rigorous statistical guarantees. Beyond raw efficiency, we demonstrate how anytime-validity enables adaptively allocating compute based upon application-specific risk thresholds, a form of resource triage impossible under classic certification frameworks. That this is achievable while also providing similar certification performance demonstrates that our approach provides a pathway for real-time, safety-critical certification deployments.

Multi-agent large language model (LLM) loops, for example a Writer that drafts and a Critic that revises, are almost always terminated by a fixed iteration cap (max_iterations). This is a syntactic kill-switch: it is blind to whether the answer is still improving, so it over-spends tokens on easy inputs and truncates hard ones. We study semantic early-stopping: the loop halts when consecutive draft embeddings stop changing in meaning (cosine distance with a patience window) and the answer's measured quality stops improving. Our work makes three contributions. First, an honest theoretical footing: we prove deterministic termination and well-definedness and machine-check these claims, while treating the convergence of the distance sequence as an empirically tested conjecture rather than a (previously over-claimed) Banach contraction. Second, a judge-efficient evaluation protocol: we generate each question's full trajectory once, replay every stopping policy over the identical drafts, and cache every LLM-judge call, yielding a strictly paired efficiency-versus-quality comparison at low cost; we further separate operational tokens (charged to a policy) from evaluation tokens (a measurement instrument). Third, an empirical study on multi-hop retrieval-augmented question answering (HotpotQA). On the 60-question test split, a judge-free semantic stopper reduces operational tokens by 38% relative to max_iterations at parity quality (Delta-IS = -0.004, p = 0.81), whereas the full quality-gated variant is counter-productive because its per-round judging dominates cost. An oracle that selects the best round attains +0.115 Information Score over every practical policy (p ~ 4e-11), reframing the problem from "when to stop" (easy) to "which round is best" (open).

Parallel test-time scaling samples many reasoning traces and majority-votes their answers, improving LLM accuracy but requiring traces to run to completion, incurring substantial computational overhead. We observe that probing partial traces at intermediate checkpoints can extract current answers without disrupting generation, revealing an evolving aggregate vote. Based on this observation, we introduce MARS, a margin-adversarial stopping rule that estimates which active traces are likely to change their answers and stops once the leader remains safe under a conservative bound on future vote movement. The rule separates two sources of uncertainty. It learns the trace-level switch probabilities that determine how much of the current margin is likely to be retained, while handling the harder question of where switching traces land through an adversarial bound calibrated from warmup traces. With true switch probabilities, MARS guarantees with high probability that the early-stopped answer matches the full-budget vote. In practice, a five-feature logistic model closely matches oracle switching behavior. Across three reasoning models and three competition-math benchmarks, MARS saves 25-47% of self-consistency tokens and 14-29% on top of DeepConf Online, a strong confidence-weighted baseline that already filters and truncates weak traces, while matching the accuracy of the corresponding full-budget baselines.

Zero-shot information extraction (IE) with large language models (LLMs) has attracted increasing attention due to its flexibility in adapting to new schemas and domains without task-specific training. Existing approaches mainly rely on monolithic prompting, each-type prompting, or multi-agent debate. However, monolithic prompting often suffers from boundary and type errors, while each-type prompting and multi-agent debate introduce cross-type conflicts, redundant agent interactions, and substantial token overhead. To address these challenges, we propose SMADE-IE, a sparse and evidence-driven multi-agent framework for zero-shot IE. SMADE-IE first employs an Adaptive Mode Selector to dynamically route inputs into either a lightweight Global Extraction Mode or a Type-Centric Extraction Mode, reducing unnecessary type selection and reasoning noise. For conflicting predictions, we further introduce an Evidence-Driven Debate mechanism that structures arguments into Toulmin-style components and performs confidence aggregation through external evidence scoring and Bayesian updates. Experimental results on 9 benchmark datasets across NER, RE, and JERE tasks show that SMADE-IE consistently outperforms existing zero-shot IE baselines while also improving token efficiency through sparse agent selection and early-stopping debate.

Gradient boosted decision trees require a stopping rule to avoid overfitting. The standard rule monitors a validation loss and stops if the loss fails to improve for a fixed patience period. However, the patience parameter has no interpretable scale and validation losses can be noisy or implicitly defined by a user-specified gradient. We propose ScoreStop, a gradient-based early-stopping rule that casts the stopping decision at each iteration as a test of the null hypothesis that the current predictor is the population risk minimizer. We use a functional score test, computed on validation data, with a statistic that is scale-invariant in the update direction, with a known asymptotic distribution under the null. Because our test uses gradients rather than loss values, the same construction applies to implicit losses such as LambdaRank, and data-dependent losses such as Cox regression via influence functions. In synthetic experiments and real-data benchmarks, we show that ScoreStop is competitive with loss-based methods.

Speculative decoding accelerates inference by having a lightweight drafter propose tokens verified in parallel by the target language model. Block diffusion drafters such as DFlash generate an entire draft block in one pass, yielding per-position marginals; DDTree uses these to build a candidate tree that maximizes expected acceptance length under a fixed node budget. We observe, however, that acceptance length is non-decreasing in budget: it always favors larger trees regardless of verification cost, offering no principled basis for budget selection. We introduce \textbf{CaDDTree} (Cost-aware Diffusion Draft Tree), a method that directly optimizes token throughput (expected tokens generated per unit time) by jointly selecting the tree structure and node budget. We model draft and verification latencies explicitly, show that the throughput objective decomposes into a per-round one-dimensional search over the budget, and prove that under a convex verification cost the throughput function is \emph{unimodal}, enabling an efficient greedy stopping rule. CaDDTree requires no offline budget search, adapting the budget each round from the current per-position distributions and verification cost. Experiments on Qwen3-4B and Qwen3-8B across eight benchmarks spanning reasoning, coding, and instruction-following tasks show that \caDDTree{} matches or surpasses DDTree with oracle budget selection on nearly all tasks.

Gradient-flow optimization is usually viewed as an algorithmic procedure for minimizing empirical loss, with training duration selected by validation or heuristic early stopping rules. We develop a statistical inference framework for gradient-flow training. We show that whenever fitted values evolve through a time-invariant positive semidefinite training operator, the output at each time is equivalent to the best linear unbiased predictor under a corresponding random-effects model. Training time then becomes a variance-component parameter governing variance reallocation from residual noise to structured signal. This turns two training decisions into inferential problems: whether training is needed becomes a variance-component test for signal beyond initialization, and how long to train becomes restricted maximum likelihood (REML) estimation of the training-time variance component. We show that the REML-guided early stopping rule selects the time at which optimized spectral losses become decorrelated from the training-operator eigenvalues. The asymptotic prediction optimality of the REML-guided early stopping time is established for fixed-design in-sample risk and random-design out-of-sample risk. Deep learning models in fixed-kernel gradient regimes provide canonical instantiations for our results. Numerical experiments and a UK Biobank proteomics application show competitive accuracy of the REML-guided early stopping time with reduced reliance on validation splits and repeated checkpoint evaluation.

In modern multimedia systems, efficient video processing is critical, especially in resource-constrained environments such as IoT-based camera networks, autonomous platforms, and wireless sensor multimedia systems. A key bottleneck in video compression and understanding is block motion estimation (ME), a process that remains computationally expensive despite the development of fast search techniques. This work introduces an Optimal Stopping Theory (OST) algorithm for block motion estimation based on the assessment of spatiotemporal differences within and across video frames. It also proposes a semantic-aware motion estimation framework that integrates Foundation Models (FMs) with the OST-based decision process. By leveraging pretrained visual models such as Vision Transformers (ViT) and the Segment Anything Model (SAM), the framework extracts semantic attention scores that indicate the importance of motion within specific spatial regions. These scores are fused with traditional distortion-based metrics, such as the Sum of Absolute Differences (SAD), to guide a hybrid stopping criterion that jointly considers motion magnitude and semantic relevance. The resulting adaptive algorithm stops early in redundant regions while continuing the search in areas where motion is semantically significant. Experiments compare the proposed solution with widely used approaches from the literature on benchmark and multimodal video datasets. The proposed method achieves a significant reduction in computation with minimal accuracy loss and improved semantic coverage. The results highlight the benefits of bridging low-level motion analysis with high-level semantic reasoning, offering a promising direction for efficient multimodal video understanding in next-generation smart systems.

Every adaptive learning system must alternate between two operations: consolidating what it already knows and expanding into new evidence. We propose \emph{Consolidation-Expansion Operator Mechanics} (OpMech), a framework that makes this structure precise. The central object is the \emph{order-gap} $\Ogap(θ; e)$, the degree to which a consolidation operator~$Q$ and an expansion operator~$P_e$ fail to commute at a given knowledge state. Because the order-gap is computable from the system's own trajectory, it serves as a real-time control signal: large values indicate that the system is still sensitive to the ordering of consolidation and expansion; once the order-gap falls and stays small, further processing is unlikely to change the outcome. Three results give the signal precise meaning: the order-gap decays along convergent trajectories; a persistently large order-gap implies the system is far from its settled state; and an order-gap-based stopping rule terminates with provable guarantees in both noiseless and bounded-noise settings. The framework applies across five domains: bandits, reinforcement learning, stochastic optimization, continual learning, and recursive language models. We give conditions under which the order-gap reliably tracks convergence in three representative cases. We develop the recursive language model application in detail, showing how OpMech replaces heuristic stopping rules and fixed recursion budgets with principled, evidence-driven alternatives.

Incorporating prior data into online reinforcement learning accelerates training but typically forces a difficult trade-off between high computational costs and long, multi-stage training pipelines. While fixed-length stabilization phases are significantly more computationally efficient than static update schedules, they require task-dependent manual tuning, risking either the waste of prior knowledge or severe overfitting. To address this, we propose SOPE, an algorithm that uses an actor-aligned Off-Policy Policy Evaluation (OPE) signal as an automated early-stopping mechanism to dynamically control the length of offline training phases. By evaluating the critic on a held-out validation split under the current policy's action distribution, SOPE halts gradient updates exactly when out-of-distribution benefits saturate, eliminating the need for manual schedule tuning. Evaluated on 25 continuous control tasks from the Minari benchmark suite, SOPE improves baseline performance by up to 45.6% while reducing the required TFLOPs by up to 22x, thus balancing the tradeoff between sample and computational efficiency. These findings demonstrate that adaptive, evaluation-driven update schedules are more effective than relying on static, exhaustive update schedules.

Early-exiting neural networks enable adaptive inference by allowing inputs to exit at intermediate classifiers, reducing computation for easy samples while maintaining high accuracy. In practice, exits can be trained sequentially by incrementally adding them to a shared backbone; however, this sequential training can cause newly introduced exits to interfere with previously learned ones, degrading the performance of earlier classifiers. We address this problem by retaining the knowledge embedded in existing exits while allowing new ones to specialize. We propose two alternative approaches that operate at different levels of the model. The first constrains learning by protecting parameters that are important for previously trained exits, while the second preserves the output distributions of earlier exits as the network adapts. These alternatives directly reflect the stability-plasticity trade-off studied in continual learning. Accordingly, we leverage \textit{Elastic Weight Consolidation} to constrain critical weights and \textit{Learning without Forgetting} to preserve output distributions. Experiments on standard benchmarks show that our approaches consistently improve early-exit performance, achieving higher accuracy over existing sequential training methods and significant performance speedups at low computational budgets.

Large Language Model ensembles improve reasoning accuracy, but only up to a performance boundary beyond which additional deliberation degrades accuracy. We introduce DASE (Deliberative Adaptive Stopping Ensemble), a stopping heuristic for iterative ensemble deliberation that commits early on genuine consensus and applies a global-frequency fallback on fragmented evidence. We make three contributions. (1) DASE produces a commit-type routing partition that generalises across benchmarks and is complementary to verbalized single-call confidence. On GPQA-Extended (N=546, 70B ensemble), the partition yields a 39.5 pp routing gap (right-wall 81.1% vs. left-wall 41.5%). On AIME 2010-2023 (N=261, 120B ensemble, 3 seeds), right-wall commits reach 98.3% accuracy vs. left-wall 72.8% (25.5 pp gap), statistically equivalent to Opus 4.6 Standard verbalized confidence at matched coverage (25.7 pp gap; bootstrap p=0.873); the two mechanisms disagree on 37% of routing assignments. (2) Adaptive stopping, not injection bandwidth, drives accuracy. On AIME-300, bandwidth accounts for only 0.3 pp (ns). On GPQA-Extended at the 120B tier, sparse injection ($\approx15$ tokens/worker/round) achieves 70.9% with a 30.7 pp routing gap; dense injection ($\approx600$ chars/worker/round) achieves 72.2% but with halved right-wall coverage and a narrower 18.9 pp gap. (3) Injection-based methods exhibit an inverted-U accuracy-vs-inference trajectory; this pattern is hypothesis-generating.

The field of Language Reasoning Models (LRMs) has been very active over the past few years with advances in training and inference techniques enabling LRMs to reason longer, and more accurately. However, a growing body of studies show that LRMs are still inefficient, over-generating verification and reflection steps. Additionally, the high-level role of each reasoning step and how different step types contribute to the generation of correct answers, is largely underexplored. To address this challenge, we introduce TRACES (Tagging of the Reasoning steps enabling Adaptive Cost-Efficient early-Stopping), a lightweight framework that tags reasoning steps in real-time, and enable adaptive, cost-efficient early stopping of large-language-model inferences. Building on this framework we monitor reasoning behaviors during inferences, and we find that LRMs tend to shift their reasoning behavior after reaching a correct answer. We demonstrate that the monitoring of the specific type of steps can produce effective interpretable early stopping criteria. We evaluate the TRACES framework on three mathematical reasoning benchmarks, namely, MATH500, GSM8K, AIME and two knowledge and reasoning benchmarks, MMLU and GPQA respectively. We achieve 20 to 50% token reduction while maintaining comparable accuracy to standard generation.

Bayesian Deep Ensembles (BDEs) represent a powerful approach for uncertainty quantification in deep learning, combining the robustness of Deep Ensembles (DEs) with flexible multi-chain MCMC. While DEs are affordable in most deep learning settings, (long) sampling of Bayesian neural networks can be prohibitively costly. Yet, adding sampling after optimizing the DEs has been shown to yield significant improvements. This leaves a critical practical question: How long should the sequential sampling process continue to yield significant improvements over the initial optimized DE baseline? To tackle this question, we propose a stopping rule based on E-values. We formulate the ensemble construction as a sequential anytime-valid hypothesis test, providing a principled way to decide whether or not to reject the null hypothesis that MCMC offers no improvement over a strong baseline, to early stop the sampling. Empirically, we study this approach for diverse settings. Our results demonstrate the efficacy of our approach and reveal that only a fraction of the full-chain budget is often required.

Large reasoning models that use long chain-of-thought excel at problem-solving yet waste compute on redundant checks. Curbing this overthinking is hard: training-time length penalties can cripple ability, while inference-time early-exit adds system overhead. To bridge this gap, we propose Step-GRPO, a novel post-training framework that internalizes dynamic early-exit capabilities directly into the model. Step-GRPO shifts the optimization objective from raw tokens to semantic steps by utilizing linguistic markers to structure reasoning. We introduce a Dynamic Truncated Rollout mechanism that exposes the model to concise high-confidence trajectories during exploration, synergized with a Step-Aware Relative Reward that dynamically penalizes redundancy based on group-level baselines. Extensive experiments across three model sizes on diverse benchmarks demonstrate that Step-GRPO achieves a superior accuracy-efficiency trade-off. On Qwen3-8B, our method reduces token consumption by 32.0% compared to the vanilla model while avoiding the accuracy degradation observed in traditional length-penalty methods.

A fundamental problem in machine learning is understanding the effect of early stopping on the parameters obtained and the generalization capabilities of the model. Even for linear models, the effect is not fully understood for arbitrary learning rates and data. In this paper, we analyze the dynamics of discrete full batch gradient descent for linear regression. With minimal distributional assumptions, we characterize the trajectory of the parameters and the expected excess risk. Using this characterization, we show that when training with any learning rate schedule and finite time horizon, the early stopped solution is equivalent to the minimum norm solution for a generalized ridge regression problem. We also prove that early stopping is beneficial for generic data with arbitrary spectrum and for a wide variety of learning rate schedules. We provide an estimate for the optimal stopping time and empirically demonstrate the accuracy of our estimate.

Reranking methods, such as Minimum Bayes Risk (MBR) decoding and Quality Estimation (QE) reranking, have been widely used in modern neural machine translation (NMT) to select an output from a set of candidate hypotheses. However, the performance gains come at the cost of high inference latency. Existing acceleration methods target MBR decoding and reduce only the reranking computation, leaving QE reranking unaddressed and candidate generation---which can be the larger computational bottleneck---largely untouched. In this work, we propose Quit (Quantifying Uncertainty for Incremental Termination), a novel early-stopping strategy for the entire generation--reranking pipeline. Quit treats candidate generation as a sequential decision-making process under uncertainty. It incrementally generates and reranks candidates, stopping when the best reranking score stabilizes. Comprehensive experiments with three NMT models across 19 language pairs show that Quit achieves end-to-end speedups of $1.47$--$2.66\times$ for MBR decoding and $3.43$--$4.12\times$ for QE reranking while preserving translation quality for nearly all external quality metrics.