Conformal Thinking: Risk Control for Reasoning on a Compute Budget

TL;DR

Conformal Thinking reframes compute-budgeted reasoning for LLMs as a risk-control problem, introducing a dual-threshold exiting mechanism consisting of an upper-threshold (confidence-based exit) and a novel lower-threshold (progress-based exit) calibrated via distribution-free risk control on a validation set. The method defines explicit correctness and efficiency losses and uses finite-sample corrections to guarantee that the realized risk stays within a user-specified budget $\epsilon$. Empirically, risk-controlled exits, especially when combining signals in an ensemble and employing the lower threshold, yield substantial compute savings while maintaining target accuracy across diverse models and tasks, with greater gains as the solvable/unsolvable mix shifts toward unsolvability. This work provides a principled, interpretable, and robust approach to adaptive reasoning budgets, enabling more efficient deployment of reasoning LLMs in resource-constrained settings.

Abstract

Reasoning Large Language Models (LLMs) enable test-time scaling, with dataset-level accuracy improving as the token budget increases, motivating adaptive reasoning -- spending tokens when they improve reliability and stopping early when additional computation is unlikely to help. However, setting the token budget, as well as the threshold for adaptive reasoning, is a practical challenge that entails a fundamental risk-accuracy trade-off. We re-frame the budget setting problem as risk control, limiting the error rate while minimizing compute. Our framework introduces an upper threshold that stops reasoning when the model is confident (risking incorrect output) and a novel parametric lower threshold that preemptively stops unsolvable instances (risking premature stoppage). Given a target risk and a validation set, we use distribution-free risk control to optimally specify these stopping mechanisms. For scenarios with multiple budget controlling criteria, we incorporate an efficiency loss to select the most computationally efficient exiting mechanism. Empirical results across diverse reasoning tasks and models demonstrate the effectiveness of our risk control approach, demonstrating computational efficiency gains from the lower threshold and ensemble stopping mechanisms while adhering to the user-specified risk target.

Figures (9)

• Figure 1: Early-stopping behavior under different target test risks.Left: Threshold values required to achieve a given target test risk vary substantially across early-stopping signals, indicating that threshold selection is signal-dependent. Right: The relative efficiency of different early-stopping methods depends on the target test risk, and no single method is uniformly most efficient across all risk levels. Results shown here use an upper-threshold-only stopping rule; introducing both upper and lower thresholds would further complicate the risk-threshold mapping.
• Figure 2: Dual-threshold early exit via risk-controlled confidence dynamics. We plot confidence trajectories as a function of token usage under Qwen3-8B on AIME questions. Left: an unsolvable instance, model confidence fluctuates and fails to reach the upper threshold; the reasoning is halted early by the parametric lower threshold, preventing unnecessary token consumption. Right: a solvable instance, where confidence steadily increases and crosses the upper threshold, triggering termination once sufficient confidence is achieved.
• Figure 3: Visualization of the proposed correctness and efficiency losses under different thresholds. Lines of different colors (purple to pink) denote different threshold curves. Numbers in the box show the correctness and efficiency loss for each threshold. The top row shows the upper-threshold correctness and efficiency loss (Eq. \ref{['eq:fpr']} and \ref{['eq:eff_loss_upper']}). Bottom two figures show lower-threshold sigmoid curves (Eq. \ref{['eq:lower_threshold_sigmoid']}) and the corresponding losses the lower-threshold correctness loss (Eq.\ref{['eq:fnr']} and \ref{['eq:eff_loss_lower']}). For a given instance, when multiple thresholds have the same correctness loss (e.g. top 3 lines in first row, all lines in the bottom row), the one with smallest efficiency loss is preferred, in order to maximize efficiency while maintaining user provided risk tolerance.
• Figure 4: Empirical verification of risk control. We plot the empirical test risk (y-axis) against the user-specified target risk $\epsilon$ (x-axis). Solid lines and shaded regions indicate the mean and standard deviation over 40 random test-validation splits. Different colors denote different early-stopping signals. The left panel (Naive) selects thresholds on the validation set without finite-sample correction, leading to frequent violations in the realized test risk exceeding $\epsilon$, particularly on false negative risk (Eq. \ref{['eq:fnr']}) controlled by the lower threshold (Eq. \ref{['eq:lower_threshold_sigmoid']}), which has more flexibility and therefore more prone to noise. The right panel (UCB) applies a probabilistic risk control procedure that accounts for validation uncertainty, guaranteeing that the test risk under the selected threshold is upper-bounded by $\varepsilon$ with high probability.
• Figure 5: Ensemble of signals improves efficiency. Under four models, we consider upper-threshold only early stopping. Given a target tolerance $\epsilon$, risk control framework picks the signal that minimizes the efficiency loss (Eq. \ref{['eq:eff_loss_upper']}), forming an ensemble of signals, which translates to superior efficiency on the test set (better accuracy v.s. token trade-off).