Think Deep, Not Just Long: Measuring LLM Reasoning Effort via Deep-Thinking Tokens

AI summary

Overview

Research area: Natural Language Processing / LLM reasoning and inference-time compute (test-time scaling), with an interpretability component based on layer-wise analysis of hidden states.

Technical level: Advanced. The method requires familiarity with transformer internals, hidden-state projections into vocabulary space, and Jensen–Shannon divergence, though the core intuition is stated simply.

Scope: The paper proposes "deep-thinking ratio" (DTR), a layer-depth-based measure of how hard a model is "thinking" per token, validates it against length- and confidence-based baselines on four benchmarks and eight model variants, and uses it to build a cheaper test-time scaling strategy called Think@$n$.

What This Paper Is About

Large reasoning models are often assumed to reason better when they generate more tokens, but a growing body of work shows that raw output length is an unreliable proxy for quality—longer chains can even signal "overthinking" and lower accuracy. This paper asks whether a signal extracted from inside the model, rather than from the surface-level token count, better predicts whether a generated reasoning trace will be correct. The authors define such a signal—tokens whose predictions keep changing in deeper layers before settling—and then use it both to predict accuracy and to select which samples to keep during parallel test-time scaling.

Key Contributions

1. Introduction of deep-thinking ratio (DTR): A measure of the proportion of "deep-thinking tokens" in a generated sequence—tokens whose prediction distributions undergo sustained revision in deeper layers before converging to the final-layer distribution—proposed as a new lens on inference-time thinking effort.
2. Empirical validation across models and benchmarks: The authors show that DTR of a generated sequence correlates positively with task accuracy across eight model variants and four reasoning benchmarks, outperforming length-based and confidence-based baselines.
3. Introduction of Think@$n$: A test-time scaling strategy that preferentially selects and aggregates samples with higher DTR, and that can early-halt unpromising generations based on DTR estimated from short prefixes.
4. A cost–accuracy result: Think@$n$ is reported to match or surpass standard self-consistency (Cons@$n$) at approximately half the inference cost.

Main Findings

• Length is negatively correlated with accuracy. Across 32 model–benchmark settings, token count shows an average Pearson correlation of $r=-0.594$ with accuracy, with the most negative values in the "strong negative" range. The authors interpret this as consistent with inverse scaling and overthinking: longer traces can reflect redundant or error-amplifying deliberation.
• DTR has the strongest average correlation. DTR achieves the highest average correlation of $r=0.687$, compared with $r=0.594$ for reverse token count, $r=0.605$ for Self-Certainty, $r=0.571$ for negative entropy, $r=0.527$ for log probability, and $r=0.219$ for negative perplexity.
• DTR is the most consistent signal. It exhibits the fewest "orange" (negative) values—2 out of the 32 model–benchmark settings tested—whereas confidence-based measures are heterogeneous, ranging from strong positive to negative correlations depending on configuration.
• Confidence metrics are only partially informative. Log probability, negative perplexity, negative entropy, and Self-Certainty show moderately positive correlations with means between $r=0.219$ and $r=0.605$, but the authors argue they conflate other factors such as overconfidence and do not reliably reflect compute effort or problem-solving effectiveness.
• A single-model illustration. In Figure 1, on GPT-OSS-120B-medium across AIME 2024/2025, HMMT 2025, and GPQA-Diamond, output token count shows a moderate negative correlation with accuracy (average $r=-0.544$), while DTR shows a strong positive correlation (average $r=0.828$).
• Not universal per setting. In Table 1, DTR is not positive everywhere: OSS-20B-medium on AIME 2024 shows $r=-0.192$, and Qwen3-30B-Thinking on AIME 2024 shows $r=-0.657$.
• Layer-depth behavior is interpretable. In the Figure 2 heatmap for GPT-OSS-120B-high, functional and templated words (e.g., "and", "is", "boxed", "<|return|>") converge at relatively shallow layers, while completions after operators (e.g., "+", "=") and answer tokens/symbols (e.g., "13", "(D)") do not settle until deeper layers.
• Hyperparameters matter, asymmetrically. Using GPT-OSS-20B-high on GPQA-D, sweeping $g \in {0.25, 0.5, 0.75}$ and $\rho \in {0.8, 0.85, 0.9, 0.95}$ shows that stricter criteria (higher $g$, lower $\rho$) reduce the range of DTR. Varying $\rho$ preserves a consistent positive slope, whereas $g$ has more impact on correlation: $g=0.25$ yields a flatter trend, while $g=0.5$ and $g=0.75$ produce more robust positive correlations. The combination $(g, \rho) = (0.5, 0.85)$ is reported as the best balance.
• Think@$n$ is Pareto-optimal in the reported comparison. On OSS-120B-medium and Qwen3-4B-Thinking, Think@$n$ matches or exceeds Cons@$n$ while cutting cost by roughly 48–50%. For example, on AIME 2025 with OSS-120B-medium, Think@$n$ reaches 94.7 accuracy versus 92.7 for Cons@$n$, at 155.4k versus 307.6k tokens.
• Short prefixes suffice. On AIME 2025, estimating DTR from only 50 prefix tokens gives 94.7 accuracy at 155.4k tokens, outperforming longer prefixes (100, 500, 1000, 2000) and matching the accuracy of using the full sequence (94.0) at far lower cost. Pass@1 is 80.0 and Cons@$n$ is 90.0 for reference.
• Length- and confidence-based selection lag. Mean@$n$ shares Cons@$n$'s cost but is the worst method; Long@$n$ performs worse than Short@$n$ with no cost savings; Short@$n$ gives only modest savings (reported reductions of 17%, 15%, 18%, 10% for OSS-120B-medium) and underperforms Cons@$n$; Self-Certainty@$n$ halves cost but underperforms both Cons@$n$ and Think@$n$ on three of the four benchmarks.

Methodology in Plain English

The authors treat a model's computation across its layers as a sequence of "guesses" about the next token. At each generation step, they take the hidden state from each intermediate layer and run it through the model's own output projection (the unembedding matrix) to get a probability distribution over the vocabulary—the same trick used in early-exit and logit-lens style analyses.

They then compare each intermediate distribution to the model's final-layer distribution using Jensen–Shannon divergence (chosen because it is symmetric and bounded). A token that already looks like the final answer in shallow layers "settles" early; a token that keeps changing until deep layers has a high divergence until late. To make "settling" strict, they track the running minimum of this divergence across layers and record the "settling depth"—the first layer where that running minimum drops below a fixed threshold $g$. If the settling depth falls inside the "deep-thinking regime" (layers at or beyond a depth fraction $\rho$ of total layers), the token counts as a deep-thinking token. DTR is simply the fraction of tokens in the generated sequence that qualify.

For the experiments, they sample 25 responses per question on AIME 2024, AIME 2025, HMMT 2025, and GPQA-diamond, using each model's developer-recommended sampling parameters and a neutral "think step by step" prompt with no imposed token budget. They set $g=0.5$ and $\rho=0.85$ and average statistics over 30 random seeds. To link effort to accuracy, they bin samples into 5 quantile bins by the effort score and compute Pearson correlation between binned effort and accuracy.

For test-time scaling, they compare aggregation methods over the same pool of $n=48$ samples with $\eta=50%$: Cons@$n$ (majority vote over all samples), Mean@$n$ (average accuracy), Long@$n$ and Short@$n$ (vote over longest/shortest samples by token count), Self-Certainty@$n$, and Think@$n$ (vote over the top half ranked by DTR). Think@$n$ and Self-Certainty@$n$ rank using only a 50-token prefix, which permits early termination of unpromising candidates; results are averaged over 10 trials.

Why This Matters

The paper argues that the field's default proxy for reasoning effort—how many tokens a model writes—is misleading, and that a cheap internal measurement can do better. If correct, this shifts how researchers evaluate reasoning traces and how systems decide when to stop spending compute.

Real-world applications (as implications of the reported results):

• Cost-efficient inference serving: Selecting and voting over high-DTR samples is reported to reach comparable or better accuracy at roughly half the token cost, which directly affects serving economics for reasoning models.
• Early termination of bad generations: Because DTR estimated from a 50-token prefix is sufficient, systems can abandon unpromising samples before they finish generating.
• Answer selection in best-of-$n$ pipelines: DTR offers a training-free ranking signal that the authors report outperforms length-based and Self-Certainty-based ranking.
• Diagnosing overthinking: DTR provides a per-token, depth-wise view of where a model is doing real work versus padding, which could support debugging of reasoning traces.

Industry relevance: The experiments span GPT-OSS, DeepSeek-R1, and Qwen3 families, and the cost figures are reported in thousands of generated tokens—the currency of inference billing. A method that halves token spend while maintaining accuracy is directly relevant to any organization deploying long-CoT models at scale.

Future Directions

• Whether DTR generalizes beyond competition math and graduate science. All four benchmarks (AIME 2024, AIME 2025, HMMT 2025, GPQA-diamond) are verifiable-answer tasks; the paper does not report results on open-ended generation, agentic tasks, or code.
• Combining DTR with other signals. Self-Certainty and DTR disagree on some settings (e.g., Qwen3-30B-Thinking, DeepSeek-R1-70B on GPQA-Diamond), and the paper does not test whether the two signals are complementary when combined.
• Using DTR during training rather than only at inference. The paper applies DTR only as a selection and early-stopping signal; it does not test whether DTR-based rewards or filtering could shape reasoning behavior, which the related work on length-filtered policy optimization suggests as a natural extension.
• Explaining the residual negative cases. DTR is negative in 2 of the 32 settings tested (OSS-20B-medium on AIME 2024 and Qwen3-30B-Thinking on AIME 2024), and the paper does not analyze why those configurations break the pattern.
• Sensitivity of the $g$ threshold. The authors report that $g$ affects the correlation more than $\rho$ and that overly strict settings ($g=0.75$ with $\rho \in {0.9, 0.95}$) are less stable, but they do not provide a principled procedure for choosing $g$ on a new model family.

Target Audience

Researchers and engineers working on LLM reasoning, test-time compute scaling, and inference efficiency will benefit most. It is also relevant to interpretability researchers interested in how predictions evolve across transformer layers, and to practitioners building best-of-$n$ or self-consistency pipelines who care about token cost. Readers without a background in transformer internals or probability divergences will find the method section demanding, though the high-level argument about length versus effort is accessible.

Authors’ abstract

Large language models (LLMs) have demonstrated impressive reasoning capabilities by scaling test-time compute via long Chain-of-Thought (CoT). However, recent findings suggest that raw token counts are unreliable proxies for reasoning quality: increased generation length does not consistently correlate with accuracy and may instead signal "overthinking," leading to performance degradation. In this work, we quantify inference-time effort by identifying deep-thinking tokens -- tokens where internal predictions undergo significant revisions in deeper model layers prior to convergence. Across four challenging mathematical and scientific benchmarks (AIME 24/25, HMMT 25, and GPQA-diamond) and a diverse set of reasoning-focused models (GPT-OSS, DeepSeek-R1, and Qwen3), we show that deep-thinking ratio (the proportion of deep-thinking tokens in a generated sequence) exhibits a robust and consistently positive correlation with accuracy, substantially outperforming both length-based and confidence-based baselines. Leveraging this insight, we introduce Think@n, a test-time scaling strategy that prioritizes samples with high deep-thinking ratios. We demonstrate that Think@n matches or exceeds standard self-consistency performance while significantly reducing inference costs by enabling the early rejection of unpromising generations based on short prefixes.