Reasoning as Compression: Unifying Budget Forcing via the Conditional Information Bottleneck

Fabio Valerio Massoli; Andrey Kuzmin; Arash Behboodi

cs.LGMar 9, 2026

Reasoning as Compression: Unifying Budget Forcing via the Conditional Information Bottleneck

Authors: Fabio Valerio Massoli, Andrey Kuzmin, Arash Behboodi

Organizations: Qualcomm AI Research

Abstract

\ac{CoT} prompting improves LLM accuracy on complex tasks but often increases token usage and inference cost. Existing ``Budget Forcing'' methods reduce cost via fine-tuning with heuristic length penalties, suppressing both essential reasoning and redundant filler. We recast efficient reasoning as a lossy compression problem under the \ac{IB} principle, and identify a key theoretical gap when applying naive \ac{IB} to transformers: attention violates the Markov property between prompt, reasoning trace, and response. To resolve this issue, we model \ac{CoT} generation under the \ac{CIB} principle, where the reasoning trace $Z$ acts as a computational bridge that contains only the information about the response $Y$ that is not directly accessible from the prompt $X$ . This yields a general Reinforcement Learning objective: maximize task reward while compressing completions under a prior over reasoning traces, subsuming common heuristics (e.g., length penalties) as special cases (e.g., uniform priors). In contrast to naive token-counting approaches, we introduce a semantic prior that measures token cost by surprisal under a language model. Crucially, the prior is queried only for token-level log-probabilities, adding negligible overhead to the training loop. Empirically, our \ac{CIB} objective prunes reasoning redundancy while preserving fluency and logic, improving accuracy at moderate compression and enabling aggressive compression with minimal accuracy drop. These gains generalize across model families and task domains, confirming \ac{CIB} as a domain-agnostic CoT compression framework.

Figures & tables

Figure 1: Left: Minimality Reward vs. Completion Length. We observe a consistent negative correlation between the completion length and the minimality reward. The shaded blue region shows the

\pm 1\sigma

band representing the spread of the information cost for the token chosen within CoTs with similar length. Right: Accuracy vs. CoT Pareto Frontier. The

\beta

parameter confers fine-grained control over the accuracy-compression trade-off. 7B prior (

\blacksquare

) yields stronger compression than a 1.5B prior (

\bullet

). Baselines: DLER ( Liu et al., 2025 ) (

\bigstar

), L3L1-EXACT ( Aggarwal and Welleck, 2025 ) (

\blacksquare

), and our L1-Exact ‡ model (

\blacktriangledown

Figure 2: Lengths Distribution. Compared to the baseline length distribution ( blue curve ), the minimality term shifts the length distribution towards shorter completions ( green curve ). The plotted distributions correspond to models with similar accuracy (see Table 1 ).

DeepScaleR-1.5B
Model	MATH500	AIME24	AIME25	Minerva	Olympiad	Avg.
	Acc./Tok.	Acc./Tok.	Acc./Tok.	Acc./Tok.	Acc./Tok.	Acc./Tok.
Baseline	$85.8$ / $3190$	$38.2$ / $9331$	$25.4$ / $8907$	$19.9$ / $6386$	$54.8$ / $5922$	$44.8$ / $6747$
L3L1-EXACT	$85.2$ / $1932$	$\mathbf{{\color[rgb]{0.545,0,0}24.4}}$ / $\mathbf{2637}$	$\mathbf{{\color[rgb]{0.545,0,0}20.2}}$ / $\mathbf{2608}$	$19.8$ / $2124$	$\mathbf{{\color[rgb]{0.545,0,0}48.8}}$ / $\mathbf{2369}$	$\mathbf{{\color[rgb]{0.545,0,0}39.7}}$ / $\mathbf{{\color[rgb]{0,0.5,0}2334}}$
ThinkPrune-I2k §	$85.5/1707$	$34.9/5095$	$-/-$	$-/-$	$54.7/3498$	$-/-$
HAPO §	$84.4/2370$	$31.4/7702$	$-/-$	$-/-$	$51.4/4571$	$-/-$

Table 1: Accuracy (

\uparrow

) and average CoT length (

\downarrow

) across different benchmarks. Bold indicates best performance within each model group. Green highlights improved average accuracy or token count vs. the baseline; red marks significant degradation. CIB subscripts denote the compression setting (

\beta^{-}\!=\!5\times 10^{-5}

\beta^{+}\!=\!1.5\times 10^{-4}

) and the prior size (1.5B or 7B). Error estimates are reported in Tables 3 – 5 in Appendix B . ∗ : Pass@1 with

n=16

for consistency, the official HuggingFace page reports pass@1(Avg@64); † : Our models; ‡ : Models we trained using the L1-EXACT length penalty ( Aggarwal and Welleck, 2025 ) ; § : Results from Wu et al. (2025) .

Figure 3: Left: Efficiency Gain. Efficiency gain of CIB across diverse benchmarks and models. Points falling in the upper half-plane (“ Golden Zone ") exhibit strictly superior efficiency, achieving higher information density with reduced computational cost. Colors denote models: CIB

{}^{DLER}_{Q_{\phi}\!=\!1.5B}

(

\bullet

), CIB

{}^{DLER}_{Q_{\phi}\!=\!7B}

(

\bullet

), CIB

{}^{DeepSc.}_{Q_{\phi}\!=\!1.5B}

(

\bullet

), CIB

{}^{DeepSc.}_{Q_{\phi}\!=\!7B}

(

\bullet

). Shapes denote tasks: MATH500 (

\bullet

), AIME24 (

\blacksquare

), AIME25 (

\blacktriangle

), Minerva (

\blacklozenge

), Olympiad (

\blacktriangledown

). Right: Information Density Profile. Token-wise surprisal evaluated against the baseline language prior. Lower surprisal indicates predictable, low-information filler.

Prior	Step Time (s)	Tokens/sec	Samples/sec
Length-based penalty (no prior model)	$21.1\,\pm\,0.3$	$3691\,\pm\,92$	$1.47\,\pm\,0.02$
$Q_{\phi}=$ 0.5B	$21.8\,\pm\,0.3$	$3677\,\pm\,97$	$1.46\,\pm\,0.03$
$Q_{\phi}=$ 1.5B	$22.5\,\pm\,0.3$	$3482\,\pm\,96$	$1.42\,\pm\,0.03$
$Q_{\phi}=$ 3B	$22.4\,\pm\,0.4$	$3489\,\pm\,99$	$1.43\,\pm\,0.03$
$Q_{\phi}=$ 7B	$23.2\,\pm\,0.4$	$3449\,\pm\,98$	$1.38\,\pm\,0.03$

Table 2: Training overhead of the semantic prior. Wall-clock comparison for DLER-1.5B. Average value and SEM over 100 steps.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

DeepScaleR-1.5B
Model	MATH500	AIME24	AIME25
	Acc. / Tok.	Acc. / Tok.	Acc. / Tok.
Baseline	$85.8_{\pm 1.6}$ / $3190_{\pm 132}$	$38.2_{\pm 1.5}$ / $9331_{\pm 116}$	$25.4_{\pm 1.5}$ / $8907_{\pm 144}$
L3L1-EXACT	$85.2_{\pm 1.6}$ / $1932_{\pm 60}$	$\mathbf{{\color[rgb]{0.545,0,0}24.4}}_{\pm 1.3}$ / $\mathbf{2637}_{\pm 15}$	$\mathbf{{\color[rgb]{0.545,0,0}20.2}}_{\pm 1.3}$ / $\mathbf{2608}_{\pm 17}$
CIB ${}^{\dagger}_{\beta^{-}\!,\,1.5B}$	$\mathbf{87.6}_{\pm 1.5}$ / $2359_{\pm 121}$	$\mathbf{40.2}_{\pm 1.9}$ / $7298_{\pm 141}$	$\mathbf{27.8}_{\pm 2.1}$ / $6866_{\pm 170}$
CIB ${}^{\dagger}_{\beta^{+}\!,\,1.5B}$	$87.2_{\pm 1.5}$ / $2017_{\pm 107}$	$38.2_{\pm 2.0}$ / $6937_{\pm 141}$	$27.6_{\pm 2.1}$ / $6757_{\pm 184}$

Appendix

Table 3: Accuracy (

\uparrow

) and average CoT length (

\downarrow

) — Math Benchmarks. Bold indicates best performance within each model group. red marks significant degradation. CIB subscripts denote the compression setting (

\beta^{-}\!=\!5\times 10^{-5}

\beta^{+}\!=\!1.5\times 10^{-4}

) and the prior size (1.5B or 7B). SEM is shown as subscript. ∗ : Pass@1 with

n=16

for consistency, the official HuggingFace page reports pass@1(Avg@64); † : Our models; ‡ : Models we trained using the L1-EXACT length penalty [ 1 ] .

DeepScaleR-1.5B
Model	Minerva	Olympiad
	Acc. / Tok.	Acc. / Tok.
Baseline	$19.9_{\pm 0.8}$ / $6386_{\pm 95}$	$54.8_{\pm 0.4}$ / $5922_{\pm 29}$
L3L1-EXACT	$19.8_{\pm 0.8}$ / $2124_{\pm 29}$	$\mathbf{{\color[rgb]{0.545,0,0}48.8}}_{\pm 0.3}$ / $\mathbf{2369}_{\pm 6}$
CIB ${}^{\dagger}_{\beta^{-}\!,\,1.5B}$	$\mathbf{20.4}_{\pm 0.8}$ / $4510_{\pm 77}$	$\mathbf{55.5}_{\pm 0.7}$ / $4472_{\pm 47}$
CIB ${}^{\dagger}_{\beta^{+}\!,\,1.5B}$	$20.1_{\pm 0.8}$ / $4147_{\pm 73}$	$55.3_{\pm 0.7}$ / $4211_{\pm 45}$

Appendix

Table 4: Accuracy (

\uparrow

) and average CoT length (

\downarrow

) — Math Benchmarks. Bold indicates best performance within each model group. red marks significant degradation. CIB subscripts denote the compression setting (

\beta^{-}\!=\!5\times 10^{-5}

\beta^{+}\!=\!1.5\times 10^{-4}

) and the prior size (1.5B or 7B). SEM is shown as subscript. ∗ : Pass@1 with

n=16

for consistency, the official HuggingFace page reports pass@1(Avg@64); † : Our models; ‡ : Models we trained using the L1-EXACT length penalty [ 1 ] .

DeepScaleR-1.5B
Model	GPQA Diamond	MMLU-Pro	LCB
	Acc. / Tok.	Acc. / Tok.	Acc. / Tok.
Baseline	$36.5_{\pm 0.7}$ / $7445_{\pm 47}$	$39.4_{\pm 0.2}$ / $5285_{\pm 16}$	$18.7_{\pm 2.6}$ / $11403_{\pm 301}$
L3L1-EXACT	$37.9_{\pm 0.7}$ / $3125_{\pm 16}$	$40.4_{\pm 0.2}$ / $2240_{\pm 5}$	$17.5_{\pm 2.5}$ / $\mathbf{4612}_{\pm 89}$
CIB ${}^{\dagger}_{\beta^{-}\!,\,1.5B}$	$\mathbf{38.4}_{\pm 0.6}$ / $4571_{\pm 98}$	$\mathbf{41.6}_{\pm 0.2}$ / $3286_{\pm 14}$	$\mathbf{18.8}_{\pm 1.2}$ / $10410_{\pm 151}$
CIB ${}^{\dagger}_{\beta^{+}\!,\,1.5B}$	$35.5_{\pm 0.6}$ / $4429_{\pm 87}$	$40.7_{\pm 0.2}$ / $3108_{\pm 14}$	$18.6_{\pm 1.1}$ / $10274_{\pm 134}$

Appendix

Table 5: Accuracy (

\uparrow

) and average CoT length (

\downarrow

) — Science, Knowledge & Code Benchmarks. Bold indicates best performance within each model group. red marks significant degradation. CIB subscripts denote the compression setting (

\beta^{-}\!=\!5\times 10^{-5}

\beta^{+}\!=\!1.5\times 10^{-4}

) and the prior size (1.5B or 7B). SEM is shown as subscript. ∗ : Pass@1 with

n=16

for consistency, the official HuggingFace page reports pass@1(Avg@64); † : Our models; ‡ : Models we trained using the L1-EXACT length penalty [ 1 ] .

Figure 4: DLER — Inference Time Compute. Pass@k accuracy for different values of

k

, with a maximum completion length of 8K tokens.

Figure 5: DLER — Inference Time Compute. Pass@k accuracy for different values of

k

, with a maximum completion length of 3K tokens.

Figure 6: DLER — Inference Time Compute. Pass@k accuracy for different values of

k

, with a maximum completion length of 2K tokens.

Figure 7: DeepScaler — Inference Time Compute. Pass@k accuracy for different values of

k

, with a maximum completion length of 8K tokens.

Panel of Figure 8 — Figure 8: Qualitative comparison on geometry reasoning. Top: Prompt. Middle: the Baseline trace is dominated by redundant “verbal parsing" of the input code and repetitive self-correction loops (highlighted in red). Bottom: The CIB trace successfully filters this syntactic noise. Notably, the information constraint induces a shift in strategy: CIB bypasses the lengthy coordinate calculation favored by the baseline, converging instead on a concise trigonometric identity.

Panel of Figure 9 — Figure 9: Qualitative comparison on arithmetic search. Top: Prompt. Middle: Baseline model engages in inefficient trial-and-error, explicitly calculating the incorrect candidate $98^{3}$ (highlighted in red) and engaging in redundant self-verification loops. Bottom: the CIB model (bottom) suppresses this exploratory computation, converging directly on the correct candidate ( $97$ ) and reducing the token count by $\sim$ 78% without loss of accuracy.

Panel of Figure 10 — Figure 10: Qualitative comparison on constraint satisfaction.Top: Prompt. Middle: Baseline trace is characterized by “verification bloat." Despite correctly deriving the constraint ( $c<16$ ) early on, the model expends tokens checking tautologies ( $8+c>8$ ) and re-verifying its own conclusions (highlighted in red). Bottom: CIB trace (bottom) retains the necessary constraint logic but eliminates the redundant self-auditing loops, trusting the derivation immediately.

Hyperparameter	Value
Optimizer	AdamW
Learning Rate	$1\times 10^{-6}$
LR Scheduler	constant
Warmup Ratio	0.03
Batch Size (Global)	128
Number of Generations per Prompt	16

Appendix

Table 6: GRPO Training Hyperparameters. All experiments share these settings unless otherwise noted.

Model	MATH500		AIME24		AMC-23		OlympiadBench
	Acc. $\uparrow$	Tok. $\downarrow$	Acc. $\uparrow$	Tok. $\downarrow$	Acc. $\uparrow$	Tok. $\downarrow$	Acc. $\uparrow$	Tok. $\downarrow$
DeepScaler-1.5B	85.8	3280	35.5	9246	74.2	6416	54.6	5974
ThinkPrune-I2k	85.5	1707	34.9	5095	74.3	2913	54.7	3498
ThinkPrune-4k	86.6	2042	35.5	6488	76.3	3839	55.7	4010
HAPO	84.4	2370	31.4	7702	70.3	4301	51.4	4571
AutoThink	84.9	1635	36.2	7201	67.8	3658	52.5	4085

Appendix

Table 7: Compression results for DeepScaler-1.5B [ 46 ] .

Dataset	Link	License
MATH500 [ 21 ]	HuggingFace	MIT
AIME 2024 [ 28 ]	HuggingFace	Copyright MAA
AIME 2025 [ 29 ]	HuggingFace	Copyright MAA
Minerva Math [ 17 ]	HuggingFace	MIT
OlympiadBench [ 9 ]	HuggingFace	MIT
AMC 2023 [ 27 ]	HuggingFace	Copyright MAA

Appendix

Table 8: The list of datasets.

Model	Link	License
DLER-R1-1.5B [ 22 ]	HuggingFace	NSCLv1 (research only)
DLER-R1-7B [ 22 ]	HuggingFace	NSCLv1 (research only)
DeepScaleR-1.5B-Preview [ 24 ]	HuggingFace	MIT
Qwen2.5-Base-0.5B [ 50 ]	HuggingFace	Apache 2.0
Qwen2.5-Base-1.5B [ 50 ]	HuggingFace	Apache 2.0
Qwen2.5-Base-3B [ 50 ]	HuggingFace	Qwen Research

Appendix

Table 9: The list of models.

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Extended chain-of-thought (CoT) traces improve LLM reasoning but incur substantial computational and memory costs. While existing CoT compression methods mitigate this by condensing thought steps into compact representations via memory tokens and retaining only these representations at inference time, the loss of fine-grained information makes subsequent steps more error-prone. To alleviate this, we propose HybridThinker, where in addition to preserved these representations, thought steps are also temporarily retained to provide fine-grained details. However, we observe that naively keeping thought steps accessible to subsequent steps during training lets the model bypass memory tokens by retrieving information directly from these steps, leaving the model's ability to compress and retrieve information through memory tokens insufficiently trained. We therefore introduce a hybrid training scheme, in which only some thought steps are directly accessible through attention to subsequent steps, while the other thought steps are masked, forcing the model to use memory tokens for compression and retrieval. Across 4 reasoning benchmarks, HybridThinker matches the uncompressed baseline, advancing the state of the art in CoT compression by 5.8 points on average accuracy with similar inference time. Ablation studies confirm that both temporary thought-step retention and the hybrid training scheme contribute to these gains.

Abstract

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