Efficient Reasoning via Constrained Optimization in Latent Space

Zhinan Hou; XingChen Li; Keyou You

cs.AISep 28, 2026

Efficient Reasoning via Constrained Optimization in Latent Space

Authors: Zhinan Hou, XingChen Li, Keyou You

Organizations: Tsinghua University

Abstract

Large Reasoning Models (LRMs) have shown remarkable reasoning capabilities, yet they still suffer from overthinking, generating redundant reasoning steps which incur substantial token consumption. Existing methods, such as suppressing reflective keywords or forcing shorter reasoning lengths, attempt to mitigate this issue but inevitably truncate necessary steps and induce underthinking, thereby compromising performance. To address this dilemma, we investigate the latent representations and observe that efficient reasoning steps naturally cluster into a concentrated region in latent space, while those deviating from this region tend to produce verbose sequences. To leverage this, we keep reasoning focused within this region via a quadratic program which projects deviating hidden states back into the region. Then we propose a novel training-free framework to achieve efficient reasoning that reduces token generation costs without sacrificing performance. Extensive experiments conducted on four models ranging from 1.5B to 14B, and across six benchmarks in math reasoning, coding, and scientific QA, validate the effectiveness of our method, up to a 12.1% improvement in accuracy while reducing generated tokens by 11.8% to 52.8%. Codes are available at \href{https://github.com/hzn18/Opt4Reasoning}{https://github.com/hzn18/Opt4Reasoning}.

Figures & tables

Figure 1 : Visualization of different reasoning steps in the latent space. The horizontal and vertical axes correspond to the first and second PCA components, respectively.

Panel of Figure 2 — Figure 2 : (a) The distribution of distances to the set $\mathcal{H}$ for trajectories with correct and incorrect answers. (b) The relation between the distance $d(c,\mathcal{H})$ and the total token length of the reasoning trajectory.

Methods	MATH								PROGRAMMING		SCIENCE
	GSM8K		MATH500		AMC2023		AIME2025		LiveCodeBench		GPQA-Diamond
	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.
DeepSeek-R1-Distill-Qwen-1.5B
Vanilla	81.5	1489	82.1	4476	64.4	7708	17.3	11811	32.1	10013	33.3	8241
CCoT	76.6	577	82.0	3832	68.8	7012	22.7	11077	30.6	9824	33.3	7592
DEER	74.2	672	72.2	2578	62.5	5139	20.2	9601	23.7	8227	29.7	7655

Table 1 : Performance comparison against baselines. For each column within a model group, the best result is bolded and the second-best is underlined .

Figure 3 : The impact of the key hyperparameters for performance in MATH500: (a) Intervention Strength. (b) Steering Layer. (c) PCA dimension.

Method	# Tokens	TPS $\uparrow$	TPR (s) $\downarrow$
DeepSeek-R1-Distill-Qwen-1.5B
Baseline	4678	5596	0.83
Ours (Online)	3074	4682	0.66 (-21.2%)
Ours (Explicit)	2812	5472	0.51 (-38.4%)
DeepSeek-R1-Distill-Qwen-7B
Baseline	3631	2659	1.36

Table 2 : Efficiency Analysis on A800 GPU.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4 : Visualization of different reasoning steps generated by Qwen3-4B. The horizontal and vertical axes correspond to the first and second PCA components, respectively.

Figure 5 : Visualization of different reasoning steps generated by DeepSeek-R1-Distill-Qwen-7B. The horizontal and vertical axes correspond to the first and second PCA components, respectively.

Figure 6 : Visualization of different reasoning steps generated by Qwen3-14B. The horizontal and vertical axes correspond to the first and second PCA components, respectively.

Figure 7 : Visualization of latent representations across different PCA dimensions. The results show that Direct CoT steps remain concentrated in a core region across various dimension pairs (e.g., Dim 0 vs 1, Dim 2 vs 3), while overthinking behaviors such as Transition and Reflection steps fall into the scattered region.

Methods	GSM8K		MATH500		AMC2023		AIME2025		LiveCodeBench		GPQA-Diamond
Methods	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.	Acc.	Tok.
Vanilla	81.5	1489	82.1	4476	64.4	7708	17.3	11811	32.1	10013	33.3	8241
Ours-MATH500	83.1	853	83.2	2949	75.2	4821	26.7	8502	32.7	7656	36.4	5651
Ours-GSM8K	82.3	707	82.3	2980	76.7	4825	23.3	8097	32.8	7575	35.8	5738
Ours-AMC23	82.6	782	82.9	2951	72.5	4609	22.2	8649	32.5	7594	35.3	5590
Ours-AIME2025	82.7	901	82.0	3294	71.6	4939	30.0	8727	33.4	8393	34.9	6149

Appendix

Table 3 : Cross-Domain and Cross-Difficulty Transferability on DeepSeek-R1-Distill-Qwen-1.5B.

Method	Reflection			Transition			Performance
Method	Word Count	TF	TF–IDF	Word Count	TF	TF–IDF	Acc.	Tok.
DeepSeek-R1-Distill-Qwen-1.5B
Baseline	28.5	9.3	13.0	8.0	2.5	4.7	82.1	4476
Ours	6.0	4.2	6.6	1.7	0.7	1.5	83.2	2949
DeepSeek-R1-Distill-Qwen-7B
Baseline	18.3	7.6	10.3	4.9	2.1	3.7	91.6	3647

Appendix

Table 4 : Semantic statistics of Transition and Reflection vocabularies.

DeepSeek-R1-Distill-Qwen-1.5B
Method	Score	Ability
Method	Rubric	Coh	Crt	Img	Pac	Ele	Inst	Voi	Dia	Flo	SDT	Ama	Emo	Pos	Pur	Ch
Baseline	13.80	0.55	1.20	1.09	1.20	0.30	0.39	0.84	1.33	0.48	1.38	0.91	0.19	16.88	14.83	0.28
Ours	14.16	0.75	1.53	1.27	1.42	0.45	0.50	0.93	1.37	0.60	1.61	1.33	0.31	14.60	13.78	0.52
DeepSeek-R1-Distill-Qwen-7B
Baseline	21.40	3.60	3.20	3.92	4.98	2.30	2.11	3.89	3.36	3.01	3.83	2.78	1.73	12.48	11.72	2.84

Appendix

Table 5 : Creative-writing performance on Creative Writing v3 benchmark. Ability abbreviations: Coh = Coherent; Crt = Creativity; Img = Descriptive Imagery; Pac = Pacing; Ele = Elegant Prose; Inst = Instruction Following; Voi = Consistent Voice & Tone; Dia = Strong Dialogue; Flo = Sentence Flow; SDT = Show-Don’tTell; Ama = Avoids Amateurish Prose; Emo = Emotional Depth; Pos = Avoids Positivity Bias; Pur = Avoids Purple Prose; Ch = Believable Characters.

DeepSeek-R1-Distill-Qwen-1.5B
Methods	AMC 2023				AIME 2025
Methods	Pass@1	Pass@4	Pass@8	Pass@16	Pass@1	Pass@4	Pass@8	Pass@16
Vanilla	64.4	85.6	92.5	95.0	17.3	33.3	40.0	40.0
Ours	75.2	91.2	96.2	97.5	26.7	36.7	40.0	40.0
DeepSeek-R1-Distill-Qwen-7B
Vanilla	86.3	94.3	95.0	95.0	26.9	58.3	63.3	66.7

Appendix

Table 6 : Pass@k Performance on AMC 2023 and AIME 2025.

Model	Method	AMC2023		AIME2025
Model	Method	Acc	Tokens	Acc	Tokens
DeepSeek-1.5B	Vanilla	64.4 $\pm$ 5.4	7708 $\pm$ 703	17.3 $\pm$ 3.7	11811 $\pm$ 625
DeepSeek-1.5B	Ours	75.2 $\pm$ 3.5	4821 $\pm$ 304	26.7 $\pm$ 4.2	8502 $\pm$ 390
DeepSeek-7B	Vanilla	86.3 $\pm$ 4.0	5846 $\pm$ 358	26.9 $\pm$ 3.9	11209 $\pm$ 780
DeepSeek-7B	Ours	91.1 $\pm$ 2.7	4083 $\pm$ 261	39.0 $\pm$ 4.0	9042 $\pm$ 631
Qwen-4B	Vanilla	95.9 $\pm$ 1.7	7588 $\pm$ 475	60.2 $\pm$ 3.7	16835 $\pm$ 944

Appendix

Table 7: Statistical reliability evaluation on AMC23 and AIME2025. We report mean accuracy, mean token count and corresponding standard deviations across datasets and model sizes.

Model	Variables	Constraints	Online (ms)	Explicit (ms)	Additional Memory (MB)
R1-1.5B	10	8	1.7	0.3	0.41
R1-7B	12	8	2.3	0.3	0.93
Qwen3-4B	12	8	2.9	0.3	0.56
Qwen3-14B	16	8	5.3	0.5	1.80

Appendix

Table 8 : Comparison of Average Solving Time. We detail the scale of the QP problem and compare the latency of the online solver with the explicit acceleration. The additional memory denotes the GPU footprint required to store the precomputed piecewise linear mappings for the explicit method.

Symbol	Description
$X_{dir},X_{van}$	The original representations of Direct CoTs and Vanilla CoTs.
$Z_{dir},Z_{van}$	The representations of Direct CoTs and Vanilla CoTs after PCA projection.
$\mathcal{H}$	The efficient reasoning.
$U$	The PCA projection matrix.
$k$	The dimension of the latent space after the PCA projection.
$M$	The number of clusters defining the efficient reasoning set $\mathcal{H}$ .

Appendix

Table 9 : Mathematical notations and their descriptions used in our method.

Figure 8 : The impact of intervention strength

\lambda

on Pass@1 accuracy and average token count across three models on the MATH500 dataset.

Figure 9 : The impact of the steering layer index on Pass@1 accuracy and average token count across three models on the MATH500 dataset.

Figure 10 : The impact of the PCA dimension

k

on Pass@1 accuracy and average token count across three models on the MATH500 dataset.

Figure 11 : The impact of the boundary shift proportion

\alpha

on Pass@1 accuracy and average token count across three models on the MATH500 dataset.

Figure 12 : The impact of the cluster number

M

(number of linear constraints) on Pass@1 accuracy and average token count across three models on the MATH500 dataset

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Large Reasoning Models (LRMs) often suffer from overthinking due to redundant verification steps. Existing approaches for mitigating overthinking, such as fast-slow thinking switching and reasoning trajectory compression, fail to make a fine-grained distinction between beneficial and redundant steps within the LRM's reasoning process, and may thus impair reasoning capability in their pursuit of efficiency. To simultaneously improve reasoning efficiency and capability, we propose EvoThink, a framework that reduces redundant verification and encourages the exploration of new reasoning paths. EvoThink comprises two key components: Self-Pruning Training (SPT), an unsupervised method that iteratively prunes redundant reasoning steps and self-trains on the concise trajectories; and Aha-Moment Preference Optimization (AMPO), which, inspired by genetic algorithms, identifies valuable failed reasoning attempts, synthesizes from-wrong-to-right aha-moment data, and optimizes the model to internalize this reasoning pattern. Extensive evaluations across mathematical reasoning and code generation benchmarks demonstrate that EvoThink not only substantially reduces inference-time token usage but also improves the reasoning capability of LRMs.

Abstract

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