Plan and Budget: Effective and Efficient Test-Time Scaling on Reasoning Large Language Models
Authors: Junhong Lin
Xinyue Zeng
Jie Zhu
Song Wang
Julian Shun
Jun Wu
Dawei Zhou
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Model
Test-Time Compute
Reasoning
Effectiveness
Efficiency
Primary area: foundation or frontier models, including LLMs

Abstract: Large Language Models (LLMs) have achieved remarkable success in complex reasoning tasks, but their inference remains computationally inefficient. We observe a common failure mode in many prevalent LLMs, overthinking, where models generate verbose and tangential reasoning traces even for simple queries. Recent work has tried to mitigate this by enforcing fixed token budgets, however, this can lead to underthinking, especially on harder problems. Through empirical analysis, we identify that this inefficiency often stems from unclear problem-solving strategies. To formalize this, we develop a theoretical model, BAM (Budget Allocation Model), which models reasoning as a sequence of sub-questions with varying uncertainty, and introduce the E3 metric to capture the trade-off between correctness and computation efficiency. Building on theoretical results from BAM, we propose Plan-and-Budget, a model-agnostic, test-time framework that decomposes complex queries into sub-questions and allocates token budgets based on estimated complexity using adaptive scheduling. Plan-and-Budget improves reasoning efficiency across a range of tasks and models, achieving up to 70% accuracy gains, 39% token reduction, and 193.8% improvement in E3. Notably, it improves the efficiency of a smaller model (DS-Qwen-32B) to match the efficiency of a larger model (DS-LLaMA-70B), demonstrating Plan-and-Budget’s ability to close performance gaps without retraining. Our code is available at https://github.com/junhongmit/P-and-B.

Published as a conference paper at ICLR 2026
PLAN AND BUDGET: EFFECTIVE AND EFFICIENT TEST- TIME SCALING ON LARGE LANGUAGE MODEL REA- SONING
Junhong Lin
Xinyue Zeng"*
Jie Zhu
Song Wang
Julian Shun!
Jun Wu Dawei Zhou²
MIT CSAIL
Virginia Tech
University of Virginia
"Michigan State University
junhonn, Jshun @mit.cdu
(xyzeng, jicz19, 9, zhoud] @vt..du wujun44msu..o
swwwvvirginia edu
ABSTRACT
Large Language Models (LLMs) have achieved remarkable success in complex rea- soning tasks, but their inference remains computationally inefficient. We observe a common failure mode in many prevalent LLMs, overthinking, where models generate verbose and tangential reasoning traces even for simple queries. Recent work has tried to mitigate this by enforcing fixed token budgets, however, this can lead to underthinking, especially on harder problems. Through empirical analysis, we identify that this inefficiency often stems from unclear problem-soll solving strate- gies. To formalize this, develop theoretical model, BAM (Budget Allocation Model), which models reasoning as a sequence of sub-questions with varying un- certainty, and introduce the E metric to capture the trade-off between correctness and computation efficiency. Building on theoretical results from BAM, we propose PLAN-AND-BAL BUDGET, model-agnostic, test-time framework that decomposes complex queries into sub-questions and allocates token budgets based on estimated complexity using adaptive scheduling. PLAN-AND- BUDGET improves reasoning efficiency across range of tasks and models, achieving up to 70 % accuracy gains, 39 % token reduction, and 193.8% % improvement in 33. Notably, it improves the efficiency of smaller model (DS- Qwen-32B) to match the efficiency of a larger model DOS-LLaMA-70BE demonstrating PLAN-AND-BUDGET 's ability to close performance gaps without retraining, Our code is available at Plan-and-Budeee
INTRODUCTION
Large Language Models (LLMs) exhibit strong generalization capabilities, enabling them to perform wide range of tasks, such mathematical problem solving (Ahn et al. 2024 Imani al. 2023) scientific question answering (Huang et al. 2024 Lu et al.) 2022a), and structured reasoning (Guo
2025 Wei al. 2022). without task-specific retraining. Recent advances in test-time com- putation, such as Chain-of-1 Thouhht (CoT) prompting ( (Wei et al. 2022), self-consistency (Wang al. 2023), and tool- aoo--uugentted inference (Chen al. | 2023). have significantly improved their performance on complex, multi-step reasoming tasks. These enhancements have paved the way for I.LMs to become increasingly deployed in high-stakes domains such as education (Golshan Academy 2023), finance (Wang et al.), law (Katz et a (0224), and scientific research (Tallor et al. 2022). where robust reasoning at inference time is critical.
al.
Despite these advances, deploying LLMs in worl-world settings introduces new challenges, particularly in scenarios that require deliberative reasoning under computational and time constraints. A key issue is the lack of calibrated reasoning behavior during inference. Although LLMs are proficient in multi-step reasoning, they often struggle to regulate how much reasoning effort is appropriate for a task. This miscalibration manifests in two major failure modes: overthinking (Sui et al. Chen et al.)
2025
Turpin et al. 2023) where models generate unnecessarily long and tangential reasoning paths,
"Equal contribution.

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Published as a conference paper at ICLR 2026
Query
Compute: 1-243-445 99.00000
Vanilla Reasoning
P
Planning
Global Budgeting
(ot2, Dz)
Sub-quration::
Attempt
(178 Insoens)
(008 minnr)
Thinking
1000--60. -00 20-20 firs, betic
Len group then in pain...
Cursnds
Scheduling
Pur lernuct of the turiperlly patters. ovaral lordl sam.
Local Budgeting
and
Skan, soi hase thrresh problen
than 2e0 tokers
Attompt
Thinking
(1-))
Attempt
(228
bij
You thisk qub-ueaiioss
Cakolltt each 1ap sollons
Fax the sorn thesetis top 40 (olon))
Alporittvely
Attempt
Verification
mustm cotfer roult
Dkay, th nam to
Thiniing
confines carler
Venttlaztion
(122 mokens)
The Brower Vacael
Solving
30 689 tokens
Anower
( 2otann)))
fuam(s big
980 tokens
Verification
J Overthinking
Thyr rialic serse Bich pair giat..
1105 iskano
Prearss Vocooll- 50,
Arssver
Effectiveness
Underthinking
Efficiency
396 tokens!!!
Our Framework
Illustration of REASONING MISCALIBRATION, Vanilla reasoning results in overthinking and wastes tokens; global budgeting results in underthinking and fails. Our method combines planning and local budgeting to guide structured, efficient reasoning, achieving the correct answer with fewer tokens.
Figure 1:
even for simple queries, incurring excessive computational cost without improving accuracy; and underthinking (Wang al.) 2025a Wei et al. 2022), where models terminate reasoning prematurely,
sacrificing correctness to conserve resources. Recent methods (Lee et al. 2025| Xu et al. 2025
Han et al. 2025) have attempted to mitigate overthinking by introducing hard token constraints (c.g.. "using fewer than B tokens" in the prompt). While these strategies may be effective on simpler tasks, they often degrade performance on complex queries by inducing underthinking, highlighting the limitations of fixed, non-adaptive approaches. To the best of our knowledge, there is limited work that has systematically addressed both overthinking and underthinking in a unified framework.
In this paper, we take the first step toward closing this gap. With a comprehensive empirical study of test-time reasoming behavior in state-of-the-arr LLMs ranging from 32B to 70B parameters, we uncover pervasive phenomenon that we term REASONING MISCALIBRATUU where models exhibit unregulated inference depth during reasoning. This miscalibration manifests as either overthinking, where the model engages in unnecessary and tangential reasoning, or under- thinking, where reasoning terminates prematurely. Our study reveals that reasoning miscalibration is frequently triggered by two types of queries: (I) riivial-uutambiiguuu queries, which elicit diffuse token distributions and lead to speculative reasoning: and (2) hard-and-rare queries, where models without meaningful convergence. These findings raise central
a failure mode
engage in shallow rial-and-ee research question: How can we characterize the internal reasoning and inference mechanisms of LLMs,, and how
guide them to allocate computation adaptively hased task complexity?
To answer this question, we reinterpret reasoning miscalibration as a resource allocation problem under uncertainty. For trivial-bur-ambiguuuss queries, although the underlying solution is simple, the model may follow multiple semantically equivalent reasoning paths, leading to redundant exploration and excessive inference depth. In contrast, for hard-and-rare queries, genuine uncertainty arises from fragile intermediate conclusions or incomplete internal knowledge, yet under fixed budgets, the model may prematurely commit incomplete reasoning trajectories. These contrasting behaviors suggest that reasoning failures do not arise merely from too little or too much computation, but from a mismatch between computational effort and the uncertainty structure of the problem. Some intermediate sub-question require substantial deliberation because ambiguity at these stages can alter subsequent reasoning trajectories and ultimately affect correctness, while others no longer have ambiguity at these stages and further reasoning yields negligible benefit. Motivated by this perspective, we introduce the Budget Allocation Model (BAM), a theoretical framework that characterizes reasoning as sequence of sub-questions with heterogeneous uncertainty reduction dynamics. BAM formalizes how fixed computational budget should be distributed across sub-questions so as to prioritize steps where uncertainty be most effectively reduced. From this viewpoint, effective
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Published as a conference paper at ICLR 2026
reasoning follows two principles: (1) reasoning should be structured, decomposing complex queries into targeted sub-questions; and (2) computation should be adaptive, allocating more budget to sub-questions with higher uncertainty.
Building on BAM, we propose a novel compute-efficient reasoning strategy, called PLAN-AND- BUDGET, which consists of two stages: Plan and Budget. In the Plan Step, the model decomposes the original query into a sequence of sub-questions, providing a soft scaffold for structured reasoning. In the Budget Step, apply simplified scheduling strategies that dynamically assign token budgets to each sub-question, guided by its uncertainty pattern, following the BAM principle. To evaluate our approach, we introduce E the Efficiency-AWe trade-off between reasoning accuracy and computational cost. Unlike conventional efficiency metrics that overlook output quality, E offers a more robust measure of inference performance.
Effectivenss Evaluation Score, which captures the
We evaluate our method through extensive experiments across four state-of-the- art LLMs, including DeepSeek-R1 Distill-Qwen-tt DeepSeek-RI Distill-Llama-70B (DS-LLaMA-70N) (Guo et al. 2025), and OpenAI o4-mini (OpenAl
(DS-Qwen-32B) (Guo et al. 2025). QwQ-32B (Team) (025).
2025) on three representative task domains: mathematical reasoning, instruction following. and agentic planning. Our method is model-agnostic-itt requires no retraining or fine-tuning, relying only on prompting and lightweight planning. Despite this simplicity. PLAN-AND BUDGET consistently improves all four LLMs across all benchmarks. We observe downstream accuracy gains of up to 70 %. token usage reductions of up to 39 %, and combined efficiency improvements (as measured by Ed of up to 193.8% over strong baselines. An especially notable case comes from the agentic planning task domain, where a smaller DS DS-Qwen-32B improves from a low E of 0.16 to 0.47 using PLAN-AND-BUDGET, closing the gap with the larger DS-LLaMA-70RB model (E 0.50) without planning. This demonstrates that uncertainty-guided planning and budgeting can act as inference-time equalizers, boosting the efficiency and competitiveness of smaller models without retraining, Together, these findings underscore the promise of principled compute allocation for more calibrated, efficient, and accessible L.LM inference.
RELATED WORK AND PRELIMINARIES
2.1 RELATED WORK
Scaling Laws. Recent work has explored how test-time computation affects LLM performance. showing that increased inference budget reduce failure rates but often suffers from diminishing
returns (Snell et al.] 2024 Wu et al., 2025 Zeng et al. 2025 Dai et al.) 2025). Methods like MCTS-Judge (Wang et al. 2025b) and EAG Mei et al. 2025) demonstrate the benefits of adaptive
computing in tasks like code evaluation and multi-hop reasoning. Unlike prior work focusing on
simply increasing compute, we investigate how allocate it efficiently through structured planning and uncertainty-awarg budgeting.
Uncertainty. Quantifying uncertainty in deep models often framed through epistemic vs. aleatoric components (Hüllermeier& & Waegeman| 2021). with techniques like MC Dropout (Gal & Ghahramani 2016), ensembles (Lakshminarrayaan et al. 2017). and evidential learning (Huang et al., (021). Recent work extends these ideas to LLMs via
consistency checks and parameter-efficient en-
Table 1: Notation Summary
sembles (Lu et al.] 2022b Tonolini al.) 2024
Mühlematter et al. 2024: Ouyang et al. 2024).
Symbol
Description
Our work builds this by using uncertainty
decomposition to guide token allocation at infer- ence time, offering a novel application of uncer- tainty for test-time efficiency.
T7
Number of sub-questions
i-th query
8iy
j-th sub-question query Jy
bej
Tokens allocated to sub sub-question Siy
Bi3
Complexity of uub-questions.
2.2 PRELIMINARIES
B
Total token budget per query Decomposition plan for query Jq Normalized complexity weight for si
m
We begin by summarizing previous work and introducing the key notations used throughout this work. Table Ilists the symbols relevant to reasoning formulation.
Wi
7.P
Decay scheduler hyperparameters
Parameter characterizing epistemic
Cjj
uncertainty reduction
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Reasoning Miscalibration in LLMs. While LLMs excel at complex reasoning tasks, they often struggle to regulate how much inference effort is appropriate per query. We refer to this phenomenon as REASONING MISCALIBRATION. It describes mismatch between task complexity and the depth of reasoning a model performs at test time.
This miscalibration presents itself in two primary modes: (1) Overthinking (Sui et al.) Chen et al. 2025 Turpin et al. 2023), where the model engages in excessively verbose or tangential reasoning even for simple queries, incurring unnecessary computational cost and introducing noise or contradic- tions; and (2) Underthinking (Wang et al. 2025a; Wei et al. 2022). where the model prematurely stops reasoning to conserve budget, often yielding incomplete or incorrect answers.
Contrary to the common belief that allocating more decoding tokens leads to better performance, we observe that excessive generation can degrade quality. In our empirical analysis, we show that longer outputs can lead models to wander within the solution space, becoming verbose, redundant, or seffinnconsistet.. Our findings suggest that REASONING MISCALIBRATION does not stem from lack of knowledge or model capacity, but rather from the model's inability to dynamically align reasoning effort with query's s evolving informational needs particclarl in response to uncertainty at each step. We leverage a foundational concept of predictive uncertainty (Hüllermeier & Waegeman, 2021 which decomposes the total uncertainty U(z) for a given input x into two distinct components:
U(x) Mepisemmie (2) + Mateatoric (1)),
where Vepistemie (x) captures uncertainty due to incomplete knowledge (and is reducible through targeted computation), while Malemtoric (a:) accounts for irreducible ambiguity noise in the input. Recent work by Falck et al. (2024) extends this decomposition to LLMs, revealing that LLMs display dynamic uncertainty profiles throughout inference. These evolving patterns offer valuable insights into both the models' reasoning processes and the quality of their generated outputs. We further demonstrate the validity of this decomposition in the L.M setting through formal analysis in Appendix [
Problem Definition. In multi-step reasoning. the decomposition above reveals crucial insight: REASONING MISCALIBRATION arises from unregulated computational effort across sub-questions with varying uncertainty levels. Some sub-problems demand greater inference depth to reduce epistemic uncertainty, while others, dominated by aleatoric uncertainty, benefit from early termination or concise solutions. Yet current LLMs lack mechanism to adaptively allocate computation across these stages. This misalignment leads to inefficiency and degraded reasoning quality. Our goal is to improve efficiency while mitigating REASONING MISCALIBRATTON.
Efficiency-Aaare Effectiveness Evaluation: -d Score. We introduce the Ed index an efficiency- aware metric that jointly captures reasoning quality and computational cost. Rather than treating token usage and accuracy separate concerns, E directly quantifies their trade-off:
A T
A² T
3 = A
Here, A denotes the average accuracy achieved across a set of queries, and T represents the average number of decoding tokens used per query. Earlier work typically measures efficiency as accuracy per token (Muennighoff et al. Lee et al. 2025) (i.e.. A/TT). By putting more weight on accuracy, E emphasizes correctness, discouraging degenerate strategies that minimize token usage at the cost of quality. In doing so, it reflects how well a model aligns its computational effort with task complexity, rewarding those who invest more where needed and conserving resources otherwise. Thus, the E score provides principled evaluation framework for assessing whether model mitigates REASONING MISCALIBRATION while maximizing reasoning efficiency. To formally characterize this efficiency-accural trade-off under adaptive computation, we now define our target optimization problem as follows.
Problem 1. LLM Reasoning Calibration
Given: (1) A set of complex queries (I1...., In), where each x; can be decomposed into a sequence of m sub-questions; and (2) A total token budget B; for each query I;. score g A² 4T, subject to the
Find: A computation strategy that maximizes the efficiency- constraint B, for each query. The objective is to allocate inference effort in a way that prioritizes correctness under limited computational resources.
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3 BUDGET ALLOCATION MODEL (BAM)
To address reasoning miscalibration in Problem | we need a principled method for allocating computation across sub-questions with varying uncertainty. As established by Falck et al. (2024). effective reasoning requires focusing effort where epistemic uncertainty is high, and limiting it where aleatoric noise dominates. Existing methods lack formal mechanism for this adaptive allocation. They often treat all reasoning steps uniformly, leading to inefficient budget use and exacerbating reasoning miscalibration. To bridge this gap, we introduce the Budget Allocation Model (BAM), a theoretical framework that models token allocation as uncertainty reduction under fixed budget. BAM provides a principled foundation for our adaptive reasoning framework presented in Section
To distribute a finite token budget B; across the sub-questions of IĄ, we adopt a Bayesian decision- theoretic formulation that aims to maximize reasoning utility by minimizing total uncertainty. While standard LLM inference is deterministic, recent theoretical work suggests that in-context learning can be viewed as implicit Bayesian inference (Falck et al., (024). We leverage this view to characterize reasoning behavior, even without performing explicit posterior sampling at test time. We assume an inverse power law governs epistemic uncertainty reduction for sub-question 8ty with token allocation biz:
Cjj kj
Mepistemic (sis) big) =
where ojj > 0 reflects the initial epistemic uncertainty and 3, 2 1 captures the complexity of reducing that uncertainty (higher Bi, corresponds to being easier to reduce the uncertainty). This formulation is motivated by established Neural Scaling Laws (Kaplan et al. [2020] [offmann et al.)
2022 Zeng al.) 2025). which demonstrate that model loss proxy for uncertainty scales power law with compute. We extend this perspective to test-time reasoning by modeling the reduction of inference error a power-law function of token allocation, where additional computation yields diminishing returns (Snell et al. 20224] Wu et al. 2025).
We model total uncertainty as the sum of the epistemic and aleatoric components:
Czj 3.g
U(sig] biz)
Malcatorie (Stz).
Here, we treat Ualeatoric as a constant with respect to by, since it reflects irreducible uncertainty that cannot be mitigated through additional inference effort. A proof of the decomposition of total uncertainty in LLMs is provided in Section C
We define the utility of resolving sub-question Sty as decreasing function of its uncertainty:
r(sij biy) (1 U(ss)) b,y))
where a is model/task-based scaling factor. The total utility for query x; is then:
Reotal r(sig buy).
(4)
The optimal budget allocation solves the following constrained optimization problem:
max BeI 1=1
Σe a
Cij
Uateatoric
(Sis)))
s.t.
Σ
S
B,.
(5)
By 1g
J=1
By introducing a Lagrange multiplier À to handle the budget constraint and solving the resulting Lagrangian, we arrive at the optional solution:
bij B, (ce Fig
(6)
(Cin F
This allocation rule reveals a unimodal relationship between by and Bej. i.e., token budget increases with complexity up to the peak, then decreases as further effort yields diminishing returns. This relationship is key to mitigating reasoning miscalibration: moderately difficult sub-questions receive more tokens avoid underthinking, while overly difficult ing. BAM thus provides a principled, relf-regulating mechanism for aligning inference effort with reasoning value. Detailed proofs are provided in Appendix and
receive fewer to prevent overthink-
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4 REASONING CALIBRATION FRAMEWORK: PLAN-AND-BUDGEN
Building directly on BAM's principle in Eq. 0 the optimal allocation distributes a query-level budget B across sub-questions by maximizing expected uncertainty reduction. In practice, however, the marginal uncertainty reduction curves are unknown, and the parameters Cij and S.y in Section 3 are not directly observable in black box LLMs. Directly estimating these quantities would require expensive sampling, defeating our efficiency goal.
Instead, we propose PLAN-AND-BUDGET as a practical instantiation that operationalizes the struc- tural insight of BAM using a lightweight surrogate. Recall that BAM prescribes allocating more computation to sub-questions with greater potential for uncertainty reduction. While the exact marginal gains are inaccessible, we observe that in multi-step reasoning tasks, early stages (e.g.. problem interpretation and strategy formation) typically exhibit higher uncertainty than later refine- ment steps. Motivated by this qualitative structure, adopt decay- based scheduling strategies that front-load computational budget toward earlier reasoning levels. Polynomial and cosine decay, in particular, approximate the non-uniform allocation shape predicted by BAM, allocating larger budgets to high-unncetainiy stages while gradually reducing allocation as reasoning stabilizes.
This schedule is budget-feasible by construction, ensuring that the total allocation respects the global constraint while preserving the non-uniform allocation structure implied by BAM.
4.1 PLANNING STEP: QUESTION DECOMPOSITION AS GUIDED SCAFFOLD
Inspired by human problem-solving strategies, we use query decomposition as a reasoning scaffold improve efficiency and focus. Our planning process has two phases:
Phase 1: Automatic Planning, A lightweight planning function P decomposes v; into an ordered
sequence of sub-questions n, and their estimated complexity scores Dj;
P(x;) (T,, D.), T (811 ,,,, Sim) D; (di1, d:2, .., dim).
Here, म; denotes the decomposition plan, sequence of sub-questions, where each 8ty is natural language prompt targeting a specific sub-problem of the query T;- The vector D; = (d.1, ... dem.) contains corresponding complexity scores, with each dg E Rso reflecting the estimated complexity of solving Sij based on LLM confidence, problem structure, or other heuristics.
The decomposition plan क; is not unique or guaranteed to be optimal, but acts as soft scaffold, plausible high-level reasoning path as prompt to guide the main LLM. The planning function P can be implemented via applying decomposition prompt lightweight LLM (see Section Hp. The resulting complexity scores dej reflect epistemic uncertainty and help estimate the computational effort required for each sub-question. These scores are then normalized
weight vector W,:
dif
Wij
dik
This normalized weight way represents the proportion of the total "complexity" that is attributed to the j-th sub-question. This weight vector then plays key role in the budget allocation mechanism, determining how the total token budget B, is distributed across the individual sub-questions.
Phase 2: Guided Reasoning. After decomposing into sub-questions (sa1,.... 8im) and allocating token budgets ba..... ... Dim. the main reasoning L.MM is guided by these sub-questions (see the prompt template in Section H. Each sub-question Siz is answered according to its allocated budget bij. yielding responses 0ij = fLLM(Sij biz), where fULM denotes the budget-constrained generation process. After all sub-q tub-questions
answered, a synthesis function S aggregates the responses, which a.m).
answers the original query z,; 3/ =
4.2 BUDGET STEP: DECAY-BASED BUDGET ALLOCATION
While our Bayesian formulation offers an optimal allocation strategy based on eeuusuinnssseeciiieee uncertainty parameters (c: and B,y), estimating these values reliably in practice is often infeasible. To bridge this gap, we introduce a family of decay-based scheduling functions that approximate uncertainty-aware budget allocation in lightweight and practical manner.

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Table 2: Decay- scheduling strategies for token budget allocation.
Strategy
Formula d
Description
Non-decay
Equal priority for all sub-questions; budget follows usj- Decreases priority linearly with j: emphasizes early steps. Stronger emphasis on early steps; steeper with higher > I. Exponentially favors earlier sub-question;; controlled by ? (0, 1).
Linear decay
m [m - j) 7
Polynomial decay
Exponential decay
Cosine annealing
0.5 (1+ cos m-
+
Smooth decay with mid-sequence flexibility; adds stability.
These functions allocate more tokens to early sub-questions based on the observation that epistemic foundational understanding and
uncertainty typically highest at the start of reasoning- strategy formation occur. Early token usage yields greater uncertainty reduction, consistent with the power law behavior of epistemic uncertainty in Equation [T] In contrast, later steps are generally narrower in scope more deterministic, and over-allocating tokens at these stages risks wasting inference effort, as additional computation cannot reduce the irreducible aleatoric uncertainty and yields diminishing returns in epistemic gain. Thus, decay functions offer a principled heuristic for prioritizing the budget where is most valuable.
Given the normalized complexity weight vector w, {Wi.,.... Wim for query x; and the total token budget B,, we allocate tokens using
Waj Ps
by
B,
Wik Pik:
where Way reflects the Lett-sttd relative complexity of sub-question j. and Pij schedule(j, m) encodes a positional prior that can emphasize earlier reasoning stages. The allocation thus combines content-driven complexity weights with a structural decay prior, preserving both sub-question difficulty and stage-dependent uncertainty considerations.
Experimental Scheduling Strategy. We ex- plore several decay strategies (Table[), each en- coding distinct prioritization schema over sub- question positions. Each strategy offers a flexi- ble way encode task-specific preferences. For instance, polynomial decay aggressively front- loads the budget, which may be beneficial in highly ambiguous tasks. Exponential decay of- fers more balanced approach for problems with both early and mid-sequence challenges. Ulti- mately, these decay functions serve as practical surrogates to our Bayesian-optimal allocation by heuristically targeting the most epistemically impactful stages of reasoning.
Allocated Iokens
Lineaz
Polynomial
Exponcsica
20
Casine
2 Sub-question Index
Figure 2: Visualization of decay functions p. We
take B 100, p 2, > 0.9, and 5 sub- questions with the same complexity as an example.
Figure 2 shows that different decay strategies yield distinct allocation patterns even under uniform complexity, with polynomial decay and cosine annealing favoring early steps, linear decay offering gradual decline, and exponential decay providing balanced based scheduling flexibly adapts token emphasis to match the structure of reasoning tasks.
-etpuutooonndmmmstttattinnn that decay-
EXPERIMENTS
We conduct extensive experiments across three types of maasoning-intensee downstream tasks to evaluate the effectiveness and efficiency of PLAN-AND-BUDGER We assess performance in terms of raw accuracy and compute-aware reasoning efficiency using our proposed E metric. In particular, we aim to answer the following questions: Q1: Does Plan-and-Ee and-Budget improve reasoning efficiency without sacrificing accuracy, compared to the baseline of using no planning (Vanilla) or applying a fixed budget (Global Budget)? Q2: How does local, uncertainty-aware budgeting perform across models, datasets, and task types, relative to uniform or global strategies? Q3: Which scheduling strategies yield the best efficiency-aud accuracy tradeoff?
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Table 3: Experiment results across different reasoning models on MATH-500. Acc denotes accuracy.
Models + DeepSeck RI- Distill-Qwen 32B
QwQB328
DeepSeck RI Distil Liama 70B
of-mini
Methods|
Ace 1
Avg. Tok.]
6
Auc 196)
Avg. Tok.)
(%8)
Arg. Tok.
8
Auc (56
Avg. Tok.]
8*
PLAN AND Broort Posned Direct
Vanilla 89. 29.76-006
21
3.83
24.88
7244722
20.44
228663
3,58
93.162039
211.20
12.20
Global Budget
89. 60:
1526.15
5.26
90.56
2565.18
18
3.20
90.80
1810
4.55
91 91844
636.41g4
13.25
Vanilla
91.04_0ax
1883
4.40
85, 20.5.3
3309
2.20
12
4.20
539.36
539.36
15.65
Global Budget
91. _1.31 1.31
1552.62,200
5.36
88.
1117
2671
2.91
92.56_a77
1661
24:3424
5,16
91
1178
14.39
Uniform
90.16_uu4
1430.70,4m.s
5.64
88, 68
2397.16,250
3.28 92.28
1575.01.20
541
91.36,uus
525.53
15.88
Weighted
90.48_nas
5.51
27s4
2479
19921
3.08
92. gave
1857.64.47
5.51
538.22
15.60
+ Lincar
90.04 Low
1336.27
6.07
88. 13 1390
2346..35
3.31
92.32
1529.98...s
5.57
90.56.u07
534.45
15.34
Exponential
90.80_nss
1389.75.4uu
5.93
87.90
2320.04,033
3.33
93.04.025
1469.29.0m
5.89
90.88,a3s
525.514m130
15.72
Polynomial
90.04-002
1371
5.91
88 28.27
2346. 17
91.92=1115
1514-42gms
5.58
90.36-443
15.55
Cosine
1365.51
5.92
88.60
2306.83,24m
3.40 92.88
1487
13,7783
5.80
91.32
522.89
15.95
5.1 EXPERIMENT SETUP
Datasets. We evaluate PLAN-AND-BB BUNDET on three representative benchmarks (dataset statistics shown in Table [5]: (1) MATH-500 (Lightman et al.] 2024), 500 math problem dataset requiring multi-step symbolic reasoning, evaluated by accuracy; (2) Naturallnstructions (Wang) al. 2022), a diverse eestuuciionfollowiiin
benchmark, evaluated using the ROUGE score; and (3) TravelPlanner (Xie et al., 2024), a challenging agentic planning task evaluated by hard constraint pass rate in a tool-free setting. This benchmark reflects the difficulty of long-horizon constraint-satiifying reasoning; prior work reports that even strong proprietary models achieve limited performance, highlighting the task's inherent complexity.
Models. We evaluate our methods on four state-of state-of-the-art the, publicly-availabe reasoning- LLMs: DeepSeek- ---ii---ll------------
(DS-Qwen-32B) (Guo et al.] 2025), QwQ-32B (Team) 2025). DeepSeek R Rit---iilll-LLMMA-----
(DS-LLaMA-70Bs (Guo et al. 2025), and OpenAl 04-mini (OpenAl 2025). These models provide strong reasoning performance while remaining publicly available and reproducible. For planning and budgeting. we use lightweight LLaMA-3. 1-B-A-3---B--Isttuuu (Grattationi et al. 2024). To ensure that it does not inadvertently contribute to final answer quality, we evaluate its standalone performance the three benchmarks and find that it underperforms specialized models: 48.7610.74 on MATH-500, 21.7210.98 on Natural Instruction,,,, and 2.91 20.28 28 on TravelPlanner. These results confirm that improvements arise from structured planning and adaptive budgeting rather than the planner model's intrinsic reasoning capability.
reassssnn LLM,
Baselines. We compare our proposed framework against several baselines: (1) Vanilla. The query is given to the LLM without planning or token constraints: (2) Global Budget. Same as Vanilla but with a token limit prompt (c.g.. "use fewer than B, tokens"); (3) Planned Vanilla Global Budget. Same as above. but with the original query and its decomposed sub-questions provided; and (4) PLAN-AND-BUDGET Our method, which decomposes the query, estimates sub-question complexity, computes a local budget allocation, and enforces these allocations through step-specific token constraints. We explore several scheduling strategies for local allocation: (a) Uniform, equal tokens per sub-question; (b) Weighted, proportional to the estimated difficulty; and (c) Linear, Polynomial, Exponential, Cosine, weighted by difficulty with additional decay (we use p = 2 and Y 0.9). A hard cutoff of 8192 tokens is applied to prevent runaway generations. We report the average and standard deviation over 5 runs for all models and baselines.
Evaluation Metrics. We report the following metrics: (1) Score (%), the original evaluation metric used in each dataset; (2) Avg. Tokens, the average number of all billed completion tokens per query, including planning, reasoning and output tokens (for open-source models, tokens before </think and final outputs: for 04-mini, the sum of reasoning and output tokens as reported in the OpenAI documentation (OpenAI 2025) ): and (3) 83 which captures the balance between correctness and computational cost.
5.2
COMPARATIVE RESULTS
We now address the questions introduced earlier by analyzing results across datasets and models.
Tables summarize our main findings. Across all datasets and model scales, PLAN-AND-BUR BUDGET consistently outperforms both the Vanilla and Global Budget baselines, achieving up to 193,8% im- provement in E", while maintaining comparable or even higher accuracy. Figure answer pass rates of QwQ-32B on TravelPlanner, grouped by difficulty. While global budgets reduce token usage,
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Table 4: Experiment results across different reasoning models NaturalInstructions.
Models
DeepSeck RI Distill Qwen 32B
QuQ:32B
DeepSeck RI Disuil Liuma 708
of-mini
Methods]
ROUGE (50) 1 Arg. Tokens J 1 ROUGE ( Aug. Tokens J 6*1
ROUGE (56) 1 Arg. Tokeus
ROUGE (5)
Arg. Tokens
Vodimaa Pasted Direct
Vanilla
43,47:a55
9681722m>
1.95
43.16
1818
1.02
43.134
894.664
47.34
460
1.81
Global Budget
42,81 10:30
287
2.33
34.77
(300
1.47
43.80
772.98
2.48
45.39
21:27
422.20g
4.88
Vanilla
42.48.c4r
200
2.10
24.24
1426
1.37
a3.40
R21.27
2.29
43.78
244.99
5.56
Global Budget
42.50:a.36
717,96gm
2.52
45.13
1265.78
1.61
42.48.a0us
42.
691.79
2.61
43.78
358.84
5.34
Uniform
41.03.ge
644.57 .4.4
2.61
g4
1.98
23.06
2.78
(4.08g331
548.74 aau
5.57
Weighted
41.29 1c:50
963.94.
2.57
44.40,am
10250222240
1.92
43.05 10.00
626.37.11...
2.96
43.72
371.85 .955
5.14
Linear
41.56-a5.0
633.79..411
2,73
44.22
44.22
1003
24.3
1.95
42.05
613M5
2.88
44.21
2044
5.37
Exponential
41.44_1e.0
650.19...2
2.64
43.9
1026.99
1.88
42
622.72.33..
2.93
43.68
364.86.188
5:23
Polynomial
600.04 .U5
2,86
995.95
2,00
13.19 poues
641.82
11:04
363.10 1..
5/18
Cosine
41,43g3m
628.20...
2.73
44.53
1000644.nas
1.98
42.83
657.933
2.79
44.36
363.05
5,42
Table 5: Experiment results on TravelPlanner. Rate denotes the hard constraint pass rate.
Models +
DeepSicek: RI Qwen 32B
QwQ-32B
DeepSeek RI-Distill Liuma- 70B
o4-mini
Methods].
Rase (90
Avg. Tokens 1. 6*
Raie (%0
Arg. Tokens 1 6 Rase (%0 1 Avg. Tokens J. 621
Rate (%0
Avg. Tokens J. 8
PAN AND Btooo Plaued Mioed
Vanilla
14. 14.33
14gars
0,14
34.8% 34.89
3432.33
0.35 26 26.22a1a2
1361,37
0.50
11,58
6559
0.086
Global Budget
13
1158.81:21
0.16
30.7
2530.04
0.37 12.3 12:00
1215.29
8.33 1m
1248.53:a07
Vanilla
2
1243.67
0,30
37.22
2669,88142
0.38 30.67
14641,02ss
0.64
12.20
1640.46:9633
0.091
Global Budget
22.562241
1241.192m4
0.41
35.2222435
3199.58gaa14
0.39 30.67
1220.41asem
7.19
1292.
+ Uniform
20.67.110
1227,99
0.35
36.00:2.7
2854.2424um
0.45 31 564222
1232,98gs4
0.81
11
11.00
1345,32
0.090
Weighted
23.33
1222,09.4m
0.45
33.89
2842.74mm
0.40 29.67 1:000
1197.32:103
0.74
10m
1353.67
0.088
Linear
19.56.244
1136.18:5s
0.34
34555
2671.70gess
0,45 31 21.67
1162.24449
11.66g1
1306.54g55s
0.103
Exponential
21.44 1156,64:1552
0.40
35.44
1226.2544.m
0.46
32.00
1187,85
0.86
1307,87:455
0.075
23.11
1148,53g11
0.47
35.004335
2511,35
0.49 32.67
1148.14;aaa
0.93
11,49
1266.11gaa
0,104
Cosine
20.22:234
1140.79g3e
0.36 36.18
2496.46:211
0.52 31.67
1173,962 in
0.85
9.79
1252,06g
0.077
they also degrade pass rates across all levels. In contrast, MAAN-AND-BUDGGE achieves both higher pass rates and lower token usage, espe-
55.22
Vasili
(34.%) 3432.33 sokees)
Pass Rate (%)
53.7
48.3
Glubal Budaat
Parnel Gilobal H5lge135.2%% 1199.8 tolens
(30.5% 2550.04 ackens)
Man-ond-Badggi
sakzne)
cially on harder queries, highlighting its ability to scale reasoning adaptively with problem com- plexity.
29.7
28.027.2
24.3
21.319 19
23 23.024.4
20
On MATH-500, our method improves 8d con- sistently by over 20% for instance, from 4.55 + 5.89 (+29.4% on DS-LLaMA-700E and from 13.25 + 15.95 (+20.3%) on 04-mini. Impor- tantly, this is achieved without compromising the accuracy. While the Global Budget baseline reduces token usage, its gains are limited due to lack of antertainty-aggann uncertainty-awwreeess. In particu- lar, we find that planning alone (Planned Global Budget) already increases efficiency by 1.,,, validating our first key principle: reasoning should be structured. This scaffolding greatly reduces speculative exploration. Moreover, E enables easy comparison across models-e.g. 04-mini consis- tently achieves the highest E3, despite having similar accuracy to other models, because it uses the fewest tokens. This underscores the importance of 3 as practical efficiency metric.
Hard Figure 3: Answer pass rate (%) grouped by the question difficulty level, with legend showing the overall pass rate (%) and average token usage. The global budget limit hurts the pass rate on all levels, while our method not only achieves a higher pass rate but also enjoys lower token usage.
Easy
Medium
A1: We achieve substantial efficiency gains with comparable accuracy. On Naturallnsttructinss, PLAN-AND-BU BUDGET improves 8 by 19.3-36.0% For example, on QwQ-32B, it improves from 1.47 - 2.00 (+36%), and on 04-mini, from 4.88 - 5.57 (+ (+14%). Although these tasks are more instruction- oriented, PLAN-AND-BI BUDGET remains beneficial. On TravelPlanner, the most open- ended and challenging benchmark, we observe the most dramatic gains: E improves from 0.16 - 0.47 (+193.89) on DS-Qwen-333 0. 1.104 (+85.7%) on 04-mini. These results highlight that the more complex the task, the greater the benefit of structure and adaptivity.
from 0.49 - 0.93 (+89.88 (9) on DS-LLaMA-70N and 0.056 +
A2: Local budgeting consistently improves efficiency. While structured planning alone improves efficiency, adding local budgeting yields significant additional gains. We observe that on MATH- 500, DS-LLaMA-708S improves Eo from 5.16 - 5.89 (+14.1%); on NaturalInstructions, QwQ-32B improves from 1.61 + 2.00 (+24.2%); and TravelPlanner. QwQ-32B improves from 0.39 - 0.52 (+33.3%). These results confirm the importance of adapting the budget to the sub-question, rather than applying global allocation.

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Published as a conference paper at ICLR 2026
A3: Front-loaded scheduling performs best on complex tasks. Among local budget schedulers, polynomial decay and cosine annealing consistently deliver the highest E on mathematical and long-form planning tasks. These strategies front-load computation, allocating more budget to early, uncertain steps where reasoning direction is established. This pattern is particularly effective on MATH-500 and TravelPlanner, where clarity the beginning of the reasoning is crucial. In contrast, on Naturallnstruutions weighted or uniform schedules usually perform well, suggesting that smooth, evenly paced reasoning suffices for tasks with clearer structure and less ambiguity.
A4: Bridging the gap between small and large models. Our method is model-agnostic: it requires no retraining or ane-ttunng. relying only on prompting and lightweight planning. We observe consistent improvements across model sizes, from small models like DS- JS-Qwen-32B and QwQ-32B to large models like DeepSeek-RR -1-70B. An especially notable result comes from TravelPlanner, where a compact model (DS (DS-Qwen-32B) originally achieved only 3 0.16, but reached 8 = 0.47 after applying PLAN-AND- AND--UGET,,,,,, on par with larger model with no planning (DS-LLaMA-70B. E = 0.50). This demonstrates that planning and budgeting can serve as powerful inference-time equalizers, closing the gap between small and large models through better compute utilization.
5.3 ROBUSTNESS AND SENSITIVITY ANALYSIS
We conduct additional studies to further stress- test the framework (Section FD.
Table 6: The average latency (seconds) break- down. The question decomposition + budgeting took 14.73 seconds average.
Early-termination baselines. We compare against the recent terminaemmnnnoon baseline, Certaindex (Fu et al.] (024) across three rep- resentative L.Mss. TLAN--ND-BUDGGE con- sistently outperforms early termination in both
Models >
Methods 1
DeepSiea-R11 Phitll-Qen-550
QuQ-32R
DeepSeck-R1 Disill-L 70B
Vanilla
335.50
756.06
777.66
Global Budget
237.83
486.48
527.39
efficiency and accuracy trade-off's (Table
PLAN-AND-BUDGHT
(14.78 206.52)
221.25
(14.73 426.88)
441.61
(14.73 466.41)
481.14
Latency breakdown. While planning intro-
duces minor overheads, total wall-clock latency decreases due reduced reasoning chains under adaptive allocation. The planning and budget allocation step contributes only 3-7% of the total end-to-end latency. while the adaptive budgeting reduces total latency by 38-44% compared to vanilla decoding, yielding a net efficiency gain (Table 6).
Metric robustness. Our primary evaluation metric 8 = A2 A2/T more strongly penalizes accuracy degradation while rewarding computational efficiency. To ensure improvements are not metric- specific, we additionally evaluate using the conventional A/T measure. Relative performance rankings remain stable across both metrics (Table 10). reinforcing that the observed gains are not artifacts of metric choice.
Budget sensitivity. When varying the prompted token budget, observe consistent changes in realized token usage, demonstrating that budget prompts effectively regulate reasoning depth under different resource regimes (Table ITp.
Planner robustness. Replacing the default planner with the stronger LLaMA 3.3-70B yields consistent improvements, indicating that gains are not tied specific planner (Table 14).
CONCLUSION
We propose PLAN-AND- lightweight test-time framework that improves LLM reasoning efficiency by combining structured planning with uncertainty token budgeting. Built on our Budget Allocation Model, PL.NN-AN--B BUDGET models reasoning as sequence of sub-questions and adaptively allocates computation based on estimated difficulty. Experiments on three different reasoning tasks show that BLLAN-AND-BUDGES achieves significant improvements in computational efficiency over strong baselines, without compromising accuracy. Although effective, our method currently requires an additional LLM call to generate the decomposition plan. In future work, aim to fine-tune and develop a dedicated planner LLM to internalize the plan-and buddet strategy, enabling end-to-end, efficient reasoning within single model.
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Published as a conference paper at ICLR 2026
REPRODUCIBILITY STATEMENT
We have taken multiple steps to ensure reproducibility. Theoretical assumptions and derivations of the Budget Allocation Model (BAM) detailed in Section []] with complete proofs provided in Section CE Experimental setups, evaluation metrics, and dataset statistics are described Section 5 including licenses for all datasets and models. We provide anonymized code repository (linked in the abstract) containing implementations of all baselines, our Plan-and-Budee framework, and scripts reproduce every table and figure. Additional details, such prompt templates and robustness and sensitivity analysis, are included in Section H and F Together, these resources allow independent verification and extensions of both our theoretical and empirical findings.
and Section G
ETHICS STATEMENT
This work relies exclusively on publicly-availaae datasets (MATH-500, NaturalInstructions. and TravelPlanner) and open-source or APl-accessible large language models (e.g., DeepSeek, QwQ, o4-mini). No human subjects, private, or sensitive data used. The proposed method improves inference efficiency by reducing unnecessary computation, which can lower environmental and financial costs of deploying LLMs. However, with any efficiency-focused technique, there is risk of misuse in high-stakes applications (e.g., medical or legal decision-making! if efficiency is prioritized over accuracy. We mitigate this risk by explicitly emphasizing correctness in our E metric and by recommending careful, task-specifer evaluation before real-world deployment. Our framework does not modify underlying model internals and thus inherits any limitations or biases present the base models.
ACKNOWLEDGEMER
We thank the anonymous reviewers for their constructive comments. This work is supported by the MIT-IBM AI Watson Lab, NSF awards #CCF-1845763, NCCF-2316235, #CCF-2403237, #IIS- 2339989, and #2406439, Google Faculty Research Award, Google Research Scholar Award, DARPA under contract No. HR001 12490370 and No. HR001 124S0013, U.S. Department of Homeland Security under Grant Award No. 17STCIN00001-000 -0--00, Amazon -Viriinia Tech Initiative for Efficient and Robust Machine Learning, Amazon AWS, Google, Cisco, V-V Commonwealth Cyber Initiative, National Surface Transportation Safety Center for Excellence, and Virginia Tech. The views and conclusions are those of the authors and should not interpreted as representing the official policies of the funding agencies or the government.
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A BROADER IMPACTS
Our work proposes lightweight test-time framework that improves LLM reasoning efficiency through structured planning and uncertainty-awart computation. This has potential positive societal impacts by reducing computational costs, improving energy efficiency, and making advanced LLM capabilities more accessible, particularly in resource-constrained settings. By narrowing the performance gap between small and large models, our method may also promote more equitable access language technologies.
However, as with any LLM inference technique, risks remain if deployed without careful oversight. More efficient reasoning pipelines could accelerate LLM integration into high-stakes applications (e.g., legal medical fecision-making) where accuracy, faimesss and robustness method does not modify model internals and inherits any limitations or biases present in the base models. Mitigation strategies include model-level auditing, task-specific evaluation, and responsible deployment practices.
critical. Our
B
LLM USAGE
Large language models (LLMs) were employed in a limited and transparent manner during the preparation of this manuscript. Specifically, LLMs were used to assist with linguistic refinement, style adjustments, and minor text editing to improve clarity and readability. They were not involved in formulating the research questions, designing the theoretical framework, conducting experiments, or interpreting results. All scientific contributions, including conceptual development, methodology, analyses, and conclusions, are the sole responsibility of the authors.
C PROOF OF UNCERTAINTY DECOMPOSITION FOR LLMs
Let E Id denote the parameters an LLM (e.g., transformer weights), and = { y/ yi)}; N represent the training dataset. The predictive distribution for new test-time input x² and output y is obtained by marginalizing over the posterior p(0) D):
la D)
D)= =
,lu"1=",00p(((D) do,
and is often approximated via Monte Carlo sampling:
1 ply" 0m), On p(01D).
plu"I=, D)
M
We define the total predictive uncertainty the Shannon entropy of this marginal predictive distribu- tion:
u(x*) H (p("" x=, D)) = H |f plu" 10" 0)p(0 D)d0
To derive the decomposition, we apply the law of total entropy, which relates the entropy of the predictive distribution p(y" aa:" D) to the expected entropy of the conditionals and the mutual information:
Hly" x=" D] = [>'"">> .0] + I(y": ;1=",, "0=",," D)-
Step-by-step Derivation:
Let p(y"""">** _ 8) be the conditional predictive distribution and ply"|=" D) be the marginal (Bayesian averaged) predictive distribution.
The total predictive uncertainty is:
u(=") = H [p("">>" [1]]"]]]]]
[D]]=- = Σ (o)g"li
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Published as a conference paper at ICLR 2026
Define aleatoric uncertainty the expected conditional entropy:
Malcatoric (x)caaori(x*) )- = 1) H(ply" ",,,,,],, 0)] = p(O1D) (-
(- EErrl ,)) 1ogpp/"0""" 01 do.
Then define epistemic uncertainty as the mutual information:
Vepistemic (x*) I[y": 00'=", D) Hly" 1=""">>", D] D) [Hly" 0]].
Combining the above, we obtain:
U(x*) Valentoric (x=) + Mepistemic (x")..
Interpretation:
(atemoori((=)) Irreducible uncertainty present in each individual model prediction, even if were known.
Mepistemic (x*); Captures model uncertainty due to limited data, reflected in disagreement across posterior samples.
In practice, following Hüllermeier & Waegeman (2021)
approximate this decomposition using
Monte Carlo estimation. Drawing M samples 0
Bx from p(0) D), we compute:
M
1
u(x)) ~ H
plu" j=,,, 0m)
M
m=1
1
Ualentoric (x*)
Σ
H 1p(y" |x" 0m)),
M
m=
Mepistemic (x") zo U(xx()))) Usteatorie
Thus, the uncertainty decomposition holds in both exact Bayesian inference and its Monte Carlo approximation, validating its use in practical LLM reasoning pipelines. In the context of our PLAN-AND-BUDGET framework, we utilize this decomposition as theoretical lens to explain why structured budgeting works. We do not perform the computationally expensive Monte Carlo sampling described above during inference; rather, we rely on the deterministic approximation that the model's greedy likely generation path serves as sufficient proxy for the underlying uncertainty distribution.
D PROOF OF LAGRANGE OPTIMALITY
aim to maximize the total utility:
m
Rintal
(ssi
big)
Σ
1
Cij
Uatentorie (aantorie(is)))
(8)
b Big
f=1
Since a and Valeatoric (siy) are constants with respect to bij. maximizing the total utility is equivalent to minimizing the following:
m
m
subject to
bi B..
(9)
Step 1: Form the Lagrangian. We define the Lagrangian:
Σ (((b)),))
Cjj
A
Σ
bij B:
(10)
Feg
3=1
f=1
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Published as a conference paper at ICLR 2026
Taking the partial derivative with respect to big and setting it to zero, obtain:
ac
cij B1
(11
+ -
A= b (11)1)
(II)
Zbig
Solving for bi gives:
B,+1 0i
CiyPij
CiySej
Pry+1
bij
(12)
Step 2: Apply the budget constraint.
Substitute into the constraint E, big = B₁:
m
Cig
Ag+
=Bi.
(13)
Let
A= (ct (a, Partz
that big = x 11(((/)))))) - Ag.
Then, the constraint becomes:
11( Bi.
(14)
This expression yields a closed-form solution for À only under the assumption of homogencous difficulty, where Pig = for all sub-questions j. Under this assumption, X 1-1/(3+1) becomes common factor across the summation, allowing us to solve for the optimal budget allocation byj analytically. In the more general case of heterogeneous difficulty, must be determined numerically to satisfy the budget constraint Σ bij B₁. In this case:
8+1
CgfB
E)(cr)) ø
bi
Σ
= )=
(15)
B
Substituting back yields:
bi B; (Gr पं
(16)
Σx ()a4
In the general case of heterogeneous Bij. the normalized form can still be written
0i = B, (Cr)
(17)
Σ (CeBBi)) PIRT
which satisfies the budget constraint E, big B,, thus completing the proof.
E ANALYSIS OF THE RELATIONSHIP BETWEEN big AND Big
We examine the behavior of the allocation function in Equation 6
(Cig FIS
bij = B, Σx (CarBes) FL
(18)
To analyze the relationship between by and Big. we focus on the numerator:
(log(Brcca)
f(B) = (3yg41) गमम = exp
(19)
Bij + 1
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Table 7:
Experiment results comparing with the early termination baseline, Certaindex Fu al. (2024) on MATH-500. Acc denotes accuracy.
Models
DeepSeck-RR Ri--isiiil Qwen-32B
Qu/Q-32B
DeepSeck-Ri Beeesscc----isill---e Di---isill-LLamaaaaae
Methods
(9) 7
Aug. Tok.
g
(76)
Avg. Tok.
(86)
Avg. Tok.
8
Vanilla
89.76
2105.12-314
3.83
84
3523,72ema:
2.04
90.44_011
2286.63-2544
3,58
Global Budget
89 29.
1526.15
1936
5.26
90.56_033
2565.18
3.20
90.80_aa2
1810.83 st.0
4.55
PLAN-AND- BLUD-EED-BUDGET BUDGET
90.04.9.4
1336.27
6.07
88. 227s
2306.83;:un
3.40
m32
1469.29.
5.89
Certaindex
13
1985.57.mm
12
5141.72gm
1.44 21.
2200.68_741s
Let us define:
g = log(Be)
so that f(8) c0())
(20)
B
We now study the behavior f(3) through the derivative of g(3):
1
1
(og(
g((B) ++1
(21)
B
(3 + 1)²
(Bg(
(22)
3(3 1)
(3 1)2
The sign of g'(3) depends on B, and is not monotonic. The function g increases initially, reaches a maximum, and then decreases. Consequently, g(3) is unimodel, and since f(ß) = e f(3) is also unimodal.
F ADDITIONAL RESULTS
F.1 ADDITIONAL BASELINES: EARLY TERMINATION
We compare PLAN-AND- BUDGET against representative anly-termination strategy (Certaindex from Pu et al. (2024)), which dynamically stops generation when confidence exceeds a predefined threshold. Unlike PLAN-AND- -AD--AND-. which redistributes budget structurally across reasoning stages, early termination reduces computation by truncating later tokens without modifying the allocation structure.
As shown in Table 7 PLAN-AND-BUTDGET consistently achieves superior efficiency-accuracy trade-off's across three L.MMs. In particular, while Certaindex reduces token usage, often sacrifices accuracy due to premature stopping (often even worse than Vanilla). In contrast, PLAN-AND-BUDGGE preserves reasoning depth in high-uncertainty stages while eliminating redundant computation in later stages, leading to more balanced improvements.
F.2 LATENCY BREAKDOWN
We analyze wall-clock latency by decomposing runtime into planning overhead and generation time. As shown in Table 6 while planning introduces modest overhead, total latency decreases due to shorter reasoning chains under adaptive allocation.
This confirms that efficiency gains translate to practical runtime improvements, not merely token-level savings.
F.3
METRIC ROBUSTNESS: 83 vs. A/T
Our primary metric is 83_ A? /T, which more strongly penalizes accuracy degradation while rewarding compute efficiency. To ensure that improvements are not metric-specific, we additionally report results under the conventional A/T measure.
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Table 8: Experiment results comparing 8 and A/T across different reasoning models on MATH-500.
Models +
Methods]
DeepSock-R1-Disil--uueeass
QwQ-32B
DeppSock-tt-Disil--lm----BB
04-mini
ARAN NIN-N DOS Punsed Direct
Vanilla Global Budget
2.04/2.41
3.96
12.20/
5.26/5.87
3.20/3:53
4.55 / 5.01
13.25 14.43
Vanilla
4.40/4.83
2.23/2.60
4,20
15.65 17.04
Glohal Budget
2.91/3.30
5.16/5.57
Unifarm
5.64 6,26
3.28 /3770
5.41 5.886
15.88 17.38
+ Weighted
5.51 6.09
3.08 /3.53
5,51
15.60 17.03
Linear
6.07 6..7
3.76
5.57/
16.94
Exponential
5.93 / 6.53
3,33/ 3.79
5,89 6.33
15.72 17.29
Polynomial
6.56
3.72
17.21
Cosine
5.92/ 6,58
3,40 3.40/3.84
5.20/6.24
15.95 17.46
Table Experiment results comparing 3d and A/T across different reasoning models on NaturalIn- structions.
Models Methods]
DeepSock- R1 Dis-Diiilll Queiill-ueea-328 QwQ-32B
DeepSock-R. -i-isill-----i-i------ Disuill-Llama-aae Llama-70B
e4-mini
PLA NO---O-O Punned Direct
Vanilla
1.95/
1.02
2.08/4.82
4.84/ 10.25 4,88 10.75
Global Budget
2.33 /.444
1.47 3.29
2.48/ 2.48/5.67
Vanilla
2.10/4.93
1.13/2.61
2.29/5.28
5,56 12.69
Global Budget
2.52/5.92
/.55
/.61/
5.34/ 12.20
Uniform
2.61 6,36
1.75 /3999
2.78/6.47
5,57 12.64
+ Weighted
2.57
1.65
2 6.87
5,14/
Lincar
2.73/6.55 6.56
1.72/3.97
2.88/6.80
5.37 12.16
Exponential
2.64 6.37
1.70/3.92
2.93/
5
Polynomial
2,86 6.99
1.70/3.87
2.91/
5,48 12.29
Cosine
2.73/6.60
1.69
2.79/6.51
5.42/ 12.22
Table 10: Experiment results comparing 3 and A/T across different reasoning models on Trav- elPlanner.
Models
DeepSock- ReepSook-R1-iisill-OOee R1-DisilllQuee--32 QwQ-32B DeepSeck-RI-Diing DiDiss-----isslllLLaa-ee
04-mini
Methods]
NO HO--HOO mexe Purove Diroct
Vanilla
0.14/
0.35 /
0.50/ 1.93
0.086 / 0.74
Global Budget
0.16/ 1.19
0.37 / 1.22
0.49 / 2.00
0.67
Vanilla
0.30/1.50
0.38 1.11
0.64/2.09
0.091
Glohal Budget
0.41 / 1.82
0.39 1.10
0.77
0.037 0.52
Uniform
0.35/ 1.68
0.45/ 1.26
0.81 /2.56
0.090 0.82
Weighted
0.45/
0.40/
0.74/2.48
0.088
Linear
0.34 1.72
0.45 / 1.29
0.86 /.772
0.104 0.89
Exponential
0.46/
Polynomial
0.47 2.01
0.49/ 1.39
0.93/
0.104 0.91
Cosine
0.367 1.77
0.52 1.43
0.85 /2.70
0.077 0.78
Tables 80 show ranking stability across all methods and datasets under both metrics. The relative ranking of Plan-and- Plan-and-Budget over baselines remains consistent under A/T, confirming that the observed gains not artifacts of metric design.
Importantly, 3 provides stronger discrimination in scenarios where A/T appears nearly indistin- guishable. For example, in several cases A/T values comparable (e.g., 6.86 vs. 6.87), while g correctly differentiates models based on accuracy preservation (e.g.. 2.96 vs. 2.88). This reflects the intended design of g which penalizes accuracy drops more heavily and prevents reward hacking (e.g., achieving high A/T by answering only a small subset of easy questions correctly).
Overall, the consistency across metrics reinforces the robustness of conclusions.
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Table 11: Token usage under varying budget limit for DeepSsek-R---isiil---------
on the MATH- 500 dataset. The total budget for a question is given by B, = Binit + Bper level d,, shrinking as Bittt and Bpe level decrease. d, is the question difficulty level E [1, 5). We see that the overall token usage is well correlated with the provided B;.
B lovel + Binit
0
25
75
100
25
1294,43256.55
1323.71:144
1430.35 .66
1503.34421..1
1.525.00
50
1302.50.24.40
1358.964300.1
1442.07 +28.38
1524,96.34.05
1550.97
75
1313.53
1400.21 434.00
1500.47 437.1
1528.84
1556.044.64.54
100
1359.06-16221
1402.721:0.43
1466.34435.83
1492.23
1572.06
Table 12: Experiment results across different reasoning models on MATH-500 (planning and budget- ing by LLaMA 3.3-70B). Acc denotes accuracy.
Models ->
DeepSeek Distill Qwen-32B
QwQ.32B
DeepSeck RI RI-Distill Sisiill-Llamm Llamm 70B
04-mini
Methods].
Ace (
Aug. Tok.j
e*
(%)
Ang. Tok.] 5
Ace (%
Avg. Tok..
8
Acc 1
Ang. Tok.., g
Panted Direet
Vanilla
89.76.u2s
2105.12gm0
3.83
84.88
3523.722mn
2.04
90.44200:
22800682
3.58
93.16:aa
711.20ga
12.20
Global Budget 89. 29.60
1526
5,26
90.566333
2566.18gm
3.20 90.80.e00
1810
4.55
91,3420
63644gx
13.25
Vanilla
90.12, aas
1635.00.4.7
4.97
85.561113
2635.961241
90.44
1799.48 19:
91
534.2ame
Global Budget
89 64-44444
1377.95 mm
5.83
87.48
2291,792as
3.34 89.44 2904
1527.37
5.24
91.042a74
578.99.m
14.32
Uniform
89 89.44_n53
1319.23
6,06
87.72400
1982,11au
12
1513.84,.u0
90.561a
15.82
Weighted
89.40 om
1320.31
6.05
88.48 jus:
2011.16:21
3.84 89.92
1513.59 suuss
5.34
90.44
535.17ass1
15.28
+ Linear
88. 88.96_0.
1294, 58 4477
6.11
87.602071
1986,5332s
3.86 89.762131
1480 a310s
5.44
89.76
525.23gsass
15.34
Exponential
89. 12=00s
1348,
5.89
88. 12412
90.04-2054
1472.0l
90.68
522.61gm1
15.73
Polynomial
89.76_.41
1263.16
6.38
88.24
3.84 89.72gnss
1467.00s.sse
5.49
89.961a47
525.00am0ar 15.41
Cosine
89. 16.
1304, 44
6,09
88
88.84
2009.66 18 3,93 90.00g007
1462.90_aus
5.54
91,1220m
$20.60g11 15.95
Table 13: Experiment results across different reasoning models on Naturallnsttruction (planning and budgeting by LLaMA 3.3-70B).
Models +
DeepSock-RR Distill- Owen 32B
QwQ-32B
DeepSeek Distilll Llama-70B
Methods]
ROOGE 6%)
Ang. Token J g
ROUGE (961
Aug. Tokess 188
ROUGE (90
Aug. Token 1 6"
ROUGE (55) 1 Aug. Token 8
Bubua Nauue Direst
Vanilla
4347:
968.172m
1.95
13.16
1818.34g2
w 1.02
43.13
398.46
2.06
47.24
40099 n
4.84
Global Budget
12.81au9
787.25 sun
2.33
44.77
1360.99
1.47
43.80=11s
772.98 ez44
2.48
15. 15.39
422.20..se
1.88
Vanilla.
12.27
844.60
2.12
M.24
1426.74.xxz
1.37
43.22
799
2.34
41.06
5.59
Budget
12.68
211.15,2330
45.13
288.78
42.69g543
672.362185
13.86
354.80.1431
5.42
Uniform
42. 10
657.50.2115
2,70
44.47
99691,5131
1.98
42.15
663.51
2.68
44 14
348.461545
5.62
Weighted
42 32
629.52
2.89
44.40zas:
1023022.001
1.92
43.14
654 351.49
2.84
44.00aa77
363.85 2366
5.32
Lineur
12 tn.
646.42
2.76
44.2
100.244:33
1.95
42.62
648 98890
2.80
14.17
364.65
5.35
Expenential
11.0
665,47.2x51
1026.89.151
43.07 2334
656.73.21.9
13.8
$1
Polynomial
42. 42.45
630622114
2,86
66.sss
995.95:144
2.00
42,45=221
641.39gmm
2.81
44.41
5.43
Cosine
11.96a04s
637.09
2.76
M4.3
1000.44mm
1.98
42.86
663.74gie
2.77
01.20
365.65
5.34
F.4 BUDGET SENSITIVITY ANALYSIS
We evaluate sensitivity to the initial budget Bini and per-level allocation Beer- As shown in Table performance seales smoothly with increasing budget, without exhibiting abrupt degradation instability.
These results confirm that PLAN-AND- BLAN-AND-BUDGEY maintains controllable and predictable behavior across resource regimes, aligning with the budget-fcasible design described in Section 4
F.5 PLANNER SENSITIVITY
To assess dependence on the planning component, replace the default planner with the stronger LLaMA 3.3-70B and re-evaluate performance. As shown in Tables 12 14 improvements persist across planners, demonstrating that gains are not tied to specific planning model.
This indicates that the allocation mechanism, rather than planner-specific behavior, drives the effi- ciency improvements.
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Published as a conference paper at ICLR 2026
Table 14: Experiment results on TravelPlanner (planning and budgeting by LLaMA 3.3-70B). Rate denotes the hard constraint pass rate.
Models
DeepScekk Rt Distill Qwen-32B
QwQ-32B
DeepSeek Distil Liama- 70B
o4-mini
Methods).
Rase (90/
Arg. Tokens L. 6"
Rate (90
Arg. Tokens 4. 6"1 Rase (%0 1 Avg. Tokens 8
Kote (%6)
Ang. Tokens & E
PLA Alo Buseer Phnned Direet
Vanilla
14.33 1430.14
0.14
34,89 34.89
2432.33
0.35 26.22 1132
1361,37
0.50
11,58
1559.65.ss
0.086
Global Budget
13.78.110
115881
0.16
30.78
2530,44,40r
0.37 24.33
1215.29
0.49
8.33
1248.53
0.056
Vanilla
21.22
1379.60,201
0.33
34.78
3691
0.33 13.1
1992,14,
0.79
12.
1610.36a514
0.089
Global Budget
2144aas
1215/1gm0
0.38
34.22+2,55
2080.284mm
0,38 34.22e054
1248.56aa
0.94
6.89
1380,11g
0.034
Uniform
23.11=244
1237,61
0.43
36.78eam
2668. iree
2668.
0.51 30.44+2.0
1149.63ge
0.81
10.78 io
1314,86:
0.088
Weighted
22.89
1208.
0.43
15
2777.30,541:
0.38
30.22
1202,07732
0.76
1333,59,3 10.67
0.085
Linear
21.89
1241,55g14
0.39
35.564255
2531,69gu
0.50 30.0021.1
1160.22gs
11.11
1299.06g
Expanential
22.22,1m
1182.50gaeas
0.42
32.89
2568.07as..
0.42 31 11.56
1156,45g343
0.86
9.56g124
1315.87.0s
0.069
Polynomial
23 23.44:212
1194
0.46
33
33.674155
2504.92491
0145 33.00:20:
1142,54ga0
0.95
11,332100
1268.
0.101
Cosine
22.22,220
1164,77,255
2446.04:12
3170
1115.86.a2e
9.67
1268.06g447
F.6 TOKEN USAGE DISTRIBUTION ON TRAVELPLANNER
In addition to the answer pass rates discussed main results, we also examine the token usage distribution across queries of varying difficulty. Figure 4 presents the average token usage and corresponding pass rates on TravelPlanner, grouped by difficulty level.
Vanilla
7000
Global Dudget
Planned Global Bodget
6000
Buddgge
60
54. 0
57.2
Vazila
(343%) 3452 33 takess)
Glohal
(30,55
Pus Rate (%5)
Batges35
6000
(16.7%
sobend)
3000
20. 20.027.2
21.319.7 23.024.4
2000
20
1000
Hard
Fasy
Medium
Hard
Figure 4: Token usage and pass rate analysis across difficulty levels on TravelPlanner. (Left) Token usage distributions. (Right) Answer pass rate (%) by difficulty level.
As expected, we observe that token usage increases with query difficulty across all models more complex tasks naturally require deeper reasoning and longer responses. However, methods using global budget constraints exhibit consistently higher token usage across all difficulty levels compared our approach MANN-NND-BUGGEE This suggests that global budgeting fails to adapt to query complexity, resulting in inefficient allocation of compute.
Moreover, global methods only consume more tokens but also suffer from reduced answer pass rates at every difficulty level. This inefficiency leads to lower overall E3, further reinforcing the advantage of our uncertainty-awart local budgeting strategy. contrast, PLAN-AND -EN----EEETT adapts computation based on sub-question difficulty, achieving better calibration of reasoning effort across both simple and complex queries.
G ADDITIONAL EXPERIMENTAL DETAILS
G.1 DATASET DESCRIPTIONS AND EVALUATION METRICS
To evaluate the general applicability of our framework, select three reasoning-heavy benchmarks spanning symbolic math, instruction following, and long-horizon planning. The dataset statistics are shown in Table [5]
MATH-500 (Lightman et al. 2024). This is a curated 500- 100-problem subset from the full MATH dataset, designed test symbolic, multi-step math reasoning. Each problem requires the model
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to interpret, manipulate, and solve high-school-lever mathematical expressions. measured using exact-match accuracy against gold answers.
Performance is
Naturallnstructions (Wang al. 2022). This a broad instruction-fol followig benchmark con- sisting of over 1600 tasks covering question swering, classification, transformation, and rea- soning. We randomly sample 500 queries from the public split for evaluation. Since an- swers are open-ended and linguistic, use the ROUGE score to measure semantic overlap with the reference answers.
Table 13: Dataset statistics. LLaMA 3.1-8B per-
formance also provided.
MATII-500
Natural Instructions
Travel Planner
Math Reasoning
Instruction Following
Agentic Planning
Task
QA Pairs Metrics
500 Accuracy
500 ROUGE
180 Pass rate
TravelPlanner (Xie et al. 2024). This is
challenging planning benchmark that simulates real-world itinerary construction under hard con- straints (e.g., timing or location compatibility) and soft common sense preferences. We focus on the sole-planning setting where all relevant knowl- edge is embedded the prompt, and no tool use is required. We evaluate on the validation set using the hard constraint pass rate, measuring whether the generated plan satisfies the minimal feasibility constraints (e.g., no overlaps or missing connections). We omit the stricter full success rate (which includes common sense and preference matching) isolate planning competence. Prior work reports that even strong proprietary models achieve limited performance, highlighting the task's 's inherent complexity.
LLaMA 3.1-8B
48.76_074
21.72
2.28
Performance
G.2 LICENSES FOR EXISTING ASSETS
All models and datasets used in this work are publicly available and used in accordance with their respective licenses:
PrepSeee-RR--istilll -wanll and DeepStea-KK--DDitilLLLLLaA.. (Guo et al.
2025 released under the DeepSeek open-source model license available at https::/
github. com deepseek wtr.eaeeeeeeeeeeeeeeeei///aaa/
QwQ-32B Team 2025 is licensed under the Apache License 2.0.
OpenAl 04-mini OOpenAl
2025
accessed via the OpenAl API under the terms service
and usage policies listed at lhtpp:: http///ppaaa..a// www. htt/pppliiiiciis//eeeeeeg--e- No model weights are released or modified.
LLaMA 3.s-st- & LLaMA 3.3-70B-Instruct: is used via Open and follows Meta's 's LLaMA 3 license available at https:/ // /tt.a/aaaaaaaaaaaaaaaaaaaaaaaaaa wwwmm/aaammiceeeeeeeeaaaaaaaaeee
All datasets (MATH-500, NaturalInstructionss, TravelPlanner) are publicly available and properly cited. They are used for evaluation purposes under academic and research-friendle terms use.
G.3 COMPUTE RESOURCES
All experiments were conducted using APl-accessible large language models, including OpenAl's o4- mini and models hosted via OpenRouter (e.g., operates entirely inference time through prompting, the computational cost is directly proportional to the number tokens generated. We report token usage for each setting in the main paper, which be used to estimate wall-clock runtime given model-specific generation rates (typically 50-80 80 tokens/sec, depending on the provider) and the parallelism used.
Since our method
All data preprocessing, prompt generation, and evaluation
performed on cloud- based virtual machine equipped with an Intel Xeon ES-2698 CPU and 500GB of main memory. No model training fine-tuning involved, and the overall compute requirements modest.
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H
PROMPT TEMPLATES
Prompt Templates for Question Decomposition
-Goal-
You are an experienced expert in domain and exam question designer. Your role to help students break down challenging math problems into series of simpler, high-level sub-questions.
We don't want too many detailed sub-questions, which are beneficial for testing students' ability in an exam. Each sub-question should build on the previous one so that, once all have been answered, the complete solution clear.
Your output should be list of sub-question with brief hints explaining the purpose of each step, but you should not reveal your internal chain-of-the of-thought either the final solution.
Instructions for Decomposition:
First, analyze the problem and identify the key ideas needed to solve it. Then, generate series of 2 to sub-questions that lead the student step by step to the complete solution. The difficulty level of the problem is presented out of 5, where 1 is easy, and is hard. Please adjust the number of sub-questions based on the level. Ideally, we want fewer sub-questions for easy problems and more sub-questions for challenging problems.
DO NOT perform reasoning. directly output those sub-questions based on your gut feelings; only output the list of sub-questions with brief hints for each.
Your answer should be a list of numbered sub-questions. Each sub-question should have brief accompanying hint that explains what the student will achieve by answering that part.
Example Decomposition:
**Problem:*** Find the remainder when (9 x 99 x 999 ... 99 ...9) is divided by 1000.
999 9's
**Levell** 3 out of
** #Dcomposedd Sub-questions:**
Compute the product modulo Hint: Simplify each term using (10 2 mod 8), noting that (10* = 0 mod 8) for k 2 3, leading to terms (-1 mod 8).
2. Compute the product modulo 125.
Hint: Recognize (103 = 0 mod 125), terms for (A > 3) become (-1 mod 125).
Calculate the product of the first two terms and combine with the remaining terms.
3. Solve the system of congruences using the Chinese Remainder Theorem.
Hint: Combine the results from modulo 8 and modulo 125 to find common solution modulo 1000.
A student has presented you with the following math problem: Problem: <problem>
Level: <level> out 5
**REMEMBER** you are not allowed to think about it, please directly generate the answer in the following:
Decomposed Sub-questions:
Prompt Templates for Question Difficulty Evaluation
You are an experienced expert in <domain> and exam question designer. Your task is to evaluate the difficulty level of given exam problem and its sub-questions by comparing
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against of benchmark questions of known levels.
Based on their levels, you will need to assign each subquestion a portion of the credits (assuming the total credit points 100 for the whole problem).
Each level reflects increasing complexity from (easiest) (most challenging). Evaluate based on the conceptual depth, steps involved in solving, required knowledge, and potential for misdirection.
Use the following benchmark examples as references:
<benchmarks>
You will be provided question and its subquestions. You will evaluate the diffi- culty level of the problem and its sub-questions.
Assuming the whole problem is worth 100 points, you assign each sub-question a portion of the score points.
Adhere to the given subquestions, and DO NOT make new subquestions.
Sum of each subquestion's 's credits MUST EQUAL to 100.
You must return the result in structured ISON format:
"problem": ("reason": "...", "evaluated peaetteddlleel::: level": level_qi _q) "1": I"reason": ... "evaluated l_vell: level_1, "credit": credit_ 1),
"2": ""reason": "...", levaluated__evee level_ 2, "credit": credit_2),
where
"reason": short explanation (up 50 words) of your level assessment.
"evaluated_levell an integer from to 5 indicating your judgment. "credit": integer between 100 indicating when the question is solved correctly, how many credit can be given.
Evaluate the level of the following question: Problem: <problem>
Sub-questions: <steps>
Output:
Prompt Templates for Vanilla Model
< dataset-specific instruction>
Please reason step by step. and conclude your answer in the following format:
<dataset specific output format>
Question: <query>
Reference: <reference> (only applicable to TravelPlanner)
Output: <think
Prompt Templates for Global Budget Model
dataset-specific instruction>
Please reason step by step, and conclude your answer in the following format:
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<dataset specific output format>
Question:
<query>
Reference: <reference> (only applicable to TravelPlanner)
Let's think step by step and use less than <budget> tokens. Output: <think>
Prompt Templates for Planned Vanilla Model
<dataset-specific instruction>
The problem is given by an overall description, difficulty level out of 5, followed by series of sub-questions as a hint. All the credit is given when you provide correct final answer for the overall problem. Please solve the question efficiently and clearly achieve as much credit as possible.
Let's start the exam. You are being given this math problem: **Problem:** ** <query>
**Reference:** <reference>(only applicable to TravelPlanner) **Level:*# <level> out of
You may think following these sub-questions feel free other methods that works the best towards getting the final answer: <decomposed>
Please provide your final answer the following format: <dataset specific output format>
Output: <think
Prompt Templates for Planned Global Budget Model
<dataset-specific instruction>
The problem is given by overall description, difficulty level out of 5, followed by a series of sub-questions as a hint. All the credit is given when you provide correct final answer for the overall problem. Please solve the question efficiently and clearly to achieve as much credit as possible.
Let's start the exam. You are being given this math problem: * **Problem:** <query>
**Reference:** <reference> (only applicable to TravelPlanner) **veell:** <level> out 5f
You may think following these sub-questions or feel free to use other methods that works the best towards getting the final answer: <decomposed>
Please provide your final answer the following format: <dataset specific output format>
Let's think step by step and use less than <budget> tokens. Output: <think>
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Prompt Templates for Planned Local Budget Model (Ours)
<dstaset-specific instruction>
The problem is given by an overall description, difficulty level out of 5, followed by series of sub-questions All the credit is given when you provide correct final answer for the overall problem.
hint.
Please solve the question efficiently and clearly achieve much credit as possible.
Let's start the exam. You being given this math problem: **Problem:** <query>
* **eference:** <reference> (only applicable to TravelPlanner)
* <level> out of 5
You may think following these sub-questions feel free to use other methods that works the best towards getting the final answer: <decomposed> (For each decomposed subquestion:) Please only think little, and directly solve it using up to <budget> words.
Please provide your final answer strictly following the format: <dataset specific output format>
Output: <think>
25


Reviews:
[{"review_id": "D8Uy8TXvhp", "reviewer": "Reviewer_V5hD", "summary": "Plan-and-Budget is a test-time framework to address \"reasoning miscalibration\" in LLMs - where models either overthink (waste tokens on simple queries) or underthink (fail on hard queries).  (1) Plan step decomposes queries into sub-questions using a lightweight LLM, estimating complexity for each; (2) Budget step allocates token budgets to sub-questions using decay-based scheduling (more tokens to early/uncertain steps). Introduces BAM - modeling uncertainty reduction as inverse power law, deriving optimal allocation as b_ij = B * (c_ij*β_ij)^(1/(β_ij+1)) / Σ(c_ik*β_ik)^(1/(β_ik+1)). Proposes E3 metric = A^2/T (accuracy squared over tokens) to jointly measure quality and efficiency.", "questions": "How do you actually estimate c_ij and β_ij in practice? Paper claims BAM is theoretical foundation but never shows these parameters can be computed. Is the planner LLM's complexity estimation reliable given its poor standalone performance (48.76% on MATH)? Can you compare token counts including planning overhead vs end-to-end latency? Does polynomial decay (best performer) actually implement BAM's allocation or is it just a heuristic that happens to work? Why does E3 use A^2 not A - this heavily penalizes any accuracy drop, is that appropriate for efficiency metric? What happens if you just use confidence-based early stopping without decomposition? Table 3 shows cosine scheduling gets 5.92 E3 vs 5.26 global budget on DS-Qwen MATH - is 12% gain worth the added complexity? How sensitive is performance to number of sub-questions (paper uses 2-5)? Can you show examples where BAM's theoretical allocation differs from decay heuristics and which performs better?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I feel like this addresses real problem - overthinking/underthinking is genuine issue in reasoning LLMs. Theoretical grounding via BAM provides principled justification beyond heuristics. E3 metric is sensible - A^2/T appropriately weights correctness over pure efficiency unlike A/T. Model-agnostic approach works across multiple architectures without retraining. Comprehensive experiments across three diverse domains with consistent improvements. Clear presentation of decay scheduling strategies (linear/polynomial/exponential/cosine). Nice result showing smaller model matching larger model efficiency. Ablations show both planning and budgeting contribute gains.", "weaknesses": "Core assumption is questionable? - uncertainty decomposition (epistemic vs aleatoric) requires Bayesian treatment but paper doesn't actually compute posterior p(theta|D), just hand-waves with \"Monte Carlo approximation\" in Appendix B without showing how this applies to deterministic transformer inference. BAM's power law U_epistemic = c/b^β is asserted not derived - why inverse power law specifically? Parameters c_ij and β_ij are never actually estimated, making Eq 6 theoretical only. Decay schedulers are admitted heuristics that don't implement BAM's optimal allocation. Gap between theory (BAM requires knowing c_ij, β_ij) and practice (use position-based decay) is huge - BAM feels like post-hoc justification for simple heuristic. Additional LLM call for planning adds overhead not fully accounted for in token counts. Complexity estimation by planner LLM (LLaMA-3.1-8B with 48.76% MATH accuracy) is unreliable - how can weak model judge difficulty? Table 3-5 show planning alone helps substantially (Planned Vanilla vs Vanilla) - suggests decomposition is doing most work, not budget allocation. Some results show minimal gains - NaturalInstructions improvements are modest (2.33-> 2.86 E3 for DS-Qwen). E3 metric has quadratic bias toward high accuracy - 90% accuracy with 1000 tokens (E3=81) beats 80% with 500 tokens (E3=128) which seems backwards for efficiency metric. No comparison to simpler baselines like adaptive stopping based on model confidence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PbSNcQy7lz", "reviewer": "Reviewer_P17L", "summary": "The paper studies reasoning miscalibration in LLMs, overthinking on easy cases and underthinking on hard ones, and proposes a test‑time framework, PLAN‑AND‑BUDGET, to allocate computation adaptively. The authors first introduce a theoretical Budget Allocation Model (BAM) that models per‑subquestion epistemic uncertainty and derives a closed‑form optimal token allocation under a query‑level budget. They then operationalize this with (i) a Plan step that decomposes a query into sub‑questions and estimates relative difficulty weights, and (ii) a Budget step that assigns local budgets using simple decay schedules (linear/polynomial/exponential/cosine). A new compute‑aware metric, balances accuracy/quality and average decoding tokens. Experiments on MATH‑500, NaturalInstructions, and TravelPlanner with four models (including DS‑Qwen‑32B, DS‑LLaMA‑70B, QwQ‑32B, o4‑mini) show consistent efficiency gains without accuracy loss; e.g., on MATH‑500, DS‑LLaMA‑70B improves (E^3) from 4.55 (Global Budget) to 5.89 (Plan‑and‑Budget), and on TravelPlanner DS‑Qwen‑32B improves from (E^3=0.16) to (0.47) (Polynomial schedule), effectively narrowing the gap to larger models.", "questions": "1. Uncertainty proxies: Which proxies were actually used in experiments (entropy drop, self‑consistency, verifier loss), and where do they enter the allocation beyond the position‑based decay? Can you report correlations between these proxies and realized per‑step utility (accuracy gains vs. tokens)? \n2. Eq. (7) and weights: Please confirm the intended formula (b_{ij}=\\frac{w_{ij}d_{ij}}{\\sum_k w_{ik}d_{ik}}B_i). Also, how are difficulty weights (w_{ij}) produced in non‑math tasks? Do they come from the same planner JSON in Appendix H? \n3. Metric sensitivity: If (A) is ROUGE (0–100) versus accuracy (0/1), how sensitive are the (E^3) conclusions to rescaling (A) to ([0,1])? Could you also report (A/T) to compare with prior \"accuracy per token\" conventions? \n4. Token comparability: Given different tokenizers/reporting, do results change when using characters, latency, or cost as (T)? A small subset analysis would clarify cross‑model claims. \n5. Planner overhead and robustness: How often does planning increase total tokens (planning+reasoning) compared to Vanilla? Please provide distributions (not only averages) and an ablation with no planning but local budgets to separate the two contributions. \n6. When does it hurt? Are there regimes where global budgeting wins (e.g., very short tasks)? Diagnostics (e.g., by difficulty bin) would be helpful. \n7. Since mainstream SOTA LLMs do not support token precision thinking effort controling, I am unaware of the importance of doing budget controllon. Please ellaborate this if possible.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Clear formulation & theory: BAM provides a principled lens on how to distribute limited compute across subproblems. \n2. Simple, model‑agnostic implementation: Works at prompt time; no training; adds a lightweight planning LLM whose contribution is controlled by planned baselines. \n3. Consistent gains across tasks/models: Large (E^3) improvements with stable or improved accuracy; front‑loaded schedules (polynomial/cosine) perform best on complex tasks. \n4. Bridging model sizes: The method helps smaller models approximate the efficiency of larger ones on TravelPlanner. \n5. Reproducibility details & prompts provided.", "weaknesses": "1. Uncertainty signal not directly validated.\n   The budgeting step is motivated by uncertainty reduction, yet the experiments do not present a correlation between used proxies (e.g., prefix‑entropy drop) and actual downstream gains, nor do they show a head‑to‑head between decay‑only vs. truly measured uncertainty allocation. An ablation that (a) measures token‑level entropy and (b) reallocates budgets on‑the‑fly based on it would strengthen the central claim. \n\n2. Metric definition vs. datasets.\n   (E^3=A^2/T) is defined using accuracy yet ROUGE is used as (A) on NaturalInstructions. The consequences of using a graded similarity metric (not bounded like accuracy) should be discussed; consider normalizing to ([0,1]) and reporting sensitivity of rankings to the choice of metric. \n\n3. Cross‑model comparability of token counts.\n   Since tokenization and API reporting differ, (T) is not strictly comparable across models; thus the statement that \"o4‑mini consistently achieves the highest (E^3)\" may partly reflect tokenizer efficiency rather than true compute efficiency. Adding characters generated, wall‑clock latency, and $ cost would provide a more robust, model‑agnostic view. \n\n4. Notation and clarity.\n   Clean up Eq. (7) and unify symbols for difficulty vs. schedule multipliers. Clarify the semantics of (\\beta) (is higher (\\beta) \"easier\" or \"more complex\"?), and restate the unimodality intuition with a small plot or example. \n\n5. Ablations on the planner.\n   While the planner’s standalone performance is low, it still shapes problem structure and budgets. Add ablations varying planner quality (e.g., weaker/stronger planner; random decomposition; noisy difficulty weights) to test robustness, and report the extra token overhead of planning vs. savings in reasoning tokens. \n\n6. Adaptive vs. fixed decay.\n   The best schedules are front‑loaded. It would be natural to choose the schedule per instance via quick signals (e.g., early prefix entropy), approaching the BAM optimum more closely. A small controller choosing among schedules could be evaluated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nGl5YCzCJL", "reviewer": "Reviewer_fET4", "summary": "This paper addresses the inefficiency of reasoning processes in large language models (LLMs), where long chains of thought (CoT) can sometimes help performance but also lead to two failure modes: overthinking and underthinking. The authors argue that these issues stem from uncalibrated budget usage—that is, the model’s inability to decide how much computation (tokens) to spend on each sub-question. To address this, the paper introduces a theoretical framework, BAM (Budget Allocation Model), which formulates an optimal trade-off between correctness and computation efficiency based on uncertainty reduction. Building on BAM, the authors propose PLAN-AND-BUDGET, a model-agnostic, test-time method that decomposes complex reasoning queries into sub-questions and allocates token budgets adaptively according to estimated difficulty. Extensive experiments across multiple open-source and closed-source LLMs (e.g., DeepSeek-R1, Qwen-32B, o4-mini) and diverse benchmarks show that PLAN-AND-BUDGET achieves comparable or improved reasoning accuracy while significantly reducing token usage.", "questions": "1. Based on my understanding of Eq. (6) in the BAM formulation, the strategy prioritizes medium-difficulty sub-questions while allocating fewer tokens to very easy or very hard ones. Could this behavior introduce bias, making the model less likely to attempt challenging reasoning steps?\n\n2. For the different baselines, are all models given the same total token budget limit? If so, how is this limit determined in practice?\n\n3. From my understanding, the method will add certain constraints in the prompt (e.g., “use less than B itokens”). This raises a question: how reliably do LLMs follow these token-limit instructions in practice?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The paper is detailed and transparent, making it easy for readers to understand and replicate the proposed approach.\n\n2. The method design is rigorous: using uncertainty as a proxy for reasoning difficulty and allocating computation where it provides the highest marginal gain is both intuitive and elegant.\n\n3. The experiments are comprehensive, covering multiple models (both open- and closed-source) and diverse benchmarks, demonstrating strong generalizability.", "weaknesses": "1. Although the method is conceptually rigorous, Table 3–5 show that the method may still hurt accuracy compared to full-length baselines.\n\n2. Some of the simpler baselines (e.g., Planned Vanilla or Global Budget) already achieve noticeable token savings with minimal performance loss. This raises the question of whether the added complexity of PLAN-AND-BUDGET always justifies its gains.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HJxygUr1qh", "reviewer": "Reviewer_Mzjt", "summary": "This paper addresses a critical problem in LLM reasoning, balancing computational efficiency and reasoning quality at test time. The authors introduce PLAN-AND-BUDGET, a lightweight, model-agnostic framework that combines structured query decomposition with adaptive token allocation. Building on a formal Budget Allocation Model (BAM), the framework distributes computation dynamically across sub-questions based on uncertainty estimates, mitigating both overthinking and “underthinking” in reasoning tasks. The study shows that PLAN-AND-BUDGET improves reasoning efficiency across multiple tasks and models without sacrificing accuracy.\n\nI personally like this paper very much. I think it addresses a very important issue in reasoning. This contribution also has the potential to connect with many real-world applications involving reasoning models.", "questions": "1. This is just a question I am curious about. Upon reviewing the tables, I noticed that for some tasks, such as TravelPlanner, the token consumption is not significantly reduced. I would be interested to hear the authors’ thoughts on whether there might be more aggressive or adaptive strategies to further reduce token usage.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. I find the motivation of this study very compelling. It addresses a fundamental issue in LLM reasoning, the need to balance reasoning depth and computational budget, and offers a promising approach grounded in uncertainty theory.\n2. The theoretical framework (BAM) provides a solid foundation, and the derivation of the E3 metric as an efficiency-aware measure is well-justified.\n3. I think the empirical evaluation is thorough, including multiple reasoning benchmarks and model scales. The reported improvements in efficiency and comparable accuracy across datasets are impressive and convincingly support the paper’s claims. \n4. The paper is well written and logically organized. The logic flow is very clear!", "weaknesses": "1. A small suggestion. The paper derives two key principles for effective reasoning: (1) reasoning should be structured; and (2) computation should be adaptive focus. For example, the paper mentions: “Inspired by human problem-solving strategies.” While I agree with both ideas and prior works are well cited in the literature, the derivation would be strengthened by citing more prior work when the strategies are introduced in the framework section, which can ground the proposed methods in solid scientific studies.\n2. I noticed that while accuracy remains comparable, the ROUGE scores in some instruction-following tasks slightly decrease. It might be useful for the authors to discuss whether this trade-off arises from task differences and how PLAN-AND-BUDGET might be adjusted for scenarios requiring richer generative outputs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jRgK5jvYzn", "reviewer": "Reviewer_f6TR", "summary": "The paper proposes \"plan and budget\", a test-time framework that improves reasoning efficiency in large language models by dynamically allocating computational effort. It addresses reasoning miscalibration by decomposing each query into sub-questions and assigning adaptive token budgets based on estimated uncertainty. The approach is a two-step method (plan + adaptive budget) that requires no retraining and is supported by the budget allocation model, a theoretical framework for uncertainty-aware token allocation. Experimental results show consistent efficiency and accuracy gains over vanilla decoding and other budget-based baselines across multiple datasets and models.", "questions": "See the weaknesses section.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The evaluation is broad and comprehensive, covering multiple datasets and model sizes, which strengthens the generality of the results.\n\n\n* The paper provides theoretical guidance through the Budget Allocation Model (BAM), offering a principled foundation for the proposed approach.\n\n\n* The writing is clear and well-structured, making the ideas easy to follow and the contributions well-presented.", "weaknesses": "- The novelty is limited. Prior work, such as [1] already explored token allocation at the request level, while this paper applies a similar idea at a finer granularity by decomposing each request into multiple sub-requests. How fundamentally different is this from existing allocation frameworks?\n\n\n\n- The paper does not justify why adaptive budgeting is necessary instead of allowing unlimited tokens with early termination for unpromising or converged reasoning chains. Would early stopping achieve similar or better efficiency without requiring explicit budget allocation?\n\n\n\n- There is no direct comparison to [1] or other related approaches that use early termination or reasoning truncation to reduce computational cost. Including these baselines would better contextualize the contribution.\n\n\n\n- The method requires an additional LLM call for the planning step, which increases the inference cost and latency. How significant is this overhead relative to the total efficiency gains?\n\n\n\n- The use of entropy as a heuristic for estimating uncertainty is not well justified. Could a learned or predictive uncertainty estimator perform better and make the framework more robust?\n\n\n\n- The approach assumes that each task can be reliably decomposed into subtasks, which may not hold for all reasoning domains. How does the method behave when such a decomposition is noisy or ambiguous?\n\n\n\n- The theoretical–practical gap is unclear. Although the paper presents formal equations, many quantities are later approximated heuristically. This raises the question: how much of the observed performance comes from the theoretical model versus empirical tuning?\n\n[1] Efficiently serving llm reasoning programs with certaindex. Fu, Yichao and Chen, Junda and Zhu, Siqi and Fu, Zheyu and Dai, Zhongdongming and Qiao, Aurick and Zhang, Hao.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)