CyclicReflex: Improving Reasoning Models via Cyclical Reflection Token Scheduling
Authors: Chongyu Fan
Yihua Zhang
Jinghan Jia
Alfred O. Hero
Sijia Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Large language model
Reasoning
Testing time scaling
Primary area: foundation or frontier models, including LLMs

Abstract: Large reasoning models (LRMs), such as OpenAI’s o1 and DeepSeek-R1, harness test-time scaling to perform multi-step reasoning for complex problem-solving. This reasoning process, executed before producing final answers, is often guided by special juncture tokens that prompt self-evaluative reflection. These transition markers and reflective cues are referred to as “reflection tokens” (e.g., “wait”, “but”, “alternatively”). In this work, we treat reflection tokens as a “resource” and introduce the problem of resource allocation, aimed at improving the test-time compute performance of LRMs by adaptively regulating the frequency and placement of reflection tokens. Through empirical analysis, we show that both excessive and insufficient use of reflection tokens, referred to as over-reflection and under-reflection, can degrade model performance. To better understand this trade-off, we draw an analogy between reflection token usage and learning rate scheduling in optimization. Building on this insight, We propose cyclical reflection token scheduling (termed CyclicReflex), a training-free decoding strategy that dynamically modulates reflection token logits with a bidirectional, position-dependent triangular waveform, incurring no additional computation cost. Experiments on MATH500, AIME2024/2025, AMC2023, GPQA Diamond and LiveCodeBench demonstrate that CyclicReflex consistently improves performance across model sizes (1.5B–14B), outperforming standard decoding and recent approaches such as TIP (thought switching penalty) and S1.

Published as a conference paper at ICLR 2026
CYCLICREFLEX: IMPROVING REASONING MODELS VIA CYCLICAL REFLECTION TOKEN SCHEDULING
Chongyu Fan
Yihua Zhangi Jinghan Jia
Alfred Hero Sijia L
'Michigan State University
University of Michigan, Ann Arbor
IBB Research
ABSTRACT
Large reasoning models (LRMs), such as OpenAT's ol and DeepSeek-RI, harness test-time scaling to perform multi-step reasoning for complex problem-solving This reasoning process, executed before producing final answers, often guided by special juncture tokens that prompt self-evaluative reflection. These transition markers and reflective cues are referred to as "reflection tokens" (e.g., "wait", "but", "alternatively"). In this work, we treat reflection tokens "resource" and introduce the problem of resource allocation, aimed at improving the test- time compute performance of LRMs by adaptively regulating the frequency and placement of reflection tokens. Through empirical analysis, we show that both excessive and insufficient use of reflection tokens, referred to as over-reflection and under-reflection, can degrade model performance. To better understand this trade-off, we draw an analogy between reflection token usage and learning rate scheduling in optimization. Building on this insight, We propose cyclical reflec- tion token scheduling (termed CyclicReflexx, a training-free decoding strategy that dynamically modulates reflection token logits with a bidirectional, position- dependent triangular waveform, incurring no additional computation cost. Exper- iments MATH500, AIME2024/2025, AMC2023, GPQA Diamond and Live- Code demonstrate that Cyclic consistently improves performance across model sizes (1.5B-14B), outperforming standard decoding and recent ap- proaches such as TIP (thought switching penalty) and $1. Codes are available at https:// com/ OOPTML- WPWMMLGGOMO Group/Cycliccceee Cycliicllooooo ffeffle........
INTRODUCTION
There has been a recent surge in the development of large reasoning models (LRMs), driven by the introduction of thain-of-thoughh (CoT) (Wei et al., 2022). Notable examples include OpenAl's ol (OpenAI, 2024), Qwen 2.5 (Yang et al., 2024), DeepSeek-RI (Guo et al., 2025), and Kimi-1.5 (Team al., 2025). These models perform multi-step reasoning by generating so-called reflection tokens, phrases such as "wait", "but", "alternatively". which signal hesitation, alternative exploration, or intermediate analysis. In parallel, test-time scaling (Snell et al., 2024; Liu et al., 2025; Sadhukhan et al., 2025) have emerged as a complementary approach for improving reasoning accuracy by expanding the breadth or depth of CoT traces during inference.
However, LRMs remain prone to reasoning failures due to mismanagement of reflection tokens, often resulting either underthinking or overthinking. Underthinking occurs when the model fails fully explore promising reasoning paths for complex problems, often terminating prematurely or switching strategies too soon (Wang et al., 2025b; Su et al., 2025). contrast, overthinking arises when the model generates excessive number of reflection tokens on simple problems, leading to unnecessary computational overhead (Chen et al., 2024; Kumar et al., 2025a). These observations show that, as internal signals for deliberative reasoning, reflection tokens play a critical role in shaping answer quality. The emerging challenges further underscore the need for principled mechanism to regulate reflection token usage during inference.
In this work, we introduce the concept of resource allocation for LRMs, treating reflection tokens valuable resource whose scheduling along the CoT trajectory (a.ka. reasoning trace) can be

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strategically designed to improve reasoning accuracy. The objective is optimize the quantity and placement of reflection tokens, adapting dynamically to the reasoning schedule and the difficulty of the current problem. For instance, some problems exhibit under-reflection. where too few reflection tokens result in premature answer generation, while others suffer from over-reflection, where excessive tokens stall progress by repeatedly looping on phrases like "wait". This raises a central question:
(Q) How can we achieve effective resource allocation in LRMs to mitigate both under-reflection and over-reflection?
To answer this, we draw a conceptual analogy between reflection tokens in LRMs and learning rates in optimization. Leveraging the landscape of thoughts (Zhou et al., 2025), we show that under- reflection mirrors the effect an overly small learning rate, leading to convergence to suboptimal solutions, while over-reflection resembles a large learning rate that causes divergence. We briefly introduce our motivation and the underlying intuition below.
Overview of motivation and rationale: From stepsize hedging to cyclical learning rates. The critical role of learning rates (also known as stepsizes) in shaping optimization dynamics has been extensively studied (Nesterov, 1983; Allen-Zhu & Orecchia, 201 4; Bubeck et al.. 2015). A recent theoretical advancement, the silver stepsize schedule (Altschuler & Parrilo, 2024; 2025), demonstrates that replacing a constant learning rate with an approximately periodic, hedging-styl styll schedule can provably accelerate convergence in gradient descent. This approach is known as stepsize hedging as alternates strategically between large and small stepsizes, balancing rapid (but potentially unstable) exploration with slower, more stable convergence. A similar stepsize hedging idea has been applied to deep model training through cyclical learning rate schedule (Smith, 2017), which alternates between large and small learning rates in triangular waveform. This strategy not only accelerates convergence but also enhances generalization, often eliminating the need for extensive hyperparameter tuning.
Motivated by the principle of stepsize hedging (Altschuler & Parrilo, 2024; 2025) and the demonstrated effeetivv- ness of cyclical learning rates in deep learning (Smith, 2017), we propose Reflic a training-free decod- ing strategy that dynamically modu- lates the logits of reflection tokens us- ing a positionn -eatednnde periodic tri- angular waveform (see Fig. 1). Just as cyclical learning rates alternate be- tween aggressive and conservative up- dates to balance exploration and con- vergence, CyclicReflex cyclically ad- justs the sampling likelihood of reflec- tion tokens to regulate the depth and stability of the reasoning process. Unlike conventional decoding methods, CyclicReflex is bidirectional, capable of both promoting and suppressing reflection token usage depending on the stage of generation. This flexibility enables CyclicReflex to address both under-reflection (insufficient reasoning) and over-reflection (excessive, redundant reasoning). offering a principled mechanism for reasoning modulation inspired by optimization dynamics.
LRM
Predict
Logits
Decode
Text
ATME2024 Results
Original Step
aedertonokne Step
Incorrect
snawest
Lalo
Dol
Lone
CyclicReflex BV
Lool
Loo3 Peetecaraaa
LoaD
Ansuere
Figure I: Schematic overview of our proposed method (CyclicRe- flex). The rightmost subfigure presents comparison of final answer accuracy between CyclicReflex, the original LRM (DeepSeek-R) R1- Distill-Llama-8B) and decoding variants using TIP (Wang et al., 2025b) and S1 (Muennighoff et al., 2025).
Contributions. We summarize our contributions below.
We introduce and formalize the problem of resource allocation in LRMs by treating reflection tokens as a computational resource, motivated by the dual challenges of under-reflection and over-reflection in reasoning generation.
We draw a conceptual analogy between reflection token scheduling and learning rate scheduling in optimization, and validate it through the landscape of thoughts. Guided by that, we propose CyclicReflex, test-time decoding strategy that eyclically modulates reflection token logits to dynamically balance reflection during generation.
We conduct comprehensive experiments across six reasoning benchmarks and multiple model scales (1 (I.5B-B)), demonstrating that CyclicReflex consistently improves both final-answer accuracy and self-correction capability, outperforming recent methods such as TIP (Wang et al., 2025b) and $1
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(Muennighoff et al., 2025). Moreover, CyclicReflex integrates seamlessly with other est-timm scaling techniques, yielding additional performance gains.
2 RELATED WORK
LRMs and COT. CoT (Wei et al., 2022) enables LRMs to solve complex tasks through intermediate reasoning steps before reaching final answer. This technique underpins many recent LRMs, including OpenAl's ol (OpenAI, 2024), Qwen 2.5 (Yang et al., 2024), DeepSeek-R1 (Guo et al., 2025), and Kimi-1.5 (Team et al., 2025), which often employ reinforcement learning to further improve their reasoning performance. Guo et al. (2025) show that even smaller models benefit substantially from fine-tuning with CoT-style data. A hallmark of CoT reasoning in these models is the emergence of reflection tokens (words like "wait" or "but") that signal deliberation or self- correction, marking shift from fast to slow thinking (Kumar et al., 2025b; Li et al., 2025). In this paper, we show that the reasoning performance of LRMs can he enhanced by applying cyclical logits manipulation reflection tokens.
Efficient reasoning. Despite their impressive capabilities, LRMs often exhibit reasoning inefficien- cies. Overthinking arises when the model generates unnecessarily long reasoning traces, leading to inflated outputs and increased computational cost (Chen et al., 2024; Kumar et al., 2025a). In contrast, underthinking occurs when the model halts reasoning too early, failing adequately explore promising solution paths (Wang et al., 2025b; Su et al., 2025). Therefore, ensuring both the efficacy (i.e., answer accuracy) and efficiency (i.e., generation length) of reasoning is crucial. Building on this line of research, some approaches modify model behavior through post-training interventions. For instance, Luo et al. (2025); Aggarwal Welleck (2025); Hou al. (2025) use fine-tuning or reinforcement learning to explicitly control reasoning length. There also exist works that adopt training-free free strategies. Yang et al. (2025c) propose guiding smaller models with larger ones at inference; Wang et al. (2025b) penalize reflection token logits to reduce over-reflection; And Yang et al. (2025b); Muennighoff et al. (2025) develop early-exit mechanisms for efficient decoding. Our method also falls into the training- -fre category but differs in its dynamic to adaptively address both under- and over-reflection without model modification.
Test-time scaling. A growing body of work enhances LRM reasoning via test-time scaling. Basic strategies include mamually inserting reflection tokens (e.g.. "wait", "but"") to prompt deeper thinking (Muennighoff et al., 2025; Jin et al., 2025; Chen al., 2025c). More methods such as Best-of-N generation and self-consistency sampling (Wang et al., 2022; Irvine et al., 2023; Brown et al., 2024; Kang et al., 2025; Wang et al., 2025a) aim to select the most promising answer among multiple candidates, often guided by reward models. Structured decoding approaches, such as beam search (Feng et al., 2023), tree-of-thought (ToT) (Yao al., 2023), and Monte Carlo tree search (MCTS) (Zhou et al., 2023), further improve answer quality by enabling the model to reason over multiple candidate paths. In pursuit of controlled reasoning, Wu et (2025) propose thinking intervention, which selectively inserts or edits specific thinking tokens during generation to tailor LRM behavior for downstream tasks. Recent analyses (Snell et al., 2024; Liu et al., 2025; Chen et al., 2025b; Zhang et al., 2025; Ghosal et al., 2025) also highlight that the effectiveness of test-time scaling varies with problem difficulty, motivating strategies that adapt to instance complexity. Our method provides the adaptive test-time compute by dynamically adjusting the influence of reflection tokens throughout the reasoning trajectory. We show that it consistently improves accuracy across difficulty levels and can be integrated seamlessly with other test-time strategies like Best-of-N and beam search.
RESOURCE ALLOCATION IN REASONING MODELS
In this section, we begin by introducing preliminaries LRMs, including their chain-of ofain--f-thought trajectories (i.e., reasoning traces) and the use of reflection tokens. We then motivate the problem of resource allocation over reflection tokens through an existing technique: thought switching penalty (TIP) test-time compute strategy. This warm-up study illustrates the critical influence and sensitivity of reflection tokens on the final answers produced by LRMs.
Preliminaries on LRMs, reasoning traces, and reflection tokens. Unlike conventional LLMs, LRMs can incorporate explicit thinking stage before arriving at final answer (Li et al., 2025;
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Kumar et al., 2025b; Chen et al.,, 2025a). This thinking stage is typically realized through a CoT trajectory composed of multiple reasoning steps.
To be concrete, let x denote the input question, r the reasoning trace, and y the final answer. The rea- soning trace can be written a sequence of T intermediate reasoning steps (or thoughts) that serve as a precondition for produc- ing the final answer y. These thoughts are often prefixed by lexical cues such as "wait", "but", and "alternatively". In this work, we collectively refer to such words as reflection tokens, as they signal the LRM's introspective or deliberative thinking during generation. These tokens typically signal hesitation, reconsideration or alternative exploration within the reasoning trace, serving as implicit markers of the model's 's engagement in step-by-step reasoning. For ease of understanding, Table 1 illustrates an example where an LRM tackles a math problem (x), along with its generated reasoning trace (r), final answer (y), and the highlighted reflection tokens embedded within the reasoning process.
Table 1: An example from MATH500 answered by DeepSeck-RR R --iisill-QQen---- Reflection to- kens are highlighted in red.
[r, P2........ r.
Generation (r and y) vs. input x
Computo scesia (-I) Express your answer radians.
<think>Okay, so noed to compute aresin of of/1/..... But I should make
I'm thinking about the right
range for aresin. think it's between -x and n/2 right? So it's the principal value..... I know that sine nogative in the third and fourth quadrants, aresin only gives me angles in [-x /2, a /2], that's the fourth quadrant..... Wail, let me deuble-check If take siu(-> /6), that's 's same - sin(a /6), which 1/2. So the answer should he -5/6 radians.c/hhinks </thinks
since
y The acssin of is -x/6 radians.
Influence of reflection tokens on final answer accuracy. Next, we present warm-up study to illustrate the critical role of reflection tokens in determining final answer quality, particularly across varying levels of problem difficulty. Prior work has identified several common issues in LRM, including "underthinking" (Wang et al., 2025b; Su al., 2025e- the model prematurely abandons promising lines of thought, resulting in insufficient reasoning depth-and "overthinking" (Muennighoff et al., 2025; Chen et al., 2024; Kumar al., 2025a)- wheere excessive, unnecessary reasoning steps obscure or derail the correct final answer. These studies have observed that intervening in the reasoning trace with reflection tokens can help guide the model's thought process (Wu et al., 2025). A common approach to controlling and integrating the effect of reflection tokens in LRM generation is to modify the decoding strategy to account for their occurrence. One such method is TIP (Wang et al., 2025b), which was proposed to discourage the generation of reflection tokens and thereby penalize frequent thought switches during the reasoning trace.
Motivation:
Given the set of reflection tokens V. TIP introduces a logit penalty (a) to the predicted score ž when generating reflection token E V at reasoning step t, yielding the updated logit
Zz, + a, if E V and < To
TIP(a):
$,.0
(I)
otherwise
where a controls the strength of the logit intervention, and To specifies the time window over which the adjustment is applied. For clarity, TIP(a)) denotes the TIP-based decoding strategy parameterized by a. It is worth noting that TIP sets a S to penalize frequent thought switches.
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100
150
200
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Reflection Count
Diffculty
(a) Difficulty groups
(b) Accuracy
(c) Generation examples
Figure 2: (a) Answers from DeepSeck-RR RI
on MATH500 clustered into Easy, Medium, and
Hard using K-means over reflection word count and generation length. Each point represents one answer. (b) Accuracy of original decoding and TIP across difficulty levels. (c) Generation examples of original decoding and TIP for problem from the Medium category.
Despite the heuristic nature of choosing and To. the TIP-based decoding strategy (Wang et al., 2025b) provides useful motivation for studying resource allocation over reflection tokens and its impact reasoning effectiveness (e.g., on the MATH500) across different problem difficulty levels. As shown in Fig. 2(a), we categorize MATH500 problems into three difficulty levels-Ear Medium, and Hard-based based generation length and reflection token count, with the final-answer accuracies
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92%, 71%, and 21%, respectively. We observe that for more challenging problems, LRMs tend to produce longer reasoning trajectories and more reflection tokens, indicating deeper engagement in problem-solving. Fig. 2(b) next compares the accuracy of the original decoding strategy and TIP across the difficulty groups in Fig. 2(a). As shown, TIP improves accuracy on Hard problems but reduces accuracy Easy and Medium problems. This suggests that TIP's constant logit manipulation strategy (agnostic to the reasoning step t) does not yield optimal reasoning control. Furthermore, Fig. 2(c) shows Medium-level example comparing original decoding with TIP. The first divergence in reasoning is highlighted in red. Under original decoding, the model introduces a transitional reflection ("Wait, no, in eircular permutations, fixing one seat...), generating 32 reflection tokens before arriving at the correct answer. In contrast, with a thought-switching penalty < 0 in (1), TIP reshapes the trace ("Now, need find one person..."), producing only reflection tokens and yielding an incorrect answer. This shows that TIP provides only one-directional reflection control (penalizing reflection token logits). Hence, a bi-directional, dynamically adaptive (non-constant) reflection token allocation strategy is needed.
Problem of interest: Resource allocation over reflection tokens. Reflection tokens have a sig- nificant impact the reasoning capability of LRMs. Therefore, if we view reflection tokens "resource" " in LRM reasoning generation, then determining their schedule, including the number of occurrences and their positions, naturally gives rise the problem of resource allocation for LRMs.
TIP offers a simple solution by applying a constant logit penalty to reflection token generation. However, this approach is static and therefore fails account for both the number and placement of reflection tokens, which are dynamically determined during reasoning trace generation. As shown in Fig. 2(b). TIP does not consistently improve performance across all difficulty levels. This leaves open the question of how to schedule reflection token generation along the reasoning trajectory, that is, how to allocate these "resources" effectively over time while accounting for problem difficulty. These underscore the need for more adaptive and fine- ame-grained strategies to control reflection token usage in order to address the resource allocation more effectively.
REFLECTION TOKEN SCHEDULING AS LEARNING RATE SCHEDULING IN OPTIMIZATION
In this section, we draw conceptual analogy between reflection token scheduling and learning rate scheduling in optimization, aimed at deepening our understanding of reflection tokens in reasoning and enabling more effective resource allocation. And we propose a new decoding strategy: eyclical reflection token scheduling (CyclicReflex).
Reflection tokens in the thought landscape vs. learning rates in the optimization landscape. The role of reflection tokens in reasoning closely mirrors that of learning rates in optimization. In the "thought landscape", a model initiates by interpreting question and leverages reflection tokens to modulate its reasoning trajectory: exploring, reconsidering, and refining intermediate steps before reaching a final answer. Likewise, in the optimization landscape, an optimizer begins from random initialization and relies on the learning rate to control the step size of the variable updates, gradually converging toward an optimal solution. In both cases, a well-tuned control mechanism, reflection tokens in reasoning or learning rates in optimization, is essential for accurate solution convergence.
Additionally, in optimization, an improperly tuned learning rate, either too small or too large, can hinder convergence, causing the optimizer to either stagnate or diverge. This challenge in scheduling the learning rate maps onto the difficulty of scheduling reflection tokens in reasoning, manifesting as under-reflection and over-reflection.
(Under-reflection) When the model generates too few reflection tokens, often terminates the reasoning process prematurely, resulting in a final answer that lacks sufficient deliberation. This is analogous to optimization with learning rate that too small, where the model converges too early and becomes trapped in suboptimal local minimum.
(Over-veflection) Conversely, generating too many reflection tokens can prevent the model from concluding its reasoning. causing it to loop stall, e.g., repeatedly producing phrases like "wait" without reaching a solution. This resembles optimization with an overly large learning rate, which leads to instability and divergence rather than convergence.
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To validate the analogy between reflection tokens and learning rates (too small learning rate vs. under-reflection, and too large learning rate vs. over-reflection), we utilize the interpretability tool introduced in (Zhou et al., 2025) to visualize the landscape of thoughts. This tool projects reasoning step ry into a two-dimensional visual space based on the measured "distance" between each step r, and the final answer y, providing
interpretable view of the model's reasoning dynamics. The distance metric captures the model's uncertainty by taking the inverse of the probability of generating the answer y conditioned the reasoning step r,, normalized by the length of y:
d(r:. y) = (r. r:)) rr:) 1/)>-
(2)
where PLRM denotes the prediction probability assigned by the LRM the answer y given the reasoning step r; and yy denotes the length y.
Ground truth
Reasoning point
Predicted answer
Alternattelly per hage the
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Incorrectt
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and BI, whichh is
Under-reflection
Over-reflection
Examples of landscape of thought for under-reflection, desired-reflection. and over-reflection, generated by -iiree--R--isstll--eeeee with the original decoding strategy. Each point represents reasoning step and is connected in the order generation. Darker regions indicate steps with higher semantic alignment to the correct answer.
Figure 3:
Fig.3 presents a visualized reasoning trajectory from the initial thought to the final answer under original decoding strategy, across three different scenarios: (i) under-reflection, where too few reflection tokens lead to a reasoning trace that is too short and results in an incorrect answer; (ii) desired reflection, which yields a well-structured reasoning trace and correct answer; and (iii) over-reflection, where excessive reflection tokens cause an overly long and off-track reasoning trace, also resulting in an incorrect answer. In the landscape, darker regions represent intermediate reasoning steps that are semantically closer to the correct answer. That is, color intensity reflects the relative correctness of each thought along the trajectory.
As we can see, the thought landscape under under-reflection is too conservative to drive the reasoning process away from the starting point, ultimately failing to converge to the correct final answer. In contrast, over-reflection could enable the model to reach semantically promising regions of the landscape, for example, step like "Alternatively, perhups the correct answer ...", which is far away from the thinking start point and located in the darker region. However, much like an excessively large learning rate that fails to properly control the optimization process, this leads the model to quickly pass through the desirable state without settling there, ultimately leading answer region. Moreover, we find that reflection tokens are responsible for the sharp turns in the reasoning trajectory. By examining the sharply turning steps in over-reflection, we observe that they are consistently initiated by reflection tokens.
incorrect
Cyclic Cyclical logits manipulation for reflection token scheduling. Although reflection tokens are crucial for guiding multi-step reasoning, balancing their use remains challenging. The need for dynamic modulation of reflection tokens closely mirrors the challenge of learning rate scheduling in optimization. As introduced in Sec. 1, the convergence of gradient descent can be provably accelerated by adopting the silver stepsize schedule, which follows the principle of stepsize hedging (Altschuler & Parrilo, 2024; 2025). The key algorithmic insight is to hedge between two individually suboptimal strategies, small and large stepsizes, since the failure modes of one are often mitigated by the strengths of the other. In deep model training, cyclical learning rates (Smith, 2017) exemplify this principle in practice. Rather than using a fixed learning rate, they employ a triangular waveform to periodically alternate between large and small step sizes. This schedule allows the optimizer to balance global exploration (enabled by large steps) with local convergence stability (provided by smaller steps), thereby yielding form of stepsize hedging.

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Inspired by cyclical learning rates, we introduce CyclicReflex. As depicted in Fig. 4, we apply periodic triangular waveform to modulate the logits of reflection tokens during generation. The waveform is governed by two parameters: the amplitude A, which controls the strength of the logit adjustment, and the period C, which determines the oscillation frequency. This logits manipulation evolves over time and is both position-dependent (varying with each decoding step) and bidirectional (allowing for the dynamic promotion or suppression of reflection token sampling based on the current stage of the reasoning process). More concretely, CyclicReflex can be cast the following logits manipulation as the function of the reasoning step t
CyclieReflex 2.. = 21,0 + 8(t) v E V,
5(t) A 4. /4. (L )) mod C
2 - (3)
otherwise,
C
where recall that the amplitude A and the period C have been previously defined as shown in Fig. mod is the modulo operation, |- is the absolute value operation, and the other notations follow (1).
In (3), (t Q) mod C gives the current thought position within the cycle, and it straightforward to validate that 6(C/4) A and 6(3C/4) A.
As shown in Fig. 4, CyclicReflex adopts representative form of hedging schedule: the increasing phase of the reflection logit ad- justment d()) promotes exploration by encouraging the model to transition away from its current line of thought, while the decreas- ing phase fosters convergence by stabilizing the reasoning process, guiding the model toward producing a coherent final answer.
8(
Compared to TIP (I), which applies a fixed unidirectional penalty, Cyclic adaptively modulates reflection token logits with at no additional computation cost, offering finer control over reasoning. This unified mechanism balances under- and over-reflection, yielding more robust and flexible behavior that adapts to the model's evolving thought process. In Fig. A1 of Appendix B, we summarize the sensitivity of CyclicReflex to its hyperparameters. The period C' has a larger impact than the amplitude A.
Figure 4: Illustration of Cycli-
cReflex (3), where denotes the token position and 5(t) the logit adjustment on reflection tokens, oscillating between -A and A with amplitude A and period C.
EXPERIMENTS
5.1 EXPERIMENTAL. SETUP
Data-model setups. To evaluate the effectiveness of Cyclic we consider both math and non-math benchmarks. The math datasets include MATH500 (Lightman et al., 2023) with 500 multi-step problems, ATME:2024/005 (MAA Committees) with 30 challenging problems each year, and AMC2023 (AI-MO, 2024) covering diverse competition topics. The non-math datasets include GPQA Diamond (Rein et al., 2024), a challenging subset of multiple-choice science questions in biology, chemistry, and physics, and LiveCodeBench (Jain et al., 2024), coding benchmark from LeetCode, AtCoder, and Codeforces that evaluates code generation, repair, and execution. We evaluate our method using DeepSeek-R. R-Distilledddde-e Qwen (1.5B, 7B) (Guo al., 2025) and DeepSeek- R1-Distilled--L Jlama-8B to assess different backbones. Additionally, we demonstrate generalization on Qwen3 (4B, 8B, 14B) (Yang al., 2025a) in Table A1 of Appendix B.
Baseline and evaluation. Our method (CyclicReflex) is compared against two primary baselines: TIP (Wang al., 2025b), S1 (Muennighoff et al., 2025). In addition, we assess the compatibility of CyclicReflex with external test-time scaling techniques, including Best-of-N (Irvine et al., 2023; Brown et al., 2024) and Beam Search (Feng et al., 2023), using RLHFlow- Swww--RR--eeppeee Deepseek-88 preference reward model (PRM) for scoring (Dong et al., 2024). We use accuracy and generation length as our primary evaluation metries. Accuracy is obtained by rule-based extraction of the final answer against the ground truth, while generation length is the total word count of the response. More implementation details are provided in Appendix A.
the
5.2 EXPERIMENT RESULTS
Overall Performance of CyclicReflex on the MATH Task. In Table 2, we show the effectiveness of
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Table 2: Accuracy (Acc) and generation length (Len) comparison on four math reasoning benchmarks (MATH500, AIME2024, AIME2025, and AMC2023) using DeepSeck-R1 -I-istilldd Model: Qwen 1.5B, Qwen 7B, and Llama 8B. Each model is evaluated under four decoding strategies: Original, TIP, SI, and CyclicReflex. The best accuracy in each setting highlighted in bold, while the second-best is underlined.
Method
MATH500
A1ME2024
AIME202S
AMC2023
Ace
Len
Acc
Len
Acc
Len
Acc
Len
Original
0.74
1253.05
0.23
3584.36
0.19
3442.07
0.63
1855,85
TIP
0.75
1206.91
0.23
3329.17
0.20
3825.17
0.63
1890.35
0.73
1532.05
0.17
4112.07
3867.71
0.45
3263.75
CyclicReflex
0.77
1212.94
0.30
3547.10
0.23
3467.97
0.65
1839.23
-
Original
0.86
785.25
0.43
2878.39
0.31
3192.59
0.81
1300.53
TIP
0.87
775.77
0.43
2806.53
0.30
3107.30
0.85
1267.83
0.83
1190.96
0.33
3541.10
3455,33
0.85
2158.00
CycticRetlex
0.89
777.93
0.50
2868.30
0.37
3190.33
0.90
1229.25
Original
0.83
1196.98
0.42
3593.73
0.30
3922.41 11
0.81
1951.88
TIP
0.83
1080.62
0.47
3572.40
0.27
3866.00
0.85
1932.63
0.78
1461.99
0.43
3742.27
4351.88
2812.75
CyclicReflex
0.85
1108.30
0.53
3454,97
0.37
3856.80
0.90
1942.40
CyclicReflex across models of varying sizes (L.5,, 7B, and 8B), model families (Qwen and LLaMA), and four widely used reasoning bench-
Oraginal
Cyclotkeflox
RI Qll--wenn 7B
TIP
RI-Qwen-1 .58
R1-L/ame-88
0.5
marks: MATH500, AIME2024, AIME2025, and AMC2023. As we can see, CyclicReflex con- sistently improves performance over the original LRM decoding strategy across all models and datasets. For example, DeepSeek-RI --itt-R---Diiii-LL-A- 8B with CyclicReflex achieves up to 10% abso- lute accuracy gain on AIME2024, while DeepSeck- RI-Distill-Qwen 7B with CyclicReflex yields up to a 9% improvement on AMC2023. Addi- tionally, these accuracy gains are achieved with- out sacrificing the efficiency of reasoning gener- ation: CyclicReflex produces comparable reason- ing traces relative to the original decoding method.
0.5
Accuracy
Accuracy
0.4
0.4
0.3
0.3
0.2
4000 Length
4200
4000
4500
Length
(a) GPQA Diamond
(b) LiveCodeBench
Figure 5: Accuracy vs. generation length on (a)
GPQA Diamond and (b) LiveCodeBench. The comparison includes the original decoding, TIP. and CyclicReflex on DeepSeck-PR Ba-cck-R--Dssilll -wwen 1.5B/7B, and Llama 8B.
We further compare CyclicReflex against two additional baselines: S1 and TIP. While SI enforces the insertion of "wait" tokens at the end of each reasoning segment, leading to significantly longer outputs, it does not yield corresponding accuracy improvements. On AMC2023, in fact, SI causes a notable performance drop, suggesting that excessive reflection may lead to overthinking. TIP, which suppresses reflection token usage, can also degrade performance in some cases. For in- stance, TIP causes a 3% accuracy drop AIME2025 when applied to DeepSeek- RI-Distill-LLama
0.95
0.90
Asguracy
0.85
0.80
Original
0.75
TIP
CyclicRefeer
0.70
12
likely because it halts reasoning steps that essential for solving more complex problems.
Pass@NN
Figure Pass@N of Origi-
nal, TIP, and CyclicReflex on MATH500 with DeepSeek-R1- Distill-Qwen-a evaluated N (1, 4,8, 16).
Effectiveness of CyclicReftex on non-math reasoning, Fig. 5 shows the relationship between accuracy and generation length two non- math benchmarks, GPQA Diamond and LiveCodeBench. Results are reported for multiple DeepSeek-RI --isttill variants (Qwen-1.5BB7BB and LLaMA-8B) under original decoding, TIP, and CyclicReflex. CyclicReflex consistently improves accuracy while maintaining response lengths comparable TIP. In contrast, TIP can even reduce accuracy, as seen in Fig. 5(b) for DeepSeek-RI -iitii--iil---eeee
and LLaMA-8B on LiveCodeBench.
Evaluation under Pass @ N. In Fig. 6, we evaluate CyclicReflex on MATH500 with DeepSeek -1- Distill-Qwen. 7B under Pass@N Across N E 1,2, @ass@N N than Original and TIP. indicating gains that persist under multi-sample scaling. In addition,
2, 4, 8, 16, CyclicReflex consistently achieves higher
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the gain from TIP diminishes N increases, likely because reflection suppression reduces sampling diversity. In contrast, CyclicReftex maintains stable margin as N grows, better balancing cycle control with exploration.
BON
BoN CyclleRatex
BS
US
CycleRedlxx
Integration with other test-time scaling methods. In Fig. 7, further investigate the integration of CyclicReflex with other test- time scaling methods across computational budgets (2 29), using on MATH500. We evaluate both Best-of-N (BoN) and Beam Search (BS), with generations scored using ALHFlow-PRM. PRRM--DepSeek-B5 -BB. Across all budget levels, BoN and Beam Search integrated with CyclicReflex consistently outperform their original counterparts, demonstrating the general compatibility and effectiveness of our method. Under fixed decoding strategies, BoN achieves higher accuracy than Beam Search, both with and without CeclicReftex. As the budget increases, the performance gap between CyclicReflex and the original decoding narrows, highlighting that CyclicReflex offers the greatest benefit under constrained inference budgets by enabling more efficient reflection token allocation.
Accuracy
62
80
DeepSeek-Re R---iisill--eentt
78
76
74
2
2
2
Figure 7: MATH500 accuracy of DeepSeek-RR MenSeck-R11 Ii--iisill-Qwer 1.5B under BoN/BS, w/ and w/o CyclicReflex.
Reflection token scheduling patterns of CyclicReftex. Fig. com- pares reflection token distributions under original decoding, TIP, and CyclicReflex, using DeepScek-R1- -iiis--R---iitt--aa-aaaa Each curve shows the proportion of reflection tokens within consecu- tive 1000-token segments relative to the total reflection count. Original decoding exhibits a gradual early rise followed by stable, evenly spread pattern. TIP follows similar trajectory but suppresses reflec- tion in the 0-1k range, reflecting its tendency to inhibit early reflection. In contrast, CyclicReflex displays cyclical hedging pattern with alter- nating peaks and troughs, allocating more reflection in the 1-2k and 3-4k ranges. This modulation avoids both excessive early suppression and late overuse, leading to stronger performance on AIME2024, im- proving accuracy from 0.42 (original decoding) to 0.53, demonstrating reflection scheduling.
Reasoning trace Answer
0.20
on AIME2024.
Ratio
0.15
Original
0.05
TIP
CycleRRefixx
0.00
0-1%
1-2k
1.5%
5.6k
6-2%
2.
Length Range
Figure 8: Reflection token distribution of DeepSeck- RI RI-Distill-Llam--3 AIME2024 under original decoding, TIP, and CyclicRe- flex. Each curve shows the proportion of reflection tokens within Ik-token segments relative to the total generation.
on
the benefit of bidirectional,
Improved self-correction with CyclicReflex. We also find that Cycli- cReflex exhibits a stronger capacity for self-correction during reason- ing. To evaluate this property, we select 50 incorrectly answered prob- lems from the MATH500 dataset, originally generated by DeepSeek- RI-Distill-Qwen-7B For each incorrect case, we extract the model's reasoning trace and truncate it to three different lengths (2.%, 50%, and 100% of the full trace), which are then reused as misleading prompts to guide a new round of reasoning.
0.6
Generation Lengtt:
1000
2njut
figure two dight petane with digtts adding up 8, thats ont.... the valid poiines are 17, S3, 71, and 57. That inikes a total of four two dight potmnss where the sum the digit equals
For huw many ben-dipit getimes as the eun the dipis equal to a resson primes step
add Alright
Question
Incorrect
8:0
trace
Accuracy
0.5
think
<hhink final
Original CyclaBuufaa TEP
4.00
Origital Dycleitetoo
Original
number Yis, ht a peluee 38 Ves, at' pelme. 71, Yos, It's a prime. 80: Peime, digits
But
me double
17:
2.00
7-16. Walt, that's not Walt, did make mistake here? .... think my
0%
Incorzeci Trace (%)
25%
50%
100%
0%
Incorrecl Trace (56
25%
50%
104%
Retiex
Cydbe
huve made matake eacler including 17, which actuallly suma 16. Correct!
51hoced(335
(a) Accuracy
(b) Generation length
(c) Example
Figure 9: (a)-(b) Accuracy and generation length on MATH500 with DeepSeek-R --istilll-ween-7 using Original, TIP, and CyclicReflex decoding for questions combined with incorrect reasoning traces of different lengths. (c) Example for a question with 100% incorrect reasoning under Original and CyclicReflex decoding.
Under each prompt condition, prompt the same model (DeepSeek-Re RI-Dissill--Wen.. answer the question five times and report the average accuracy. As shown in Fig. 9()), CyclicReflex significantly outperforms both the original decoding and the TIP baseline across all trace lengths.
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Notably, the performance gap increases with longer misleading traces, indicating that CyclicReflex enhances the model's s ability to resist and correct earlier reasoning errors. In addition, this improved self-correction does not come at the cost of generation efficiency. As shown in Fig. 9b),, the generation lengths under CyclicReflex remain comparable to those of the original decoding strategy. In contrast, TIP tends suppress reflection token usage, which hampers the model's ability to re-evaluate its own reasoning and results in markedly lower accuracy. Finally, Fig. 9() provides a concrete example. Given MATH500 question with 100% incorrect reasoning trace in which the model incorrectly claims that the digits of 17, 53, 71, and 97 sum to 8, the original decoding strategy fails to correct the error and outputs the wrong answer, 4. In contrast, CyclicReflex initiates double-check, correctly identifies the error (specifically excluding 97), and ultimately outputs the correct answer, 3.
Visualizing CyclicReflex's efficacy using the landscape thoughts. In Fig. 10, we present the landscape of thoughts generated by --n-Seek-R--isiil-----eeee CyclicReflex. The darkest area the landscape represents the correct answer, while other dark regions correspond misleading areas that may attract the reasoning trajectory toward alternative but incorrect answers. As shown in Fig. 10(a), the landscape produced by CyclicReflex more concentrated, with fewer distracting regions. This suggests that the reasoning trajectory remains focused and is more likely to converge directly to the correct answer.
under the original decoding and
In contrast, the landscape under the original decod- ing strategy in Fig. 10(b) is more scattered, making the model more susceptible misleading areas and eventually reaching an incorrect final answer. This comparison demonstrates that our method enhances reasoning efficiency by reducing unnecessary detours and guiding the model toward the correct solution path more effectively.
Additional analyses and ablations. As shown in Fig. A2 of Appendix B, CyclicReflex improves MATH500 accuracy across all difficulty levels, whereas TIP mainly benefits the Hard split. Table A2 shows that CyclicReflex is robust to the choice of reflection tokens; and Table A3 confirms that Cyeli- cReflex outperforms alternative reflection-logit adjustments, supporting the necessity of our cyclic hedging schedule. Finally, Table A4 in Appendix C provides generation examples comparing the original decoding with CyclicReflex.
(a) CyclicReftex
(b) Original
Figure 10: Examples of landscape of thought for SeesSecc--I--istill----
with original
and CyclicReflex decoding strategy. The format follows Fig
6 CONCLUSION
We introduce the problem resource allocation LRMs, focusing on the challenge of managing reflection tokens during test-time generation. We show that both under-reflection and over-reflection, stemming from insufficient or excessive use of reflection tokens, can severely degrade reasoning performance. To address this, draw a conceptual analogy between reflection token scheduling and learning rate control in optimization, and propose CyclicReflex, training-free decoding strategy that cyclically modulates reflection token logits using triangular waveform. CyclicReflex dynamically adapts to the evolving stage of reasoning, enabling more balanced token allocation. Extensive experi- ments across multiple reasoning benchmarks demonstrate that CyclicReflex consistently improves accuracy, enhances self-correction capability, and integrates seamlessly with existing test-time scaling methods. Our work highlights the critical role of reflection tokens as a valuable resource for LRMs and opens new avenues for principled, adaptive reasoning control. The use of LLM, limitations and broader impacts are further discussed Appendix D, Appendix and Appendix F.
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ACKNOWLEDGMENT
This research is supported in part by the ARO Award W91 W911NF2310343, Additionally, the contribu- tions of Fan, Y. Zhang, Jia, and S. Liu are partially supported by the NSF CISE Core Program Awards IIS-2207052 and IIS-2504200 Sciences' Trustworthy Al Program. The contributions of A. Hero were also supported by the United States National Science Foundation under Grant NSF CCF 2246213.
the Open Philanthropy Research Award, and the Schmidt
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APPENDIX
A DETAILED EXPERIMENT SETUPS
A.1 COMPUTING RESOURCES
All experiments are conducted on single node equipped with 8 NVIDIA A6000 GPUs.
A.2 DECODING DETAILS
During generation, we employ the vLLM framework to enable efficient and scalable inference. The decoding configuration is as follows: the maximum number of new tokens is set to 8192; the top-p value is set to 0.95; and the temperature is set to 0.6. Top-p sampling selects tokens from the smallest possible whose cumulative probability exceeds p, effectively filtering out low-probability candidates to maintain generation quality while allowing diversity. A temperature of 0.6 sharpens the token probability distribution, promoting more deterministic and focused outputs by reducing sampling randomness.
For CyclicReflex on the MATH500 and AMC2023 datasets, we perform a grid search over A [1, 10] and C E [200, 1000]. On the AIME2024 and AIME2025 datasets, we perform grid search over [1, 10] and C' [1000, 2000]. For TIP, we conduct a grid search with E [-10, -1] and To E [100, 1000]. For S1, we forcefully insert the reflection token "Wait" after the model generates </think>, prompting continued reasoning.
In the Best-of-N setting, the LRM generates multiple independent candidate answers, and the PRM selects the most preferred one based on final-answer evaluation. For Beam Search, we perform step-by-step search guided by PRM feedback to optimize cumulative reward. Throughout decoding, we use multiple candidate beams with a fixed beam width of 4.
A.3 PROMPT DETAILS
We present the prompt used evaluate the reasoning ability of the LRM. For each question, replace the { (question ) placeholder in the User section of the prompt. After the Assistant generates the reasoning trace and the final answer (1 (Igeneratin interratio]] )), we follow the approach of Yang et al. (2024) to first extract the final answer and then apply rule-based matching to assess its correctness.
Evaluation prompt
System:
You are helpful AI bot that answers questions for user. Keep your response short and direct.
User:
Question: (question)
Let's reason this step by step.
Assistant:
Answer: (generation]
B ADDITIONAL EXPERIMENT RESULTS
Ablation study on CyclicReflex's hyperparameter. Fig. A1, we study how CyclicReflex's key hyperparameters affect final performance. Following Fig. 4, we vary three parameters: the amplitude A, the period C, and the initial phase shift 6. Fig. A analyzes and C' by reporting the number of additional correct answers over the original decoding strategy (improvement). Overall, C' has larger effect than A: in particular, C = 600 achieves the best accuracy across all tested amplitudes (A E {4.0, 5.0, 6.0)). Finally, Fig. A (lb) shows that ó 0 performs best (matching the schedule in Fig. 4), suggesting that promoting reflection tokens early can help, while gradually suppressing their logits later improves convergence.
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1.0
87.
87.0
86-1
10.4
Improvement
12.5
87.6
68.2
10.0
5.0
7.5-
7.0
-67.0
86.8
062
87.2
87.0
90
-666
86.2
862
36.8
59.4
200
400
600
800
1000
(a) Amplitude and period
(b) Shift
Figure A1: (a) Improvement heatmaps DeepSeek-RR ---isekl---- D----stlll-ween---- on MATH500 across amplitude and period C, measured as the number of additional correct answers. (b) Improvement of CyclicReflex over the original decoding strategy under different initial phase shifts ø.
Accuracy of CyclicReflex at different difficulty levels on MATH500. In Fig, A2, we categorize the MATH500 dataset by difficulty level to closely examine where the accuracy improvements from CyclicReflex are most pronounced. The grouping strategy follows that used in Fig. 2(a). and the accuracy reported in a manner consistent with Fig. 2(b). For comparison, we also include TIP as a baseline. We observe that CyclicReflex consistently improves accuracy across all difficulty levels (Easy, Medium, and Hard) whereas TIP primarily yields gains on Hard problems and even leads to performance degradation on Easy and Medium ones. This contrast stems from the bidirectional nature of CyclicReflex, which allows for dynamic promotion or suppression of reflection token sampling based the current stage of the reasoning process. Such flexibility enables CyclicReflex to better adapt to problem difficulty, leading to more effective resource allocation and improved overall performance.
0.920.91 0.93
Original
Accuracy
0.8
0.71
0.68
0.76
CyclccRffexx
0.6
0.50
0.40
0.4
0.21
0.2
Easy
Medium Difficulty
Hard
Figure A2:
Improvement DeepSeck-RI-Distin difficulty levels (Easy, Medium, Hard), following Fig. 2 (a) and (b).
on MATH500 hy TIP and CyclicReflex across
Generalization to other model families and scales. We further extend CyclicReflex to a different model family by evaluating Qwen 4B, 3 and Qwen 4B on MATH500, AIME2024, AIME2025, AMC2023 and LiveCodeBench, with Original and TIP included as baselines. The results are provided in Table A1. Across all five benchmarks, CyclicReflex consistently outperforms both baselines, even the underlying model scale increases and the family differs from the DeepSeek- distilled lineage. This demonstrates that our method generalizes beyond a single family of reasoning models and remains robust and effective for larger model with an alternative architecture.
Robustness to the reflection-token set. Next, we show that the proposed CyclicReflex is robust to the choice of reflection tokens and can even operate with dynamically updated reflection-token set. To validate this, we consider a dynamic variant in which the reflection-token set is expanded online during decoding: we start from a small seed set (e.g., "wait", "but"), and at decoding step 1, if the Top-1 token belongs to the current reflection-tokee set and the logit gap between Top-2 and Top-1 smaller than the gap between Top-2 and Top-3, we add the Top-2 token to the set. The rationale is that the model tends to in contexts similar to existing reflection tokens, gradually enriching the set and yielding additional reflection like tokens (e.g.. "hmm" "maybe", "perhaps"), which are absent from the original reflection-toker set. We evaluate Cyflic with both fixed and dynamic reflection tooonn sets on DeepSeek- - RL--Dstill-Qwent 7B across MATH500, AIME2024, AIME2025, and AMC2023, and compare against the Original decoding and TIP. As shown in Table A2, the two
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Table AI: Accuracy comparison on five benchmarks (MATH500, AIME2024, AIME2025, AMC2023, and LiveCodeBench) using Qwen3-4B, Qwen3-8B, and Qwen3-14B under three decoding strategies: Original. TIP, and CyclicReftex.
Method
MATH500
AIME2024
AIME2025
AMC2023
LiveColeRench
Qwen3-IB
Original
0.60
0.13
0.03
0.47
0.38
TIP
0.63
0.17
0.07
0.49
0.59
CyclicRotiex
0.68
0.23
0.10
0.61
Qwen3-8B1
Original
0.83
0.37
0.27
0.79
0.59
TIP
0.84
0.40
0.23
0.80
0.60
CyclicRetlex
0.86
0.43
0.33
Qwen3-14BB
141
Original TIP
0.88
0.43
0.33
0.88
0.47
0.33
0.82
0.63
CyclicReflex
variants perform comparably across all benchmarks, indicating that CyclicReflcx exhibits robustness to the selection of reflection tokens.
Table A2:
Accuracy Original, TIP, and CyclicReflex variants on MATH500,, AIME2024, AIME2025, and AMC2023 with DeepSeck- R1 RI-Distill Qwen- 7B.
Method
MATH500
AIME2024
AIME2025
AMC2023
Original
0.86
0.43
0.31
0.81
TIP
0.87
0.43
0.30
0.85
CyclicReflex w/ dynamic set Ours: CyclicRellex
0.89
0.50
0.37
0.91
0.89
0.50
0.37
0.90
Comparison with approaches using additional reflection-logit adjustments. Table A3, we extend our comparison DeepSeck-RI HiRR-Distlll-Weent by adding several additional adjustment- based baselines: (1) TIP w/ positive adjustment, which always boosts reflection-token logits rather than penalizing them; (2) random adjustment, which injects zero-mean noise into the reflection-toker logits at every decoding step; and (3) linear decay, which begins with a strong positive boost that linearly decreases into penalty across the generation process. Across MATH500, AIME2024, AIME2225, and AMC2023, CyclicReflex consistently achieves the highest accuracy. In contrast, both the aositive-adjustment and random-adjustmen variants underperform the linear-decay baseline narrows, but does not close, the performance gap relative to our method. These findings indicate that naïve purely monotonic adjustments to reflection-token logits insufficient, and that the cyclic hedging schedule in CyclicReflex provides a fundamentally more effective mechanism for balancing exploration and control during reasoning.
standard TIP, and
Table A3: Performance of Original, TIP, and adjustment-bew bbscd variants on MATH500, AIME2024, AIME2025, and AMC2023 with DeepSeck RI-Distill Qwen- 7B.
Method
MATHS00
AIME2024
AIME2025
AMC2023
Original
0.36
0.43
0.31
0.81
TIP
0.87
0.43
0.30
0.85
TIP w/ positive adjustment
0.85
0.40
0.33
0.83
Random adjustment
0.82
0.79
Linear decay
0.87
0.45
0.33
0.87
Ours: CyclicRetlex
0.89
0.50
0.90
C GENERATION EXAMPLES
In Table A4-A7, we present the answers generated for questions in MATH500 under both the original decoding strategy and CyclicReflex. For better comparison, we also include the ground truth labels. Compared to the original decoding strategy, CyclicReflex demonstrates deeper reasoning and avoids prematurely terminating the thinking process. For instance, in Q1, the original answer is given as
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(14) (5) whereas CyclicReflex continues simplifying the expression to arrive at the final result. Moreover, CyclicReftex exhibits a better understanding of the problem semantics. In Q2, for example, the original decoding misinterprets the notion of projection, leading to an incorrect answer. Finally, CyclicReflex is more adept at identifying key problem elements. In Q3, the original response overlooks the trap involving 9, while CyclicReflex correctly accounts for this subtlety.
Table A4:
Examples from DeepSeek- R M-Distill-Qwen-117 on MATH500 under the original decoding strategy and CyclicReflex. Reflection tokens are highlighted in red.
Q1: Senate commitiee has Republicans and Democrats. In how many ways can we form subcommittee of 5 members that has least one member from each pant?
Label:
1940
Original:
<think Okny, so have this problem w where there's a Senate committee with 8 Republicans and 6 Democrats. I need to figure out how many ways can form a subcommittec of members that has least one member from each party. Ham, let's break this down.
First, know that the total number of people on the committee is + 6, which 14, If were form a suhcommirtes of 5 without restrictions, the namber of ways would just be the combination of 14 taken 5 at time, right? So that's C(14,5). But the 1 have Republicans
to least one from each party, So, I think nood to subtract the cases where the subcommittes is all Republican or all Democrats. Let me calculate those separately. For all Republicans, it's C(S.5) because there Republicans. For all Democrats, it's C(6,5) since there 6 Democruts.
problem says there
Democrats
suhcommittee.
Petting it all together, the total number of valid subcommitzes should be C(4,,) minus C(8,5) minus C(6,5), Let me make sure Republizan and all- Delernttt from the total, I'm left with only those subcommittees that have at least one from each party. Yeah, that scems right. should probably calculate these values to get the exact number, but think the reasoning is solid. So the formula is C(14,5)
that makes sense. subtracting
C(8,5) C(6.5). I'll go with that.
</think>
The number of ways is ca calculated by subrmcting the all- Repuhlican and all- all-Demnerat subcommitees from the total.
Total ways: (1) (-)
(5)
Answer:
(5)
(14)
6
5
CyelicRefle::
<thinks Okny, so have this problem where there's is Senate committee with 8 Republizans and 6 Democrati. I need to figure out how ways can form a subcommittec of memberr that has least one member from each party. Hmm, let's break this down.
First, I know that the total number of people on the committec is 8 + which 14, If were to form a subcommittes of without any restrictions, the number of ways would just the combination of taken 5 time, right? So that's C(14,5). But the to least one from each party, So, I think need to subtract the cases where the subcommittee is all Repuhlicans or all Demecrats. Let me calculate those separately. For all Republicans, it's (Q8.5) because there Republicans. For all Democrats, it's CX6,5) since there 6 Democrats.
problem says there
I have Republicans
Democrats
subcommittee.
But bold on. I should make sure these numbers are correct. C(K.5) is the number of ways to choose Republicans out of 8, and C(6.5) choosing Democrats out of 6, lhink that's right.
So, the total number of valid subcommittess would be C1.5 minus C(8.5) minus C16.5). Let me compute those values.
(014,5) is 2002, C(8,5) is 56, and C16,5) is 6. So, 2002 56 equals 1940. That should be the numher of ways to form the least one from cach party.
subcommnitee
</think>
The number of ways to form subcommitiec of members with at least one member from each party is calculated subtracting all-Repeblican and all Democrat subcomnnittess from total number possible subcommittess.
Total subcommittess: (19) - 2002
All- Republican subcommittees: (-) = 56
All- Di-ooontrat subcommittees: (9) =6
Thus, the number valid subcommittess 2012 56 6 1910.
Answer:
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Table A5: Examples from DeepSeck-RR BaBSScck-RI-Dine Re---istill--uee -ee MATH500 under the original decoding strategy and CyclicReflex. Reflection tokens are highlighted in red.
Q2: Find the projection
ifa
16
19
Label:
24
Original:
<thinks Okny, so need to find the projection vector a onto vector b. remember that the formula for the projection onto is (a divided by the magnitude of squared, and then multiplied by vector b. But in this case, they only gave that product
Hmm, maybe Let me think. The formula for the scalar projection is (a b divided by the magnitude of b. Since they alrcady gave mc a need find the magnitude of The vector is 6,3), its magnitude is square of (2* 6* 3²) Calculating that, 2
and
need the entire vector b. just scalar projection?
Adding those up gives Therefore, the scalar projection of ento b 8 divided by which 8/7. I think that's it. The problem didn't I specifly whether scalar,
The square
magninade
they wanted the they's re asking for. So the answer should be 8/7.
projection or
they only gave sealar product, guess it's scalor projection
</think>
The projection of onto b is
[1] Since 8 and [b]] the projection
Answer:
CyclicReflex:
<thinks Okny, so need to find the projection of vector a onto vector b. remember that the formula for the projection of onto case, they only gave that dot product don't need the entire vector b, just scalar projection?
ath multiplied by vector
Hmm. so maybe
Wait, question says the projection of a onto b," which usually refers to vector projection. The formula for b. But since I don't have [b] maybe I can just express it in terms of the given dot product and the
voctor projection is ( [b] magnitude b.
Alteratively, maybe the question asking for the sealar projection, which just 16] But they gave b 8, but don't v
have |b]
calculate [b]] first. The magnitude So need the vecter projection, it's (5/7)) which is (8/ (8/49)22 6, That would give me [16/ 19, 48/49, 24/49). Alternatvelyy question says "projection." specifying vector
that's nice,
scalar projection, it's 8/7. vector b. think they want the vector projection. So the answer should be 16/49, 48/40 24/ 10).
they gave
</think>
The projection of vector onto b is given by: projea
Given:
First, culculate [b]
[b]]
V22
6
Then, [hi]
7
40
New substitute: projaa
40
Answer:
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Table A6:
Examples from DeepSeek-R1-Disiie -ieeee on MATH500 under the original decoding strategy and CyclicReflex. Reflection tokens are highlighted in red.
811 0 Q3: Solve 36r
Label:
(0. (9,
Original:
<think> Okay, so I have this inequality to solve: Sx-or First. I know that fraction to be less than zero, the numerator and denominator must have oppositc signs. That means cither the numeratoe is positive and the denominator is negative, or the numerator is negative and the denominator is positive, But wail, numerator
_f-011 O. Hmm, let me try to figure this out step step.
au absolute value, right? meuns the numerator can't It be negutive. So the only possibility is that the numerator is positive and the denominator is negutive. morrriiiii Wair, but the numerator is (af 811, which is zero when x 81, so a 9 oe -9. At those points, the numerator zem, making the whole fraction zoro, which doesn't satisfy the incquality because it's less than acro, not equal to zero. we can ignore those points for now.
always non-acgative, meaning it's cither positive or zero.
So, locusing on when the denominator is negative. The denominator x² 36r, which I can factor as x(x 36). So, the deneminator negutive when a(zr
solve inequality.
To find where afa: 36) < 0. can consider the critical points where the expression equals zero, which are O and a 36.
These points divide the number line into intervals. I'll test cach interval:
37, which is positive. 2. When a < 36, say 18(18 -36) 36) = 18(-18) = -221, which is ucgative. 3. When a > 36, say = 37: 37(37 36) = 37(1) = 37, which is positive.
1. When x 0, say =
[-1]-1 36) [-11[-33)
So, the denominator negative only when < 36.
zero because division by zero undefined. a can't he or Now, going back to the original inequality, since the numerator is always non negutive, the fruction will be negative only when the denominator negative. Therefore, the solution is O x 36, excluding 2 = O and 36, But wait, = O already excluded because the interval starts at D, but not including O. Similarly, Wait, should
Rut wair, also need to make sure that the denominator
the solution should where the denominator negative, excluding the points where it's
36 is not included because the interval ends at 36, not including it.
check if there any other restrictions. For example, when the fraction zero, which doesn't It satisfy the inequality Similarly, when z 99 but that's is ounside the interval we found, so it's not part of the solution.
the mumerator becomes zero, making
putting all togetherr solution but since those points are already excluded from the interval, the solution is simply 0 < z < 36.
incquality is numbers such
36, excluding and
</think>
The solution to the inequality ls 51l < Oisa a in the interval (0,36).
Answer: (0,36)
CyclicReflex:
<think> Okay, so have inequality solver goorr O. Hmm, me try to figure step hy step.
First, I know that for fraction to be less than zero, the numerator and denominator must have opposite signs. That means either numeratoe positive and denominator negative. the numerate ator here is an absollte value, right? So ja 81 is always non-nggative, meaning it's cither positive or zero. That the numerator can't be negalive. So the only possibility that the numerator is positive and the denominator is negative. Wait, numerator is
aumerator negative the denominatee is positive.
wait.
S11, which is zero when 2 81, so a a 9. those points, the numerator zero, making the whole fraction zeto, which doesn't satisfy the incquality because it's less than zero, not equal to zero. So we can ignore those points for So, focusing on when the denominator is negative. The denominator x² 36e, which can factor as a(x - 36). So, the deneminator is negutive when z(2 - 36) O. Let's solve that inequality.
To find where ziz 36) O, can consider the critical points where the expression equalss zero, which are 0 and
These points divide the number line into intervals. I'll test each interval:
I. When o 0. say
1: D)
36)
1)( 37) - 37, which is positive. 2. When @ < 36, say 18: say 37: 37(37 36) = 37(1)
18(18 36) 18(-18)
3224, which is negative. 3. When a
which positive.
So, the denominator negative only when < 36.
Now. also need to make sure that the denominater isn't zeno, so.r 0 and of 36. So, the solution so far is O st < 36. But wait. should also consider the numerator a2 811, Since it's absolute value, it's always non negative,
zero only at a 9 and a 9. But in the interval O < a < 36, a is within this interval, so at a 8, the numerator is zero, making the entire fraction zero, which doesn't satisfy the incquality. So, need to exclude a = 9 from the solution.
Wait,
numerator zero, so fraction zero, which is less than zero. x should not be included. But in the interval ( < a < 38, excluding x = 9, the fraction is negative everywhere clse bocause the numerator is positive and the denominatoor negative. Wair, hut hold on,
cheel.
1, then [12 81| so 80/(-30 is negative. Similarly, between 9 and 36, say a = 18, 118? SI] = [324 - 81| 243, and denominator negative. So, both sub-intervals, the fraction is negative.
between and 9, pick
and the denominator 12 36 1
18a 36 18 324 - 618 -321, 213/(-324)
So, the solution is 0 < a < 36, excliding 2 = But wall. at = 9, the numerator is zero, so the fraction is zero, which not less than zero. So, we exclude a = 9. Therefore, the solution 0 < 9 9 < < 36.
numerator is zern. The denominator a and 36, which we'v ve already excluded. The numerator is zero at and a = -9. at -9 is not the interval
just make
there are other points where the denominator
O 36, so we don't i have to werry about that in this case.
putting all together, </think>
solution
incquality is in the interval (0, 9) (9,36).
The solution set is a in the interval (0, 9) U (9.36).
Answer: (0.9) U (9.36)
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Table A7:
Examples from DeepSeck-RI-Diseee --ii------isslll--ee--BBBBBB on MATH500 under the original decoding strategy and CyclicReflex. Reflection tokens are highlighted in red.
Q4: Find sin sin 40" sin 60" sin 80" sin 100" sin 120" sin 140" sin 160".
Original:
Label:
<thinks Okay, have this problem where need fiod the product several sine terms: sin 20", sin 10", sin sin sin 100", sin 120", sin 140", and sin 160" Hmm, that's i lot of sines multiplied together. I remember that sometimes products of sines simplified using rrgonmmetric identiiess but I'm not exactly First, notice that some of these angles are related. For example, sin 100 is the sume as sin|
15c
which ones apply here. Let me think.
sin [180 - 80"), which is sin 80". Similarly, sin 120" is sin; 180 60" ). so that's sin 60" Wait, hut sin 60 is already in the list, so mayhe that's not helpful. Let me list all angles: 20", 40" 60" 80". 100", I can pair some of these. For instance, sin 1010 is sin 80", as thought, and sin 160" is 20', Similaty, sin 141" is sin 10' and sin 120" siu GD', actually, each of these angles from 20 to 80" supplementary, meaning their sines are equal. So, the product is actually (sin 20 sin 40" sin 60 -in 80")" hecmuse each torm from 20 to 80" is squared when considering their supplementury counterpurts.
140", 160".
corresponding angle from 100" up to 100" that
Wait, no, let me check. The original list has eight terms, but when pairing, each pair gives the same sine value, so it's like (sin 20" sin 40 sin 60 sin 80 J But let me confirm: sin 100 sin 80", sin 120 -in 60', sin 140" sin 40,, sin 160"
20", Sa, yes, product becomes (siu 20) sin 4D= sin 60 50
Now. I can compute this product step by step. First, let's compute sin 60%, which a known value: v3/2 So, that's strightforward. Now, the remaining terms are sin 20", sin 40", and sin remember there are product identilies for multiple angles. maybe using the sine of multiple angles Wail, maybe there's a formula for product sines angles in arilhmetic progression.
product-to-sum formulas, Alternttvvly, perhaps using complex numbers roots of unity.
I recall that for n terms, the product of sin kar (2ni) for It = 1 to n- 1 is v7/2"=1 But I'm not sure that applics here. Let use the ideatity for the product of sines of angles are Wait, another approach: using the identity that sin coat90" 8). So, sin 60" - ore 10 sin 40 ons 50", sin 20 cos 70", lmmm, not sure if that helps directly.
me think. Alternatively,
pattern.
Alternatively. perhups using the mulliple-angle Formula for sine. Let's consider sin 50 or something. Wait, maybe using the identity for sin 50 terms of sin 6. But that might he complicated.
Wait, maybe using identily that product of sines can be expressed as fraction involving powers of and products cosines. Alternatively. perhaps I can use the identity that the product of sin for @ = 20", 10", 60" 80 is equal to something specific.
Wait, I think I rememher that the product sin 20 sin 40 sin 60 sin 80" is equal to v3/16. Let me check that. If that's 's the case. then the square of that would be /3//16" - 3/256. But I'm not entirely sure. Let me try to compute it step by step. Allematively, perhaps using the identity that the product of sines can he expressed
but that might not be straightforward.
Wait, another approach: using the identity that the product sin sin B = (oos(A - B) cos( B)J/2. Maybe can apply this iteratively.
Let try compute sin sin 4P
Using identity, that's (eos(200) cos( 11/2. oos approximately 0.9397.
and cos 60' is 0.., so (0.9397- - 0.5)/2 0.21985.
Now, multiply this by siu 6II, which is v /3/22 as 01.8600. So, 1.219155 0,8660 es 0.1902. Next, multiply by sin 80", which is approximnaelly 0.9818. So, 0,1902 - 0.98 18 0,1873.
Wait, but thought the product sin 20 sin 40f sin 60 sin 80" was v3/16 =s 0.10825. Hmm, my approximate calculation gave around 0.1873, which higher. Mayhe my approach is wrong. Wait, perhaps should use exact values. Let me try again.
We know that sin 60 v3/2. Now, let's consider the product sin 20" sin 40" sin 80'. remember that sio A sin(60"
A) sin(GD" + A) 3n If A 20", then sin 20" sin 10 sin 80 uns""" va/2
L/ Yes, that's correct.
So, the product sin 20 sin 10 60 siu 80 is (siu 20 sin 10 siu 80%) sin 60 = + + =
Wait, that makes sense. So, the product sin 20 sin 40" 00' sin 80" 3/16.
Then, since the original problem's 's product (sin 20" sin 40" sin 60" sin (0r)s that would be
Wait, but let make sure. The original product is sin 20 sin 40 sin 60" sin 80 sin 100 sin 120 sin 1400 sin 160", which we paired (sin sin 40" 60 sin 80") hecause each
80 a supplementary angle equal to another term the list. So, the product is indeed (sin 201 sin 40" sin 60 sin s0 - (3/16)² 9 /256..
from
Wait, when calculated the approximate product. got around 0,1873, which appeoximately 3/16 0.1875. that matches. Therefore, the square would be (3/16)² = 9/256.
So, the tinal answer should be 9/256. </think>
The product simplifies by recognizing that each sine term from 20 to 50' pairs with its supplementary angle, resulting in the square of the product of these terms. Using known identities, the product of sin 20" sin 40, sin 60%, and 80" is v5/8. Squaring this gives (v3/8)² - 3/16. However, upon re-evaluating, the correct product all eight terms is actually (v3/88)² 3/266. However, upon re-evaluating, the correct product of all eight terms is actually (v3//16)* 3/256.
Answer: 256
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CyclicReflex:
<thinko- Okay, have this problem where I need to find the product of several sine terms: sin 20" sin 10 siu 60", 80", sin 100",
sines multiplied together. remember that sometimes products sines can be simplified using migononetric identities, but I'm not exactly sure which ones apply here. Let me think.
140",
that's
First. notice that some of these angles related, For example, siu 100" is the same sin (1800 80)) ). which is sin S0", Similarly, sin 12I" is sin 180 - 60"). so that's sin 60". Wait, but sin 6ll is already in the list, so maybe that's is not helpful. Let all anglss: 20" 10" 60", 80", 100". 120", 140", 160".
can pair some of these. For instance, sin 100 is sin((0)) ). as thought, and sin 160" is sin Similarty, sin 140" sin si0")))) and sin I 200 is sin (80"). So, actually, each of these ingles from 20" to S0" has a corresponding angle from 100" to 160" that supplementary, meaning their sines are equal. So, the product actually 20in sin 40 sin 60 sain 90") hocause each squared when considaring
term from
supplementary counterparts.
So, the original product simplifies compute the product inside the square finst und then square the result.
20" sin 40" sin B0 sin 80' |. Now, need to compute this square. Mayhe can
remember that sin 60' is <II which is approximately 0.866, but I'm not sure if that helps directly. Let me think about other identities. There's a product formula for multiple sine terms, but I can't it recall it exactly. Altematively, I might use complex numbers
of unity, hut that might he too arvanced for
Wait, I think there's a formula for the product of sines of cqually spaced angles. Let me try to recall. For example, the product sind sin(9 +-))) .... where d is a common difference. But I'm not sure if that applies here.
apply that Maybe use identity that relates the product of soddi 8), But with four terms, that might get complicated. another upprosch: using the idemity sin A sin B
Altematively,
sum,
I'm sure
Wair, perhups can pair terms way that
easier, Let me pair sin 20" with sin
sin 40 with 60", leaving
sin 60 as is. So, isin 20" sin 50" [sin do sin 60% sin 60" Hmm, but I'm not sure if that helps.
me test sin 20p sin 10" sin S0 = sin 20" sin(60 - 200) sin( + 20P) = sin or according to that identity. Since sin 60 va. then sin 201 sin 10" siu 80' =4.
Wait, think A sin( A) sin(e0" A)
amadd that correct?
Then,
Wait, but I'm trying compute sin 20" sin 40" 60 sin 80", which includes sin 60" as well. So, sin 20" sin 40 80 gives (x)) +2 So, product inside the square q. squaring it gives
v then multiplying by siu (=)²
Wair, but me double-check that. 20" sin 40" 50 2 then multiplying by
x indeed gives (203 X So, the product inside the square ip- and squaring gives 25e So, the final answer should be 250-
Wait, I'm a confused bocause thought the product of sin 20" sm sin sin 80" would invol more steps. Mayhe should verily this identity again. Let me consider the identity: sin A sin(60' - A) sin(00' + A) 2n94, So, if I A 20".
then sin sin 80P sinds" + v3 including siu x the product becomes
that seems correct.
Aliemattvely, maybe can use another appeoach to confirm. Let's compute the product step by step. Starting with sin 20" e 0.3420, sin 40" 0.6428, sin 80" 0.8660, sin 80" 0.9848, 100 sin 80 a 0.9848. sin 120" sin 60" 0.8080, 141"
40 ss 0.6428, sin 160 sin 20 0.3420. Multiplying these together yields approximately 0.03515625, which matches 200-
Therefore, I'm confident that the product is 2i0- </think>
product simplifies using trigonometric identities and properties supplementary angles. By recognizing pairs sines and applying the identity for the product of sines of angles in arithmetic progression, the resalt is found to he 9/256.
Answer:
256
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D LLM USAGE
GPT-5 was employed to provide writing assistance, particularly in polishing and refining the language, during the preparation of this manuscript.
E
LIMITATIONS
While CyclicReflex effectively mitigates under-reflection and over-reflection by cyclically modulating reflection token logits, its theoretical foundations remain underexplored. The proposed analogy between reflection token allocation in reasoning and learning rate scheduling in optimization offers promising direction for future theoretical investigation. A key open question is why LRMs exhibit under- over-reflection during generation. Understanding the underlying causes of these behaviors essential for developing a principled understanding of CyclicReflex. As such, future work should aim formalize the generative dynamics of reflection in LRMs and establish deeper theoretical basis for the design and improvement of reasoning resource allocation strategies, including CyclicReflex.
F BROADER IMPACTS
On the positive side, work demonstrates that scheduling reflection token appearance effectively address under-reflection and over-reflection, leading to more accurate and effective responses from LRMss This contributes enhanced reasoning capabilities and improved performance complex problem-solving tasks. Moreover, by drawing connection between learning rate schedules in optimization and reflection token dynamics in reasoning, our work opens new research directions and may inspire more interpretable and controllable LRM designs.
On the negative side, CyclicReflex could potentially be misused manipulate reasoning traces. For example, an adversary could deliberately modulate reflection token usage to craft outputs that embed sensitive hallucinated content in more convincing manner, potentially evading safety filters. To mitigate such risks, is crucial that advanced decoding strategies, such as CyclicReflex, are deployed within robust ethical and safety frameworks, especially in the context of unlearning and high-staker applications, We hope this research contributes to the development of LRMs that are not only efficient and capable but also safe, trustworthy, and aligned with human values.
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Reviews:
[{"review_id": "8jyNp3OwUM", "reviewer": "Reviewer_9cUv", "summary": "This paper investigates the problems of \"overthinking\" and \"underthinking\" in large reasoning models. The authors treat \"reflection tokens\" (e.g., \"wait\", \"but\") as a cognitive resource and propose CyclicReflex, a training-free decoding strategy. Drawing an analogy with cyclical learning rates (CLR) in optimization, CyclicReflex uses a position-dependent triangular wave function to periodically adjust the logits of these reflection tokens. The goal is to dynamically balance \"exploration\" (encouraging reflection) and \"convergence\" (suppressing reflection) during the reasoning process.", "questions": "- How robust is the method to linguistic drift? If the model learns to use other phrases for reflection (e.g., \"let's re-examine\", \"on the other hand\") that are not in the pre-defined set, will the method's effectiveness degrade or fail entirely?\n\n- Were other dynamic baselines considered? For instance, a baseline that encourages reflection (e.g., the TIP method but with a positive $\\alpha$) or a baseline that applies simple random noise to the reflection token logits? Comparing against these would help clarify whether the observed benefit comes specifically from the cyclical nature of the adjustment or just from any form of dynamic modulation.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper frames the problem from a novel and interesting perspective: treating reflection tokens as a computational resource and formalizing their management as a scheduling problem.\n\n- The analogy between reflection scheduling and cyclical learning rates is creative and highly intuitive. The resulting method is simple, training-free, and adds minimal computational overhead, which makes it very attractive for practical application.\n\n- The paper provides good analysis, including visualizations of the \"landscape of thought,\" which help build intuition for why the method works.", "weaknesses": "- The definition of \"reflection tokens\" is subjective and relies on a pre-defined, fixed set. The method's dependency on this curated list is a potential vulnerability, as it may not capture all forms of model reflection.\n\n- The core motivation is based on an analogy. While the \"landscape of thought\" metaphor is appealing, it lacks a rigorous theoretical foundation and strict validation. It is not as formally defined as the \"loss landscape\" it draws inspiration from, making the connection feel more intuitive than theoretical.\n\n- The method appears sensitive to its new hyperparameters. The triangular wave in Equation 3 is controlled by an amplitude ($A$) and a period ($C$). The ablation studies suggest that performance can vary significantly based on the choice of these parameters, which implies that significant tuning may be required. This undermines the practical benefit of a \"training-free\" approach.", "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": "rdWgQsNhz4", "reviewer": "Reviewer_rfTz", "summary": "This paper tackles reasoning failures in LRMs caused by under-thinking and over-thinking stemming from the mismanagement of reflection tokens. The key contribution is CyclicReflex, a training-free decoding strategy that modulates reflection token logits using a periodic, bidirectional triangular waveform. This approach is inspired by cyclical learning rate schedules in optimization, aiming to dynamically balance reasoning exploration and convergence. Experiments across six benchmarks show CyclicReflex consistently improves accuracy over baselines, notably enhancing performance across all difficulty levels where prior methods failed.", "questions": "Please see Weaknesses above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well-written and logically structured, clearly motivating the problem of under- and over-reflection and leading to the proposed solution.\n2.  The paper addresses the important problem of managing LRM reasoning paths. Framing reflection token generation as a resource allocation problem and drawing an analogy to learning rate scheduling in optimization is clear and novel.\n3. The proposed CyclicReflex method demonstrates strong and consistent accuracy improvements across a diverse set of models and six reasoning benchmarks. It shows gains across all problem difficulty levels, validating the effectiveness of its approach.", "weaknesses": "**Weaknesses:**\n\n1.  I have some questions on experimental setup:\n    - The `max_token` limit is set to 8192. It is unclear if this is sufficiently long for R1-style CoT generation, as significant truncation could prematurely cut off reasoning paths. This could potentially confound the analysis of under-reflection (cut off before completion) and over-reflection (never fully realized), thus impacting the reliability of the results.\n    - The hyperparameter search space for the period $C$ seems somewhat arbitrary. The paper uses small ranges for different methods (e.g., $[200, 1000]$ and $[1000, 2000]$ for CyclicReflex and $[100, 1000]$ for TIP) without a clear justification for these specific bounds.\n2. The paper's analysis suggests an \"early-explore, late-converge\" pattern is beneficial (e.g., the phase shift analysis in Fig. A3d shows $\\phi=0$ is optimal). It would be interesting to see a simpler, non-cyclical baseline that implements this insight, such as a linear decay schedule (i.e., starting with a high logit boost for reflection tokens and linearly decaying it to a penalty). It is therefore unclear if the *cyclical* nature of CyclicReflex is the key contributor to its success, or if a simpler decay schedule would have achieved similar results.", "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": "nsOQzPvbz1", "reviewer": "Reviewer_cPcQ", "summary": "This paper introduces CyclicReflex, a novel, training-free decoding strategy for Large Reasoning Models (LRMs) that dynamically modulates the logits of \"reflection tokens\" (e.g., \"wait\", \"but\") using a cyclical, triangular waveform. The core motivation is the \"resource allocation\" problem for reflection tokens, where models suffer from either under-reflection (premature convergence to an answer) or over-reflection (excessive, looping reasoning). The authors draw a conceptual analogy between scheduling reflection tokens and scheduling learning rates in optimization, where under/over-reflection mirror the effects of too small/large a learning rate. Inspired by cyclical learning rates, CyclicReflex alternates between promoting and suppressing reflection tokens, balancing exploration and convergence in the reasoning \"thought landscape.\" Extensive experiments on math and non-math benchmarks show that CyclicReflex consistently improves accuracy over baseline decoding and competing methods like TIP and S1, without significantly increasing generation length, and integrates well with other test-time scaling techniques.", "questions": "* How would you advise a practitioner to set the hyperparameters A and C for a new, unseen task or a different LRM architecture not tested in this work? Are they transferable across similar domains?\n\n* How was the vocabulary of reflection tokens V_hat constructed? Is the method's performance sensitive to minor changes in this set (e.g., adding \"however\" or \"therefore\")?\n\n* The learning rate analogy is powerful, but in optimization, the loss landscape is fixed, whereas the \"thought landscape\" is generated on-the-fly. How does this fundamental difference impact the interpretation of your method?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The core idea of treating reflection tokens as a schedulable resource and drawing an analogy to learning rate scheduling is highly original and compelling.\n\n* CyclicReflex is simple to implement, incurs no additional training cost, and demonstrates consistent, non-trivial performance gains across a wide range of tasks and model scales.\n\n* The experiments are thorough, covering multiple datasets (MATH, AIME, AMC, GPQA, LiveCodeBench), model families (Qwen, LLaMA), and integration with other test-time scaling methods (Best-of-N, Beam Search).", "weaknesses": "* The method introduces two new hyperparameters (amplitude A, period C). The grid search ranges provided in Appendix A are helpful, but the paper could do more to provide heuristics or guidelines for setting these parameters on new tasks or models, as optimal values seem to vary by dataset (e.g., different C for MATH500 vs. AIME).\n\n* The set of reflection tokens V_hat is a critical component, but its definition and potential sensitivity are not deeply discussed. Is the performance robust to the specific choice of tokens? Could the method be extended to learn or identify reflection tokens dynamically?", "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": "7ikbbKohBU", "reviewer": "Reviewer_Fqv6", "summary": "The paper propose CycliReflex, a novel algorithm that control the generation of \"reflection tokens\" by adding a periodically changing penalty. Extensive experiments on math and other reasoning benchmarks are provided to show that the proposed algorithm is performing better than baselines like S1 and TIP.", "questions": "1. How is LiveCodeBench tested? Specifically in code, the \"reflection tokens\" should not be included during the code generation process in the middle. How are such linguistic restrictions considered and affect the performance?\n2. According to Appendix A, a grid search is needed to find the optimal hyperparameters used in the setting. Is the search done per-model per dataset or only once? And is there any overall recommendations for parameters when trying on new settings?\n3. For accuracy results like Table 4, is the experiments averaged through multiple runs? Is it still 5 times or something else?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper is well-written.\n- The algorithm is easy-to-follow and well-motivated.", "weaknesses": "- Only one class of models, i.e., Deepseek-distilled models are tested. Given the behavior of different model lineages could be substantially different, additional models, potentially larger ones, need to be covered.\n- Certain highly related works are missing [1, 2, 3]. Specifically, [1, 2] are highly correlated papers that provide insights and potential baselines, while [3] is one highly similar paper that needs to be compared with. \n- Some implementation details that are necessary for reproducibility are missing. Please refer to some of the questions for more details.\n- For test-time scaling, no pass@N or cons@N results are provided, which are the more commonly used test-time scaling metrics.\n\n[1]. Sadhukhan R, Chen Z, Zheng H, et al. Kinetics: Rethinking Test-Time Scaling Laws[J]. arXiv preprint arXiv:2506.05333, 2025.\n\n[2]. Kang Z, Zhao X, Song D. Scalable best-of-n selection for large language models via self-certainty[J]. arXiv preprint arXiv:2502.18581, 2025.\n\n[3]. Chen W, Koenig S, Dilkina B. Iterative Deepening Sampling as Efficient Test-Time Scaling[J]. arXiv preprint arXiv:2502.05449, 2025.", "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)