Beyond Prompt Count: How Data Shapes Transfer in On-Policy Distillation

Jiaxuan Wang; Jiafei Lyu; Yuchen Cai; Siye Wu; Pengyuan Wang; Jiashun Liu; Xiang Cheng; Kai Yang

cs.AISep 29, 2026

Beyond Prompt Count: How Data Shapes Transfer in On-Policy Distillation

Authors: Jiaxuan Wang, Jiafei Lyu, Yuchen Cai, Siye Wu, Pengyuan Wang, Jiashun Liu, Xiang Cheng, Kai Yang, +3 more

Organizations: State Key Laboratory of Novel Software Technology, Nanjing University · School of Intelligence Science and Technology, Nanjing University · LLM Department, Tencent · RUC

Abstract

On-policy distillation (OPD) trains students using teacher feedback on their own sampled responses, yet how prompt choice shapes transfer across teacher-student pairs remains poorly understood. We systematically study prompt quantity, source, and selection across RL- and SFT-continuation pairs and cross-model settings. We find that OPD can be highly prompt-efficient: a few prompts can approach large-pool performance, with four DAPO prompts matching the observed mathematics score of 3,840 DeepMath prompts. However, prompt utility is relational rather than intrinsic: changing only the teacher can reverse the relative effectiveness of mathematics and code prompts. To characterize these transfer differences, we analyze parameter and functional changes across prompt supports and model pairs. Functional alignment with the teacher varies across supports and target tasks; in continuation pairs, teacher-aligned prediction changes can coexist with weak parameter alignment. Continued OPD on effective supports can restore performance after unfavorable transfer. Finally, targeted selection does not consistently outperform uniform random sampling, and filtering out a source that performs poorly alone yields no consistent gain across three paired support draws. Overall, our results distinguish prompt efficiency from prompt interchangeability and show that effective data choice depends on the teacher-student pair and target capability, with random sampling providing a competitive baseline in the studied settings.

Figures & tables

Figure 1: Prompt efficiency does not imply prompt interchangeability. A single prompt can induce substantial transfer, but outcomes vary with prompt identity and composition. Moreover, a few prompts from one source can match the observed mathematics score of thousands from another. These examples motivate our study of prompt count, source, teacher–student pairing, and selection. In (b),

P_{1}

–

P_{3}

denote the individual code prompts described in Appendix C.4 , while

P_{1}+P_{2}

denotes joint OPD using the first two prompts.

Table 1: Teacher–student pairs and OPD sources. The first three teachers are trained by us; the others are publicly released models. We list constituent data sources; support construction, including mixtures and source filtering, is detailed in Appendix B.3 .

Figure 2: Prompt-count scaling across six teacher–student pairs. Stars mark full-pool OPD; S/T denote the initial student and teacher. Circled points in (d) attain the same observed Math score with 4 DAPO versus 3,840 DeepMath prompts. Training budgets are comparison-specific; full results appear in Appendix C .

Figure 3: Data-source preferences across teacher–student pairs. (a) Source sensitivity with a 30B Instruct teacher. (b) Preference reversals between Math-RL and Code-RL teachers with the same initial student. (c) Cross-domain transfer from the Nemotron teacher. Avg is the unweighted mean of the four displayed benchmark groups; side annotations report differences from DeepMath. Darker and lighter shading mark the highest and second-highest OPD scores within each column.

Data	Tokens (M)	Avg
GSM8K	1.19	44.01
+ token match	16.14	44.84
DeepMath	16.63	51.02

Table 2: Token control.

M=48

; Avg averages Math, GPQA, HE+, and LCB.

Model / OPD support	Math	GPQA-D	HumanEval+	LCB v6	IFEval	IFBench	BFCL-v3	Avg 6
Student (init.)	45.83	45.52	81.48	27.86	83.71	30.42	63.91	53.61
Teacher	48.18	45.27	79.57	30.21	86.02	39.96	70.81	56.17
DeepMath	48.28	47.35	81.86	29.71	83.90	29.04	64.25	54.65
Textbook	46.46	51.14	83.23	28.93	83.50	27.88	62.66	54.69
Eurus Code	48.49	61.81	81.17	31.93	83.13	28.67	66.50	57.63
IF	41.51	46.46	68.29	22.07	78.35	42.56	53.25	48.67

Table 3: Data-source preferences with a multi-domain RL teacher. Qwen3-4B-Inst-Mix

\rightarrow

Qwen3-4B-Instruct-2507;

M=48

. Teal marks the best and second-best OPD scores; mixture construction is detailed in Appendix B.3 .

Figure 4: Parameter and functional alignment. (a) Parameter and all-domain functional cosines for 43 checkpoints; each line joins the same endpoint. (b) Teacher-direction projection

\alpha

versus functional cosine on Math probes, grouped by model pair. (c) Functional cosine versus accuracy: 30B-to-4B Math (left) and Mix-RL-to-4B GPQA (right). Definitions: Appendix E.1 .

Figure 5: GPQA accuracy and output-length changes. All supports contain

M=48

prompts; changes are relative to each pair’s initial student. Inset: mean output tokens.

Figure 6: Redirecting unfavorable transfer. (a) CKA relative to full-pool OPD. (b) Bidirectional hidden-state replacement. (c) Performance recovery through continued distillation; vertical ticks mark native references. (d) Functional versus parameter alignment to native endpoints. Reference checkpoints and unequal Code update budgets are specified in Appendix E.5 .

Figure 7: Continuation data shapes recovery at matched update counts. 30B Instruct

\to

Qwen3-4B, with 15 updates per stage and a fresh Adam optimizer in every second-stage run; token budgets are not matched. Orange and blue indicate stage-2 GSM8K and DeepMath, respectively. (a) AIME24/25 validation mean@2; solid/dashed lines start from GSM8K/DeepMath checkpoints. Gray step-0 markers show the shared starting scores of

44.17

50.83

. (b) Formal Avg 4 (%), with the y-axis truncated at

30\%

, grouped by stage-1 source; annotations show the gains from using DeepMath rather than GSM8K in stage 2.

Method	Math	GPQA-D	HE+	LCB	Avg 4	GPU $\cdot$ h
$M=8,\quad C=384$
Uniform random	31.09 $\pm$ 0.50	39.44 $\pm$ 1.24	62.40 $\pm$ 0.32	17.67 $\pm$ 0.86	37.65 $\pm$ 0.38	0
Stratified random	30.36 $\pm$ 0.39	38.19 $\pm$ 0.58	61.74 $\pm$ 0.46	17.93 $\pm$ 0.41	37.06 $\pm$ 0.25	1.1
Semantic diversity	30.21	39.46	62.88	17.36	37.48	$<0.1$
Hard selection	28.96	38.07	63.57	18.21	37.20	1.1
Shortest	28.75	39.08	61.28	16.86	36.49	2.2

Table 4: Prompt selection within DeepMath. JustRL-1.5B

\rightarrow

DS-1.5B. Random selectors report mean

\pm

sample SD over three sampled supports. Bold marks the column best within each

M

. Costs estimate selection-only GPU hours;

0

denotes no GPU screening.

Figure 9: Source composition and a four-prompt control. 30B Instruct

\to

Qwen3-4B, 25 updates. DeepMath/GSM8K composition sweep (circles,

M=48

) and DeepMath4-only control (diamonds,

M=4

). Within each seed, the control reuses exactly the four DeepMath prompts in the corresponding 4/44 mixture. Error bars and bands show sample SD across selection seeds 42/43/44.

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Math / OOD	Code (4B)	Direct-code
Samples per question ( $k$ )	16 / 8	8	16
Temperature	1.0	1.0	0.6
Top- $p$	1.0	1.0	0.95
Top- $k$ / min- $p$	$-1$ / 0.0	$-1$ / 0.0	$-1$ / 0.0
Max. generation length	16,384	16,384	4,096

Appendix

Table 11: Evaluation configurations. Lengths are in tokens; all settings report mean@

k

and pass@

k

, with

k=16

for Math and

k=8

for OOD.

Source	$M$	AIME24	AIME25	HMMT Feb	HMMT Nov	Math	GPQA-D	HE+	LCB
Initial	–	22.29	19.79	11.04	8.75	15.47	42.74	77.06	17.71
Teacher	–	62.71	56.46	34.38	39.79	48.33	52.65	83.92	20.07
DeepMath	1	55.21	54.17	32.29	37.29	44.74	51.70	82.39	29.93
DeepMath	48	60.21	53.12	33.33	38.96	46.41	53.22	83.23	27.21
DeepMath	3,840	61.25	55.83	32.92	37.08	46.77	52.15	85.14	29.79

Appendix

Table 12: Math-RL continuation. Math GRPO-500 to Qwen3-4B ( Figure 2 a ), with 3,840 rollouts per OPD run. Math reports mean@16; other tasks report mean@8.

Source / support	$M$	HE+	MBPP+	LCB	Code macro
Initial	–	76.60	65.15	18.14	53.30
Teacher	–	86.36	71.03	28.00	61.80
Eurus Code (ID 5038)	1	79.04	68.39	26.43	57.95
Eurus Code	48	86.66	70.14	28.57	61.79
Eurus Code	3,840	86.05	70.90	28.14	61.70
Eurus Code (full)	22,618	86.74	71.06	27.86	61.89

Appendix

Table 13: Code-RL continuation. Code GRPO-300 to Qwen3-4B ( Figure 2 b ), at OPD step 100; all scores are mean@8. The principal

M=1

support is ID 5038 (

B

in Section 5 ); rows below the separator give additional single-prompt experiments.

Source / reference	$M$	HE+	MBPP+	LCB	Code macro
Initial: direct-code SFT-554	–	44.32	52.88	11.25	36.15
Teacher: Code GRPO-400	–	58.12	58.47	15.96	44.18
Open-R1 Code	1	58.82	58.74	15.96	44.51
Open-R1 Code	256	59.98	59.51	15.57	45.02
Open-R1 Code	3,840	59.68	59.62	16.61	45.30
Open-R1 Code (full)	6,519	58.73	58.80	15.50	44.34

Appendix

Table 14: Direct-code RL continuation. Code GRPO-400 to Qwen3-1.7B-Base after direct-code SFT at step 554 ( Figure 2 c ). OPD endpoints are at step 50; all scores are mean@16.

Source	$M$	AIME24	AIME25	HMMT Feb	HMMT Nov	Math	GPQA-D	HE+	LCB
Initial	–	27.50	25.63	12.50	11.25	19.22	35.92	54.34	13.50
Teacher	–	53.96	39.38	21.67	23.75	34.69	40.47	64.56	19.36
DAPO	1	42.71	30.42	16.88	17.08	26.77	38.13	60.98	16.29
DAPO	2	49.58	36.25	20.83	20.83	31.87	38.64	62.27	16.71
DAPO	4	47.71	36.04	22.08	24.38	32.55	41.10	63.87	17.50
DAPO	8	51.04	35.00	21.25	22.71	32.50	40.91	64.10	18.64

Appendix

Table 15: JustRL continuation. JustRL-DeepSeek-1.5B to DeepSeek-R1-Distill-Qwen-1.5B ( Figure 2 d ), at OPD step 100. Math reports mean@16; other tasks report mean@8. Full denotes the available source pool; DeepMath

M=8

is the random support in the prompt-count sweep.

Source	$M$	AIME24	AIME25	HMMT Feb	HMMT Nov	Math	GPQA-D	HE+	LCB
Initial	–	22.29	19.79	11.04	8.75	15.47	42.74	77.13	17.71
Teacher	–	75.21	61.67	42.50	56.25	58.91	54.86	82.39	33.14
DeepMath	1	46.0	46.7	24.6	31.9	37.29	48.3	82.4	24.3
DeepMath	48	56.46	45.63	29.38	39.79	42.81	52.15	82.32	26.79
DeepMath	3,840	53.8	48.3	30.2	38.1	42.60	52.0	83.7	27.0
DAPO	1	49.4	40.4	26.2	31.5	36.88	49.7	83.5	26.3

Appendix

Table 16: Cross-model distillation. Qwen3-30B-A3B-Instruct-2507 to Qwen3-4B ( Figure 2 e ), at OPD step 15. Math reports mean@16; other tasks report mean@8. Full-pool endpoints at other training durations appear in Table C.2 .

Source	$M$	AIME24	AIME25	HMMT Feb	HMMT Nov	Math	GPQA-D	HE+	LCB
Initial	–	13.33	10.00	5.00	5.63	8.49	29.10	59.83	12.36
Teacher	–	75.21	61.67	42.50	56.25	58.91	54.86	82.39	33.14
DeepMath	1	27.3	24.0	16.0	14.2	20.4	30.1	65.5	16.1
DeepMath	8	33.1	26.7	15.0	15.2	22.5	32.5	65.8	19.4
DeepMath	48	35.0	26.9	14.2	17.1	23.3	36.2	66.7	18.1
DeepMath	3,840	36.0	30.8	14.8	16.7	24.6	32.7	68.9	18.7

Appendix

Table 17: Cross-model distillation. Qwen3-30B-A3B-Instruct-2507 to the official Qwen3-1.7B checkpoint ( Figure 2 f ), not the SFT-554 initialization in Table C.1 . Math reports mean@16; other tasks report mean@8.

Math GRPO-500 $\rightarrow$ Qwen3-4B
Source	Step	AIME24	AIME25	HMMT Feb	HMMT Nov	Math	GPQA-D	HE+	LCB
DeepMath	25	62.92	54.58	32.29	38.75	47.14	52.78	84.91	29.86
DeepMath	50	64.79	56.04	34.38	41.46	49.17	54.29	84.45	28.07
Qwen3-30B-A3B-Instruct-2507 $\rightarrow$ Qwen3-4B
DeepMath	25	55.0	48.8	27.3	37.9	42.2	48.4	82.8	25.8
DeepMath	50	56.9	49.0	31.7	38.8	44.1	48.4	83.5	26.6

Appendix

Table 18: Additional full-pool OPD endpoints. Math reports mean@16; other tasks report mean@8. Math-RL runs use batch size 1,024, with 25,600 and 51,200 rollouts at OPD steps 25 and 50, respectively.

Figure 10: Support size and learning dynamics. All runs use the Qwen3-4B Math-RL continuation pair. (a–b) Training-time AIME24 and AIME25 validation accuracy (mean@2). (c) Mean rollout response length, in thousands of tokens. (d) Teacher–student discrepancy, reported as logged mean K1 on a logarithmic scale. These validation traces are distinct from the final mean@16/mean@8 evaluations in the complete result tables.

Figure 11: Teacher and prompt-source effects during Qwen3-4B OPD. Math-T and Code-T denote Math GRPO-500 and Code GRPO-300; Math48 and Code48 denote the fixed DeepMath and Eurus Code supports. (a) AIME25 validation with Math48. (b) TACO validation with Code48. Both report training-log mean@2. (c) Mean rollout response length. (d) Teacher–student discrepancy, reported as logged mean K1 (log scale). Panels (a–b) use different validation tasks; the common-benchmark source-preference reversal is shown separately in Figure 3 b .

	Math-RL teacher		Code-RL teacher
Support / training seed	DeepMath48	Code48	DeepMath48	Code48
42 / 42	54.30	46.84	44.94	50.75
43 / 42	54.20	46.62	44.79	51.27
44 / 42	54.23	46.17	45.22	50.51
42 / 43	53.87	46.37	44.69	50.31
42 / 44	54.26	46.66	44.67	50.42

Appendix

Table 19: Seed robustness of teacher-dependent source preferences. Avg 4 (%) across support-selection and training seeds.

Data	Tokens (M)	Math	GPQA	HE+	LCB
GSM8K	1.19	31.15	46.84	81.25	16.79
+ token match	16.14	31.56	48.04	82.32	17.43
DeepMath	16.63	42.81	52.15	82.32	26.79

Appendix

Table 20: Benchmark-level supervision-token control.

M=48

; tokens are cumulative supervised tokens in millions, and GSM8K token matching to DeepMath is approximate.

Figure 12: Additional source-dependent functional diagnostics. (b) 30B-to-4B. (c) Nemotron. (d) Mix-RL. Upper plots report Math functional cosine and teacher-direction coefficient $\alpha$ ; lower matrices report task-specific KL closure (%). Closure uses teacher-first conditional KL on the initial student’s frozen top-128 support, not a task-score recovery rate. Blue and red indicate negative and positive closure, respectively; colors saturate at $-100\%$ and $100\%$ .

Model / OPD support	GPQA-D	HumanEval+	LCB v6
(1) Math-RL: Qwen3-4B $\to$ Qwen3-4B
Initial (Compact)	42.74 / 870	77.06 / 325	17.71 / 557
Teacher (GRPO-500)	52.65 / 2,619	83.92 / 700	20.07 / 2,670
DeepMath M48 (Compact)	53.22 / 5,590	83.23 / 2,328	27.21 / 9,567
(2) Qwen3-30B-A3B-Instruct-2507 $\to$ Qwen3-4B
Initial	42.74 / 870	77.13 / 325	17.71 / 557

Appendix

Table 21: Accuracy and output length for seven teacher–student pairs. Each cell gives mean@8 accuracy (%) / mean output tokens;

s

denotes the OPD update step.

Figure 13: Code recovery in parameter and function space. PCA views of the initial student (

I

), degraded single-prompt endpoint (

A^{\prime}

), recovered endpoint (

R

), and native Code-M48 endpoint (

N

). Native Code-M48 is trained directly from

I

; recovery continues from

A^{\prime}

on an effective 48-prompt support. From

A^{\prime}

R

, the reported distance to

N

increases from

1.46

1.69

in parameter space but decreases from

1.03

0.29

in function space. Axis percentages report explained variance; arrows connect endpoints rather than tracing intermediate updates.

Figure 14: Training trajectories under single-prompt and two-prompt supports. All runs use the same Qwen3-4B initialization and Code GRPO-300 teacher. $A^{\prime}$ uses prompt 15240; M2 jointly uses prompts 15240 and 187 from initialization. (a) Code accuracy: the unweighted mean of mean@8 over HumanEval+, MBPP+, and LCB-v6. Markers for M2 and prompt 187 alone show only their step-100 endpoints. (b) Functional cosine to the teacher’s change. (c) Normalized functional-update magnitude of $A^{\prime}$ , $\lVert v_{\mathrm{OPD}}\rVert/\lVert v_{T}\rVert$ .

Figure 15: Parameter displacements after bidirectional support switching. Cosines compare FP32 endpoint-minus-original-initialization vectors for the four paths in Figure 7 . Every endpoint has 30 total updates; $G$ denotes GSM8K and $D$ denotes DeepMath. The $G\to D$ versus $D\to D$ comparison uses the step-30 reference, unlike the step-15 Math reference in Figure 6 (d) .

Selection seed	Mixed	Filtered	Shared prompts
	DeepMath / DAPO / GSM8K	DeepMath / DAPO / GSM8K
42	17 / 15 / 16	21 / 27 / 0	32
43	13 / 15 / 20	22 / 26 / 0	28
44	20 / 13 / 15	31 / 17 / 0	33

Appendix

Table 22: Mixed-pool support composition and overlap. Shared counts prompt instances selected by both arms under the same random ordering, not shared sampled responses.

Benchmark	Selection seed	Mixed	Filtered	$\Delta$
Math average	42 / 43 / 44	43.2 / 40.3 / 38.8	41.4 / 39.3 / 40.7	$-1.8$ / $-0.9$ / $+1.9$
	Mean $\pm$ SD	$40.7\pm 2.3$	$40.5\pm 1.0$	$-0.3\pm 2.0$
AIME24	42 / 43 / 44	55.8 / 54.2 / 49.2	56.5 / 51.3 / 49.8	$+0.6$ / $-2.9$ / $+0.6$
	Mean $\pm$ SD	$53.1\pm 3.5$	$52.5\pm 3.5$	$-0.6\pm 2.0$
AIME25	42 / 43 / 44	50.4 / 48.1 / 43.8	44.4 / 46.7 / 46.9	$-6.0$ / $-1.5$ / $+3.1$
	Mean $\pm$ SD	$47.4\pm 3.4$	$46.0\pm 1.4$	$-1.5\pm 4.6$

Appendix

Table 23: Math results for mixed-pool and source-filtered sampling. 30B Instruct

\rightarrow

Qwen3-4B,

M=48

, 25 optimization steps; selection seeds 42/43/44 share a fixed training seed. Scores are mean@16 (%); Math average is the unweighted mean of the four benchmarks.

\Delta

denotes paired Filtered

-

Mixed differences; means, SDs, and differences use unrounded scores.

Benchmark	Selection seed	Mixed	Filtered	$\Delta$
OOD average	42 / 43 / 44	55.2 / 54.9 / 53.9	55.2 / 53.7 / 54.6	$0.0$ / $-1.2$ / $+0.7$
	Mean $\pm$ SD	$54.7\pm 0.7$	$54.5\pm 0.8$	$-0.2\pm 0.9$
GPQA-D	42 / 43 / 44	51.5 / 50.7 / 50.0	51.0 / 50.4 / 51.2	$-0.4$ / $-0.3$ / $+1.2$
	Mean $\pm$ SD	$50.7\pm 0.7$	$50.9\pm 0.4$	$+0.2\pm 0.9$
HE+	42 / 43 / 44	87.0 / 87.0 / 86.1	88.1 / 85.3 / 86.1	$+1.1$ / $-1.8$ / $+0.1$
	Mean $\pm$ SD	$86.7\pm 0.6$	$86.5\pm 1.4$	$-0.2\pm 1.4$

Appendix

Table 24: OOD results for mixed-pool and source-filtered sampling. 30B Instruct

\rightarrow

Qwen3-4B,

M=48

, 25 optimization steps; selection seeds 42/43/44 share a fixed training seed. Scores are mean@8 (%); OOD average is the unweighted mean of GPQA-D, HE+, and LCB.

\Delta

denotes paired Filtered

-

Mixed differences; means, SDs, and differences use unrounded scores.

Method	HE+	MBPP+	LCB	Code Macro	GPU $\cdot$ h
Uniform random	88.26 $\pm$ 0.32	71.33 $\pm$ 0.34	27.50 $\pm$ 0.43	62.36 $\pm$ 0.22	0
Semantic diversity	88.03	71.83	28.14	62.67	0.1
Hard selection	88.03	71.56	28.14	62.58	6.9
T–S disagreement	87.58	70.92	27.86	62.12	8.6
Cost-D-opt	88.42	71.17	27.64	62.41	48.6

Appendix

Table 25: Prompt selection within Eurus Code. Code-RL 4B

\to

Qwen3-4B;

M=48

C=2{,}304

, 100 updates. Scores (%) use mean@8; Code Macro averages the three benchmarks. Uniform random reports mean

\pm

sample SD across three support draws. Bold marks column-best scores. Costs cover selection only;

0

denotes no GPU screening.

Method	Math	GPQA-D	HE+	LCB	Avg 4	GPU $\cdot$ h
Uniform random	40.56 $\pm$ 1.07	49.87 $\pm$ 0.55	86.16 $\pm$ 0.52	25.74 $\pm$ 0.52	50.58 $\pm$ 0.61	0
Stratified random	41.79 $\pm$ 1.13	50.65 $\pm$ 1.03	86.81 $\pm$ 1.47	26.81 $\pm$ 0.36	51.52 $\pm$ 0.83	5.7
Semantic diversity	40.21	50.44	86.20	27.36	51.05	$<0.1$
Hard selection	40.00	49.87	87.65	26.64	51.04	5.7
Shortest	35.05	46.53	83.61	20.21	46.35	5.7

Appendix

Table 26: Prompt selection within DeepMath L6. 30B Instruct

\to

Qwen3-4B;

M=48

C=2{,}304

, 25 updates. Scores (%) use mean@16 for Math and mean@8 otherwise; Avg 4 averages the four benchmarks. Random selectors report mean

\pm

sample SD across three support draws. Bold marks column-best scores. Costs cover selection only;

0

denotes no GPU screening.

Method	Selection seed	$E(\mathcal{S})$ (%)	Cosine	$\mathrm{Avg}_{4}$ (%)
Uniform random	42	29.7109	0.9670	37.5227
Uniform random	43	29.7557	0.9565	38.0760
Uniform random	44	23.4674	0.9722	37.3466
Stratified random	42	21.7985	0.9833	37.0561
Semantic diversity	42	23.5230	0.9721	37.4771
Hard selection	42	39.0314	0.9606	37.2021

Appendix

Table 27: Per-support gradient representativeness and transfer. JustRL–DeepMath,

C=384

M=8

. Each row is one support in Figure 8 , not a mean across selection seeds.

E(\mathcal{S})

and

\mathrm{Avg}_{4}

are percentages; cosine is dimensionless.

DeepSeek 1.5B
	Training cost				Performance (%)
Method	$F/U$	Generated	Loss	Time	Math	GPQA-D	HE+	LCB	Avg 4
Fresh-only	67/67	128.19	128.19	590.5	32.55	39.02	63.64	18.57	38.45
Fresh-only	20/20	39.13	39.13	178.8	31.09	38.83	63.80	18.36	38.02
Uniform token replay	17/68	32.78	57.64	205.1	32.92	39.65	62.73	19.21	38.63
Full-batch reuse	17/68	32.60	130.40	226.0	32.14	40.15	64.94	18.50	38.93

Appendix

Table 28: Performance and training cost with trajectory reuse.

F/U

denotes fresh rollout batches / optimizer updates; generated and loss tokens are in millions (including repeated exposures for loss), and time is in minutes. Math reports mean@16 and the other benchmarks mean@8; Avg 4 is their unweighted mean.

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Uni-Opd Token-Level Supervision Student-Generated Trajectories Prompt Engineering

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On-policy distillation (OPD) applies token-level teacher supervision to student-generated trajectories, but this supervision is not always reliable. Existing methods use local confidence or teacher-student agreement to weight, filter, or truncate the sampled trajectory. These signals do not directly determine whether the teacher can continue a student prefix to a correct answer, and trajectory-level interventions can conflate one rollout's unreliability with low expected training value of its prompt. We define prompt-level teacher continuation reliability

R

as the teacher's probability of reaching a correct answer from a student prefix, averaged over prefixes and trajectories induced by the current student. Oracle experiments show that high-

R

prompts yield larger OPD gains and that descending-

R

training outperforms random and ascending orders on a fixed prompt pool. Because estimating

R

requires many teacher continuations, we use the maximum ROUGE-5 F1 between one independent student rollout and verifier-correct same-prompt teacher trajectories. Across ten equal-frequency bins of this actual score, mean

R

rises monotonically, showing that the proxy separates coarse reliability levels. ReOrder-OPD sorts prompts by the proxy, then draws independent on-policy training trajectories for vanilla OPD. It improves every matched aggregate comparison across Qwen3 and Gemma4 mathematics settings and Qwen3 code settings. Gains in all six FiRe-OPD and ExOPD settings show that prompt ordering complements within-trajectory supervision.

Abstract

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