ReLaDrift: Recurrent Language Generation via Drifting for Test-Time Scaling

Yuki Otsuka; Daisuke Oba; Naoaki Okazaki

cs.CLOct 16, 2026

ReLaDrift: Recurrent Language Generation via Drifting for Test-Time Scaling

Authors: Yuki Otsuka, Daisuke Oba, Naoaki Okazaki

Organizations: Institute of Science Tokyo · AIST · NII LLMC

Abstract

Drifting Models generate a sample with a single function evaluation, making them attractive for fast parallel language generation, but their one-step formulation does not support iterative refinement. We propose Recurrent Language Generation via Drifting (ReLaDrift), which uses a shared Drifting Model to generate an initial draft from noise and iteratively refine it in a continuous latent space. We train this recurrence by sampling the number of rounds and applying the drifting loss only at the final round, stopping gradients through preceding rounds so that only one round requires backpropagation. Unlike the one-step Drifting Model, whose accuracy does not improve when iterated, ReLaDrift enables a trade-off between test-time computation and generation quality, as diffusion and flow language models do. On Sudoku Hard and GSM8K, it achieves competitive or superior performance compared with these iterative-generation approaches across a range of inference budgets. Our code is available at https://github.com/Yuki-Ots/ReLaDrift

Figures & tables

Figure 1: Accuracy vs. the number of function evaluations (NFE) on Sudoku Hard (top) and GSM8K (bottom). Results use a single generation trajectory and EMA weights, with mean

\pm

standard deviation over three seeds for Sudoku. Models are evaluated at 20,000 updates (Sudoku) and 250,000 updates (GSM8K); sampling settings are detailed in Tables 1 and 3 .

Figure 2: ReLaDrift in the source formulation. Round 0 (Initial) transports the noise

\epsilon

to the first draft

d_{1}

through the input projection

W_{\mathrm{noise}}

, and every later round (Refine) transports the current draft through

W_{\mathrm{draft}}

; all rounds share the network

f_{\theta}

and receive the prompt latent

P

and the round index

r

(Sec. 4.2 ). The frozen decoder is applied once to the last draft

d_{N}

; at test time, the number of rounds

N

is the NFE and may exceed the training depth

R_{\max}

(Sec. 4.4 ). The marks

\operatorname{sg}

and the loss

\mathcal{L}

apply only during training, where

N=R\sim\operatorname{Uniform}\{1,\ldots,R_{\max}\}

and the drafts that enter a round are treated as constants, so the loss backpropagates through the terminal round only (Sec. 4.3 ). The positives and the negatives of the loss are not shown.

Method	NFE=1	4	5	8	16	32	64	128	180
AR (sequential)	Greedy: $0.5\pm 0.1$ ; sampling: $0.5\pm 0.2$
MDLM	$0.6\pm 0.2$	$8.1\pm 1.0$	$11.6\pm 1.6$	$15.9\pm 1.9$	$22.7\pm 2.6$	$27.1\pm 2.1$	$30.1\pm 2.2$	$32.2\pm 1.6$	$32.6\pm 2.6$
Duo	$4.0\pm 2.1$	$55.2\pm 6.7$	$56.8\pm 5.6$	$60.6\pm 5.4$	$61.9\pm 4.9$	$62.9\pm 4.7$	$61.8\pm 4.8$	$62.9\pm 4.8$	$61.5\pm 4.9$
S-FLM	$6.7\pm 1.1$	$33.0\pm 2.9$	$34.9\pm 3.2$	$38.0\pm 3.0$	$40.6\pm 3.2$	$41.9\pm 3.2$	$42.5\pm 3.4$	$43.0\pm 3.3$	$43.2\pm 3.3$
FMLM+	$2.4\pm 0.4$	$67.0\pm 2.4$	$71.2\pm 1.5$	$76.8\pm 1.4$	$77.5\pm 2.1$	$78.5\pm 1.4$	$78.1\pm 2.0$	$78.5\pm 1.5$	$77.2\pm 1.4$
FRM (Euler)	$6.4\pm 0.9$	$54.9\pm 1.0$	$60.6\pm 0.8$	$66.7\pm 1.1$	$71.8\pm 1.5$	$74.2\pm 1.6$	$75.2\pm 1.7$	$75.8\pm 1.7$	$75.9\pm 1.6$

Table 1: Sudoku Hard accuracy (%) across inference budgets. Exact-match accuracy on 2k validation puzzles after 20k training updates (mean

\pm

standard deviation over three seeds). ReLaDrift uses the Source formulation with

R_{\max}=5

. The Drifting Model is iterated by decoding and re-encoding its output between rounds (Appendix A.15 ). FRM and DFM are our implementations (Appendix A.10 ). Bold indicates the best accuracy at each NFE.

Input formulation (Default: Source)
Setting	NFE=1	4	5	8	16	32	64	128	180
Default	$7.3\pm 1.0$	$78.9\pm 0.6$	$82.5\pm 0.8$	$88.4\pm 0.4$	$92.1\pm 0.4$	$93.1\pm 0.6$	$93.5\pm 0.7$	$93.9\pm 0.7$	$94.0\pm 0.7$
- Additive	$7.1\pm 2.2$	$77.6\pm 2.6$	$81.5\pm 3.1$	$86.2\pm 3.6$	$89.5\pm 3.1$	$90.5\pm 3.0$	$91.0\pm 2.8$	$91.6\pm 2.5$	$91.6\pm 2.8$
- Conditional	$7.4\pm 0.9$	$77.6\pm 1.7$	$81.1\pm 1.1$	$86.1\pm 1.2$	$89.6\pm 1.8$	$90.7\pm 1.7$	$91.2\pm 2.0$	$91.6\pm 2.3$	$91.7\pm 2.2$
Training depth $R_{\max}$ (Default: $5$ )
- $3$	$5.7\pm 2.7$	$66.2\pm 3.4$	$68.5\pm 2.9$	$71.4\pm 2.1$	$72.6\pm 2.0$	$72.9\pm 1.9$	$73.1\pm 1.9$	$73.1\pm 2.0$	$73.2\pm 2.0$

Table 2: Ablation of ReLaDrift on Sudoku Hard. Exact-match accuracy (%; mean

\pm

standard deviation over three seeds) under the settings of Table 1 . The default Source formulation uses

R_{\max}=5

and serves as the reference for both the input-formulation and training-depth comparisons.

Method	NFE
	1	4	5	8	16	32	64	128	256	512	1024
AR (sequential)	72.3
MDLM	0.3	1.3	1.9	5.2	12.1	18.8	27.2	28.7	29.7	32.0	32.1
Duo	0.5	5.2	6.8	12.5	23.1	30.6	34.7	34.9	36.3	37.1	35.9
S-FLM	0.5	10.5	13.3	16.2	17.7	19.9	20.2	20.5	20.3	20.4	20.8
FMLM+	0.0	2.6	4.9	10.1	20.5	24.9	26.0	25.6	25.5	25.5	25.5

Table 3: GSM8K test accuracy (%) across inference budgets. Accuracy on 1,319 test problems after 250,000 training updates (single training seed; one generated program per problem). ReLaDrift uses

K=12

given positives and

R_{\max}=5

. The Drifting Model is iterated by decoding and re-encoding its output between rounds (Appendix A.15 ). AR, MDLM, Duo, and S-FLM use

T=0.1

; FMLM+ uses

\arg\max

. Bold indicates the best parallel generator at each NFE.

Figure 3: GSM8K test accuracy versus the wall-clock time of generating the 1,319 test programs on one H200 (log scale; batches of 64 problems; settings of Table 3 ). The labels are the NFE; the time of ReLaDrift includes the encoder and the decoder.

Number of given positives $K$ (default 12)
Setting	NFE
	1	4	8	16
Default	13.2	27.3	29.0	29.4
- $4$	10.4	22.7	24.4	24.9
- $1$	1.0	8.7	10.7	12.2
Training depth $R_{\max}$ (default 5)

Table 4: Ablation of ReLaDrift on GSM8K. Test accuracy (%) after 250,000 training updates. The default uses

K=12

R_{\max}=5

, a contextualized CodeT5 encoder, and training-time verification with a positive queue. Embedding-table variants replace the contextualized encoder with context-free representations. Additional settings and results across all NFE are provided in Appendix A.14 .

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	NFE=1	4	5	8	16	32	64	128	180
$(2,1)$ , $n=2$ (default)	7.3 $\pm$ 1.0	78.9 $\pm$ 0.6	82.5 $\pm$ 0.8	88.4 $\pm$ 0.4	92.1 $\pm$ 0.4	93.1 $\pm$ 0.6	93.5 $\pm$ 0.7	93.9 $\pm$ 0.7	94.0 $\pm$ 0.7
$(2,1)$ , $n=4$	7.6 $\pm$ 2.2	78.1 $\pm$ 1.4	81.7 $\pm$ 1.3	87.4 $\pm$ 1.5	91.1 $\pm$ 1.2	92.1 $\pm$ 1.1	92.6 $\pm$ 1.2	92.7 $\pm$ 1.2	92.9 $\pm$ 1.4
$(1,1)$ , $n=2$	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0
$(1,1)$ , $n=4$	6.5 $\pm$ 1.7	76.0 $\pm$ 2.6	80.7 $\pm$ 2.2	88.0 $\pm$ 1.2	93.5 $\pm$ 0.6	95.9 $\pm$ 0.4	97.1 $\pm$ 0.3	98.0 $\pm$ 0.2	98.3 $\pm$ 0.2

Appendix

Table 5: Sudoku Hard accuracy (%) at 20,000 updates of the source formulation with the drift-field coefficients

(\lambda^{+},\lambda^{-})\in\{(2,1),(1,1)\}

and

n\in\{2,4\}

generated samples (2,000 validation puzzles, one candidate per puzzle, EMA weights, mean

\pm

standard deviation over three seeds, round embedding clamped to

\phi(\min(r,4))

; the other settings as in Section 5.1 ). The first row is the ReLaDrift Source row of Table 1 .

Item	ReLaDrift (ours)	S-FLM	FMLM+	FRM (our implementation)	DFM (our implementation)
Sequence, vocabulary	180 tokens, vocabulary 12, fixed prefix 91	same	same	same	same
State	continuous vector (12-dim one-hot code)	learned embedding on the sphere	one-hot + Gaussian noise	one-hot + carry	one-hot + Gaussian noise (linear interpolant $x_{t}=t\,x_{1}+(1-t)\,x_{0}$ )
Encoder / decoder	one-hot code / inner product with the codes + argmax	learned embedding on the sphere / $\arg\max$ of the output layer	one-hot / argmax	one-hot / argmax	one-hot / argmax
Transformer	9 layers, width 512, 8 heads, FFN 2,048, RoPE, dropout 0.1, round embedding $\phi(r)$ added to the input	DiT on the sphere, 8 layers, width 512, 8 heads	DiT, 8 layers, width 512, 8 heads, two-time adaLN	DiT, 8 layers, width 512, 8 heads, adaLN	DiT of FMLM+, 8 layers, width 512, 8 heads, two-time adaLN ( $t$ , $r-t$ )
Parameters	28.39M	28.56M	28.61M	28.57M	28.61M
Optimizer	AdamW 3e-4 (warmup 2,500, then constant), $\beta=(0.9,0.999)$ , weight decay 0, clip 1	same (official)	same (official)	same; optimizer and EMA restarted at Stage B	same

Appendix

Table 6: Sudoku settings of ReLaDrift and the compared methods. AR, Duo, and MDLM are trained with the implementations in the S-FLM repository and its settings (the same 8-layer, width-512 DiT ( Peebles and Xie, 2023 ) , 28.6M parameters; the AR model is causal and has no adaLN, 25.2M). DFM (Drift Flow Matching) is our implementation following its paper and official code (commit b8f287c ): the drift field, the law of the time pairs, the loss, and the Euler sampler are those of the official code, and the solution of each puzzle is paired with

n_{g}=2

source draws as in Appendix D of its paper. Duo and MDLM use ancestral sampling at temperature 1, and AR is evaluated with greedy decoding and with sampling at temperature 1. S-FLM uses its truncated, adaptive noise schedule; all methods drop the last, incomplete batch of each epoch (128 of the 48,000 puzzles; the permutation is redrawn at every epoch, so the dropped puzzles differ), whereas the official loaders keep it; with this batching, the official code never refits the adaptive schedule, so we added a patch at run time, without modifying the repository, so that the schedule is refitted (first refit at update 1,051). The single-round Drifting Model uses the ReLaDrift column without the round embedding, the draft projection, and the refinement output projection (28.37M parameters) and runs one round. All methods use dropout 0.1 and bf16 mixed precision.

Item	ReLaDrift (ours)	AR	S-FLM	FMLM+
Positives	first $K$ of the 12 solutions (fixed) + up to 4 verified samples (positive queue)	12 (problem, solution) pairs per problem, shuffled	same	same
Tokenizer, length	CodeT5 (vocabulary 32,100), 576	SmolLM-135M (vocabulary 49,153), 512	same	same
Latent	CodeT5 contextualized embedding, 512 dim (frozen)	discrete tokens	learned embedding on the sphere	one-hot + Gaussian noise
Transformer	DiT blocks of S-FLM (bidirectional, no adaLN), 12 layers, width 768, 12 heads, FFN 3,072, RoPE, round embedding $\phi(r)$	DiT, 12 layers, width 768, 12 heads (causal)	S-arch, 12 layers, width 768, 12 heads	DiT, 12 layers, width 768, 12 heads, two-time adaLN
Parameters	core 87.4M (+ frozen encoder 35.3M, decoder 42.2M = 164.9M)	160.5M	163.0M	168.0M
Optimizer	AdamW $\beta=(0.9,0.999)$ , weight decay 0, clip 1 (same as the compared methods)	AdamW $\beta=(0.9,0.999)$ , weight decay 0 (= Adam), clip 1 (official)	same	same

Appendix

Table 7: GSM8K settings. All methods use the same 1,588,781 problems, batch size 512, learning rate 3e-4 after 2,500 warm-up updates (then constant), 250,000 updates, dropout 0.1, and the EMA decay

\min(0.9999,(u+1)/(u+10))

of the official S-FLM implementation. Duo (167.9M) and MDLM (167.9M) are trained with the same code base and settings as S-FLM; S-FLM, Duo, and MDLM sample with 1,024 steps in the S-FLM paper ( Deschenaux and Gulcehre, 2026 ) . All methods use bf16 mixed precision. The single-round Drifting Model uses the ReLaDrift column without the round embedding, the draft projection, and the refinement output projection (core 86.2M parameters). Only ReLaDrift and the single-round Drifting Model use a verifier during training.

Condition	NFE=1	2	4	5	8	16	32	64	128	180
Source (main)	7.3 $\pm$ 1.0	54.7 $\pm$ 0.3	78.9 $\pm$ 0.6	82.5 $\pm$ 0.8	88.4 $\pm$ 0.4	92.1 $\pm$ 0.4	93.1 $\pm$ 0.6	93.5 $\pm$ 0.7	93.9 $\pm$ 0.7	94.0 $\pm$ 0.7
Additive	7.1 $\pm$ 2.2	54.8 $\pm$ 3.9	77.6 $\pm$ 2.6	81.5 $\pm$ 3.1	86.2 $\pm$ 3.6	89.5 $\pm$ 3.1	90.5 $\pm$ 3.0	91.0 $\pm$ 2.8	91.6 $\pm$ 2.5	91.6 $\pm$ 2.8
Conditional	7.4 $\pm$ 0.9	54.3 $\pm$ 3.3	77.6 $\pm$ 1.7	81.1 $\pm$ 1.1	86.1 $\pm$ 1.2	89.6 $\pm$ 1.8	90.7 $\pm$ 1.7	91.2 $\pm$ 2.0	91.6 $\pm$ 2.3	91.7 $\pm$ 2.2
Single-round Drifting Model	5.1 $\pm$ 0.8	4.9 $\pm$ 0.8	4.9 $\pm$ 0.8	5.0 $\pm$ 0.8	4.9 $\pm$ 0.8	4.9 $\pm$ 0.8	4.9 $\pm$ 0.8	4.9 $\pm$ 0.8	4.9 $\pm$ 0.8	4.9 $\pm$ 0.8
Source, w/ verifier	7.0 $\pm$ 1.0	56.4 $\pm$ 0.6	80.0 $\pm$ 1.0	83.7 $\pm$ 0.7	88.4 $\pm$ 0.7	92.0 $\pm$ 0.6	93.0 $\pm$ 0.6	93.4 $\pm$ 0.8	93.6 $\pm$ 0.8	93.7 $\pm$ 0.8
Source, $(1,1)$	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0	0.0 $\pm$ 0.0

Appendix

Table 8: Sudoku Hard accuracy (%) of all ReLaDrift conditions at 20,000 updates (2,000 puzzles, one candidate, EMA, mean

\pm

standard deviation over three seeds, round embedding clamped to

\phi(\min(r,R_{\max}-1))

). Unless noted: source formulation,

(\lambda^{+},\lambda^{-})=(2,1)

n=2

R_{\max}=5

, no verifier. At NFE

\geq 2

, the single-round Drifting Model is iterated, with its decoded output fed back in place of the noise (the dashed line of Figure 1 ; Appendix A.15 ).

Condition	Round embedding	NFE=8	16	32	64	128
Source	extrapolated	39.4 $\pm$ 40.2	46.4 $\pm$ 41.4	54.4 $\pm$ 47.2	57.1 $\pm$ 49.5	58.0 $\pm$ 50.2
	clamped	88.4 $\pm$ 0.4	92.1 $\pm$ 0.4	93.1 $\pm$ 0.6	93.5 $\pm$ 0.7	93.9 $\pm$ 0.7
Additive	extrapolated	38.6 $\pm$ 40.3	33.5 $\pm$ 19.0	49.0 $\pm$ 26.8	55.6 $\pm$ 29.6	58.4 $\pm$ 32.3
	clamped	86.2 $\pm$ 3.6	89.5 $\pm$ 3.1	90.5 $\pm$ 3.0	91.0 $\pm$ 2.8	91.6 $\pm$ 2.5
Conditional	extrapolated	74.7 $\pm$ 18.5	87.6 $\pm$ 1.5	89.5 $\pm$ 2.5	90.1 $\pm$ 2.7	90.9 $\pm$ 3.4
	clamped	86.1 $\pm$ 1.2	89.6 $\pm$ 1.8	90.7 $\pm$ 1.7	91.2 $\pm$ 2.0	91.6 $\pm$ 2.3

Appendix

Table 9: Sudoku Hard accuracy (%) at 20,000 updates with the round embedding extrapolated or clamped for NFE

\geq 8

(same checkpoints; mean

\pm

standard deviation over three seeds). For

R_{\max}=8

and

R_{\max}=16

, the two coincide up to NFE=8 and NFE=16, respectively.

	Round embedding	NFE=8	16	32
$K=1$	extrapolated	11.1	13.3	14.7
	clamped	10.7	12.2	12.9
$K=4$	extrapolated	24.0	23.4	24.9
	clamped	24.4	24.9	25.2
$K=12$	extrapolated	29.0	29.4	30.1
	clamped	29.0	29.4	29.4

Appendix

Table 10: GSM8K test accuracy (%) of ReLaDrift at 250,000 updates with the round embedding extrapolated or clamped for NFE

\geq 8

(EMA, single; same checkpoints and noise).

Setting	NFE=1	4	5	8	16	32	64	128	256	512	1024
$(2,1)$ , verifier, $K=12$ (default)	13.2	27.3	27.4	29.0	29.4	29.4	29.4	29.5	29.5	29.4	29.4
without verifier and queue	4.5	17.0	17.3	18.5	20.0	20.3	20.8	20.7	20.7	20.6	20.7
$(1,1)$ , verifier (stopped at 40,000 updates)	0.0	0.0	0.0	0.0	0.0	0.0	0.0	–	–	–	–
$\tau^{+}=0.1\times$ median	10.3	24.0	24.5	24.6	26.3	26.8	26.8	27.0	27.0	27.0	27.0
$\tau^{+}=1\times$ median (40,000 updates)	0.0	0.2	0.2	0.2	0.4	0.4	0.4	–	–	–	–
CodeT5 embedding table	6.7	15.1	16.0	16.7	17.7	18.0	18.2	18.1	18.1	18.1	18.2

Appendix

Table 11: GSM8K test ablations at 250,000 updates (%; EMA, single; round embedding clamped to

\phi(\min(r,R_{\max}-1))

;

R_{\max}=5

unless noted). Unless noted, the source formulation with

K=12

(\lambda^{+},\lambda^{-})=(2,1)

, the attraction bandwidth

\tau^{+}=0.01\times

the median distance between distinct positives (Appendix A.3 ), the verifier, and the positive queue. The single-round Drifting Model (trained in the same way) is iterated at NFE

\geq 2

, with its decoded and re-encoded program fed back in place of the noise (the dashed line of Figure 1 ; Appendix A.15 ). The

(1,1)

runs (

n=2

and

n=4

) were stopped at 40,000 updates after collapsing to 0.0% at every NFE

\leq 64

at 20,000 and 40,000 updates. The

\tau^{+}=1\times

median run was trained for 40,000 updates only; its row gives the values at 40,000 updates (0.0% at every NFE at 20,000 updates). The control row keeps the round embedding and all projections of ReLaDrift but is trained with

R_{\max}=1

; its refinement projections are never trained, so only NFE=1 is reported. The columns to the right of the vertical line (NFE

\geq 32

) lie outside the few-step regime.

Updates	AR	S-FLM	FMLM+	default	w/o ver.	$(1,1)$	$\tau^{+}=0.1$
				ReLaDrift ( $K=12$ ), NFE=16
20k	48.1	0.7	3.1	9.1	6.7	0.0	5.5
40k	56.7	2.4	6.7	16.1	11.9	0.0	12.7
60k	59.8	4.7	9.0	19.6	13.9	–	17.2
80k	62.5	5.9	11.2	22.1	16.6	–	20.4
100k	62.5	8.0	16.9	22.8	17.6	–	20.6

Appendix

Table 12: GSM8K test accuracy (%) over training (EMA, single). AR and S-FLM sample with temperature

T=1

(AR sequential, S-FLM 32 steps), and FMLM+ uses 32 rounds with argmax; ReLaDrift (

K=12

) is at NFE=16 with the clamp. “w/o ver.” is

(\lambda^{+},\lambda^{-})=(2,1)

without the verifier and the queue, “

(1,1)

” is

V^{+}-V^{-}

with the verifier, stopped at 40,000 updates, and “

\tau^{+}=0.1

” uses

\tau^{+}=0.1\times

median instead of

0.01\times

median; the progress of

K=1

and

K=4

is in Table 13 .

$K$	Updates	NFE=1	2	4	5	8	16	32	64
1	20k	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
	40k	0.0	0.3	0.2	0.3	0.4	0.4	0.5	0.5
	60k	0.2	0.3	0.5	0.6	0.8	1.1	1.1	1.1
	80k	0.3	0.4	1.0	1.0	1.1	1.8	2.1	2.4
	100k	0.3	0.5	1.3	1.4	2.4	3.5	4.5	5.2
	120k	0.3	0.9	2.4	3.1	3.9	5.1	6.1	6.5

Appendix

Table 13: GSM8K test accuracy (%) of ReLaDrift (Source,

(\lambda^{+},\lambda^{-})=(2,1)

, verifier and positive queue) at every checkpoint and NFE with the clamped round embedding (EMA, single).

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