EOS Early Rejection (EOSER)

• EOSER is a conditional computation mechanism that uses partial evidence to halt low-utility processing, achieving significant FLOP reductions in autoregressive reasoning.
• In masked diffusion language models, EOSER attenuates EOS token probability to prevent premature sequence termination, stabilizing full diffusion decoding.
• Extended EOSER applications include multi-stage in-flight rejection, OOD detection, and approximate Bayesian computation, offering efficiency gains across various domains.

Searching arXiv for papers on EOS Early Rejection and related MDLM/LLM early rejection mechanisms. EOS Early Rejection (EOSER) denotes a class of early-halting mechanisms that stop, prune, or reweight computation before natural completion when partial evidence already indicates low utility, premature termination, or an unpromising trajectory. In current arXiv usage, the term appears in multiple adjacent senses: in autoregressive reasoning, it can mean emitting an effective end-of-search decision for low-quality beams on the basis of partial Process Reward Model (PRM) scores (Cheshmi et al., 4 Aug 2025); in masked diffusion LLMs (MDLMs), it can mean suppressing premature $\texttt{<EOS>}$ during denoising so that full diffusion-style decoding remains stable (Yang et al., 28 Sep 2025). Related work extends the same early-rejection principle to diffusion-model termination and padding design (Liu et al., 16 Jun 2026), synthetic data generation (Chowdhury et al., 13 May 2026), out-of-distribution detection (Peng et al., 6 Feb 2026), and approximate Bayesian computation (Cao et al., 2024), while also showing that in some diffusion LLMs EoS tokens themselves function as hidden scratchpad state and therefore cannot be treated as inert padding (Breckner et al., 5 Mar 2026).

1. Definitions and operational variants

The common structure across EOSER formulations is conditional computation: a model observes a partial state, computes an inexpensive or intermediate signal, and then either continues expensive processing or halts early. What differs across papers is the object being rejected. In PRM-guided reasoning, the rejected object is a beam or trajectory. In MDLM decoding, the rejected object is an early $\texttt{<EOS>}$ hypothesis. In cascaded OOD detection, the rejected object is an input sample. In ABC, it is a parameter proposal. This suggests that EOSER is best understood as a control policy over incomplete computations rather than as a single decoding trick.

Setting	Operational meaning of EOSER	Representative reported effect
PRM-guided LLM reasoning	Effective EOS for beams with low partial PRM score	Up to $1.4\times$–$9\times$ inference FLOP reduction
MDLM full diffusion decoding	Step-dependent attenuation of EOS probability	Countdown $0.00\%\rightarrow 41.80\%$ at $L=256, S=64$
Diffusion LLM EoS workspace	Warning against pruning EoS positions as dead padding	Sudoku improves by $\sim 10\%$ from 1 to 128 EoS tokens
VoidPadding	Reserve $\texttt{[EOS]}$ for termination and $\texttt{[VOID]}$ for padding	$+17.84$ points over original on Dream-7B-Instruct
Multi-Stage In-Flight Rejection	Stop low-quality generations at checkpoints	$\texttt{<EOS>}$0–$\texttt{<EOS>}$1 token reduction
Cascaded Early Rejection	Halt OOD inputs at signal or semantic gates	Computational overhead reduced by $\texttt{<EOS>}$2
GP-based ejMCMC	Reject proposals before simulation	$\texttt{<EOS>}$3–$\texttt{<EOS>}$4 fewer simulations in the ODE example

The technical consequence of this breadth is that EOSER has no single canonical mathematical form. Some variants define a binary continuation policy $\texttt{<EOS>}$5; others define a gate $\texttt{<EOS>}$6; others act by multiplicatively attenuating EOS probability with a step-dependent coefficient $\texttt{<EOS>}$7. The unifying criterion is that a cheap or partial signal is used to reduce expected compute before the full computation would otherwise terminate naturally (Cheshmi et al., 4 Aug 2025, Yang et al., 28 Sep 2025, Chowdhury et al., 13 May 2026, Peng et al., 6 Feb 2026, Cao et al., 2024).

2. PRM-guided early rejection in autoregressive reasoning

In "Accelerating LLM Reasoning via Early Rejection with Partial Reward Modeling" (Cheshmi et al., 4 Aug 2025), EOSER is formulated for PRM-guided beam search over reasoning trajectories. A trajectory is written as

$\texttt{<EOS>}$8

where each step $\texttt{<EOS>}$9 is a token block. For beam $1.4\times$0, a completed step receives final step reward $1.4\times$1, while the first $1.4\times$2 tokens of that step receive partial reward

$1.4\times$3

The core hypothesis is that a PRM is also a Partial Reward Model, with

$1.4\times$4

where $1.4\times$5 is monotonic increasing and $1.4\times$6 is zero-mean, $1.4\times$7-sub-Gaussian noise. Early rejection then becomes a mid-step EOS decision: after generating only the first $1.4\times$8 tokens of the next reasoning step, beams are ranked by $1.4\times$9, the top $9\times$0 are kept, and all others are rejected and never expanded further.

Algorithmically, the rule is thresholded ranking. Let $9\times$1, and let $9\times$2 be the $9\times$3-quantile of the current partial rewards. Then beam $9\times$4 is kept iff $9\times$5; otherwise it is terminated. In EOSER terms, this is equivalent to emitting EOSER at token $9\times$6 of the step. The continuation policy is

$9\times$7

The baseline and the Early Rejection variant are presented as Algorithm 2 and Algorithm 3, respectively, with the latter maintaining beam width $9\times$8 by keeping $9\times$9 parents and expanding each into $0.00\%\rightarrow 41.80\%$0 children.

The theoretical guarantee concerns the probability of killing the optimal beam. If $0.00\%\rightarrow 41.80\%$1, the minimal expected partial-reward gap is

$0.00\%\rightarrow 41.80\%$2

and the paper proves

$0.00\%\rightarrow 41.80\%$3

The bound decays exponentially in $0.00\%\rightarrow 41.80\%$4, so larger partial prefixes $0.00\%\rightarrow 41.80\%$5 make premature termination of the optimal beam increasingly unlikely. In an i.i.d. token-contribution toy model, the correlation between partial and final rewards is

$0.00\%\rightarrow 41.80\%$6

and the empirical correlations reported for math PRMs are already $0.00\%\rightarrow 41.80\%$7 at $0.00\%\rightarrow 41.80\%$8 and $0.00\%\rightarrow 41.80\%$9 at $L=256, S=64$0, with a plateau beyond $L=256, S=64$1.

The reported empirical effect is substantial. On SAT-MATH (AGIEval), MATH-500, and AIME 2024, early rejection achieves $L=256, S=64$2–$L=256, S=64$3 reduction in inference FLOPs with no loss in task accuracy when using MathShepherd-Mistral-7B, and $L=256, S=64$4–$L=256, S=64$5 with Skywork-PRM-1.5B while maintaining or slightly improving accuracy. The behavior is model-dependent: for exploratory models such as Qwen-2.5-3B, EOSER can improve both accuracy and compute at small beam widths, whereas for more deterministic models such as Llama-3.2-3B the dominant effect is compute reduction.

3. EOSER in masked diffusion LLMs

In "Taming Masked Diffusion LLMs via Consistency Trajectory Reinforcement Learning with Fewer Decoding Step" (Yang et al., 28 Sep 2025), EOSER has a different but closely related meaning. The target model class is the masked diffusion LLM, trained to denoise masked sequences in parallel with bidirectional attention. The paper identifies a failure mode in full diffusion-style decoding: LLaDA pretraining replaces $L=256, S=64$6 with $L=256, S=64$7, so training sequences contain a large number of EOS tokens, inducing a strong prior for EOS. In early denoising steps, when token confidence is low, EOS receives much higher average confidence than non-EOS tokens, causing the model to emit EOS near the tail of the sequence far too early. This prematurely semantically terminates the canvas and derails the reverse process.

EOSER addresses that failure by attenuating EOS probability with a step-dependent coefficient. For a uniform scheduler with $L=256, S=64$8 denoising steps,

$L=256, S=64$9

and the modified token distribution is

$\sim 10\%$0

The method changes only EOS probability; it does not change masking, unmasking, or generation order. Under uniform scheduling, the reported hyperparameters are $\sim 10\%$1 and $\sim 10\%$2. Under the Ascending Step-Size (ASS) scheduler, where decoding proceeds in approximately $\sim 10\%$3 steps, EOSER uses a decoded-token-fraction schedule and $\sim 10\%$4, $\sim 10\%$5.

This EOSER makes full diffusion-style decoding viable. In the training-free comparison at $\sim 10\%$6, $\sim 10\%$7, full diffusion without EOSER collapses on Countdown with $\sim 10\%$8 accuracy, whereas EOSER reaches $\sim 10\%$9. On GSM8K the same settings move from $\texttt{[EOS]}$0 to $\texttt{[EOS]}$1. The pattern is similar, though weaker, on MATH500 and Sudoku. With ASS and few-step decoding, EOSER remains decisive: at $\texttt{[EOS]}$2, EOSER+ASS reaches $\texttt{[EOS]}$3 on Countdown, $\texttt{[EOS]}$4 on GSM8K, $\texttt{[EOS]}$5 on MATH500, and $\texttt{[EOS]}$6 on Sudoku, whereas full diffusion+ASS is frequently near zero.

The paper further embeds EOSER inside CJ-GRPO, a rollout-consistent RL algorithm for MDLMs. Here the task dependence becomes explicit. EOSER+full diffusion is highly effective on planning tasks such as Countdown and Sudoku, including $\texttt{[EOS]}$7 on Sudoku at $\texttt{[EOS]}$8 for CJ-GRPO+EOSER, and $\texttt{[EOS]}$9 on Countdown and $\texttt{[VOID]}$0 on Sudoku at $\texttt{[VOID]}$1 for CJ-GRPO+EOSER+ASS. On GSM8K and MATH500, however, EOSER+full diffusion underperforms semi-AR baselines, and the paper interprets this as evidence that sequential reasoning remains better served by semi-autoregressive decoding. EOSER is therefore not presented as a universal replacement for block-wise decoding, but as a mechanism that restores the viability of full diffusion-style decoding, especially for tasks with more parallel or bidirectional structure.

4. EoS as workspace, padding, and semantic termination

A central complication for any EOSER design is that EoS tokens can be computationally active. "Diffusion LLMs can think EoS-by-EoS" argues that, in diffusion LLMs with visible EoS padding and bidirectional attention, the hidden representations of EoS tokens serve as a hidden scratchpad (Breckner et al., 5 Mar 2026). Increasing generation length improves reasoning even when the number of semantic tokens stays constant or decreases, because the extra positions are filled with EoS tokens that remain visible to the model. In controlled prompting with fixed numbers of masks and fixed decoding steps, adding EoS tokens still improves accuracy. For LLaDA1.5 and Dream on Addition and Entity Tracking, appending 4 EoS tokens is described as the simplest way to achieve close to the best possible performance; on Sudoku, LLaDA1.5 improves by $\texttt{[VOID]}$2 absolute accuracy between 1 and 128 EoS tokens, and Dream improves by $\texttt{[VOID]}$3. Activation patching of 16 trailing EoS tokens shifts outputs toward counterfactual prompts, showing that EoS hidden states causally carry problem-specific information. A practical implication drawn in the paper is that one should never prune below $\texttt{[VOID]}$4 trailing EoS tokens unless there is strong evidence that the model does not use EoS as scratchpad.

"VoidPadding: Let [VOID] Handle Padding in Masked Diffusion LLMs so that [EOS] Can Focus on Semantic Termination" addresses a different but adjacent issue: EOS overflow in MDLM instruction tuning when repeated $\texttt{[VOID]}$5 is used as padding (Liu et al., 16 Jun 2026). The paper formalizes how repeated EOS padding creates a positionwise length prior, so that on a fully masked canvas the model’s EOS probabilities $\texttt{[VOID]}$6 track the response-length CDF $\texttt{[VOID]}$7. Under large-block decoding, EOS is first committed in the high-confidence tail and then propagates leftward as committed EOS tokens condition earlier positions. VoidPadding replaces repeated EOS padding with repeated $\texttt{[VOID]}$8,

$\texttt{[VOID]}$9

reserves exactly one semantic $+17.84$0, always masks that EOS during training, and bans $+17.84$1 at inference by setting its logit to $+17.84$2. The resulting $+17.84$3Termination rule stops decoding once no masks remain before the leftmost visible $+17.84$4, discarding everything to its right.

The empirical effect is large. On Dream-7B-Instruct, VoidPadding improves the block-size-averaged four-task mean by $+17.84$5 points over the original model and $+17.84$6 points over RainbowPadding, while reducing decoding NFE by $+17.84$7 on average. In the LLaDA ablation, enabling $+17.84$8Termination on the original EOS-padded model slightly hurts mean score while reducing NFE by $+17.84$9, and on RainbowPadding it is nearly accuracy-neutral with $\texttt{<EOS>}$00 mean NFE reduction; on VoidPadding, by contrast, $\texttt{<EOS>}$01Termination increases mean score by $\texttt{<EOS>}$02 and reduces mean NFE by $\texttt{<EOS>}$03. This suggests that EOSER depends critically on whether EOS is a clean semantic terminator or a mixed padding-and-termination token.

Taken together, these two papers imply that EoS semantics are architecture- and training-dependent. In some diffusion models, EoS positions are active latent workspace and should be preserved during active diffusion. In MDLMs trained with repeated EOS padding, however, EOS can be overburdened with a length prior and must be disentangled from padding before it becomes a reliable early-stopping signal.

5. Multi-stage and cascaded early rejection beyond decoding

The same early-rejection principle appears outside token decoding. "Know When To Fold 'Em: Token-Efficient LLM Synthetic Data Generation via Multi-Stage In-Flight Rejection" formulates in-flight rejection as a sequential decision process over partial trajectories (Chowdhury et al., 13 May 2026). Generation is divided into stages with validators $\texttt{<EOS>}$04 operating on the partial trajectory $\texttt{<EOS>}$05. If a validator fails, decoding stops immediately and the sample is discarded. The paper proves that any non-trivial discard policy reduces expected token consumption, with larger savings when rejection happens earlier, and models the conditional utility process $\texttt{<EOS>}$06 as a martingale to argue that early rejection does not bias the expected utility of retained samples. Across five instruction-tuned models and seven reasoning benchmarks, MSIFR yields $\texttt{<EOS>}$07–$\texttt{<EOS>}$08 token reduction as a standalone method and up to $\texttt{<EOS>}$09 when combined with early-exit methods, while preserving or improving evaluation accuracy.

"Halt the Hallucination: Decoupling Signal and Semantic OOD Detection Based on Cascaded Early Rejection" presents a layerwise halting architecture in vision (Peng et al., 6 Feb 2026). Inputs pass through stages $\texttt{<EOS>}$10, and each stage has a gate

$\texttt{<EOS>}$11

with global rejection $\texttt{<EOS>}$12 if any gate fires. The first gate, Structural Energy Sieve (SES), uses a Laplacian-based high-frequency score to catch physical anomalies near the input; later gates use Semantically-aware Hyperspherical Energy (SHE) to detect semantic OOD while removing feature-magnitude bias. On CIFAR-100, CER reduces computational overhead by $\texttt{<EOS>}$13, decreases average FPR95 from $\texttt{<EOS>}$14 to $\texttt{<EOS>}$15, and improves AUROC to $\texttt{<EOS>}$16. The paper can be read as a non-generative EOSER analogue in which EOS is replaced by explicit halting gates over depth.

"Using early rejection Markov chain Monte Carlo and Gaussian processes to accelerate ABC methods" instantiates early rejection in stochastic simulation (Cao et al., 2024). The expensive step is synthetic-data simulation inside ABC-MCMC. ejMCMC fits a GP to the discrepancy $\texttt{<EOS>}$17 and uses a conservative lower predictive quantile

$\texttt{<EOS>}$18

to define a pseudo-acceptance stage and an early-rejection upper bound $\texttt{<EOS>}$19 before simulation. Only proposals passing that screen are simulated and subjected to the full Metropolis-Hastings correction. In the ODE example, ejMCMC reduces the number of simulations by $\texttt{<EOS>}$20–$\texttt{<EOS>}$21 relative to OejMCMC at similar posterior accuracy, and with $\texttt{<EOS>}$22 the early-rejection efficiency ranges from $\texttt{<EOS>}$23 at $\texttt{<EOS>}$24 to $\texttt{<EOS>}$25 at $\texttt{<EOS>}$26, exposing an explicit efficiency-accuracy trade-off. The chain targets a modified posterior $\texttt{<EOS>}$27, and the paper gives conditions under which this converges to the standard ABC posterior in $\texttt{<EOS>}$28.

These extensions show that EOSER is not confined to language-model termination. It is a broader conditional-computation paradigm: compute a partial signal, construct a principled gate, and stop expensive continuation whenever the expected value of continuation is too low.

6. Limitations, misconceptions, and open problems

A persistent misconception is that EoS tokens are always safe to prune. The diffusion-LM evidence directly contradicts that view. In LLaDA1.5 and Dream-v0, extra EoS tokens improve reasoning even when decoding steps and semantic-token budgets are fixed, and patching only EoS hidden states can move outputs toward counterfactual answers (Breckner et al., 5 Mar 2026). Any EOSER mechanism that removes EoS positions during active diffusion therefore risks deleting part of the model’s internal computation. This risk is largest on hard tasks such as Sudoku and entity tracking, exactly where hidden workspace is most valuable.

A second misconception is that early rejection uniformly improves performance. The literature instead shows marked task dependence. PRM-based EOSER is validated primarily on math reasoning and relies on the assumption that partial rewards are monotonic predictors of final rewards; the paper explicitly cautions that domains such as code with backtracking, creative writing, and multimodal tasks may violate that assumption (Cheshmi et al., 4 Aug 2025). MDLM EOSER is highly effective on planning tasks but underperforms semi-AR decoding on GSM8K and MATH500, and its behavior depends on $\texttt{<EOS>}$29, $\texttt{<EOS>}$30, scheduler choice, and denoising step count (Yang et al., 28 Sep 2025). CER depends on natural-image spectral statistics and stable intermediate feature geometry (Peng et al., 6 Feb 2026). MSIFR depends on conservative validators and can over-reject trajectories that might have recovered later (Chowdhury et al., 13 May 2026). ejMCMC depends on GP fidelity and pays the price of a modified target distribution unless additional conditions hold (Cao et al., 2024).

The current research frontier is therefore not merely more aggressive pruning, but better semantics for halting. Several directions are already explicit in the literature: monitoring what is stored in EoS hidden states and reconciling explicit chain-of-thought with hidden EoS-by-EoS computation (Breckner et al., 5 Mar 2026); decoupling padding from semantic termination through tokens such as $\texttt{<EOS>}$31 or related length-modeling mechanisms (Liu et al., 16 Jun 2026); and developing more principled few-step full-diffusion decoding strategies and hybrid diffusion/AR schemes tailored to task structure (Yang et al., 28 Sep 2025). A plausible implication is that future EOSER systems will be increasingly model-aware: they will reserve hidden workspace where needed, separate padding from termination, and apply early rejection only when the intermediate signal is calibrated to the continuation dynamics of the underlying architecture.