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Consider an original counting puzzle: three boxes contain four
beads each, then five extra beads are added.

2
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The correct total is seventeen, but an unfinished answer still
contains gaps in its reasoning and final result.

3
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Imagine some words becoming visible while other positions remain
masked, as a response gradually takes shape.

4
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A later, more complete version can contain clues that the
earlier noisy view has not yet made explicit.

5
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These token boxes are an illustrative schematic, not a
reproduced benchmark question or an actual model decoding trace.

6
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Can self-generated later information train a diffusion model to
improve its earlier prediction decisions?

7
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A diffusion language model generates by iterative denoising,
beginning with a response whose token positions are masked.

8
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Each step predicts masked-position distributions, rather than
only the next left-to-right token.

9
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Confident predictions become visible; other positions stay
masked until later steps improve their reliability.

10
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The current noisy state and clean final response are different
moments along one sampled generation trajectory.

11
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Arbitrary-order generation permits conditioning on later
information, rather than only a left-to-right prefix.

12
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During training, a completed self-generated response supplies
privileged teacher-specific context.

13
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The student first samples its own complete trajectory,
preserving earlier noisy states and the final generated
response.

14
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For an earlier denoising step, its input remains that current
noisy state together with the original prompt.

15
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The teacher view selectively reveals tokens from the completed
response at positions that are currently masked.

16
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Revealing a subset supplies clues while leaving prediction work
for the teacher distribution.

17
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Both roles start from the same diffusion model, but different
views inform their next-step predictions.

18
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Future information comes from its sampled response, not time
travel or external knowledge of unseen test answers.

19
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The objective compares student and teacher distributions at the
current denoising step.

20
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Only selected confident positions among the currently masked
tokens actively determine the next state in that step.

21
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The divergence loss is applied to that subset, rather than
treating every position as a left-to-right continuation.

22
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The teacher supplies dense vocabulary-level guidance, giving
more information than the final correctness reward alone.

23
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The default selects supervised masked positions by teacher
confidence, supported by the reported ablation.

24
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Summing this guidance over the sampled trajectory aligns the
objective with the iterative denoising process.

25
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The completed response used for guidance is generated by the
student itself, keeping the trajectory on-policy.

26
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The default recipe samples until a correct final answer appears,
or reaches its budget of eight attempts.

27
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Task-specific checks select useful future information; Sudoku
uses a score threshold rather than binary answer matching.

28
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Training follows the whole sampled trajectory, rather than one
artificial forward pass over a static response.

29
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The default fixes the initial teacher policy, providing a stable
reference as student parameters change.

30
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Teacher information comes from enriched context, while model
updates apply to the learner.

31
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The experiments use LLaDA Instruct, an eight-billion-parameter
diffusion model, on four math and planning tasks.

32
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Tasks include GSM eight K, MATH five hundred, Countdown, and
four-by-four Sudoku, with retained benchmark settings.

33
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Training uses low-rank adaptation and reverse KL, retaining a
quarter of eligible tokens for teacher context.

34
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Training length is two hundred fifty-six tokens; evaluation
lengths depend on the task category.

35
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Evaluation occurs every twenty-five steps before five hundred
one, reporting the best evaluated checkpoint.

36
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The comparison specifies the model, distillation recipe, task,
generation length, and checkpoint-selection rule.

37
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Consider GSM eight K at length two hundred fifty-six, using the
eight-billion-parameter LLaDA Instruct diffusion model.

38
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The base scores seventy-six percent; diffusion GRPO reaches
seventy-nine point eight percent.

39
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Diffusion on-policy self-distillation reaches eighty-one
percent, exceeding that baseline in this specific setting.

40
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Other Table one tasks and generation lengths differ, so the
comparison remains tied to this configuration.

41
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At MATH five hundred length five hundred twelve, diffusion GRPO
reports thirty-nine point two.

42
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The proposed method reports thirty-seven point eight for that
same task and generation length.

43
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The reported improvement therefore does not imply that the
proposed method wins every individual cell in the table.

44
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This chart preserves the precise model, baseline, generation
length, and scope of the reasoning comparison.

45
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Optimization-step counts also describe how quickly dense teacher
guidance can improve the learner.

46
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GSM eight K lists seven thousand seven hundred baseline updates
versus four hundred twenty-five for d-OPSD.

47
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Baseline counts come from its original report: these are
optimization steps, rather than matched wall-clock timings.

48
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Dense supervision is linked to efficiency, while each training
step still spends computation on sampled trajectories.

49
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Return to the bead example: a checked later response guides
earlier masked prediction states.

50
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Ordinary generation evaluates the trained model, whose
parameters improve through training-specific privileged context.

51
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The method pairs self-generated future context with step-aligned
supervision, tailored to iterative diffusion reasoning.