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D-DOIT: Training-free Adaptation of Discrete Diffusion via Doob's h-Transform

Jieke Wu, Qijie Zhu, Weimin Wu, Zeqi Ye, Minshuo Chen, Han Liu

arXiv:2610.04938v1cs.LGcs.AI

TL;DR

Reward-guided sequence design seeks task-level scores, but adapting masked diffusion to generic black-box rewards without retraining or reward gradients remains challenging. D-DOIT reweights frozen reverse transitions using a discrete Doob h-transform, and outperforms training-free guidance baselines across DNA design and protein inverse folding. Its highest-quality setting carries substantial recurring inference cost.

  • Problem

    Adapting a frozen masked diffusion model to generic black-box rewards using only inference-time sampling remains an open question, as existing approaches may need reward gradients or costly search.

    Source: §1 · p. 2

  • Method

    D-DOIT applies a discrete Doob h-transform that reweights the frozen masked diffusion reverse kernel using reward values, with prediction-head completions and finite-candidate selection as practical approximations.

    Source: §1 · p. 2

  • Results

    Across DRAKES HepG2, Ctrl-DNA, and protein inverse folding, D-DOIT outperforms training-free guidance baselines; protein inverse folding reaches 68.1% joint stability–structure success.

    Source: §1 · p. 2, §5 · p. 10

  • Takeaways & Limitations

    D-DOIT provides a training-free mechanism for reward-guided sequence generation, although its highest-quality results require a larger compute budget.

    Source: §5 · p. 10

    The highest-quality setting incurs substantial recurring inference cost, leaving the reward–compute trade-off a practical challenge.

    Source: §5 · p. 10

Abstract

from arXiv · show

We propose D-DOIT (Discrete Doob-Oriented Inference-time Transformation), a training-free and efficient adaptation method for discrete diffusion models with generic rewards. D-DOIT formulates adaptation as sampling from a reward-tilted target distribution and realizes this transport through Doob's h-transform of the discrete diffusion reverse kernel, using only reward values rather than reward gradients. Unlike continuous diffusion, masked discrete diffusion samples categorical token-reveal transitions rather than continuous state updates. D-DOIT derives the corresponding discrete Doob's h-transform, which guides sampling by reweighting reverse transition probabilities instead of adding a drift correction. To make this transformation practical, D-DOIT avoids expensive future rollouts. At each guided step, D-DOIT samples candidate next states, uses the model prediction head to complete each candidate into a clean sequence, evaluates each completion with the reward oracle, and resamples the next state with probabilities proportional to the rewards. An optional late-stage best-of-K refinement further improves sample quality by branching trajectories only near the end of denoising, avoiding the $K$-fold cost over the full trajectory. Empirically, across regulatory DNA design and protein inverse folding benchmarks, D-DOIT outperforms training-free guidance baselines. It improves enhancer activity and cell-type specificity while preserving sequence naturalness, and achieves the highest success rate in protein inverse folding.

1 Introduction

D-DOIT adapts frozen masked discrete diffusion to sequence-level rewards by reweighting categorical reverse transitions, without reward gradients or backbone updates. Across DNA design and protein inverse folding, it improves on training-free guidance baselines, with quality–compute and biological-metric trade-offs.

1 Introduction

  • Reward-guided sequence design targets task-level scores such as activity and specificity, but training-based adaptation updates the backbone and often needs reward gradients, while training-free alternatives may require differentiable rewards or costly search.The paper asks whether a frozen masked diffusion model can instead be adapted to generic black-box rewards by changing only inference-time sampling.Source: §1 · pp. 1–2
  • D-DOIT derives a discrete Doob h-transform that multiplicatively reweights categorical token-reveal transitions by candidate downstream values, rather than adding the score or drift correction used in continuous diffusion.The exact transform is distinguished from the practical approximations built on it.Source: §1 · p. 2
  • To avoid future rollouts, D-DOIT completes finite candidate next states with the prediction head, queries their rewards, and selects with Doob-weighted probabilities; optional best-of-K branching is limited to late denoising.It uses reward values without updating the diffusion backbone, and late branching avoids multiplying the cost of the full trajectory.Source: §1 · p. 2
  • 6.19 predicted activity exceeds training-based DRAKES on Pred-Activity, 3-mer Corr, and JASPAR Corr, while D-DOIT-Q remains slightly lower on ATAC-Acc; protein inverse folding reaches 68.1% joint stability–structure success.On Ctrl-DNA, D-DOIT-Q raises ∆R to 0.119 with a motif-preservation trade-off, while endpoint search captures much of the gain at matched wall-clock budget.Source: §1 · p. 2
  • Across HepG2 enhancer design, cell-type-specific enhancer design, and protein inverse folding, D-DOIT improves over reported training-free guidance baselines while retaining trade-offs in naturalness, motif structure, and inference budget.The highest-quality results are not presented as a matched-cost replacement for every high-budget or training-based method.Source: §1 · pp. 2–3

2 Preliminaries

Masked discrete diffusion supplies a frozen reverse chain, and the section defines a reward-tilted target together with an exact Doob h-transform that transports the chain toward it. The transform reweights categorical transitions, but its exact-target guarantee requires the true downstream reward value.

2.1 Masked Discrete Diffusion Models

  • Masked diffusion independently replaces clean tokens with a mask in the forward process, while a learned reverse chain samples from the all-mask state to a clean sequence.D-DOIT keeps the reverse-model parameters fixed and changes transition probabilities at inference time.Source: §2.1 · pp. 3–4

2.2 Reward-tilted Target and Doob’s h-transform

  • The target tilts the frozen model’s clean-sequence law by reward, and the discrete h-function is the conditional downstream reward weight for a partially denoised state.At clean time, the h-function reduces to the reward weight; at intermediate times, it represents reward-to-go under the frozen reverse chain.Source: §2.2 · p. 4
  • Unlike continuous Doob guidance’s additive score or drift correction, the discrete transform multiplicatively reweights categorical reverse-transition probabilities.Masked diffusion has categorical, irreversible token-reveal dynamics rather than a Euclidean score field.Source: §2.2 · pp. 4–5
  • With the true downstream value ratio at each step, locally reweighting the frozen reverse kernel yields the reward-tilted clean-time distribution exactly.The guarantee concerns the exact normalized kernel, not the prediction-head implementation.Source: §2.2 · pp. 4–5

3 D-DOIT: A Discrete Sampler Based on Doob’s h-transform

D-DOIT turns the exact discrete Doob h-transform into a practical sampler for frozen masked diffusion by estimating candidate values with clean completions and selecting among finite candidate states. It optionally branches trajectories late and offers two operating points with different inference-budget allocations.

3 D-DOIT: A Discrete Sampler Based on Doob’s h-transform

  • D-DOIT estimates downstream values with the prediction head, uses finite candidates to approximate the transformed kernel, and optionally branches late to hedge irreversible token reveals.The prediction-head completion replaces future rollouts, and late branching applies best-of-K refinement only after a chosen split update.Source: §3 · p. 5, §3.1 · pp. 6–7

3.1 D-DOIT Algorithm

  • For each candidate state, the prediction-head h-estimator uses a one-step clean-completion value instead of the exact future rollout value.The completion distribution fixes revealed tokens and independently samples masked positions from coordinate-wise clean-token posteriors; multiple completions can reduce Monte Carlo variance without removing rollout-value discrepancy.Source: §3.1 · p. 6
  • D-DOIT samples M candidate next states and selects among them using estimated-value weights; proposal and completion temperatures control different stages.With a smoothed proposal (τ > 1), the weighted rule is a biased proposal approximation; the operating points omit the importance-ratio correction needed to retain the original target under that proposal.Source: §3.1 · pp. 6–7
  • Late best-of-K refinement keeps the highest-reward clean output among branches created after a split, but its terminal selection changes the output law and cannot repair earlier reveals.The K-fold multiplier is paid only after the split; with K = 1 or a split at L, the method reduces to D-DOIT.Source: §3.1 · p. 7

3.2 Two Configurations: D-DOIT-LBOK-S and D-DOIT-Q

  • D-DOIT-LBOK-S uses a small candidate budget with late refinement, while D-DOIT-Q allocates more inference budget to candidate evaluation and/or split-time refinement.Both are operating points of the same sampler and differ only in budget allocation.Source: §3.2 · p. 7

4 Experiments

Across enhancer design and protein inverse folding, D-DOIT improves reported reward-guided outcomes, with trade-offs in motif preservation, structural fidelity, and inference cost. The experiments also examine completion estimation and reward-oracle robustness.

4.1 Task 1: Regulatory DNA Sequence Design

  • 6.19 Pred-Activity, 91.7% ATAC-Acc, 0.894 3-mer Corr, and 0.923 JASPAR Corr: D-DOIT-Q beats every training-free baseline on every Table 2 metric.It exceeds training-based DRAKES on Pred-Activity, 3-mer Corr, and JASPAR Corr, while ATAC-Acc is 91.7% versus 92.5%; near-TDS-speed LBOK-S matches TDS activity (4.56 vs 4.58) while improving the other reported metrics.Source: §4.1 · p. 8, Table 2 · p. 8

4.2 Task 2: Cell-type-specific Enhancer Design

  • D-DOIT-LBOK-S raises three-cell-line specificity (∆R) from 0.002 to 0.033 over TDS and Motif Corr from 0.573 to 0.648 at near-TDS wall-clock.D-DOIT-Q reaches ∆R = 0.119 while trading motif preservation for reward; matched-time endpoint search also improves specificity, and Q’s mean differences from Frozen Best-of-708 are not claimed statistically significant.Source: §4.2 · pp. 8–9, Table 3 · p. 9

4.3 Task 3: Protein Inverse Folding

  • 68.1 ± 3.5% joint success across 12 test backbones, three seeds, and 128 sequences per backbone and seed, versus published SMC at 63.6% and TDS at 62.9%.D-DOIT has the highest predicted stability and joint success among six methods, while TDS has better structural fidelity; the comparison supports transfer to protein inverse folding, not experimental protein validation.Source: §4.3 · p. 9, Table 3 · p. 9

4.4 Compute–quality Trade-off

  • With similar model budgets, Frozen Best-of-15 uses 15 rather than 1,777 oracle evaluations and runs faster (97.8 versus 327.8 seconds), but has lower Pred-Activity (2.219 versus 4.491).LBOK-S is the near-TDS-runtime point at 1.16× in the main timing study; Q allocates more compute to quality, and absolute timings depend on each study’s timing protocol.Source: §4.4 · pp. 9–10

4.5 Completion Quality and Search Diagnostics

  • Increasing completions per candidate from nh = 1 to 2 and 3 raises final guidance reward from 5.272 ± 0.057 to 5.764 ± 0.033 and 5.978 ± 0.026, with 41% and 81% additional wall time.The default remains nh = 1; direct and sequential completions have similar h-value distributions, with Wasserstein-1 0.060–0.124.Source: §4.5 · p. 10

4.6 Oracle Robustness

  • Enformer, CNN, and MLP reward predictors each improve all four metrics over unguided sampling, and held-out rescoring preserves the gains.Pred-Activity is 4.491, 2.203, and 1.544 for Enformer, CNN, and MLP, respectively, versus 0.371 for the frozen model; evaluator scales are incomparable, and these controls do not establish experimental biological activity.Source: §4.6 · p. 10

5 Conclusion and Discussion

D-DOIT applies a Doob h-transform to frozen masked discrete diffusion and outperforms training-free guidance baselines across DRAKES HepG2, Ctrl-DNA, and protein inverse folding, though its highest-quality results require more compute. Its related-work comparison distinguishes kernel-level h-transform guidance from fine-tuning, intermediate-latent selection, and heuristic guidance.

5 Conclusion and Discussion

  • Across DRAKES HepG2, Ctrl-DNA, and protein inverse folding, D-DOIT outperforms training-free guidance baselines, while its highest-quality results require a larger compute budget.The conclusion also frames Doob-style kernel reweighting as a general framework for guiding discrete generative models with sequence-level rewards.Source: §5 · p. 10
  • Practical approximations can lose the exact-target guarantee, late branching cannot revise revealed tokens, biological benchmarks require experimental validation, and the highest-quality setting incurs substantial recurring inference cost.The paper identifies the reward–compute trade-off as a practical challenge.Source: §5 · p. 10

A Related Work

  • Prior reward-guided discrete-model methods include backbone-updating training approaches and inference-time steering methods, which do not directly tilt the reverse kernel toward an optimal proposal.The passage surveys approaches including RL-style fine-tuning, sequential Monte Carlo, posterior sampling, and iterative refinement.Source: A Related Work · p. 17
  • D-DOIT uses an executable estimator of the categorical reverse-kernel h-transform, unlike DRAKES fine-tuning, soft value-based latent selection, or heuristic masked-diffusion guidance.The authors do not claim that local finite-candidate selection itself is unique to D-DOIT.Source: A Related Work · p. 17
  • On regulatory DNA benchmarks, D-DOIT competes without policy training, trading modest inference-time compute for reward gains while preserving near-base per-step cost.The comparison contrasts D-DOIT with DRAKES, Ctrl-DNA, and a concurrent tree-search method.Source: A Related Work · p. 17

B Proof of Theorem 2.1

The proof establishes that reward tilting the pretrained reverse-chain path law induces a normalized Doob-transformed reverse kernel. It also contrasts this categorical kernel reweighting with continuous diffusion’s additive score or drift correction.

B Proof of Theorem 2.1

  • The reward-tilted path law induces the Doob-transformed reverse kernel, whose normalization follows from the tower property and the definition of h.The proof treats identities on the pretrained reverse-chain support.Source: B Proof of Theorem 2.1 · p. 18
  • Because the fully masked terminal state is deterministic, its h-value is Zβ and the transformed terminal distribution remains the pretrained terminal distribution.This terminal-state result is stated for the fully masked terminal setting.Source: B Proof of Theorem 2.1 · p. 18
  • The separate tempered-reference result changes the target unless τ = 1 and η = 1/β, and the practical proxy, windowing, and terminal selection do not inherit its exactness.The proposition concerns a different reference law and does not identify the executed estimator with an exact Doob value.Source: B Proof of Theorem 2.1 · p. 19

C Continuous-time Doob’s h-transform on Score-based Diffusion

  • In continuous diffusion, Doob guidance shifts the reverse drift or score additively, whereas masked discrete diffusion uses multiplicative reweighting of categorical transition probabilities.The continuous construction requires sufficient regularity of h for ∇log h to exist; the discrete state has no Euclidean score field.Source: C Continuous-time Doob’s h-transform… · p. 20

D On the Prediction-head h-estimator

The prediction-head h-estimator replaces future rollouts with clean-sequence completions, but it generally differs from the rollout value. The appendices specify the baseline masked sampler and transition behavior, and account for model NFE and reward evaluations separately.

D On the Prediction-head h-estimator

  • The prediction-head estimator completes each candidate from its current masked state, while the exact h-value averages terminal reward over future reverse-chain rollouts.The completion law generally differs from the rollout marginal; more completion samples converge to the head expectation, not generally to the rollout value.Source: D On the Prediction-head h-estimator · p. 21
  • The estimator’s completion distribution matches the full rollout marginal only under a sufficient invariance condition on masked-coordinate predictions across reachable later contexts and times.For attention-based masked discrete diffusion models, this invariance need not hold, so the mismatch can persist with arbitrarily many completions.Source: D On the Prediction-head h-estimator · p. 21

E Standard Masked Discrete Diffusion Sampling

  • The standard sampler predicts clean tokens, reveals still-masked positions according to the schedule, and freezes tokens once revealed.The baseline algorithm uses a linear schedule, though other monotone schedules can be substituted.Source: E Standard Masked Discrete Diffusion… · p. 23

F Masked Diffusion Transition Details

  • In the masked transition details, an already revealed token stays fixed, while a masked token is either kept masked or revealed as its original clean value.Conditioned on the clean sequence, the one-step reverse kernel factorizes across coordinates.Source: F Masked Diffusion Transition Details · p. 24

G Per-output NFE Breakdown

  • Per-output NFE counts sequence-level model forward passes, including candidate head passes in the guidance window; reward evaluations are counted separately.The accounting depends on active branch multiplicity and guidance-window size, and batching does not reduce sequence-level NFE.Source: G Per-output NFE Breakdown · p. 25

H Detailed Sampler Pseudocode

The sampler applies guided updates only within a chosen reverse-step window, with standard transitions elsewhere. Optional best-of-K branching duplicates trajectories after a split and selects the highest-reward clean copy at the end.

H Detailed Sampler Pseudocode

  • After the split, each trajectory is duplicated into K copies, and the highest-reward clean copy in each group is selected at the end.Final reward selection changes the output law; copies share tokens revealed before the split.Source: H Detailed Sampler Pseudocode · p. 26
  • At each outer-loop update, D-DOIT uses a guided step inside the selected guidance window and the standard transition outside it.The reverse state index is mapped from the update index before choosing the transition.Source: H Detailed Sampler Pseudocode · p. 26
  • The practical guided update uses a proposal pass and is specified as a single-completion update without proposal-ratio correction.Setting K = 1 or splitting at the final update disables the optional branching refinement.Source: H Detailed Sampler Pseudocode · p. 26

I Additional Analyses

The additional analyses test D-DOIT against matched-time controls, vary guidance oracles and evaluators, and diagnose compute and selection behavior. They show that similar backbone-evaluation budgets need not imply similar oracle work or runtime.

I Additional Analyses

  • At M = 8 and ω = 8, mean ESS is 1.143 ± 0.002 and soft resampling reaches final guidance reward 5.237 ± 0.051, versus 5.154 ± 0.027 for direct argmax.The top-weighted candidate is selected on 95.72% of guided steps, so selection is near-greedy but not deterministic.Source: I Additional Analyses · p. 27
  • Moving the suffix-search split from update 105 to 0 raises final guidance reward from 5.272 ± 0.057 to 5.890 ± 0.028, while increasing wall time from 326.4 ± 0.5 to 793.6 ± 2.2 seconds per job.Late branching searches a diverse suffix conditional on a fixed prefix and cannot revise pre-split reveals.Source: I Additional Analyses · p. 27

J.1 Dataset and Training Details

  • The HepG2 setup uses a frozen pretrained model, two Enformer-based oracles, and four metrics spanning activity, accessibility, 3-mer composition, and motif correlation.The held-out oracle scores activity; sampling uses the finetuning oracle, and the evaluation follows the DRAKES protocol.Source: §J.1 · p. 28, §J.2 · p. 28

J.3 Nearest-neighbor and Endpoint-search Controls

  • At similar wall-clock cost, D-DOIT-Q improves activity, accessibility, and 3-mer correlation over Frozen Best-of-732, while endpoint selection has higher JASPAR correlation (0.957 vs. 0.923).Q also has higher means on all four metrics than SVDD-PM; the clearest separation is ATAC-Acc (91.67% vs. 39.84%).Source: §J.3 · p. 30, Table 6 · p. 30
  • With five seeds and 640 outputs per seed, branched M=9 uses 1,970 diffusion evaluations per output, close to Best-of-15’s 1,920, but requires 1,777 reward evaluations.Across a 640-output job, batching combines 1,137,280 scored sequences into 1,290 vectorized oracle calls.Source: §J.3 · p. 30

J.4 Guidance-oracle and Evaluator Robustness

  • All three guidance oracles improve held-out activity and accessibility over the frozen model, with Enformer highest on those metrics and MLP highest on 3-mer correlation.When the same sequences are rescored by held-out Enformer, CNN, and MLP predictors, D-DOIT exceeds the frozen model within each evaluator; score scales differ across evaluators.Source: §J.4 · p. 30

J.5 Additional Experimental Analysis

The section reports DRAKES comparisons and held-out checks, then documents the Ctrl-DNA setup and baseline screen. The final-test checks preserve the distinction between selected configurations and reported evaluation results.

J.5 Additional Experimental Analysis

  • D-DOIT-Q exceeds TDS by +1.61 Pred-Activity, +51.7 percentage points ATAC-Acc, +0.077 3-mer Corr, and +0.124 JASPAR Corr.Against DRAKES, Q is higher on Pred-Activity (6.19 vs. 5.61), 3-mer Corr (0.894 vs. 0.887), and JASPAR Corr (0.923 vs. 0.911), but lower on ATAC-Acc (91.7% vs. 92.5%).Source: §J.5 · pp. 30–31

J.6 Held-out Final-test Check for D-DOIT-LBOK-S

  • On fresh final-test seeds, D-DOIT-LBOK-S exceeds TDS thresholds on ATAC-Acc, 3-mer Corr, and JASPAR Corr, with similar mean Pred-Activity.The reported comparison does not imply a statistical equivalence test.Source: §J.6 · p. 31, Table 10 · p. 32

J.7 Held-out Selection for High-budget D-DOIT-Q

  • D-DOIT-Q was selected on held-out eval seeds 42–44 and evaluated once on final-test seeds 100–102; its held-out check passes three DRAKES targets but misses ATAC-Acc.Table 11 identifies the targets as Pred-Activity, 3-mer Corr, and JASPAR Corr; ATAC-Acc is the missed target.Source: §J.7 · p. 31, Table 11 · p. 32

K.1 Dataset and Training Details

  • Ctrl-DNA uses a 200 bp enhancer corpus covering HepG2, K562, and SK-N-SH, with cell-specific reward oracles and a frozen masked discrete diffusion model.The same checkpoint is used for all three target cell lines and both operating points.Source: §K.1 · p. 33

K.2 Evaluation Metrics and Baselines

  • Ctrl-DNA reports specificity as target-minus-average-off-target reward, motif correlation against cell-specific natural enhancers, and set-level Shannon entropy as diversity.The high-budget and low-budget configurations are selected on eval seeds and then reported on separate final-test seeds.Source: §K.2 · p. 33, §K.1 · p. 33

K.3 Training-free Baseline Tuning Screen

  • The bounded one-seed screen of 13 CG, SMC, and TDS configurations found CG near zero reward movement, SMC with small gains and degraded motif/diversity, and TDS weak on composite specificity.The screen documents tuning provenance rather than replacing final-test baseline rows.Source: §K.3 · p. 34, Table 12 · p. 34

K.4 Additional Experimental Analysis

On Ctrl-DNA, D-DOIT improves specificity over the reported training-free guidance rows while retaining diversity, with distinct trade-offs between the two operating points. A matched-time endpoint control yields similar mean specificity, but the small difference is not claimed significant.

K.4 Additional Experimental Analysis

  • D-DOIT-LBOK-S improves ∆R from 0.002 to 0.033 and Motif Corr from 0.573 to 0.648 over TDS, at about 0.94× its rerun wall-clock.D-DOIT-Q reaches ∆R = 0.119 ± 0.063 and Motif Corr = 0.498 ± 0.070; it remains behind regLM on ∆R but exceeds the listed remaining training-based and evolutionary baselines on Motif Corr.Source: §K.4 · pp. 34–35
  • Across all three cell lines, D-DOIT-Q occupies the high-∆R, high-diversity region, while SMC and TDS lose diversity to approximately 1.5 Shannon entropy.Both D-DOIT operating points retain diversity around 1.98 across cells; LBOK-S has higher Motif Corr than Q while achieving lower specificity.Source: §K.4 · p. 35, Figure 3 · p. 35

K.5 Wall-clock-matched Endpoint Control

  • At matched time, Frozen Best-of-708 reaches ∆R = 0.108 ± 0.0281 versus Q’s 0.119 ± 0.063, and Motif Corr = 0.490 ± 0.165 versus 0.498 ± 0.070; differences are not claimed significant.The measured times are 5.51 and 5.43 seconds per output, respectively.Source: §K.5 · p. 35

L Ablation Study

The ablations examine how guidance-window width, guidance strength, temperatures, candidate budget, completion count, and selection rule relate to utility, naturalness, and cost. The findings are diagnostic sweeps, with several settings explicitly distinguished from the final held-out quality configuration.

L Ablation Study

  • The sweeps identify separate controls: guidance-window width exhibits a naturalness phase transition, completion temperature affects prediction-head completion quality, and candidate budget mainly controls utility.Larger M and earlier best-of-K splits also increase candidate evaluations and oracle-scored completions; Q trades greater recurring inference cost for quality.Source: L Ablation Study · p. 37

L.1 Effect of Guidance Window l∗

  • At l∗ = 128, 3-mer Corr reaches 0.894, 14× the value at l∗ = 64; a sharp increase occurs from 0.21 at l∗ = 96 to 0.65 at l∗ = 112.Utility improves across the sweep without a sharp transition, and the authors use l∗ = 128 as the default.Source: §L.1 · pp. 37–38, Figure 4 · p. 37

L.2 Effect of Guidance Strength ω and ESS Collapse

  • At ω ≥ 4, ESS collapses to near 1; 95% of guided steps have ESS < M/4 at ω = 4, and all do at ω ≥ 16.The diagnostic thresholds depend on the reward distribution and ω/β, so they are not transferable unchanged to other configurations.Source: §L.2 · p. 38, Figure 5 · p. 38

L.3 Effect of Proposal and Completion Temperatures τ, γ

  • Increasing τ from 2.5 to 5.0 raises Pred-Activity from 3.89 to 4.07 and ATAC-Acc from 53.7% to 69.4%, while reducing γ from 1.2 to 0.6 raises averaged 3-mer Corr from 0.846 to 0.905.The strongest γ-sweep setting reaches 3-mer Corr 0.949 ± 0.006 across three seeds; the final held-out Q configuration is a separate, larger-budget setting.Source: §L.3 · p. 38, Figure 6 · p. 39

L.4 Effect of Candidate Budget M

  • Reducing M from 32 to 8 drops Pred-Activity by 31% and ATAC-Acc by 29%, while 3-mer Corr drops 9% and JASPAR Corr less than 1%.The sweep reports that naturalness changes comparatively little as candidate budget varies.Source: §L.4 · pp. 38–39, Figure 7 · p. 39

L.5 Completion Count and Matched-state Proxy Diagnostics

  • Increasing completion count nh from 1 to 2 raises final guidance reward by 0.492 at 41% additional wall time; nh = 3 raises reward by 0.706 at 81% additional wall time.In the matched-state diagnostic, direct-head and sequential-completion pooled means are similar, but W1 grows from 0.060 to 0.124 as fixed bases rise from 0.78% to 66.34%.Source: §L.5 · pp. 39–40, Table 16 · p. 40, Table 17 · p. 40

L.6 Soft Selection versus Argmax

  • As ω rises from 1 to 8, mean ESS falls from 1.877 to 1.143 and top-candidate selection rises from 77.53% to 95.72%; at ω = 8, soft selection reward is 5.237 ± 0.051.At ω = 1, soft selection has lower reward than argmax (4.894 ± 0.055 versus 5.154 ± 0.027); low ESS indicates local weight concentration, not final-sequence diversity or target accuracy.Source: §L.6 · p. 40, Table 18 · p. 40

L.7 Split Time and Reward Trajectories

The split-position ablation shows a reward–runtime trade-off for branching, while trajectory diagnostics show diversity in unresolved suffixes and nonmonotone completion-proxy progress.

L.7 Split Time and Reward Trajectories

  • 5.890 ± 0.028 final guidance reward at update 0 costs 793.6 ± 2.2 seconds per job, versus 5.272 ± 0.057 and 326.4 ± 0.5 seconds at update 105.The earlier split also gives a best-of-4 gain of 1.605 ± 0.019 versus 0.959 ± 0.020; late branching cannot revise tokens fixed before the split.Source: §L.7 · p. 40, Table 19 · p. 41
  • Suffix Hamming distance is 0.710 ± 0.010 with 25.43% of bases fixed and 0.661 ± 0.011 with 90.38% fixed.Distances are measured only on positions still masked in the shared partial state, so they indicate suffix diversity rather than variability in early reveals.Source: §L.7 · p. 41, Table 20 · p. 41
  • D-DOIT’s completion-proxy score rises from 2.067 at update 1 to 2.328 at update 32, falls to 1.758 at update 96, then reaches 3.670 at update 128.Frozen sampling declines from 2.053 to 0.371; the argmax-completion score is neither the guidance reward nor the rollout h-function, and does not guarantee future reward.Source: §L.7 · p. 41, Table 21 · p. 41
  • From update 32 onward, argmax-completed D-DOIT sequences are 100% unique, with mean pairwise Hamming distance increasing from 0.726 to 0.749.This batch-level completed-sequence diagnostic is distinct from diversity conditional on a shared partial state and does not imply guidance universally preserves diversity.Source: §L.7 · p. 41

M Experimental Details of Protein Inverse Folding

D-DOIT is evaluated for protein inverse folding using a fixed sampling protocol and predictor-based metrics. It leads six methods on three reported measures, while TDS retains better structural fidelity; the comparison is not matched-compute or experimental validation.

M Experimental Details of Protein Inverse Folding

  • The evaluation covers 12 test backbones, with three sampling seeds and 128 sequences per backbone and seed; M = 52, K = 8, and nh = 1 were fixed before testing.D-DOIT values summarize variation across seed-level metrics, not across sequences or backbones.Source: M Experimental Details of Protein… · p. 42
  • D-DOIT has the highest median Pred-ddG (0.870 ± 0.046), ddG-positive fraction (74.8 ± 3.1%), and joint success (68.1 ± 3.5%) among six methods.Joint success exceeds the published SMC and TDS means by 4.5 and 5.2 percentage points, respectively.Source: Table 22 · p. 42, M Experimental Details of Protein… · p. 42
  • TDS has better structural fidelity: median scRMSD is 0.834 Å versus D-DOIT’s 0.872 Å, and 94.4% versus 92.5% of sequences are below 2 Å.Joint success requires both positive predicted ddG and scRMSD below 2 Å; the reported measures are predictor-based, not experimental stability or folding measurements.Source: M Experimental Details of Protein… · p. 42
  • Other methods’ entries and uncertainties come from DRAKES Table 2 rather than reruns, so the shared oracle and metrics support a literature comparison, not a matched-compute or matched-implementation claim.The evaluation extends computational assessment to protein sequence design but does not establish experimentally measured stability or folding.Source: M Experimental Details of Protein… · p. 42, Table 22 · p. 42

N Licenses for Existing Assets

  • The work uses pre-existing assets for non-commercial academic research and does not redistribute DRAKES code or checkpoints.The DRAKES repository had no top-level software license in GitHub metadata at access time; other named assets have specified licenses or data-use terms.Source: N Licenses for Existing Assets · p. 43
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