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Budgeted Quotient-Residual Guidance for Frozen Pocket-Conditioned Molecular Diffusion

Xinyu Wang, Jinbo Bi, Minghu Song

arXiv:2609.31222v1cs.LG

TL;DR

Pocket-conditioned molecular diffusion updates ambient coordinates, while many lead-optimization objectives are defined on quotient features and raw local quotient gradients can be dormant at sampler scale. QRG lifts those directions into bounded corrections scaled by frozen-sampler motion. Frozen-backbone evaluations report task-dependent quality–runtime gains, with important boundaries in target-level variation and supporting docking diagnostics.

  • Problem

    Molecular samplers update ambient coordinates, but lead-optimization objectives often depend on quotient features, and direct invariant-energy gradients can be numerically dormant at finite sampler steps.

    Source: §1 · pp. 1–2

  • Method

    QRG lifts quotient covectors into metric-horizontal ambient directions and bounds their correction length by the frozen sampler’s own update norm.

    Source: §1 · p. 2

  • Results

    Frozen-backbone QRG improves the quality–runtime trade-off on ligand-generation/editing tasks, with clearest gains on fragment growing and scaffold hopping and a LOCAL-QRG gain on linker design.

    Source: §1 · p. 3, §5 · pp. 9–10

  • Takeaways & Limitations

    QRG offers a practical, scoped route to quotient-aware inference for frozen molecular samplers without changing or retraining the backbone.

    Source: §1 · p. 2, §5 · p. 9

    Paired target analyses reveal heterogeneity, docking is only supporting evidence, and side-chain decoration exposes a budget-transfer failure mode.

    Source: §2 · p. 4, §5 · p. 10

Abstract

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Pocket-conditioned molecular diffusion updates ambient atom coordinates, but many lead-optimization objectives are expressed on quotient features such as distances, contacts, and anchored substructures. We introduce budgeted quotient-residual guidance (QRG), an inference-time correction that makes these quotient objectives active without retraining the molecular generator. QRG lifts quotient covectors to metric-horizontal ambient directions and delivers them through a trust budget set by the frozen sampler's own step norm: quotient geometry chooses the direction, while sampler motion bounds the scale. We derive the horizontal lift, closed-form sampler-budget update, KL/kinetic interpretation around a frozen reverse step, equivariance conditions, and a product-budget split for budget-capped section and residual controls. Controlled quotient tasks confirm that sampler-relative delivery activates signals that raw local quotient gradients leave dormant. On frozen TargetDiff backbones, official seed-0 CBGBench ligand-generation/editing sweeps show practical quality-runtime gains: Local-QRG improves validity from 0.815 to 0.864 on fragment growing, 0.664 to 0.707 on scaffold hopping, and 0.681 to 0.712 on linker design, while PredNext-QRG improves fragment/scaffold and remains near-neutral on linker. Novelty remains 1.000 and diversity is preserved in the matched multi-seed molecular slice, giving task-dependent improvements without sampler retraining or backbone modification. Overall, QRG provides a lightweight route to quotient-aware inference for frozen molecular samplers with explicit runtime accounting.

1 Introduction

Pocket-conditioned diffusion evolves ambient coordinates while many lead-optimization objectives depend on quotient features, whose raw gradients can be dormant at sampler scale. QRG uses quotient geometry to choose a correction direction and the frozen sampler’s motion to bound its scale; frozen-backbone evaluations report task-dependent quality–runtime gains.

1 Introduction

  • Ambient–quotient mismatch arises because molecular samplers update coordinates while objectives such as fragment geometry, scaffold preservation, linker closure, and pocket contacts depend on quotient features.Direct invariant-energy gradients can be directionally meaningful yet numerically dormant at finite sampler steps, while rollout lookahead requires repeated denoiser queries.Source: §1 · pp. 1–2
  • QRG lifts quotient covectors to metric-horizontal ambient directions and bounds the correction norm by the frozen sampler’s own update norm.Expensive rollout is a teacher or diagnostic; deployed variants use a predicted next state or a local quotient surrogate.Source: §1 · pp. 2–3, Figure 1 · p. 2
  • The evaluation integrates QRG with frozen TARGETDIFF on CBGBENCH-style lead-optimization tasks and reports quality, runtime, control budgets, paired target statistics, and corrected docking diagnostics.Side-chain decoration is reported separately as a budget-transfer stress test because it changes the output family to narrow protein-local packing.Source: §1 · p. 2
  • The formulation derives a closed-form sampler-relative correction, a KL-budgeted quotient-descent interpretation for Gaussian reverse kernels, and a kinetic trust-region rule for deterministic samplers.The theory also covers equivariance under isometric group actions and a product-budget split for section and residual controls.Source: §1 · pp. 2–3
  • Frozen TARGETDIFF experiments report validity gains on fragment growing and scaffold hopping, with LOCAL-QRG improving linker design and PREDNEXT-QRG remaining near-neutral there.Novelty and diversity are preserved in the matched multi-seed molecular slice; paired target-level analyses show heterogeneity, and docking is a corrected supporting diagnostic.Source: §1 · p. 3

2 Background

The background defines quotient representations and situates frozen-sampler guidance among diffusion and symmetry-aware molecular generation methods. It also frames multi-task evaluation and cautions against overreading molecular-quality and docking metrics.

2 Background

  • A quotient representation identifies states in the same equivalence class, with pairwise distances illustrating rigid-motion invariance and pocket-relative or substructure-restricted distances giving task-specific examples.The text defines invariant functions and equivariant maps under group actions, and notes molecular coordinates’ rigid-motion and graph atom-index symmetries.Source: §2 · p. 3
  • Plug-and-play guidance modifies a diffusion sampler step with a correction from an auxiliary objective, while this paper’s setting requires delivering invariant objectives in ambient molecular coordinates.The background describes denoising reverse-time updates and cites classifier guidance and later guidance for inverse problems and restoration.Source: §2 · p. 3
  • Equivariant molecular diffusion builds symmetry into neural architectures, whereas this work asks how to use quotient information at sampling time with a frozen backbone.The background presents QRG as complementary to symmetry-aware training and cites molecular diffusion applications and Orbit Diffusion.Source: §2 · p. 3
  • CBGBENCH frames de novo generation and lead optimization as binding-graph completion across linker design, fragment growing, scaffold hopping, and side-chain decoration.The background argues that evaluating across tasks matters because a method limited to one custom subset may be less useful.Source: §2 · pp. 3–4
  • Docking scores are supporting diagnostics rather than standalone evidence of generation quality because raw scores can depend on size, strain, coordinate frames, and post-processing.The paper separates its main generation claim from corrected raw-pocket docking diagnostics.Source: §2 · p. 4

3 Budgeted quotient-residual guidance

QRG separates quotient-space direction from delivery scale: it lifts quotient covectors into horizontal ambient directions, then caps their delivery using the frozen sampler’s motion. The section extends this rule to separately budgeted controls, practical direction-evaluation variants, and equivariance conditions.

3 Budgeted quotient-residual guidance

  • The method is a local first-order quotient-lifted trust-region update at the level of a frozen sampler; variants differ in where they evaluate the quotient direction.Figure 1 summarizes the deployed step in algorithm form.Source: §3 · p. 4

3.1 Quotient differential and horizontal lift

  • The metric-horizontal lift maps a quotient covector to an ambient direction orthogonal to motions in the quotient differential’s kernel, giving the steepest quotient-descent direction under the metric.The pullback alone does not specify how far the frozen sampler should move.Source: §3.1 · p. 4

3.2 Sampler-relative trust-region delivery

  • The sampler-relative correction is a unique optimizer of the local trust-region problem when its budget is active, with full-budget descent set by sampler motion rather than the raw quotient-gradient norm.For Gaussian reverse kernels the budget has a local KL interpretation; for deterministic samplers it is a kinetic-energy budget.Source: §3.2 · p. 5

3.3 Section-residual product budget

  • The product budget separately caps section and quotient-residual directions, defining motions that generally cannot be represented by one-branch scalarization.The section branch stabilizes anchors or preserved atoms; the residual branch uses the quotient-horizontal lift.Source: §3.3 · pp. 5–6

3.4 Instantiating the quotient direction

  • LOCAL-QRG computes residuals from a local-radius feature subset, while PREDNEXT-QRG evaluates them using the sampler’s one-step prediction and applies correction every two denoising steps.A rollout teacher is a diagnostic, not the deployed method; the trust-region rule itself does not require rollout.Source: §3.4 · p. 6

3.5 Equivariance of the lift and delivery

  • The lift and delivered correction respect an isometric group action when the quotient chart, metric, and frozen sampler satisfy the stated invariance and equivariance conditions.This does not claim that the frozen backbone itself becomes perfectly equivariant.Source: §3.5 · p. 6

3.6 What the method is not

  • QRG is a bounded inference-time correction, not a retrained backbone or an unbounded external optimizer, and its quality–runtime trade-off is reported with controls.The experiments include section-only, sham-QRG, and rollout-teacher controls.Source: §3.6 · p. 7

4 Experimental setup and results

The experiments first test quotient guidance on controlled tasks, then evaluate frozen TargetDiff variants on CBGBench-style ligand-generation and editing tasks. They also compare alternative controls and use side-chain decoration to stress-test budget transfer.

4.1 Controlled quotient tasks: does trust-region delivery activate the signal?

  • Budgeted residual guidance raises success from 0.159 to 0.246, 0.117 to 0.310, and 0.079 to 0.371 across the Gaussian, orbit-point, and toy-molecule tasks.Raw local quotient guidance remains essentially unchanged from the base sampler under the same denoising schedule.Source: §4.1 · p. 7, Figure 2 · p. 8, Table 1 · p. 8

4.2 Budgeted inference on frozen pocket-conditioned samplers

  • Across the matched multi-seed slice, novelty remains 1.000 and diversity stays close to the base sampler; paired target-level results show heterogeneity, including a linker interval crossing zero.LOCAL-QRG's fragment paired mean is +0.049 with interval [0.019, 0.078].Source: §4.2 · p. 8

4.3 Anti-reparameterization controls

  • Deployed guidance occupies a practical runtime regime, and rollout guidance is unnecessary for the observed validity gains; shuffled residuals are weaker than LOCAL-QRG but close to PREDNEXT-QRG.The authors avoid claiming the shuffled-residual control alone rules out every scalarized-guidance explanation.Source: §4.3 · p. 8

4.2 Budgeted inference on frozen pocket-conditioned samplers

  • LOCAL-QRG validity rises from 0.815 to 0.864 on fragment growing, 0.664 to 0.707 on scaffold hopping, and 0.681 to 0.712 on linker design.PREDNEXT-QRG improves validity from 0.815 to 0.829 on fragment growing and 0.664 to 0.683 on scaffold hopping, but is near-neutral on linker design at 0.680 versus 0.681.Source: §4.2 · p. 7, Table 2 · p. 9

4.4 Side-chain decoration as a budget stress test

  • Official-run norm accounting shows delivered control below sampler step scale, with LOCAL-QRG section and residual components spending a small bounded budget relative to base motion.This is a kinetic-budget diagnostic, not a measured path KL.Source: Figure 3 · p. 9
  • On side-chain decoration, validity falls from 0.873 unguided to 0.848, 0.837, and 0.828 for PREDNEXT-QRG, LOCAL-QRG, and SECTION-ONLY; sparse LOCAL-QRG reaches 0.869.This narrow-packing task is treated as a budget stress test, not part of the ligand-generation win table.Source: §4.4 · pp. 8–9

5 Conclusion and limitations

The conclusion presents QRG as a bounded inference-time correction for frozen molecular diffusion, with gains on the main ligand-generation and editing tasks. The related work distinguishes this approach from architectural changes and places its evaluation within benchmark and control traditions.

5 Conclusion and limitations

  • The scoped evidence supports a quality–runtime trade-off on frozen-backbone ligand tasks, clearest for fragment growing and scaffold hopping, with a LOCAL-QRG gain on linker design.The conclusion also identifies heterogeneous target statistics, mixed supporting docking diagnostics, and side-chain decoration as a budget-transfer failure mode.Source: §5 · pp. 9–10

A Expanded related work

  • Unlike equivariant diffusion architectures, QRG is an inference-time correction applied to a frozen model.The related work frames this as complementary to approaches that change or retrain the generator.Source: A Expanded related work · p. 14
  • QRG reads its Gaussian reverse-step radius as a local one-step kinetic or KL budget, rather than an arbitrary ambient learning rate.The paper distinguishes this local interpretation from path-space KL formulations.Source: A Expanded related work · p. 14
  • CBGBENCH combines lead-optimization tasks, while benchmark work cautions against over-interpreting raw docking scores from invalid geometries.The paper uses this benchmark context to bound its claims.Source: A Expanded related work · p. 14

B.1 Proof of proposition 1

The proof states that the lower bound is attained by the normalized negative direction and that this direction is orthogonal to the vertical subspace.

B.1 Proof of proposition 1

  • The lower bound is attained at u⋆ = −h / ∥h∥M, and h is M-orthogonal to every vertical direction in ker J.The supplied proof passage identifies the attaining direction and its orthogonality property.Source: §B.1 · p. 15

B.2 Proof of proposition 2

These proof passages establish the sampler-budgeted update and its local KL or kinetic interpretation, then address split budgets, equivariance, and the scale mismatch behind dormant guidance.

B.2 Proof of proposition 2

  • The quotient-descent optimizer is u_quo = −B_t h_t / ∥h_t∥_M_t, with B_t set to ρ_t times the sampler-step norm.For nonzero lifts, the trust-region update controls motion by the sampler-relative budget rather than raw quotient-gradient magnitude.Source: §B.2 · p. 15, §B.6 · p. 16

B.3 Proof of proposition 3

  • For Gaussian kernels with equal covariance, the mean-shift cost is a conditional KL; deterministic samplers admit the corresponding kinetic trust-region interpretation, not a measured path KL.The trust radius is ρ_t ∥v_t∥_M_t.Source: §B.3 · p. 15

B.4 Proof of proposition 4

  • Independent branch budgets vary the correction direction across the cone spanned by non-collinear branches, unlike fixed scalarization, whose direction is budget-independent.No fixed scalar recovers the split directions across multiple branch-allocation ratios unless the branches are collinear.Source: §B.4 · p. 15

B.5 Proof of proposition 5

  • Under metric preservation and sampler equivariance, the quotient covector and trust radius are preserved, so the correction transforms equivariantly.The proof substitutes these identities into the sampler-budgeted update.Source: §B.5 · p. 16

B.6 Why the trust-region update addresses dormancy

  • Dormancy reflects scale mismatch, not a useless direction: raw guidance can be arbitrarily small, while trust-region delivery sets motion near ρ_t ∥v_t∥_M_t.The normalized lift can remain the horizontal steepest quotient direction even when the raw update is small.Source: §B.6 · p. 16

C.1 Toy quotient tasks

The toy tasks test whether sampler-relative trust-region delivery activates quotient-guidance signals that remain dormant under raw local delivery. The accompanying molecular protocol, controls, and diagnostics define the scope and interpretation of the evaluations.

C.1 Toy quotient tasks

  • The toy tasks compare base sampling, dormant local quotient guidance, and budgeted residual guidance on quotient MMD and invariant template errors.The key comparison is whether trust-region delivery activates a direction that is inert under raw local delivery.Source: §C.1 · p. 16

C.2 Molecular generation protocol

  • Molecular experiments match guided variants and the base sampler by target, seed, and sample contract, reporting quality, runtime, and control-budget metrics.The matched multi-seed slice supports novelty, diversity, development-ablation, sham-control, and teacher-diagnostic summaries; runtime excludes contaminated co-run measurements when clean single-run measurements are available.Source: §C.2 · p. 16

C.3 Partial-result policy

  • Only completed task–method aggregates support main-text quantitative claims; incomplete rows are labeled by completed-part count and treated as bounded evidence.A row is complete when planned shards have written run_summary.json and are included in the aggregate script.Source: §C.3 · p. 16

C.4 Variant definitions

  • The reported variants include base and section-only controls, PREDNEXT-QRG every two denoising steps, local-radius LOCAL-QRG, a rollout teacher, and a shuffled-signal sham.The rollout teacher is reported only as a teacher/oracle point, while SHAM-QRG tests whether a mismatched residual signal can produce a gain.Source: §C.4 · pp. 16–17

C.5 Docking diagnostic

  • Docking comparisons use target-balanced paired rows of raw AutoDock Vina score, where negative mean differences favor the guided variant over base.The diagnostic is separated from the main validity claim because raw docking can be confounded by molecular size, strain, and conformation validity.Source: §C.5 · p. 17

D.1 Frontier figure

  • Figure 4 compares quality–runtime trade-offs across matched seeds, with the richest ablation set for fragment and base, PREDNEXT-QRG, and LOCAL-QRG for scaffold and linker.The figure complements the official seed-0 aggregate with novelty and diversity preservation across matched seeds.Source: Figure 4 · p. 17

D.2 Generation metrics

  • Generation summaries track validity, uniqueness, QED, diversity, novelty, seconds per sample, and control/base ratio.The main text reports validity, runtime, novelty, and diversity; scripts recompute summaries from generated outputs.Source: §D.2 · p. 17

D.3 Development ablation for the cheap surrogate

Development and target-level analyses characterize the cheap quotient surrogate and the variability behind aggregate outcomes. The claim boundary remains task-dependent, with side-chain and docking results treated as bounded diagnostics rather than universal gains.

D.3 Development ablation for the cheap surrogate

  • The development-slice fragment ablation reports that the local-radius surrogate outperforms sampled-pair residual surrogates at comparable runtime while preserving diversity.This is development evidence, not an additional official benchmark claim.Source: Table 3 · p. 17, §D.3 · p. 18

D.4 Paired target-level validity

  • LOCAL-QRG has the clearest fragment target-level validity signal, while scaffold intervals are wider and linker remains task-dependent.Fragment mean delta is +0.049 with interval [0.019, 0.078]; LOCAL-QRG linker mean delta is +0.030 with an interval crossing zero.Source: §D.4 · p. 18, Table 4 · p. 18

D.5 Target-level heterogeneity

  • Across 504 task–method–target rows, LOCAL-QRG has the strongest fragment distribution, scaffold has positive central tendency with wider tails, and linker remains task-dependent.The quantiles show that aggregate gains coexist with heterogeneous pockets, including a small number of large negative and positive shifts.Source: §D.5 · p. 18, Table 5 · p. 18

D.6 Claim matrix

  • The claim matrix identifies official seed-0 fragment and scaffold validity gains at modest overhead, LOCAL-QRG linker gains, and no novelty or diversity collapse in the matched multi-seed slice.It treats side-chain over-control and task-dependent quality–runtime trade-offs as bounded claims, and docking as mixed rather than universal binding improvement.Source: §D.6 · p. 18

D.8 Side-chain budget stress test

  • Default ligand-generation budgets reduce side-chain validity relative to unguided sampling; a sparse low-budget rescue nearly closes the gap but does not beat unguided in the full sweep.This stress test supports a task-dependent budget-mismatch claim.Source: Figure 5 · p. 19

D.9 Corrected docking diagnostic

  • The corrected raw-pocket docking diagnostic shows its clearest signal on scaffold hopping and mixed results on the other tasks.Values are paired mean differences versus base with bootstrap intervals; negative values indicate better raw AutoDock Vina scores.Source: Table 7 · p. 19

E.1 Broader impact

  • Generated molecules are computational candidates inside benchmark pockets, not validated binders, drugs, or therapeutic leads; biological activity is not claimed.The paper separates generation validity from docking diagnostics and says use should require independent chemistry review, safety screening, and experimental validation.Source: §E.1 · p. 20

E.2 Compute resources

The evaluation uses frozen TARGETDIFF and CBGBENCH assets with explicit shard, runtime, and query accounting. The code supplement does not redistribute upstream assets, which remain subject to their original terms.

E.2 Compute resources

  • Official seed-0 molecular sweeps use 20-target shards, 100 samples per target, batch size 2, and seed 0, with wall-clock seconds per sample aggregated by generated-sample count.Contaminated local co-runs are excluded when clean single-run measurements are available.Source: §E.2 · p. 20

E.3 Denoiser-query accounting

  • Denoiser-query accounting distinguishes deployed PREDNEXT-QRG and LOCAL-QRG from rollout-teacher diagnostics; guidance opportunities are not hardware-level FLOP measurements.The deployed variants avoid the rollout-teacher query pattern.Source: §E.3 · p. 20, Table 10 · p. 21

E.4 Existing assets and licenses

  • The code supplement includes QRG implementation and evaluation scripts but not upstream benchmark data or checkpoints, which users must obtain under original terms.The cited upstream licenses are MIT for TARGETDIFF code, GPL-3.0 for CBGBENCH code, and CC BY 4.0 for the CBGBENCH paper.Source: §E.4 · p. 21

F Reproducibility notes

The supplement provides a minimal code package and runner workflow, while full regeneration depends on obtaining upstream CBGBENCH and TARGETDIFF assets.

F Reproducibility notes

  • The code package contains QRG implementation, a frozen TARGETDIFF/CBGBENCH runner wrapper, metric and runtime scripts, and an environment file, but no data, checkpoints, outputs, logs, figures, or precomputed tables.Full regeneration requires upstream CBGBENCH/TARGETDIFF assets under their original terms.Source: F Reproducibility notes · p. 21
  • Reproduction uses the frozen TARGETDIFF checkpoint, CBGBENCH task configuration, official split, and guidance parameters; aggregation recomputes metrics from shard summaries and generated SDFs.Official sweeps use 20-target shards, 100 samples per target, and batch size 2.Source: F Reproducibility notes · p. 21

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