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The Geometry of Sequential Learning: Lie-Bracket Prediction of Transfer Order

John Sweeney

arXiv:2606.24993v1cs.LG

TL;DR

Sequential learning makes source order consequential, while evaluating all N! curricula is impractical. The paper predicts pairwise order effects with gradient-update commutators and extends the scores to scalable tournament rankings. Experiments report strong pairwise and many-domain results across post-training and domain-adaptation settings.

  • Problem

    Choosing among sequential source orders matters for target loss, but exhaustive search over N sources requires evaluating N! curricula.

    Source: §1 · p. 1

  • Method

    The paper projects a Lie-bracket commutator of gradient updates onto the target gradient for pairwise order prediction and uses a shared-reference tournament for many-source ranking.

    Source: §1 · pp. 1–2

  • Results

    The planner reaches 98.1% and 98.9% pairwise accuracy at k=1 for instruction-SFT and DPO, respectively, and recovers the best of all 3! schedules in 87.5% of trials.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The bracket primitive supplies both local pairwise order information and a scalable ranking method for many-domain schedules.

    Source: §1 · p. 2

    The clean derivation applies to memoryless gradient-style updates; AdamW requires an augmented-state commutator theory that remains future work.

    Source: §4.5 · p. 9

Abstract

from arXiv · show

Sequential learning is order-dependent: from Pile-style next-token domain adaptation to instruction-SFT and DPO, N candidate sources induce N! possible curricula. We show that the local order effect is governed by a computable geometric quantity, the Lie-bracket commutator of gradient update fields, yielding a pairwise score for whether A->B or B->A is better for a target domain. The pairwise bracket primitive also defines a Lie-Bracket Tournament: with a shared theta_0 target-gradient reference, Hessian symmetry gives Borda/row-sum scores from one Hessian-vector product per source, O(N) dot products, and an O(N log N) sort, without materializing the O(N^2) edge matrix. Empirically, the planner reaches 98.1%/98.9% pairwise accuracy at k=1 for instruction-SFT/DPO, remains at 73.1%/72.2% at k=20, and preserves the original pretraining-domain evidence with 82.4-92.0% accuracy across four LLMs and 91.1% on diffusion. At curriculum scale, it recovers the best of all 3! schedules in 87.5% of trials, ranks 85 Stack programming-language source domains for a Python target in the 99th sampled percentile, and reaches the 99.0-99.6th sampled percentile on 56 MMLU subjects, sharply above the reported descending gradient-norm baseline. These results reframe sequential learning as a geometric tournament problem: commutators provide both local pairwise order information and a scalable primitive for many-domain schedules.

1. Introduction

Sequential learning makes source order consequential, but exhaustive curriculum search grows to N! schedules. The paper uses gradient-update commutators to predict pairwise order effects and extend them to scalable curriculum ranking.

The problem

  • For N sources, exhaustive order testing requires N! evaluations, making curriculum selection the central practical obstacle.Sequential pipelines may combine instruction tuning, preference tuning, and adaptation or evaluation on a target distribution.Source: §1 · p. 1

Geometric view

  • Noncommuting dataset updates trace different parameter-space trajectories in opposite orders, with a Lie bracket governing the leading-order effect.The target-gradient projection of the bracket gives a directional score for which order yields lower target loss.Source: §1 · p. 1

Lie-Bracket Tournaments

  • With a shared θ0 target-gradient reference, Hessian symmetry yields Borda scores using one HVP per source, linear score computation, and an O(N log N) sort.The tournament avoids materializing the pairwise matrix for row-sum ranking; exact permutation scoring remains available for small N.Source: §1 · p. 2

Evidence and scope

  • 87.5% of trials recover the best 3! schedule; Stack ranks at the 99th sampled percentile, and 56 MMLU subjects reach the 99.0–99.6th sampled percentile.The reported results also include top-two recovery in 96.0% of trials.Source: §1 · p. 2
  • 98.1% and 98.9% pairwise sign accuracy at k=1 for instruction-SFT and DPO, respectively, remain above 72% in both settings at k=20.The introduction also reports 81.5% accuracy at k=20 and 65.3% at k=50 over 1,020 Llama trials.Source: §1 · p. 2

Pairwise planner

  • A drift-matched Trotter estimator uses gradients and Hessian-vector products in a chosen trainable parameter subset, with stakes and confidence scores.It evaluates the target gradient at a reference point approximating the common first-order drift without running either complete order.Source: §1 · p. 2

Step-size selection

  • The paper develops an η-autopilot that selects step sizes from pilot data by balancing detectability against higher-order sign-stability constraints.This is intended to avoid manual per-model tuning.Source: §1 · p. 2

2. Preliminaries

The preliminaries define a local gradient-step model and show how noncommuting updates produce a second-order target-loss order effect. They then motivate drift-matched scoring and projected computation in trainable parameter subsets.

2.1. Notation and Setup

  • The setup compares held-out target losses after the two source orders, using a single-step update as a local surrogate for short fine-tuning phases.The paper denotes k as gradient steps per domain and evaluates both orders from the same initialization.Source: §2.1 · pp. 2–3

2.2. Commutators and Lie Brackets of Update Operators

  • Under local smoothness assumptions, the endpoint difference is driven to second order by the bracket vector, with an O(η3) remainder.The result applies to projected updates when gradients and Hessians are restricted to a fixed parameter subset.Source: §2.2 · p. 3

2.3. From Parameter Commutators to Target Loss Differences

  • The target-loss difference is approximated by η2 times the target-gradient projection of the bracket, whose sign predicts the better order.Accuracy rises when evaluation is restricted to pairs with large |ω̂|, which are more stable across seeds.Source: §2.3 · pp. 3–4

2.4. Reference-Point Estimators

  • The Trotter reference approximates the shared first-order drift, and all three target-gradient estimators are deployable without running both orders.Performance varies by model and operating point: the single-point Trotter estimator is most reliable in harder settings, while trapezoid averaging can help some models.Source: §2.4 · p. 4

2.5. Projected Computation in a Trainable Subspace

  • Projected computation restricts gradients and HVPs to trainable parameters; this subspace restriction makes the planner compatible with parameter-efficient fine-tuning.Validation within selected trainable subsets includes 80.9% sign accuracy for Llama-3.2-1B tuning its last eight transformer layers.Source: §2.5 · pp. 4–5

3. Method: Transfer-Order Planner

The planner outputs an order, stakes, and confidence from gradients and Hessian-vector products. Its many-source extension turns pairwise bracket preferences into a tournament ranking, with a shared-reference Borda method for scalable scoring.

3. Method: Transfer-Order Planner

  • The pairwise planner outputs a predicted order, stakes, and confidence; its compute model predicts a practical crossover around k ≈3–4 against trying both orders.The planner uses three gradients and two HVPs per pair, and measured wall-clock results match the predicted crossover.Source: §3 · p. 5

3.1. Lie-Bracket Tournaments for N Sources

  • The tournament uses Hessian symmetry to derive pairwise edges from source gradients and HVPs; small-N schedules can instead be scored as complete permutations.The large-N shared-reference tournament and drift-matched pairwise planner use the same bracket vector but different target-gradient references.Source: §3.1 · p. 6
  • For large N, shared-reference Borda ranking uses one HVP per source and O(N log N) sorting without materializing the pairwise matrix.The shared θ0 reference makes edges N-independent and enables exact row-sum factorization; pair-specific Trotter references would break that factorization.Source: §3.1 · p. 6
  • Stakes estimate order-effect magnitude, while alignment-based confidence tracks sign reliability; the η-autopilot selects step sizes from pilot triples using detectability and sign-validity constraints.The paper recommends confidence gating, with validation for low-confidence predictions.Source: §3.1 · pp. 6–7

4. Experiments

Experiments test bracket-based order prediction across post-training, pretraining-domain adaptation, diffusion, long horizons, and many-domain schedules. Results extend from pairwise accuracy to curriculum rankings, with additional evaluations of compute, optimizer scope, and capability preservation.

4.1. Experimental Setup

  • The setup spans instruction-SFT, offline DPO, four LLMs on pretraining-domain adaptation, class-conditioned diffusion, and large-N domains including MMLU, Stack, and Dolly task categories.Pairwise metrics include sign accuracy and regret; many-domain schedules are compared by percentile against sampled random curricula.Source: §4.1 · p. 7

4.2. Post-Training: Instruction-SFT and DPO

  • At k=1, sign accuracy was 98.1% for instruction-SFT and 98.9% for DPO; with the initial predictor fixed, k=20 accuracy was 73.1% and 72.2%, respectively.The DPO evaluation is offline preference optimization; online sampling and policy-optimization effects are left for future work.Source: §4.2 · pp. 7–8, Table 1 · p. 8

4.3. Pairwise Pretraining-Domain Adaptation and Long Horizons

  • On Llama-3.2-1B, the predictor computed once at θ0 retained 81.5% accuracy at k=20 and 65.3% at k=50, above random, while the original pretraining-domain and diffusion experiments validate the same primitive across model families.Table 2 covers Pile-style next-token domains and class-conditioned diffusion domains; its caption specifies mean ± std over five seeds.Source: §4.3 · p. 8, Table 3 · p. 8, Table 2 · p. 8

4.4. Many-Domain Scheduling by Lie-Bracket Tournament

  • 87.5% of three-domain trials recovered the best schedule; at N=56, MMLU rankings reached the 99.0–99.6th sampled percentile, while N=85 Stack rankings reached the 99th percentile for Python.The MMLU comparison is against 500 sampled random curricula and a descending gradient-norm baseline; Stack reached the 69th percentile against that baseline.Source: §4.4 · p. 8, Table 4 · p. 8
  • For N=3, the complete pairwise surrogate recovered the best schedule in 175/200 cases (87.5%) and placed it in the top two in 192/200 cases (96.0%); the scalable Borda variant recovered the best order in 79.0%.Only 1.5% of quadruples showed cyclic intransitivity, concentrated near indifferent pairs.Source: §4.4 · p. 8
  • The pairwise bracket is an N-independent primitive: each edge asks the same local geometric question, and Algorithm 2 converts the resulting tournament into a curriculum without enumerating permutations, enabling scalable scheduling.Large-N percentiles are relative to 500 sampled random curricula.Source: §4.4 · pp. 8–9, Table 4 · p. 8

4.5. Compute, Optimizer Scope, and Control

  • The planner is slower than brute force at k=1, faster by k=5, and about 13× faster than brute-force pilots at k=20; a local bracket-direction control improved over shared drift in 179/204 cases (87.7%).The control beat both sequential endpoints in 95/204 cases (46.6%).Source: §4.5 · p. 9
  • The clean derivation applies to memoryless gradient-style updates, whereas AdamW is stateful; the unmodified predictor reached 57.1% at AdamW k=20, and the augmented-state theory remains future work.The same target-domain formulation treats a protected capability loss as the target and allows rehearsal or safety data to be included as tournament sources.Source: §4.5 · p. 9

5. Related Work

Related work connects curriculum scheduling, sequential transfer, gradient diagnostics, and Hessian-vector products to the paper’s geometric account of domain order. The authors distinguish their deployable pairwise predictor and scalable tournament from prior bracket projection and gradient-similarity approaches.

5. Related Work

  • Curriculum learning and sequential transfer study how training order or domain emphasis affects optimization and generalization (Choi et al., 2023; Kumar et al., 2020).The paper identifies Rukhovich et al. (2025) as closest to its geometric view.Source: §5 · p. 9
  • Rukhovich et al. (2025) use the same target-gradient bracket projection as a descriptive local criterion; this work adds a held-out predictor, stakes and confidence, and a shared-reference tournament for many-domain ranking.The paper also distinguishes its validation across model families and settings from the single bilingual-pretraining setting it attributes to that work.Source: §5 · p. 9
  • On Llama-3.2-1B, the cosine baseline reached 77.5% accuracy, 16.2pp below the Lie-bracket score; the added second-order information uses Hessian-vector products without forming the Hessian.The discussion relates HVPs to influence-function and geometric numerical perspectives.Source: §5 · p. 9

6. Conclusion

The conclusion presents Lie-bracket commutators as a geometric basis for predicting sequential transfer order. It reports pairwise prediction and many-domain ranking across several adaptation settings, with tournament ranking avoiding exhaustive schedule enumeration.

6. Conclusion

  • The directional score projects HBgA − HAgB onto the target gradient to choose between two domain orders and estimate stakes and confidence.The conclusion frames this as a geometric approach to sequential learning based on Lie-bracket commutators of gradient update fields.Source: §6 · p. 9
  • For many domains, a shared-reference variant forms a Lie-Bracket Tournament with one HVP per source, avoiding enumeration of N! schedules.The tournament turns the bracket primitive into a curriculum-ranking procedure.Source: §6 · p. 9
  • Across pretraining-domain adaptation, instruction-SFT, DPO, and diffusion adaptation, the primitive predicts pairwise order effects, composes into sequence-level rankings, and is faster than brute-force pilots at practical horizons.Source: §6 · p. 9

Impact Statement

The impact statement links the planner to reducing the need to test multiple training orders. It presents geometric curriculum learning as a broader direction while noting that optimizer-scope evaluations inform when further validation is prudent.

Impact Statement

  • The planner is intended to support ordering decisions without running both orders, reducing unnecessary GPU hours in resource-constrained settings.The impact statement also notes associated energy consumption and carbon footprint from testing multiple orders.Source: Impact Statement · p. 10
  • Confidence gating indicates when predictions can be used directly versus when validation is warranted.Source: Impact Statement · p. 10
  • The authors describe the work as a step toward a geometric theory of curriculum learning, with optimizer-scope evaluations clarifying when additional validation is prudent.Source: Impact Statement · p. 10

A.2. Proof of Proposition 2.3

The proofs connect order-dependent target loss to the source-order displacement and justify evaluating the target gradient at a shared reference point. They also establish that the endpoint orders lie within O(η²) of that reference.

A.2. Proof of Proposition 2.3

  • The Proposition 2.3 proof uses the mean value theorem and an O(η²) displacement with an O(η³) remainder to establish the local order-effect approximation.Table 5 splits triples by low-sample-domain involvement and reports sign accuracy for the deployable trap-local estimator.Source: §A.2 · p. 12, Table 5 · p. 13

A.3. Proof of Lemma 2.4

  • Lemma 2.4 obtains the reference-point forms by substituting θref; the endpoint orders are O(η²) from that point because the leading corrections are O(η²) and the remainder is O(η³).Source: §A.3 · p. 13

A.4. Proof of Proposition 2.5

  • For Proposition 2.5, a bounded target Hessian makes gradient replacement error O(η²), which contributes O(η⁴) when paired with the O(η²) order displacement.Source: §A.4 · p. 13

B. Oracle Endpoint Integration Baselines

The oracle baselines integrate the target gradient along the segment between the two order endpoints, but require both endpoints and are therefore not deployable for order selection. Simpson’s rule adds a midpoint evaluation yet is nearly indistinguishable from the trapezoid oracle.

B. Oracle Endpoint Integration Baselines

C. Robustness to Domain Sample Size

The additional evaluations test robustness across sample sizes, step sizes, baselines, training horizons, and parameter subsets. Results remain informative beyond the shortest horizon and final-layer setting, while accuracy depends on step size and declines at longer horizons.

C. Robustness to Domain Sample Size

  • Sign accuracy remains comparable for triples involving low-data domains and those using only high-data domains, supporting robustness to domain sample size.The low-data examples include Books3 with 61 available samples and YoutubeSubtitles with 77.Source: C. Robustness to Domain Sample Size · p. 13

D. Additional Ablations

  • The base first-order estimator is computed alongside the Trotter and trap-local predictors, but the tables report gradient cosine similarity as the main first-order comparator.Source: D. Additional Ablations · p. 13

D.1. η-Autopilot

  • The η-autopilot balances detectability against sign validity, estimates a conservative noise floor, and clamps the selected step size using directional and loss-level ceilings.Its entanglement correction compares high-coupling probe effects with within-domain pilot triples.Source: §D.1 · pp. 14–15

D.2. η Sweeps (Robustness Check)

  • Across η sweeps, effect magnitudes increase with η as predicted, while sign accuracy varies substantially, motivating principled step-size selection.Source: §D.2 · p. 15

D.3. Gradient Cosine Similarity Baseline

  • 77.5% sign accuracy for the cosine baseline versus 93.6% for the Lie-bracket method shows first-order alignment is informative but insufficient for reliable order prediction.The cosine baseline also exceeds the 50% random level.Source: §D.3 · p. 16

D.4. Multi-Step Generalization

  • At k≥10, accuracy degrades as higher-order trajectory effects accumulate, although the fixed k=1 predictor remains above random at k=20.The predictor is held fixed at θ0 across the tested horizons.Source: §D.4 · p. 17

D.5. Long-Horizon Validation (k = 50)

  • 65.3% sign accuracy at k=50 accompanies a 54.7% mean-regret reduction relative to random ordering, using the fixed k=1 predictor.Source: §D.5 · p. 17

D.6. Expanded-Subspace (Multi-Layer) Validation

  • 80.9% sign accuracy across the expanded 8-layer trainable subset, with 22.3% regret reduction, validates the predictor beyond final-layer tuning.The evaluation uses layers 8–15, with o proj and down proj matrices in each layer.Source: §D.6 · p. 17

D.7. AdamW Robustness

The appendix tests the bracket predictor under stateful optimization, approximate curvature, longer horizons, broader parameter subspaces, and sequence-level scheduling. Results support its scheduling and geometric predictions while identifying optimizer state and curvature precision as important boundaries.

D.7. AdamW Robustness

  • AdamW remains a partial extension: the SGD-derived predictor is weak at short horizons, while the exploratory augmented-state score reaches 59.6% accuracy and 29.8% regret reduction at k=10.The theory assumes memoryless gradient descent; a faithful AdamW commutator should include optimizer state (θ, m, v), which remains future work.Source: §D.7 · pp. 17–18

D.8. Finite-Difference HVP Ablation

  • Finite-difference HVP approximation degrades accuracy by 4–12pp across k, indicating sensitivity to curvature precision despite comparable computational cost to exact HVPs.The comparison uses Qwen2.5-1.5B, 50 triples, η=0.0026, and ϵ=10−5; FD-HVP uses two gradient passes instead of one double-backward.Source: Table 9 · p. 18, §D.8 · p. 18

E.1. Theory-Validation Figures

  • The theory-validation appendix reports large-N percentiles against 500 sampled random curricula, not against all N! possible orders.This qualifies the interpretation of the large-domain results.Source: §E.1 · p. 18

E.2. Full Long-Horizon Table

  • With the predictor fixed at θ0, long-horizon degradation reflects accumulated higher-order trajectory effects rather than predictor recomputation.Table 10 reports the full multi-seed validation over 1,020 trials; the accompanying scaling figure reports mean absolute loss difference growing linearly with η2.Source: §E.2 · p. 18, Table 10 · p. 18, Figure 1 · p. 19

E.3. Multi-Layer Subspace Validation

  • The multi-layer Trotter predictor reaches 80.9% sign accuracy and 22.3% regret reduction across 1,020 trials, showing the signal is not confined to a tiny final-layer subspace.On a 50-triple eager-attention probe, both post-attention-only and expanded attention-input projections obtain 92.0% sign accuracy.Source: §E.3 · p. 19, Figure 2 · p. 19

E.4. Sequence-Level N=3 Scheduling

  • Exact tournament scoring recovers the best of all 3! schedules in 87.5% of cases and places it in the top two in 96.0%.Across 40 quadruples, the scalable Borda variant reaches 79.0% top-1 and 94.5% top-2; exact scoring has mean Spearman correlation 0.929 and mean regret reduction 93.5%.Source: §E.4 · p. 19

E.5. Large-N Tournament Details

Large-N evaluations compare bracket-derived schedules with sampled curricula, while additional experiments test edge stability, post-training transfer, runtime, and local bracket correction. The reported results support high-percentile scheduling and show that the bracket can also guide a local correction.

E.5. Large-N Tournament Details

  • For 56 MMLU subjects, the bracket tournament reaches the 99.0–99.6th sampled percentile, while descending gradient-norm ordering falls below the first percentile.Percentiles compare against 500 sampled random curricula, not all N! schedules.Source: §E.5 · p. 19, Table 11 · p. 20
  • Across N=5, N=7, and N=10 runs, all 41 overlapping edges are bit-for-bit identical at float32 precision, with pairwise sign accuracy of 80%, 76%, and 73%, respectively.The N=10 test has p = 0.0025 over 45 pairs; sequence-level aggregation, rather than edge changes, varies with N, horizon, and target.Source: §E.5 · pp. 19–20

E.6. Instruction-SFT and DPO Details

  • Instruction-SFT sign accuracy is 98.1% at k=1 and 73.1% at k=20; DPO accuracy is 98.9% at k=1 and 72.2% at k=20.DPO is evaluated as offline preference optimization, not online RLHF.Source: §E.6 · p. 20

E.7. Wall-Clock Measurements

  • The planner is not cheaper than brute force at k=1, but becomes substantially cheaper at horizons relevant for pilot fine-tuning.Table 12 reports wall-clock measurements in seconds.Source: §E.7 · p. 20, Table 12 · p. 20

E.8. Bracket-Control Experiment

  • With c=0.5, bracket control improves over the uncorrected shared-drift update in 87.7% of cases and beats both sequential endpoints in 46.6%.The result is from Llama-3.2-1B Pile-domain triples.Source: §E.8 · p. 20
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