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LARC: Low-Rank Adaptive Residual Connections for Learning in Frozen Models

Junyi Zou, Avrova Donz

arXiv:2609.40063v1cs.LGcs.CL

TL;DR

LARC asks whether feedback updates retained in a compact residual improve later predictions, beyond the learned starting policy. It implements a slow initialization and private fast adaptation on a frozen model. Development results show lower query error after feedback updates, while the continuous-integration replay favors reset over retained online updates.

  • Problem

    The paper asks whether retaining parameter updates from feedback improves a later read, and under which learning conditions.

    Source: §1 · p. 2

  • Method

    LARC implements a low-rank hidden-state correction whose factors start from a learned slow state and adapt in a private fast state.

    Source: §1 · p. 2

  • Results

    Real-feedback updates reduce expected query error on held-out development groups, whereas the task-matched continuous-integration study favors reset over retained online updates.

    Source: §1 · p. 2, §9 · p. 15

  • Takeaways & Limitations

    Feedback responsiveness, starting-policy quality, and sustained online benefit require separate measurements.

    Source: §1 · p. 2

    Episodes begin from trained factors, so their starting behavior includes slow learning; numerical state is retained while the observed prompt remains fixed.

    Source: §2.1 · p. 3

Abstract

from arXiv · show

Low-Rank Adaptive Residual Connections (LARC) give a frozen model a compact numerical state that can learn from feedback. The map $h+BAh$ adds a low-rank correction to a hidden representation. A slow state $\rho$ learns starting factors across tasks; a private fast state $\Phi$ copies them, changes with feedback, and resets to the trained initialization. This report specifies an input-side realization of the numerical policy carrier in Memory-Mediated Learning Architecture and examines its factor-space dynamics and learning lifetime. We study a rank-4 input residual with 12,288 trainable parameters on a frozen MiniCPM5-1B-SFT substrate. In a four-candidate program-selection task, two feedback-gradient steps reduce expected query execution error by 24.65 and 36.65 percentage points relative to resetting to the respective trained static and post-adaptation initializations. These development results cover 16 parameter groups and three paired training seeds. A direct support-loss selection rule is much more accurate, reaching 0.78125% error. In a repository-balanced chronological replay of public continuous-integration jobs, retaining online updates raises half-Brier loss from 0.1274 to 0.1808. A fixed follow-up intervention records same-batch non-descent and inconsistent future benefit from shrinking updates. Together, the algebra and measurements distinguish residual capacity, adaptation relative to a starting point, and usefulness on later decisions.

1 Introduction

LARC tests whether a compact residual can retain feedback in its parameters to improve later predictions. The study separates that question from the quality of the trained starting policy and from sustained benefit on later decisions.

1 Introduction

  • Feedback can arrive during an ongoing task, and the next decision can use it through context, stored records, or parameter changes.The paper asks whether retaining parameter updates improves a later read, and under which learning conditions.Source: §1 · p. 2
  • LARC adds a low-rank hidden-state correction whose factors start from a learned slow state and adapt as a private fast state.Resetting restores the bound trained initialization, removing fast updates while preserving slow learning.Source: §1 · p. 2
  • LARC realizes MMLA’s numerical policy carrier with a hidden-to-hidden residual, while the frozen language-model mixer supplies the experimental computation.The residual interface and its learning lifetime define the realization; the mixer is a separate architectural choice.Source: §1 · p. 2
  • Retaining real-feedback updates reduces expected query error on held-out development groups, but the advantage between training objectives varies across seeds and the symbolic rule remains more accurate.In the task-matched continuous-integration study, preserving online updates is harmful relative to reset.Source: §1 · p. 2
  • The report specifies the residual’s gradient path, reset and resume operations, factor-space dynamics, and comparisons of static versus post-adaptation training.It also compares real and sham feedback with keep and reset reads.Source: §1 · p. 2

2 LARC as a Learning State

LARC adds a low-rank correction at a hidden-state boundary and separates a learned slow reference from private, episode-local fast adaptation. Its training and execution procedures preserve the gradient path through the frozen mixer while distinguishing reset from trajectory restoration.

2.1 Residual interface

  • The residual reads learned linear combinations of hidden coordinates and writes a correction into the full representation, preserving an identity path; rank limits the correction subspace, not the complete computation.The evaluated interface uses rank 4 and hidden dimension 1536; its factor entries require 49,152 bytes in FP32 and 12,288 multiply-accumulates per token.Source: §2.1 · pp. 2–3

2.2 What changes, and on which timescale?

  • The slow state holds learned factors and a version; each episode privately copies that initialization into a fast state, and reset restores the bound initialization.The trained factors can both be nonzero, so the episode’s starting behavior already includes slow learning; numerical state can persist while the prompt stays fixed.Source: §2.1 · p. 3

A trainable residual before the frozen mixer

  • Zeroing B makes the initial forward map an identity, while an episode reset later restores the trained factors.The paper’s LARC residual is distinct from the optimizer acronym Layer-wise Adaptive Rate Control.Source: §2.1 · p. 2, Figure 2 · p. 3

2.3 Two timescales and two outer objectives

  • Static training evaluates query loss at the slow initialization, whereas adapted training evaluates it after two support updates using a first-order MAML-style estimator.The estimator measures the query gradient at the adapted state and passes it to corresponding slow factors without differentiating through the support updates.Source: §2.3 · p. 4

One slow version, independent fast trajectories

2.4 Gradient flow, batching, and restoration

  • The frozen mixer must remain differentiable with respect to its input so gradients can reach the residual factors; reset restores the slow initialization, while resume restores the current fast trajectory.Batch rows select their episode’s own factors, and checkpoints preserve the state needed to continue that trajectory.Source: §2.4 · pp. 4–5

3 Properties of the Residual and Its Updates

The section characterizes how input-side low-rank residuals relate to LoRA and how their factorization shapes representational capacity and learning dynamics. It also gives first-step behavior under identity initialization and conditions that constrain descent.

3.1 Relation to LoRA: placement and learning lifetime

  • The effective correction has rank at most r and spans only directions in the image of the following linear map; full row rank makes every rank-at-most-r correction representable.The evaluated residual precedes the full nonlinear mixer, so this linear equivalence does not generally merge it into an arbitrary internal LoRA site.Source: §3.1 · p. 5

3.2 Identity initialization and the first learning step

  • E[C1] = -ηrσ2G_C,0 when A0 has independent zero-mean entries of variance σ2 and the gradient depends on the factors only through C0 = 0.With both factors zero, plain SGD leaves this bilinear branch stationary; the first-step calculation does not determine numerical steps under outer optimization, clipping, or mixed precision.Source: §3.2 · pp. 5–6

3.3 Rank controls capacity, not displacement

  • Rank four bounds each instantaneous residual to rank four, but the difference between two learned residuals can have rank eight.Rank alone guarantees neither bounded factor norms nor a uniform downstream Lipschitz constant.Source: §3.3 · p. 6

3.4 Equal forward maps can have different learning dynamics

  • Rescaling factors can preserve the same forward map while changing subsequent SGD updates, so the product BA alone generally cannot reproduce the next factor-space update.Initialization scale, factor norms, optimizer state, and step size are part of the learning specification.Source: §3.4 · p. 6
  • A strict decrease for a nonzero gradient requires smoothness along the step and 0 < η < 2/L; rank supplies neither condition.A decrease on the arrived support batch also does not determine loss on later jobs.Source: §3.4 · p. 6

4 Measuring Feedback Adaptation

The evaluation crosses real or permuted feedback with keeping or resetting the fast state before the query. It uses controlled query reads to distinguish the effect of retaining updates from the effect of feedback binding.

4 Measuring Feedback Adaptation

  • Both feedback conditions take the same number of updates from the same trained initialization, then either keep the fast factors or reset to their own slow state before the query.The query receives no feedback transcript, inner optimizer, live inner random stream, or fast-dependent cached representation.Source: §4 · pp. 6–7

Identify what the feedback writes into the fast state

  • Figure 4 crosses feedback binding with state retention: G measures retaining real-feedback state against reset, while D compares that effect with independently permuted feedback.Both branches execute two updates, and reset occurs immediately before the query.Source: Figure 4 · p. 7, Identify what the feedback writes into… · p. 7
  • The G and D contrasts differ from the outer-objective contrast: useful adaptation can occur under similar outer objectives, while real updates can still be harmful.A positive difference between real and permuted updates can coexist with harmful real updates.Source: Identify what the feedback writes into… · p. 7

5 Grouped Program-Selection Study

The study evaluates a rank-4 input LARC residual on a frozen MiniCPM5-1B-SFT substrate in grouped four-candidate program selection. It compares feedback updates and reset reads, finding benefits from retaining real-feedback state while the symbolic support rule remains much more accurate.

5.1 Model and task construction

  • Episodes present four complete programs, four support examples, and 20 disjoint query inputs; query outcomes are excluded from inner feedback and the policy prompt.Development covers 16 independent parameter groups and 64 recipe episodes.Source: §5.1 · p. 7

5.2 A concrete feedback update

  • Execution losses favor below-average-error candidates, while the factor update is shaped by the model Jacobian linking residual factors to candidate energies.On query reads, updated factors change candidate probabilities; resetting instead reads the trained starting policy.Source: §5.2 · p. 8

5.3 Analysis and selection rule

  • The primary endpoint averages recipes within parameter groups, weights task families equally, and pairs three training seeds; the preregistered rule selects by worst-seed min(G, D).Intervals use 10,000 paired bootstrap resamples of complete parameter groups, conditional on the three trained seeds.Source: §5.3 · p. 8

5.4 Results

  • 24.61% mean error for adapted training versus 29.30% for static training, a paired improvement of 4.69 points with a 99% interval of [1.19, 8.88].The mean advantage reverses for one seed: improvements are 1.45, −10.46, and 23.09 points.Source: §5.4 · p. 8, Table 3 · p. 9
  • Both objectives yield positive feedback benefits: static training obtains G = 24.65 and D = 28.36 points, while adapted training obtains G = 36.65 and D = 35.77 points.Source: §5.4 · p. 8
  • 0.78125% error for the symbolic support rule, considerably below either learned policy.The rule selects directly from the same support execution losses available to the gradient-based policy.Source: §5.4 · p. 9, Table 5 · p. 9
  • 15.49% versus 16.04% greedy query error for static and adapted models, respectively, differs in ordering from their expected-risk comparison.Probability shifts among complete programs can improve expected risk without changing the most probable program.Source: §5.4 · p. 10, Table 6 · p. 10

6 Delayed Feedback in Public CI Workflows

The study evaluates task-matched policies and memory in a repository-local chronological replay of public CI jobs. Retaining real-feedback updates raises loss relative to reset and HEDGE4, while outperforming permuted feedback.

6.1 Chronological replay and task-matched training

  • Predictions are made at job start, before same-timestamp completions are processed; arrived labels update the policy using the prompt saved at prediction time.This ordering prevents later outcomes or memory contents from entering an earlier prediction.Source: §6.1 · p. 10
  • The audit queue contains 3,321 jobs from 72 groups, but shared training includes later pandas jobs, making this a repository-local retrospective evaluation rather than a global-calendar prospective deployment.The queue comprises 83 NumPy jobs from one commit and 3,238 pandas jobs from 71 commits.Source: §6.1 · p. 10
  • Each selected static initialization receives one epoch of equally weighted, task-matched training; three fixed slow states each undergo 670 additional updates before audit.Training uses no-memory and causal-memory views of each job.Source: §6.1 · p. 10

6.2 Prediction quality and retained state

  • 0.1808 full-path half-Brier loss versus HEDGE4’s 0.1024; retaining real-feedback state has G = −0.0534 versus reset, though real feedback beats permuted feedback by D = 0.1497.The full path’s gain over HEDGE4 is −0.0784 and negative for every seed.Source: Table 7 · p. 11, §6.2 · p. 11
  • Keep raises pandas commit-weighted loss from 0.2248 to 0.3343, while slightly improving the single all-success NumPy commit from 0.0300 to 0.0273.Across label strata, losses are lower on failures and cancellations but higher on successes and other outcomes; these strata do not identify which update caused the pattern.Source: §6.2 · pp. 11–12, Table 8 · p. 12
  • HEDGE4, reset, keep, and permuted remain in that loss order under all three aggregations, so global commit- or job-weighting does not remove observed retention harm.The weighting-sensitivity rows reuse the same saved predictions and are descriptive.Source: §6.2 · p. 12, Table 9 · p. 12

7 Separating Update Magnitude from Parameter History

A fixed-prefix intervention varies whether fast parameters persist across batches and whether the realized update is full-sized or reduced. Smaller carried updates do not consistently improve future loss, and the original update is locally non-descending on the first pandas batch.

7 Separating Update Magnitude from Parameter History

  • CARRY preserves fast parameters across feedback batches, whereas LATEST restores the bound slow initialization; TENTH scales the realized FP32 parameter difference by 0.1.RESET always reads the bound slow initialization.Source: §7 · p. 12
  • Reducing the carried update yields mean gain −0.0237, with per-seed gains 0.0730, −0.1459, and 0.0017; all four active paths remain worse than RESET and HEDGE4.At the smaller increment, removing accumulated parameter history improves loss by 0.0976 on average and for every seed; at the original increment, that effect changes sign across seeds.Source: §7 · p. 12, Table 10 · p. 12

7.1 What happens on the first feedback batch?

  • After two updates, all three original-update seeds show local retained-state gain below −0.01 on the same 37 newly starting jobs, which later resolve to 24 successes and 13 cancellations.This is a descriptive local marker; the first cumulative crossing occurs after 2, 421, and 25 feedback batches across the three seeds.Source: §7.1 · p. 13
  • At the first pandas batch, the original update increases same-batch cross-entropy for all three seeds; the smaller increment decreases it for two seeds and increases it for one.The matched pre- and post-update losses use the same saved prompts, labels, masks, positions, and batch-mean normalization.Source: Table 11 · p. 13, §7.1 · p. 13
  • Across the diagnostic prefix, smaller CARRY has lower average support loss before and after updates yet worse mean future half-Brier loss; the selected update batch and later prediction population differ.Feedback delay and class composition are plausible mechanisms, but their individual causal contributions require different interventions.Source: §7.1 · p. 13
  • Figure 6 separates full-replay retention, first-batch same-batch CE change, and the fixed-prefix loss difference from shrinking updates; the latter panels use distinct populations and losses.The first-batch panel marks positive CE changes as non-descent.Source: Figure 6 · p. 13, §7.1 · p. 13

8 Implementation, Cost, and Reproducibility

The implementation makes episode state and update lifetimes explicit while preserving a frozen substrate. Its recorded costs include substantial non-kernel work, and reproducibility depends on saving more than factor matrices.

8 Implementation, Cost, and Reproducibility

  • Recorded wall times include model loading, evaluation, CPU work, saving, and cleanup, so Table 12 reports end-to-end entry costs rather than pure GPU kernel times.The CI total also includes stopped preflights, an interrupted partial replay, and the completed evaluation.Source: Table 12 · p. 14
  • The episodic ledger records 11,581 mixer forwards, 6,195 head forwards, and 5,386 backward calls; its 49,025,792 physical positions and 19,521,408 effective positions measure different operation totals.These totals are neither mixer-input tokens alone nor FLOP-equivalent work or independent training examples.Source: §8 · p. 14, Table 13 · p. 14
  • Reproducibility artifacts separately record the frozen substrate, semantic adapter, residual placement, trained factors, optimizer state, and random-number state; access to the base model and adapter is required to execute trained residuals.The package also records source manifests and checkpoint identities for historical residual implementations.Source: §8 · p. 14

8.1 The state boundary in an implementation

  • Begin copies factors and initializes private update state, Update replaces fast factors using arrived feedback, and Read scores the current prompt and candidates.Functional SGD requires no momentum tensors, but factors, step count, and private random stream remain episode state.Source: §8.1 · p. 14
  • After LARC changes the representation, cached mixer outputs or key–value state from another factor version evaluate a different function, so the mixer is recomputed with current factors.The implementation preserves masks and positions and uses activation checkpointing to trade computation for backpropagation memory.Source: §8.1 · pp. 14–15
  • Delayed-feedback restoration must retain the original prediction prompt and pending-label linkage; rebuilding from current memory would train on a different observation.Saving only factor matrices cannot determine which feedback belongs to earlier reads.Source: §8.1 · p. 15

9 Discussion

The program-selection study finds measurable adaptation from feedback in a trained rank-4 input residual, while the CI replay finds harm from retaining online updates. The evidence remains bounded to the evaluated setting and leaves update-control questions open.

9 Discussion

  • In the four-program setting, feedback written into the trained rank-4 input residual reduces expected query execution error with the query read otherwise fixed.The direct support-loss rule remains much more accurate, and a fixed worst-seed stability rule selects static training despite its higher mean endpoint error.Source: §9 · p. 15
  • In the repository-local retrospective CI queue, the repository-balanced result favors trained reset over retained online state, while HEDGE4 remains stronger than the full neural path.The follow-up records same-batch non-descent and no consistent future benefit from shrinking updates; causes remain unresolved.Source: §9 · p. 15
  • Evidence covers one frozen language-model substrate, one rank, finite candidate decisions, and three training seeds; comparisons with alternative ranks, insertion boundaries, and adapter types remain unrun.The results do not measure a performance advantage over those alternatives, and the CI intervention reuses an observed audit population.Source: §9 · p. 15
  • A learning-state specification must capture both current factors and optimizer state because the product BA describes only the current correction, not the next update.A controller using arrived-batch fitting and later outcomes remains a proposed next study requiring independent task groups and existing comparators.Source: §9 · p. 15

10 Relation to Existing Methods

LARC connects compact residual adaptation to slow initialization and feedback-driven fast weights, while using external feedback and explicit state resets in its experiments.

Residual learning and compact adaptation

  • LoRA parameterizes low-rank weight corrections [10], while LARC applies the factors as a hidden-state residual; the paper identifies their linear relationship and insertion boundary.Residual networks preserve a reference path [8], ReZero gates residual branches with a zero-valued scalar [2], and adapters train compact additions to frozen networks [9].Source: §10 · p. 15

Fast weights and learning an initialization

  • Fast weights provide an intermediate timescale [1], while MAML learns initial parameters for rapid adaptation [6]; LARC applies a first-order estimator to its factors.The estimator omits support-loss Hessians [12], and ordinary query-risk training supplies the matched static objective.Source: §10 · p. 15

Related low-rank adaptation objectives

  • Unlike Block et al. [4] and ABMLL [16], LARC fixes the pretrained substrate and learns a deterministic pair of input-residual factors.Its experiments measure feedback-retention effects through real/sham and keep/reset interventions.Source: §10 · p. 15

Learning during inference

  • Unlike self-supervised test-time training [15], TTT layers [14], and Titans [3], LARC uses external execution feedback or completed-job labels and resets to isolate their effects.Within MMLA, its numerical policy state is separate from explicit stored records, which retain a distinct memory lifetime.Source: §10 · p. 15

11 Conclusion

LARC uses a low-rank residual with trained starting factors and private feedback updates, but the reported results distinguish adaptation from whether retaining updates helps later decisions.

11 Conclusion

  • Two feedback-gradient steps reduce expected query execution error relative to both trained reset baselines, while direct support-loss selection remains more accurate; retained CI updates increase later loss.The conclusion also reports local non-descent in the CI diagnostic.Source: §11 · p. 16

A Derivation of Factor-Space Dynamics

  • Factor updates induce state-dependent preconditioning, so identical products BA can have different subsequent updates; optimizer moments add coordinate dependence.The stated zero-residual gradient result assumes no factor regularizers or factor-dependent stochastic operations; the smoothness bound is conditional and does not contradict observed finite-step non-descent.Source: A Derivation of Factor-Space Dynamics · p. 17

B Episodic Learning Procedure

  • Each paired episodic run shares its substrate, data, support feedback, and query targets, then compares two support-risk updates before updating the slow factors.Six trajectories are completed before development predictions; real/sham feedback is crossed with keep/reset, and predictions are saved before query targets are opened.Source: B Episodic Learning Procedure · p. 17

C.1 Finite parameter support

  • The finite task support uses four candidate programs per parameter group, with target recipes generating support and query outputs and the target index withheld.Support and query inputs are disjoint; the historical exclusion uses effective parameter tuples, while pretraining overlap is unknown.Source: §C.1 · pp. 17–18

C.2 What reset removes

  • Episodic reset restores trained ρ and clears fast-state optimizer and cached representations, whereas the CI diagnostic resets fast parameter history but preserves memory and its prescribed random stream.The CI diagnostic reads the post-update state until the next batch, unlike the always-reset control or resetting explicit memory.Source: §C.2 · p. 18

C.3 CI observations, labels, and baseline parameters

  • The CI predictor receives seven job fields and predicts eventual status from completed-history cases; noncompleted jobs and invalid timestamps are excluded, and labels map into four classes.The neural path scores label strings, while HEDGE4 predicts four probabilities directly; its expert mixture weights use accumulated prediction-time half-Brier loss from arrived feedback.Source: §C.3 · p. 18

C.4 Data and artifact access

The artifact materials document source identities, but public end-to-end rerunning depends on access to artifacts that are not publicly hosted.

C.4 Data and artifact access

  • The program-selection initialization package is assembled locally and is not publicly hosted; later ρ926 checkpoints are distinct, and continuation states remain in the private archive.The authors invite readers to contact them for access; exact artifact identities do not establish public hosting, and the frozen upstream model retains its own distribution terms.Source: §C.4 · p. 19, Table 15 · p. 19
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