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Local and Global Recursive Ridge Regression for Online Residual Correction of Frozen Multivariate Forecasters under Label Delay

Takumi Fujimoto, Hiroaki Nishi

arXiv:2610.29604v1cs.LG

TL;DR

Under strict label delay, frozen forecasters can be corrected from residuals, but retained state can constrain deployment. DRiC uses recursively updated ridge regressions on leading DCT coefficients, locally or globally, with a residual-scale guard. DRiC-L8 reduces MSE more than ELF on average with 15–17 times less state, though ELF is more accurate on Traffic.

  • Problem

    Under strict label delay, a multi-horizon target arrives only after the horizon has passed, and closed-form correctors may require substantial retained state per channel.

    Source: §1 · pp. 1–2

  • Method

    DRiC recursively fits ridge regressions on leading DCT coefficients of the next residual block, using forecast and latest-input coefficients, with local or global statistics and a residual-scale guard.

    Source: §1 · p. 2

  • Results

    DRiC-L8 reduces MSE by 14.55% on average over 126 conditions versus 12.98% for ELF, with 15–17 times less state, although ELF is more accurate on Traffic.

    Source: §1 · p. 2

  • Takeaways & Limitations

    DRiC-L offers the stronger average accuracy–state trade-off, while DRiC-G is the most accurate method at or below its state on Electricity and Traffic.

    Source: §1 · p. 2

    Evaluation covers one horizon (H = 24), three moderate-size trained backbones, and seven datasets; foundation-model forecasters and longer horizons were not evaluated.

    Source: §6 · p. 12

Abstract

from arXiv · show

A frozen multi-horizon forecaster can be corrected online from its own residuals. However, under label delay the target of a block arrives only after the whole horizon has passed, and the corrector's retained state grows with the number of channels. Aiming to combine a small retained state with a large error reduction, we propose DRiC (DCT Ridge Corrector), which fits a ridge regression of the leading discrete cosine transform (DCT) coefficients of the next residual block on those of the frozen forecast and of the latest input block, with sufficient statistics updated recursively. Its local variant, DRiC-L, keeps one such regression per channel; its global variant, DRiC-G, pools all channels into one regression, whose statistics do not depend on the channel count. A guard tracks a running scale of the frozen forecast's residual for each channel; it skips updates on blocks far outside that scale and clips the issued correction. We evaluate on seven datasets (four ETT series, Weather with its missing-value codes filled, Electricity, and Traffic) with DLinear, PatchTST, and iTransformer backbones, against 10 naive, gradient-trained, and closed-form methods. DRiC-L with eight DCT coefficients (DRiC-L8) reduces the mean squared error (MSE) by 14.55% on average over 126 conditions, against 12.98% for ELF, the most accurate comparator, while retaining 15-17 times less state, although ELF is more accurate on Traffic. DRiC-G retains 14.6 kB on the 321-channel Electricity data and 18.9 kB on the 862-channel Traffic data and is the most accurate method at or below its state on both. However, 5 to 7 of the 10 comparators, all with at least 10 times its state, are more accurate there. On the unfilled Weather file, DRiC-L8 without the guard has on average 8.0 times the frozen forecast's MSE, whereas with the guard it reduces the MSE by 7.94%. Code: https://github.com/keiotakmin/dric.

1 Introduction

Under strict label delay, the paper seeks accurate online residual correction with little retained state. It introduces DRiC, with local and global regressions and a guard for recursive updates.

1 Introduction

  • Strict label delay means a multi-horizon target is complete only H steps after its forecast, while corrector state can constrain deployment on gateways or servers handling many streams.Under this delay, closed-form correctors are strong reference methods.Source: §1 · pp. 1–2
  • DRiC predicts the leading K DCT coefficients of the next residual block from the frozen forecast and latest input block, using fixed-size per-channel statistics.The stated per-channel state is a (2K + 2) × (2K + 2) Gram matrix and its right-hand side, independent of lookback, horizon, and stream length.Source: §1 · p. 2
  • The guard skips blocks whose residual power greatly exceeds a channel’s running residual scale and bounds the issued correction using that same scale.It uses the frozen forecast’s residual as its reference, so the decision does not depend on what the corrector has learned.Source: §1 · p. 2
  • DRiC-L keeps one regression per channel, whereas DRiC-G pools channels into one Gram matrix whose statistics do not grow with channel count.Only the guard’s one-per-channel scale grows with channel count in DRiC-G.Source: §1 · p. 2
  • DRiC-L8 reduces MSE by 14.55% on average over 126 conditions versus 12.98% for ELF, with 15–17 times less state; ELF is more accurate on Traffic.DRiC-G is the most accurate method at or below its state on Electricity and Traffic, although 5 to 7 larger-state comparators are more accurate there.Source: §1 · p. 2

2 Related Work

Related work includes gradient-trained and closed-form test-time correctors, predeployment corrections, and robust recursive estimation. DRiC is distinguished by its complete-block updates and shared-statistics option.

2 Related Work

  • TAFAS [1], PETSA [2], COSA, FAC [4], and ORCA [13] adapt frozen forecasters using gradients or replay, while the comparison protocol updates methods only after complete target blocks.The paper does not compare with DynaTTA.Source: §2 · p. 2
  • Post-training corrections [14], UEC-STD [15], and CRC [16] are fitted before deployment and use no labels after deployment, unlike DRiC’s online updates from delayed labels.DRiC’s guard decides during deployment whether each block and channel enters its statistics.Source: §2 · p. 3
  • ELF [5], RLS-Bank [6], and EPOC [7] are closed-form correctors with different regression inputs and statistics; DRiC instead uses fixed-size recursive statistics and complete target blocks.ECHO [17] stores records for a growing-set ridge regression, while EFC [18] combines ridge regression with a trained recurrent corrector and uses partially elapsed horizons.Source: §2 · p. 3
  • Robust adaptive filtering can threshold updates using a running error-scale estimate; DRiC’s guard applies such a threshold to the frozen forecast’s residual and also bounds output.The guard keeps one scale per channel.Source: §2 · p. 3
  • Global models fit all series in a group, and DRiC-G applies this choice to online residual correction with regression statistics shared across channels.Those statistics do not grow with channel count.Source: §2 · p. 3

3 DRiC: DCT Ridge Corrector

DRiC corrects frozen multi-horizon forecasts using residual feedback that arrives only after each target block is complete. Its local and global recursive ridge variants predict residual DCT coefficients, with a shared guard limiting updates and issued corrections.

3.1 Setting and feedback order

  • Each block is forecast before its target is complete; the corrector uses only earlier blocks and updates its state after the target arrives.The residual is measured against the frozen forecast, independent of the correction issued.Source: §3.1 · pp. 3–4

3.2 Residual representation

  • The corrector represents forecasts, input blocks, and residual targets with leading DCT coefficients, then maps predicted coefficients back to the horizon.The input features include the newest input sub-block; the representation uses J = 8 basis columns.Source: §3.2 · p. 4

3.3 Recursive ridge regression

  • Exponentially weighted ridge regressions update sufficient statistics after admitted blocks; DRiC-L fits per channel, while DRiC-G pools channels into one regression.The global map uses all input sub-blocks, and its shared statistics decay once when at least one channel is admitted.Source: §3.3 · pp. 4–5

3.4 Guard

  • A per-channel guard skips blocks whose frozen-forecast residual power exceeds a threshold based on a running scale, and clips the issued coefficients to that scale.The standard Weather example shows it skipping blocks with missing-value codes in their targets and most later blocks distorted by codes in their input windows.Source: §3.4 · p. 5

3.5 Blending and retained state

  • Both variants use a shared blending weight, while DRiC-G’s regression state is channel-count independent and its guard scales add 8 bytes per channel.The global regression takes 9,912 bytes; total retained state is 14.6 kB on Electricity and 18.9 kB on Traffic.Source: §3.5 · p. 6

4 Experimental Setup

The evaluation covers seven datasets under a chronological split, with complete forecast-target blocks scored after a one-horizon delay. Three frozen forecasters and ten comparator methods receive the same forecasts, targets, and origins in each condition.

4.1 Data and protocol

  • Evaluation uses four ETT series plus Weather, Electricity, and Traffic, with L = 96, H = 24, and forecast origins H samples apart.Weather’s main-view copy forward-fills its missing-value codes; the standard file is reported separately.Source: §4.1 · p. 7

4.2 Frozen forecasters

  • DLinear, compact PatchTST, and iTransformer are evaluated across three seeds, legacy and refit bases, yielding 126 conditions.Each method receives identical forecasts, targets, and origins within a condition.Source: §4.2 · p. 7

4.3 Comparators

  • Comparators include naive forecasts, gradient-trained adapters, ORCA, RLS-Bank, ELF, and EPOC; all issue before targets arrive and update only after complete blocks.ORCA issues no correction until its replay buffer fills, which takes the first 125 evaluation blocks.Source: §4.3 · pp. 7–8

4.4 Correctors and reporting

  • Correctors use λ = 1, a 128-block half-life, and τ = 10; results report MSE reduction against the frozen forecast and retained state.The reduction is averaged over 126 conditions and 12 dataset–backbone panels; the authors make no significance claims from paired-win counts.Source: §4.4 · p. 8

5 Results

DRiC-L8 leads the overall accuracy comparison with substantially less retained state than ELF, while DRiC-G offers a channel-independent alternative whose accuracy varies by dataset. Component tests identify input features and the guard as consequential design choices, and the section reports per-block computation times.

5.1 Accuracy and retained state

  • 14.55% mean MSE reduction over 126 conditions for DRiC-L8 versus 12.98% for ELF; DRiC-L8 has lower MSE in 99/126 conditions, but ELF wins Traffic in 16/18 (15.93% against 13.00%).DRiC-L8 retains 19.8 kB on ETT and 2.18 MB on Traffic, against ELF's 302 kB and 37.1 MB, respectively.Source: §5.1 · p. 8, Table 2 · p. 9
  • DRiC-G lies on the accuracy–state frontier in 8 panels, including all six Electricity and Traffic panels, but is not on the frontier in any of the three ETT panels.In the six Electricity and Traffic panels, the next-smallest method is EPOC, with 10 times DRiC-G's state on Electricity and 21 times on Traffic.Source: §5.1 · pp. 8–9

5.2 Components

  • Adding a local regression to DRiC-G raises reduction but makes state grow with channel count; the local and global versions are complementary operating points.Adding the global regression before the local changes mean reduction by −0.01 points at K = 4 and −0.82 points at K = 8, while adding about 10 kB.Source: §5.2 · p. 9, Table 3 · p. 11
  • Removing the latest input block and last input value changes mean reduction by −3.36 points at K = 8 and −4.04 at K = 4, the largest component effect.Using all four input blocks instead of the latest changes reduction by −1.02 and +0.02 points while multiplying Traffic state to 8.55 MB and 2.36 MB; older blocks add state without adding accuracy.Source: §5.2 · pp. 9, 11, Table 3 · p. 11
  • Reducing the DCT component count from K = 8 to K = 4 changes reduction by −1.37 points and cuts state by about a factor of three.The component count therefore trades accuracy for state.Source: §5.2 · p. 11, Table 3 · p. 11
  • Removing the guard changes mean reduction by −1.75 points for K = 8, −1.37 for K = 4, and −0.34 for the global version.The guard raises reduction on filled Weather by +9.61 points for K = 8; on standard Weather, unguarded DRiC-L8 reaches −700.38% while the guarded version reduces error by 7.94%.Source: §5.2 · p. 11, Table 4 · p. 12

5.3 Computation

  • On one Intel Xeon Silver 4310 CPU thread, issuing and updating a block takes 0.31 and 6.38 ms for DRiC-L8, and 0.35 and 4.04 ms for DRiC-G.The timing uses synthetic inputs with the datasets' channel counts; comparators were not timed.Source: §5.3 · p. 11

6 Discussion and Limitations

The discussion presents the latest-block rationale as untested and notes that correction gains depend on the backbone. The guard and evaluation scope also impose trade-offs and boundaries.

6 Discussion and Limitations

  • Adding older input blocks hardly changed accuracy with K = 4 and lowered it with K = 8, so the latest-block explanation remains unestablished.The authors say the results are consistent with the explanation, not proof of it.Source: §6 · p. 12
  • On Traffic with PatchTST, the guard reduced accuracy by −2.08 points for DRiC-L8 and −5.08 points for DRiC-G.Larger τ reduces this cost but weakens protection on the standard Weather file.Source: §6 · p. 12
  • DRiC-L8’s reduction was 17.70% with PatchTST but 11.17% with iTransformer, so the gains vary with backbone.The authors note that PatchTST lacks instance normalization, while iTransformer normalizes each window.Source: §6 · p. 12
  • Evaluation covers one horizon (H = 24), three moderate-size trained backbones, and seven datasets; foundation models and longer horizons were not evaluated.The study also did not test whether online updating is needed for the reported accuracy.Source: §6 · p. 12

7 Conclusion

The conclusion asks whether online residual correction can combine small retained state with substantial error reduction under label delay. It presents DRiC-L8 and DRiC-G as different state–accuracy trade-offs, with dataset-specific qualifications.

7 Conclusion

  • The paper frames its question as whether an online residual corrector can combine small retained state with large error reduction under label delay.Source: §7 · p. 12
  • DRiC-L8 reduces MSE more than ELF on average with 15–17 times less retained state, although ELF is more accurate on Traffic.The conclusion describes this as the outcome of using leading DCT coefficients of the forecast and latest input block with fixed-size recursive statistics.Source: §7 · p. 13
  • On the standard Weather file, guarded DRiC-L8 reduces error by 7.94%, while unguarded DRiC-L8 has 8.0 times the frozen forecast’s MSE on average.The file’s missing-value codes drive the unguarded result; the guard’s cost includes a lower reduction on Traffic with PatchTST.Source: §7 · p. 13
  • DRiC-G is the most accurate method at or below its state on Electricity and Traffic, but 5 to 7 larger-state comparators are more accurate there.Those comparators retain at least 10 times DRiC-G’s state.Source: §7 · p. 13

A Sensitivity to the settings

The sensitivity analysis reports that mean MSE reduction generally stays within specified ranges across one-at-a-time settings, while the guard threshold presents a Weather-specific trade-off.

A Sensitivity to the settings

  • Mean reduction stays within 12.71–13.45% for K = 4, 13.97–14.79% for K = 8, and 8.78–9.38% for the global variant across most tested settings.Exceptions are λ = 100 and τ ≥80 for DRiC-L4 and DRiC-L8, and λ ≥1000 for DRiC-G.Source: A Sensitivity to the settings · p. 15, Table 5 · p. 15
  • Although τ = 20 for DRiC-L8 and τ = 40 for DRiC-L4 slightly raise mean reductions, the authors retain τ = 10 because loosening the guard lowers standard-Weather reduction for DRiC-L8 and DRiC-G.The table reports mean MSE reduction over 126 conditions, with the standard Weather file in parentheses.Source: A Sensitivity to the settings · p. 15, Table 5 · p. 15

B Frozen forecasts

This section identifies the frozen-forecast MSE comparison and its evaluation scope, but the supplied passages do not provide the table’s values.

B Frozen forecasts

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