Downside-Controlled Online Forecast Combination under Delayed and Revised Outcomes

Minkyoung Kim; Hyunjung Byun; Yohan Lee; Beakcheol Jang

cs.LGSep 24, 2026

Downside-Controlled Online Forecast Combination under Delayed and Revised Outcomes

Authors: Minkyoung Kim, Hyunjung Byun, Yohan Lee, Beakcheol Jang

Organizations: Graduate School of Information, Yonsei University, Seoul, 03722, South Korea

Abstract

Post-hoc correction adjusts a forecaster that cannot be retrained, such as a foundation model, but a correction fitted where errors are stable can hurt where they shift. We aim for downside control: not much worse than the starting forecast. We combine the frozen forecaster, a static corrector and an online corrector on the simplex, using only losses that mature after the horizon. Across seven benchmarks and four base models, two of them foundation models, the worst deterioration over 28 pairs at the main horizon is 0.15% and gains reach 11.5%. On day-ahead load for seven European bidding zones it lowers mean MSE in all seven zones, while single correctors raise mean MSE by up to 102% where the published forecast is most accurate. Three empirical conditions on expert speed, stream length and outcome alignment, each fixed by a documented failure, delimit its scope. Learning from the provisional outcome improves four zones on the settled one; learning on the settled outcome restores all seven.

Figures & tables

Region	Decides	Never touches
Training split	Base model and corrector parameters	Gate weights, evaluation
Held-out fit region	Online corrector’s initial fit	Gate weights, evaluation
Held-out warm slice	Gate’s initial weights, by replaying the update	Corrector parameters, evaluation
Held-out early-stopping tail	Correctors’ stopping epoch	Gate weights, evaluation
Test stream	Gate and online-corrector state, from matured losses only	Any parameter or threshold

Table 1: Exact data-use specification: what each region decides, and what it never touches.

Base model	Dataset	Static		Online		Held-out		Gate, $K{=}2\hphantom{{}^{\dagger}}$		Gate, $K{=}3\hphantom{{}^{\dagger}}$
DLinear	ETTh1	$-0.53$ $\pm$	$0.075$	$-1.40$ $\pm$	$1.250$	$+0.20$ $\pm$	$1.568^{\dagger}$	$-1.95$ $\pm$	$0.926$	$\mathbf{-2.13}$ $\pm$	$0.901\hphantom{{}^{\dagger}}$
	ETTh2	$-0.09$ $\pm$	$0.139^{\dagger}$	$\mathbf{-1.62}$ $\pm$	$0.889$	$-0.76$ $\pm$	$0.868^{\dagger}$	$-0.86$ $\pm$	$0.894^{\dagger}$	$-0.72$ $\pm$	$0.901^{\dagger}$
	ETTm1	$-0.52$ $\pm$	$0.049$	$\mathbf{-1.42}$ $\pm$	$0.873$	$-0.16$ $\pm$	$0.996^{\dagger}$	$-0.25$ $\pm$	$0.947^{\dagger}$	$-0.30$ $\pm$	$0.718^{\dagger}$
	ETTm2	$-0.87$ $\pm$	$0.146$	$\mathbf{-7.61}$ $\pm$	$3.265$	$-7.46$ $\pm$	$3.216$	$-7.59$ $\pm$	$3.274$	$-7.53$ $\pm$	$3.347\hphantom{{}^{\dagger}}$
	Weather	$\mathbf{-4.90}$ $\pm$	$1.220$	$+0.40$ $\pm$	$1.029^{\dagger}$	$+5.20$ $\pm$	$1.470$	$-0.24$ $\pm$	$0.550^{\dagger}$	$-4.76$ $\pm$	$1.216\hphantom{{}^{\dagger}}$
	ECL	$\mathbf{-2.59}$ $\pm$	$0.936$	$-0.95$ $\pm$	$0.598$	$-0.64$ $\pm$	$0.520$	$-1.08$ $\pm$	$0.595$	$-2.48$ $\pm$	$0.931\hphantom{{}^{\dagger}}$

Table 2: MSE change versus frozen base model in percent, computed from the run means of the two MSEs, all origins, printed as mean

\pm

standard deviation of the five per-run changes with one more decimal than the mean. The held-out corrector has the static architecture and is not a library expert.

K{=}2

is the base model with the online corrector;

K{=}3

adds the static corrector. Negative is better, bold is the row’s best, a dagger marks a change smaller in absolute value than the standard deviation of the five per-run changes.

	Chronos-Bolt				TimesFM
Dataset	Static		Gate, $K{=}3$		Static		Gate, $K{=}3$
ETTh1	$-0.23$ $\pm$	$0.001$	$-0.80$ $\pm$	$0.265$	$-0.28$ $\pm$	$0.003$	$+0.04$ $\pm$	$0.062^{\dagger}$
ETTh2	$-0.21$ $\pm$	$0.049$	$+0.15$ $\pm$	$0.081$	$-0.14$ $\pm$	$0.004$	$-0.86$ $\pm$	$0.628\hphantom{{}^{\dagger}}$
ETTm1	$-0.68$ $\pm$	$0.001$	$-10.55$ $\pm$	$0.288$	$-0.36$ $\pm$	$0.007$	$-3.15$ $\pm$	$0.546\hphantom{{}^{\dagger}}$
ETTm2	$-0.68$ $\pm$	$0.009$	$-11.53$ $\pm$	$0.217$	$-0.67$ $\pm$	$0.025$	$-10.46$ $\pm$	$0.217\hphantom{{}^{\dagger}}$
Weather	$-7.97$ $\pm$	$0.104$	$-8.61$ $\pm$	$0.464$	$-5.38$ $\pm$	$0.057$	$-5.54$ $\pm$	$0.054\hphantom{{}^{\dagger}}$

Table 3: The static corrector alone and the combination on frozen zero-shot foundation models. MSE change versus base model in percent, mean

\pm

standard deviation over five runs, all origins; the standard deviation carries one more decimal than the mean so that no entry rounds to zero. A dagger marks a change smaller in absolute value than that standard deviation.

Dataset	TAFAS		PETSA		Gate, $K{=}3$
ETTh1	$-1.44$ $\pm$	$1.488$	$-1.34$ $\pm$	$1.350$	$\mathbf{-2.13}$ $\pm$	$0.901\hphantom{{}^{\dagger}}$
ETTh2	$\mathbf{-2.50}$ $\pm$	$2.125$	$-2.08$ $\pm$	$2.077$	$-0.72$ $\pm$	$0.901\hphantom{{}^{\dagger}}$
ETTm1	$-0.76$ $\pm$	$0.703$	$\mathbf{-1.05}$ $\pm$	$0.325$	$-0.30$ $\pm$	$0.718\hphantom{{}^{\dagger}}$
ETTm2	$+7.29$ $\pm$	$4.171$	$-2.09$ $\pm$	$3.383$	$\mathbf{-7.53}$ $\pm$	$3.347\hphantom{{}^{\dagger}}$
Weather	$+0.71$ $\pm$	$1.919$	$-1.45$ $\pm$	$1.458$	$\mathbf{-4.76}$ $\pm$	$1.216\hphantom{{}^{\dagger}}$
ECL	$+1.68$ $\pm$	$0.504$	$+2.88$ $\pm$	$0.530$	$\mathbf{-2.48}$ $\pm$	$0.931\hphantom{{}^{\dagger}}$

Table 4: Accounted comparison on DLinear. MSE change versus base model in percent, computed from the run means of the two MSEs, printed as mean

\pm

standard deviation of the five per-run changes with one more decimal than the mean; update surface and outcome assumption are part of the comparison.

Block set	Method	Mean rank	Mean	Improved	Worst
28 pairs	Frozen base model	3.661	$0.00$	n/a	$+0.00$
$\mathrm{CD}=0.886$	Static	1.893	$-2.24$	28/28	$-0.09$
$p<0.001$	Gate, $K{=}2$	2.661	$-2.43$	21/28	$+0.47$
	Gate, $K{=}3$	1.786	$\mathbf{-3.99}$	25/28	$+0.15$
14 trained	Frozen base model	4.286	$0.00$	n/a	$+0.00$
$\mathrm{CD}=1.630$	Static	2.214	$-2.17$	14/14	$-0.09$

Table 5: Friedman mean ranks with the Nemenyi critical distance (CD), mean MSE change in percent and pairs improved, all origins. Worst is the largest deterioration over the block set, in percent.

Zone	Static		Online		Held-out		Gate, $K{=}2$		Gate, $K{=}3$		$w_{0}$
HU	$-41.08$ $\pm$	$0.19$	$-57.48$ $\pm$	$2.02$	$-55.84$ $\pm$	$2.23$	$-57.48$ $\pm$	$2.02$	$-57.34$ $\pm$	$2.08$	0.000
PT	$-2.84$ $\pm$	$0.29$	$+1.11$ $\pm$	$2.03$	$+2.76$ $\pm$	$2.10$	$-0.13$ $\pm$	$0.24$	$-2.42$ $\pm$	$0.27$	0.217
HR	$+0.09$ $\pm$	$0.52$	$+35.18$ $\pm$	$7.64$	$+38.50$ $\pm$	$7.97$	$+0.00$ $\pm$	$0.00$	$-0.33$ $\pm$	$0.22$	0.494
BE	$+0.52$ $\pm$	$0.29$	$+20.55$ $\pm$	$5.03$	$+23.01$ $\pm$	$5.48$	$-0.02$ $\pm$	$0.04$	$-0.09$ $\pm$	$0.14$	0.549
DK	$+1.00$ $\pm$	$2.11$	$+84.92$ $\pm$	$21.44$	$+101.96$ $\pm$	$25.10$	$+0.00$ $\pm$	$0.00$	$-0.48$ $\pm$	$0.77$	0.637
DE	$-25.10$ $\pm$	$0.47$	$-20.51$ $\pm$	$2.57$	$-16.28$ $\pm$	$2.72$	$-12.78$ $\pm$	$4.38$	$-25.00$ $\pm$	$0.46$	0.003

Table 6: L1, learn and score on the provisional outcome. MSE change versus the TSO forecast in percent, mean

\pm

standard deviation over five runs, all origins. Worst is the largest deterioration over zone-run cells.

	Gate, adaptive			Gate, split			TSO, adaptive
Zone	Cov	Width	Winkler	Cov	Width	Winkler	Cov	Width	Winkler
HU	0.941	0.796	0.922	0.964	0.892	0.971	0.883	0.914	1.138
PT	0.792	0.784	1.449	0.680	0.642	1.841	0.789	0.791	1.467
HR	0.888	0.489	0.695	0.888	0.497	0.699	0.887	0.490	0.696
BE	0.896	0.613	0.789	0.915	0.653	0.795	0.896	0.613	0.790
DK	0.888	0.229	0.367	0.853	0.205	0.383	0.888	0.229	0.367

Table 7: Intervals at the 90% level on the TSO forecast, learn and score on the provisional outcome, five-run mean, all origins. Coverage closer to 0.90 is better; lower width and Winkler are better. Under the two gates the run-to-run standard deviation is at most 0.00383 in every cell except Hungary’s Winkler scores, where it is 0.0128 under the adaptive gate and 0.0109 under split calibration; the TSO forecast and the split-calibrated width carry no run variation.

Zone	Revision (%)	L2		L3, $D{=}0$		L3, $D{=}30$		Scalar	Static, L2
HU	0.59	$-53.23$ $\pm$	$1.84$	$-52.74$ $\pm$	$1.91$	$-46.85$ $\pm$	$0.86$	$+33.36$	$-41.53$ $\pm$	$0.20$
PT	2.22	$-0.10$ $\pm$	$0.21$	$-9.30$ $\pm$	$0.14$	$-9.24$ $\pm$	$0.16$	$-12.69$	$-0.22$ $\pm$	$0.24$
HR	2.60	$-0.63$ $\pm$	$1.17$	$-15.59$ $\pm$	$0.45$	$-15.67$ $\pm$	$0.33$	$-4.35$	$-0.82$ $\pm$	$2.47$
BE	3.18	$+0.16$ $\pm$	$0.40$	$-9.37$ $\pm$	$4.07$	$-6.50$ $\pm$	$1.01$	$-9.03$	$+0.44$ $\pm$	$0.82$
DK	3.87	$+0.43$ $\pm$	$0.49$	$-12.18$ $\pm$	$0.69$	$-11.16$ $\pm$	$0.49$	$+66.43$	$+1.78$ $\pm$	$1.03$
DE	5.93	$-25.04$ $\pm$	$0.62$	$-53.05$ $\pm$	$3.47$	$-43.71$ $\pm$	$4.84$	$-38.71$	$-25.13$ $\pm$	$0.62$

Table 8: Scored on the settled outcome, mean

\pm

standard deviation over five runs. Revision is the mean absolute relative difference between the two outcome versions. Every column from L2 onward is the MSE change versus the TSO forecast in percent: L2 and the two L3 columns are the gated combination under the corresponding outcome protocol, Scalar is a deterministic post-hoc per-hour-of-day ratio with no run replication, and Static, L2 is the static corrector.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Term	Meaning
Frozen	A forecaster its user cannot retrain; its parameters are never updated in this study
Base model	The frozen forecaster the correction layer wraps, written $E_{0}$ ; on the load data it is the forecast of the transmission system operator (TSO)
Corrector	A module that alters the base model’s forecast: the static corrector $E_{1}$ (a trust-region adapter fitted once on the training split), the online corrector $E_{2}$ (updated from matured errors during the stream) and, outside the library, the held-out corrector and the intercept corrections (running means of matured errors, plain or exponentially weighted)
Expert	A forecast the gate holds weight on: the base model $E_{0}$ , the static corrector $E_{1}$ and the online corrector $E_{2}$ ; called a corrector ( $E_{1}$ , $E_{2}$ ) when its mechanism rather than its weight is discussed
Gate, Gate $K{=}n$	The rule that sets the weight on each expert from the losses that expert has already accrued; $K$ is the number of experts, $K{=}3$ the reported configuration
Combination, gated combination	The single forecast the gate’s weights produce; the row labelled Gate in every table

Appendix

Table S1: Terms used throughout, with the symbol or the count each one names.

Dataset	Channels	Train	Val	Test	Freq.
ETTh1/h2	7	8,640	2,880	2,880	Hourly
ETTm1/m2	7	34,560	11,520	11,520	15-min
Weather	21	36,887	5,270	10,539	10-min
ECL	321	18,412	2,632	5,260	Hourly
Exchange	8	5,311	760	1,517	Daily
ILI	7	966 rows, Section S10 of the supplement			Weekly

Appendix

Table S2: Dataset statistics and split sizes as used, after windowing at

L{=}384

H{=}96

Item	Setting
Form	$y=\hat{y}+\delta\cdot A(\hat{y},x)$
Trust region	$\lVert A\rVert_{\infty}\leq 1$ via tanh
Radius $\delta$	$0.01$ on ETT, $0.1$ otherwise
Optimizer	Adam, learning rate $10^{-4}$
Network	Depth 2, width 128
Fitted on	Training split, base model frozen

Appendix

Table S3: Settings of the static expert.

Method	Fitted on	Updates in the stream	In the library
Frozen base model	Not fitted here	No	$E_{0}$
Static corrector	Training split	No	$E_{1}$
Online corrector	Held-out fit region	Yes	$E_{2}$
Held-out corrector	Held-out fit region	No	No
Gate, $K{=}2$	Carries $E_{0}$ and $E_{2}$	Weights only
Gate, $K{=}3$	Carries $E_{0}$ , $E_{1}$ and $E_{2}$	Weights only

Appendix

Table S4: The methods compared in Section 5 of the main text, what each is fitted on, and whether it is an expert in the gate’s library. The documentation accompanying the code maps each name to its identifier in the result files.

Dataset	Ljung–Box $p$ (lag 24), max	Lag-1 $\rho$	Ridge $R^{2}$
ETTh1	$3.9\times 10^{-5}$	0.089	$-0.744$
ETTh2	$3.1\times 10^{-4}$	0.108	$-0.815$
ETTm1	$5.5\times 10^{-12}$	0.225	$-0.366$
ETTm2	$6.3\times 10^{-6}$	0.109	$-0.575$
Weather	$8.2\times 10^{-3}$	0.675	$-0.149$

Appendix

Table S5: Residual predictability of a frozen DLinear base model on the held-out fit region. The Ljung–Box column is the maximum

p

over channels and five runs; lag-one

\rho

is the mean over channels and runs; ridge

R^{2}

is the channel median averaged over runs. Ljung–Box rejects whiteness everywhere; ridge

R^{2}

from per-sample statistics is negative everywhere.

Dataset	$z{=}0$ MSE	Ceiling, 50 steps	Per-run ceiling, 2,000 steps					Mean	SD	$\lVert z\rVert$ , 2,000 steps
ETTh1	0.6080	$3.76\pm 1.03$ %	26.55%	20.49%	19.50%	23.92%	20.76%	22.24%	2.92	$79.939$ $\pm$	$4.571$
ETTh2	0.2998	$0.63\pm 0.23$ %	4.32%	6.58%	2.66%	6.48%	6.94%	5.40%	1.84	$104.867$ $\pm$	$12.760$
ETTm1	0.3272	$0.14\pm 0.19$ %	0.95%	8.32%	5.93%	0.42%	0.07%	3.14%	3.75	$223.934$ $\pm$	$36.245$
ETTm2	0.1055	$0.01\pm 0.02$ %	0.01%	0.01%	0.16%	1.17%	1.97%	0.66%	0.88	$198.110$ $\pm$	$53.313$
Weather	0.2715	$0.80\pm 0.76$ %	8.48%	8.99%	14.45%	1.56%	0.90%	6.88%	5.67	$175.624$ $\pm$	$77.876$

Appendix

Table S6: Conditioning ceiling from per-sample oracle probing on the last third of the held-out fit region (737 origins on ETTh1 and ETTh2, 3,329 on ETTm1 and ETTm2, 1,454 on Weather), with the corrector fitted on the first two thirds (1,327, 5,992 and 2,616 training origins plus an early-stopping tail). Base MSE at

z{=}0

on the probe region; ceiling at 50 steps, mean

\pm

standard deviation over five runs. Extended-budget ceiling at 2,000 steps run by run with the five-run mean and standard deviation, and

\lVert z\rVert

at 2,000 steps, mean

\pm

standard deviation. The 5% threshold was fixed before running.

Method	Mean	Worst pair	Worst cell	Improved
Static corrector	$-2.24$	$-0.09$	$+0.08$	28/28
Equal weights	$-1.54$	$+9.06$	$+11.74$	24/28
Intercept correction	$+18.02$	$+33.08$	$+33.19$	0/28
Exponentially weighted errors	$+13.07$	$+22.18$	$+28.26$	0/28
Gate, $K{=}2$	$-2.43$	$+0.47$	$+0.64$	21/28
Gate, $K{=}3$	$\mathbf{-3.99}$	$+0.15$	$+0.46$	25/28

Appendix

Table S7: Classical weighting against the static corrector and the gate, MSE change versus the frozen base model in percent; pair and zone entries are five-run means. A, equal weights and intercept corrections on the 28 pairs, all origins, basis of Table 2 of the main text; worst cell is the maximum over the 140 pair-run cells. B, fixed and error-driven weight rules on the 28 benchmark pairs and the seven load zones, scored on matured origins, unlike Tables 2, 3, 5 and 6 of the main text.

Zone	Static		Equal		Intercept	EW errors	Gate, $K{=}3$
HU	$-41.08$ $\pm$	$0.19$	$-42.77$ $\pm$	$1.53$	$-71.11$	$-70.21$	$-57.34$ $\pm$	$2.08$
DK	$+1.00$ $\pm$	$2.11$	$+6.04$ $\pm$	$3.55$	$+23.37$	$+1{,}440.94$	$-0.48$ $\pm$	$0.77$
HR	$+0.09$ $\pm$	$0.52$	$+4.04$ $\pm$	$2.27$	$-2.73$	$+0.45$	$-0.33$ $\pm$	$0.22$
BE	$+0.52$ $\pm$	$0.29$	$+0.91$ $\pm$	$0.69$	$-4.36$	$+14.25$	$-0.09$ $\pm$	$0.14$
DE	$-25.10$ $\pm$	$0.47$	$-19.81$ $\pm$	$0.56$	$-48.88$	$-30.77$	$-25.00$ $\pm$	$0.46$
PT	$-2.84$ $\pm$	$0.29$	$-3.49$ $\pm$	$0.63$	$-14.98$	$+8.47$	$-2.42$ $\pm$	$0.27$

Appendix

Table S8: Equal weights and intercept corrections on the TSO forecast under L1, basis of Table 6 of the main text. Mean

\pm

standard deviation over five runs; the intercept corrections have no random source.

Dataset	Chronos-Bolt	TimesFM
ETTh1	$-0.67\pm 0.269$	$+0.32\pm 0.117$
ETTh2	$+0.47\pm 0.099$	$-0.87\pm 0.680$
ETTm1	$-10.77\pm 0.292$	$-3.10\pm 0.188$
ETTm2	$-11.53\pm 0.217$	$-10.46\pm 0.216$
Weather	$-5.33\pm 1.499$	$-3.02\pm 0.280$
ECL	$-0.77\pm 0.115$	$-1.57\pm 0.045$

Appendix

Table S9: The two-expert gate on the 14 foundation-model pairs. MSE change versus base model in percent, mean

\pm

standard deviation over five runs, all origins, on the basis of Table 3 of the main text; the standard deviation carries one more decimal than the mean.

Dataset	DLinear	PatchTST	Chronos-Bolt	TimesFM
ETTh1	Online (0.621)	Static (0.467)	Online (0.443)	Static (0.485)
ETTh2	Online (0.689)	Static (0.548)	Static (0.484)	Static (0.419)
ETTm1	Static (0.623)	Static (0.435)	Online (0.958)	Online (0.470)
ETTm2	Online (0.986)	Online (0.962)	Online (0.964)	Online (0.998)
Weather	Static (0.898)	Static (0.759)	Static (0.741)	Static (0.629)
ECL	Static (0.956)	Static (0.915)	Static (0.880)	Static (0.841)

Appendix

Table S10: Expert receiving the largest mean gate weight, with that weight, five-run mean. On four cells the two correctors sit close enough that which of them receives the largest weight changes with the run; the result files flag those cells.

Base model	Dataset	$m$	Frozen		Static		Gate, $K{=}3$		Frozen		Static		Gate, $K{=}3$
			MASE						RMSSE
DLinear	ETTh1	24	$0.999$ $\pm$	$0.013$	$0.994$ $\pm$	$0.012$	$0.977$ $\pm$	$0.008$	$0.932$ $\pm$	$0.006$	$0.930$ $\pm$	$0.006$	$0.922$ $\pm$	$0.003$
	ETTh2	24	$1.123$ $\pm$	$0.006$	$1.122$ $\pm$	$0.006$	$1.089$ $\pm$	$0.009$	$0.898$ $\pm$	$0.006$	$0.898$ $\pm$	$0.007$	$0.895$ $\pm$	$0.008$
	ETTm1	96	$0.845$ $\pm$	$0.006$	$0.841$ $\pm$	$0.006$	$0.835$ $\pm$	$0.005$	$0.820$ $\pm$	$0.003$	$0.818$ $\pm$	$0.003$	$0.819$ $\pm$	$0.001$
	ETTm2	96	$0.868$ $\pm$	$0.025$	$0.862$ $\pm$	$0.024$	$0.804$ $\pm$	$0.004$	$0.706$ $\pm$	$0.013$	$0.703$ $\pm$	$0.013$	$0.679$ $\pm$	$0.002$
	Weather	144	$0.594$ $\pm$	$0.012$	$0.558$ $\pm$	$0.004$	$0.560$ $\pm$	$0.005$	$0.471$ $\pm$	$0.002$	$0.459$ $\pm$	$0.002$	$0.460$ $\pm$	$0.002$

Appendix

Table S11: MASE and RMSSE against a seasonal naive forecaster of period

m

, mean

\pm

standard deviation over five runs. MASE is the mean of the five per-run values. RMSSE is the root of the run-mean MSE. The standard deviation printed beside it is that of the five per-run roots, whose mean sits below the printed RMSSE by at most 0.013 on Exchange and by less than the last digit elsewhere.

Dataset	Base model	MSE	Static		Held-out		Gate, $K{=}3$
ETTh1	Seasonal naive	0.5122	$-0.74$ $\pm$	$0.001$	$-20.10$ $\pm$	$0.172$	$\mathbf{-21.40}$ $\pm$	$0.257$
	Smoothing	1.1014	$-0.96$ $\pm$	$0.001$	$-59.33$ $\pm$	$0.745$	$\mathbf{-60.04}$ $\pm$	$0.709$
ETTh2	Seasonal naive	0.3905	$-0.77$ $\pm$	$0.010$	$-22.68$ $\pm$	$2.043$	$\mathbf{-23.06}$ $\pm$	$2.086$
	Smoothing	0.4059	$-1.10$ $\pm$	$0.004$	$-27.85$ $\pm$	$0.447$	$\mathbf{-28.51}$ $\pm$	$0.388$
Weather	Seasonal naive	0.3167	$-9.76$ $\pm$	$0.063$	$-35.73$ $\pm$	$0.462$	$\mathbf{-37.89}$ $\pm$	$0.220$
	Smoothing	0.3380	$-13.41$ $\pm$	$0.066$	$-42.90$ $\pm$	$2.488$	$\mathbf{-44.30}$ $\pm$	$2.489$

Appendix

Table S12: Frozen statistical base models, mean

\pm

standard deviation over five runs, the standard deviation at one more decimal so that no entry rounds to zero. Change is the MSE change in percent against the base model in the same row; the static corrector, the held-out corrector and the combination are each run alone.

Constant	Grid point	MSE change (%)
Trust-region radius $\delta$	$0.5\times$	$+0.19$ $\pm$	$0.076$
	$1\times$ (reported)	$+0.15$ $\pm$	$0.081$
	$2\times$	$+0.04$ $\pm$	$0.119$
Warm slice length	$H+100$	$-0.08$ $\pm$	$0.026$
	$H+200$ (reported)	$+0.15$ $\pm$	$0.081$
	$H+400$	$+0.15$ $\pm$	$0.054$

Appendix

Table S13: Sensitivity of the gate, mean

\pm

standard deviation over five runs. A, one-at-a-time perturbation of the layout constants on ETTh2 with Chronos-Bolt, gate

K{=}3

. B, early-stopping tail across the foundation-model pairs, gate

K{=}3

. A and B give the MSE change versus the frozen base model in percent. C, gate learning rate, MSE.

	Native quantiles			Adaptive tracker
Dataset	Cov	Width	Pinball	Cov	Width	Pinball
ETTh1	0.759	1.066	0.152	0.817	1.226	0.159
ETTh2	0.739	0.833	0.132	0.764	0.938	0.137
ETTm1	0.746	0.902	0.137	0.797	1.021	0.141
ETTm2	0.757	0.669	0.100	0.775	0.808	0.105
Weather	0.741	0.576	0.088	0.776	0.693	0.090

Appendix

Table S14: Intervals at a matched 80% level on frozen Chronos-Bolt, with the mean pinball loss over the nine deciles. Coverage closer to 0.80 and lower pinball are better. Neither method carries a random component on a frozen zero-shot forecaster, so the five runs coincide and no dispersion is reported.

Zone	Gate, adaptive		Gate, split		TSO, adaptive
HU	$\mathbf{0.06316}$ $\pm$	$0.001486$	0.06981 $\pm$	$0.001036$	0.09910
PT	$\mathbf{0.09742}$ $\pm$	$0.000127$	0.10313 $\pm$	$0.000152$	0.09871
HR	$\mathbf{0.04708}$ $\pm$	$0.000068$	0.04711 $\pm$	$0.000066$	0.04716
BE	$\mathbf{0.05774}$ $\pm$	$0.000035$	0.05794 $\pm$	$0.000042$	0.05777
DK	$\mathbf{0.02282}$ $\pm$	$0.000108$	0.02308 $\pm$	$0.000111$	0.02286
DE	$\mathbf{0.07294}$ $\pm$	$0.000220$	0.07361 $\pm$	$0.000247$	0.08553

Appendix

Table S15: Mean pinball loss over the nine deciles, same three methods and basis as Table 7 of the main text. Lower is better. Gate columns print mean

\pm

standard deviation over five runs; the TSO forecast and its split-calibrated width carry no run variation.

Zone	Verdict	Miss prov.	Miss settled	Defect h
DE, HU, PT, HR, DK, IT, BE	Accept	$\leq 0.0001$	$\leq 0.0001$	0
NL	Defect	0.0000	0.0001	96
CH	Defect	0.0000	0.0000	20
ES, SK, FR, RO, FI, LV, CZ, ME	Gap	$\leq 0.0062$	$\leq 0.0000$	$\leq 10$
AT, SI, PL, GR, NO, SE	Missing settled	$\leq 0.0041$	$\geq 0.0209$
GB_GBN, DK_1	Missing settled	0.0000	$\geq 0.7574$

Appendix

Table S16: Accepted zones and the cause of each rejection. Missing fractions are over the common grid; defect hours are forecast hours above three times load.

	Held-out	Warm slice	Tail	Fit region
$D{=}0$ , six zones	297	201	30	66
$D{=}0$ , Belgium	295	201	30	64
$D{=}30$ , six zones	297	231	30	36
$D{=}30$ , Belgium	295	231	30	34

Appendix

Table S17: Held-out layout. A, load streams in daily origins: the warm slice is

1+D+200

origins and the early-stopping tail 30 origins, the fit region the remainder; Belgium’s held-out split is two origins shorter than the other six zones’. At

D{=}90

the slice alone would take 291 origins, so that configuration is refused (Section 7.2.2 of the main text). B, benchmarks by horizon: held-out split and the fit region remaining after the warm slice and the tail.

Zone	Mean settled/prov.	Revision (%)	MAPE vs prov.	zMSE vs prov.	zMSE vs settled
DE	1.0573	5.930	3.104	0.0508	0.2023
HU	0.9942	0.592	3.846	0.0870	0.0722
PT	1.0003	2.225	2.773	0.0497	0.0990
HR	1.0235	2.598	2.088	0.0226	0.0795
DK	1.0352	3.870	1.017	0.0089	0.0608
IT	1.0940	9.401	1.973	0.0135	0.1576

Appendix

Table S18: Outcome versions and forecast quality per accepted zone, whole window. Revision is the mean absolute relative difference between the two outcome versions, the column Table 8 of the main text also carries; MAPE vs prov. is the forecast’s own mean absolute percentage error against that version and zMSE is the mean squared error on per-channel z-scaled series. The entries describe the data and not a fitted model, so they carry no run dispersion.

Split, $L$ , $H$	Slice	Updates	Warm		Uniform		Uniform $-$ warm
6:2:2, 36, 24	77	53	$-6.47$ $\pm$	$1.845$	$-6.85$ $\pm$	$2.044$	$-0.38$ $\pm$	$0.214$
6:2:2, 36, 36	71	35	$-14.32$ $\pm$	$0.700$	$-15.82$ $\pm$	$0.696$	$-1.50$ $\pm$	$0.140$
6:2:2, 36, 48	66	18	$-18.98$ $\pm$	$0.965$	$-20.73$ $\pm$	$0.980$	$-1.75$ $\pm$	$0.241$
6:2:2, 104, 24	77	53	$-8.28$ $\pm$	$0.943$	$-9.07$ $\pm$	$1.022$	$-0.79$ $\pm$	$0.183$
7:1:2, 36, 24	33	9	$-1.00$ $\pm$	$0.775$	$-1.00$ $\pm$	$0.764$	$-0.00$ $\pm$	$0.015$

Appendix

Table S19: Short-series warm start, five runs; the three MSE columns print mean

\pm

standard deviation. The last column is the uniform column minus the warm column, negative meaning the warm start is worse.

Horizon 192
Dataset	Base model	Held-out	Gate, $K{=}3$
Exchange	DLinear	$+90.12\pm 18.635$	$+23.03\pm 14.989$
ETTh2	DLinear	$+3.24\pm 2.833$	$+1.01\pm 1.616$	$\dagger$
ETTh2	PatchTST	$+2.23\pm 2.746$	$+0.11\pm 1.543$	$\dagger$
ETTh1	PatchTST	$+9.60\pm 2.087$	$+0.02\pm 0.281$	$\dagger$
ETTm1	DLinear	$+1.95\pm 0.984$	$-0.50\pm 0.758$	$\dagger$

Appendix

Table S20: Combination against frozen base model at horizons 192 and 336, MSE change in percent, five-run mean with the standard deviation of the five per-run changes after

\pm

. Negative is better. A dagger marks a gate change smaller in absolute value than that standard deviation.

Expert learning rate	Expert alone	Four-expert gate
0.005 (published default)	$-0.11$	$+1.22$
0.0005	$-0.83$	$-0.40$
0.00005	$-0.09$	$-0.63$

Appendix

Table S21: Expert speed and the fourth expert, MSE change versus the frozen base model in percent. A, post-hoc diagnosis on ETTh2 with DLinear: slowing the added expert improves that expert on its own and restores the gate’s downside control at the same time. B, the intercept correction as a fourth expert, five-run means: worst pair over the 28 benchmark pairs, worst zone and mean over the seven zones; bases of Tables 2 and 6 of the main text. A dash marks a configuration that was not run.

Rule	Mean	vs gate	Better	Worst
A. Benchmarks, 36 pairs
Hedge (the gate)	$-13.44$	–	–	$-0.17$
Fixed-share, $\alpha=1/T_{m}$	$-13.49$	$-0.05$	19/36	$-0.24$
Fixed-share, $\alpha=0.01$	$-12.02$	$+1.41$	6/36	$-0.16$
Bernstein aggregation	$-13.29$	$+0.15$	17/36	$-0.00$
B. Load, 21 zone-protocol cells

Appendix

Table S22: Aggregation rules replayed over the same experts, MSE change versus the frozen base model in percent at the time-averaged weight. A, the 36 benchmark pairs at horizon 96, the 28 main-grid pairs and the 8 classical-base-model pairs of Table S12 ; B, the 21 load zone-protocol cells. Pair and cell entries are five-run means; vs gate is the mean gap to the replayed Hedge gate in percentage points; better counts pairs or cells below the gate.

Block set	Pair-block $p$	Dataset-block $p$	Gate rank	CD	Separations
All base models	$3.1\times 10^{-8}$	$0.008$	1.571	1.773	gate vs base model
Trained base models	$0.001$	$0.038$	1.857	2.306	gate vs base model
DLinear only	$0.246$	$0.246$	3.143	3.405	none

Appendix

Table S23: Friedman test with seven dataset blocks: rank of the three-expert gate, Friedman

p

, Nemenyi critical distance (CD) at the 5% level, and the separations that remain, beside the pair-block values of Table 5.

Explore similar work

Aug 5, 2026cs.LG

When Do Corrective Features Help? An Agent for Corrective Feature Discovery on Black-Box Forecasters

Frozen pretrained forecasters often fail in structured, recurring ways that are costly to repair through fine-tuning. We study corrective feature discovery: mining interpretable features of a frozen forecaster's residual to drive a lightweight post-hoc corrector. Prior automated feature engineering models the data-generating process; corrective features instead model the model-failure process. We present CRAFTER (Corrective Residual Agent with Feature-based Temporal Exploration and Reasoning), which keeps the backbone frozen and mines its residual with two complementary generators: a compositional search over the raw input channels, and a large language model (LLM) that proposes named feature combinations, binary flags, and short executable code. A single validation-grounded gate accepts or rejects every candidate regardless of its origin, and a validation-selected corrector applies the accepted features or leaves the forecast unchanged. This source-agnostic pipeline also allows prior feature-engineering systems to be evaluated under identical conditions, making CRAFTER an instrument for attributing forecast improvements to the feature source alone. Across six public datasets and six frozen backbones, CRAFTER surpasses every dedicated feature-engineering system at every feature budget, roughly doubling the improvement achieved by the corrector alone and reducing the error of the weakest backbones by up to 27%. These gains are robust across different LLM backends and persist even when applied on top of fine-tuned backbones.

Fangxin Wang, Ziyi Zhang, Diyi Zhuang +4

University of Illinois Chicago · Texas A&M University · Massachusetts Institute of Technology +1

Feature Engineering

Ask PDF

Jun 23, 2026cs.LG

Adapt Only When It Pays: Budgeted Decision-Loss Priority for Delayed Online Time-Series Adaptation

Online time-series forecasters receive labels only after horizon-dependent delays, while every adaptation step spends limited compute. We study when an online learner should update, not how to adapt at every opportunity, and introduce ADOWIP: a residual-adapter framework with sealed delay queues, exact budget accounting, and auditable update telemetry. Its main scheduler is an observed decision-loss priority gate that updates only after feedback is revealed, when downstream loss, optionally penalized by prediction MSE, exceeds a calibrated empirical quantile and budget remains. We prove hard-budget feasibility, projected-OGD regret for a convex linear accepted-update subproblem, and stability plus conditional finite-sample gate-selection statements. On public ETT capacity-planning tasks, a frozen calibration/evaluation split selects a gate that lowers held-out decision loss against always, fixed-period, and drift-triggered exact-update baselines under matched compute. Secondary threshold/load-index ETT suites are mixed: 33 of 41 selected contrasts clear the stricter cross-artifact Holm family, and the 8 nonpassing rows are explicitly excluded from primary claims. The same protocol improves an external UCI Bike capacity proxy with 20/0 held-out wins, and a fixed gate passes three full-year Capital Bikeshare station-rebalancing contrasts. Probe-based and finance experiments remain negative, delimiting the current scope of decision-prioritized adaptation.

Xibai Wang

NeuroQuant Labs Limited Hong Kong SAR, China

Time Series Forecasting

Ask PDF

Jul 22, 2026cs.LG

Expert-Guided Forecast Editing for Time-Series Foundation Models

Time-series foundation models can forecast across heterogeneous domains without task-specific training, but their forecasts are fixed once produced and cannot directly incorporate task-specific expert feedback. We study expert-guided forecast editing: a frozen foundation model generates candidate future trajectories, and an expensive expert evaluator scores them to guide forecast revision. Under a tight query budget, two natural strategies sit at opposite ends: best-of-

N

purely exploits the foundation model's predictive distribution, while optimization approaches mostly explore the forecast horizon as an unstructured high-dimensional vector. Each extreme is individually sub-optimal. We introduce \textbf{DEFT}, an expert-guided forecast editing framework that balances the two by first exploiting the foundation model's predictive samples in a decomposed trend--seasonal space, then exploring around them via component-wise refinement. DEFT queries the expert only on complete trajectories, then reuses scores for the trend and seasonal components that appeared in the queried recombinations. This lets each expert query provide structured component-level feedback while keeping the foundation model frozen. We compare DEFT against direct search approaches, including best-of-

N

, cross-entropy methods, and Bayesian optimization, under matched expert-query budgets. Across two forecasting benchmarks consisting of 78 datasets, three time-series foundation models, four feedback types, and seven query budgets, DEFT consistently improves the effectiveness of expert guidance. A molecular-dynamics case study further suggests that the same principle extends to more physically grounded feedback, supporting the hypothesis that sparse test-time guidance should be spent balancing prior exploitation with structured exploration.

Hung Le, Minh Hoang Nguyen, Manh Nguyen +2

Deakin Applied Artifical Intelligence Initiative Deakin University, Australia

Time Series Foundation Models Data-Driven Forecasting Bayesian Optimization

Ask PDF