Compose the Horizon: Parameter-Shared Blockwise Forecasting under a Sequential-Compute Trade-off

Zeyan Li; Weiheng Li; Shengda Zhuo; Libing Chen; Min Chen; Yin Tang

cs.LGOct 17, 2026

Compose the Horizon: Parameter-Shared Blockwise Forecasting under a Sequential-Compute Trade-off

Authors: Zeyan Li, Weiheng Li, Shengda Zhuo, Libing Chen, Min Chen, Yin Tang

Organizations: Shanghai Jiao Tong University Shanghai, China · Jinan University Guangzhou, China · The University of Chicago Chicago, IL, USA · South China University of Technology Guangzhou, China

Abstract

Most long-horizon forecasters emit the complete future in one pass. We ask whether a compact operator can instead be reused to emit contiguous future blocks, trading stored parameters for sequential computation. Recursive Functional Decomposition (RFD) factorizes the horizon mapping into three parameter-shared block predictions. The blocks occupy disjoint future intervals and are concatenated, not added as trend, seasonal, or residual components, and each emitted block joins the rolling context that conditions the next application. In controlled three-seed experiments, removing forward feedback hurts all four evaluated datasets while stopping gradients through it changes little, so the useful signal is the evolving forecast state itself; reapplying the shared operator helps consistently, but by less than widening it. The model uses 0.08--0.46M parameters, yet its sequential channel-wise execution is slower and more memory-intensive than an iterative baseline at high channel count. In a frozen long-context evaluation against ten strong baseline families, RFD records lower test error in 23 of 24 cells; the comparison is cross-protocol, with RFD using 512/720-step context and the baselines using 96.

Figures & tables

Figure 1. One-shot widening versus parameter-shared blockwise rollout. Direct prediction grows capacity through a wider single pass; RFD reuses one compact gated operator over $K{=}3$ stages, appending each emitted block to the rolling context and concatenating the blocks into the forecast. Two side-by-side panels. Left: direct prediction maps the history through stacked wide Transformer blocks to the full horizon in one pass, with capacity growing through width. Right: RFD applies a small shared gated-Transformer operator three times; each application emits an H/3-step block, each block is appended to a rolling context that conditions the next stage, and the blocks are concatenated and re-anchored into the forecast within 0.08–0.46M shared parameters.

Figure 2. Traffic rollout (

L{=}512

H{=}192

); each stage appends one 64-step block. Four plots show one Traffic example. The first compares the complete blockwise forecast with the ground truth; the next three show the forecast growing from one to three contiguous blocks as the rolling context advances.

Figure 3. The RFD construction. The normalized history is right-aligned into 48-step patches; one shared recursive operator (shared linear head and one gated pre-LN block, identity at initialization) is applied $K{=}3$ times, each application emitting an $H/3$ -step block that re-enters the rolling context; the blocks are concatenated and re-anchored, and training minimizes the weighted prefix TFMAE. Pipeline diagram in three sections. Left: the normalized history is patched into 48-step segments and linearly embedded with positions into per-channel patch tokens. Middle: a shared recursive operator applied at stages k = 1 to 3, containing a shared linear head and one gated pre-LayerNorm Transformer block with four-head attention and a feed-forward network, initialized as the identity, emitting one horizon block of H/3 steps per stage. Right: the three blocks are concatenated along time into the normalized forecast, re-anchored with the normalization statistics, and the parameter footprint compares RFD's 0.08–0.46M parameters with ReNF's 16.4M on ETTm1.

Dataset	RFD		WPMixer		TimeMixer		PatchTST		iTransformer		DLinear		FEDformer		SparseTSF		FITS		TimeXer		TQNet
Dataset	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.122	0.268	0.078	0.214	0.074	0.209	0.081	0.220	0.077	0.213	0.109	0.252	0.104	0.250	0.091	0.235	0.088	0.230	0.077	0.216	0.076	0.211
ETTh2	0.190	0.341	0.192	0.346	0.202	0.353	0.201	0.353	0.195	0.349	0.219	0.364	0.204	0.350	0.260	0.408	0.235	0.383	0.204	0.354	0.190	0.342
ETTm1	0.054	0.172	0.053	0.173	0.053	0.172	0.054	0.174	0.056	0.177	0.066	0.188	0.064	0.196	0.055	0.179	0.053	0.174	0.056	0.176	0.053	0.172
ETTm2	0.120	0.255	0.122	0.257	0.123	0.260	0.123	0.261	0.133	0.270	0.125	0.261	0.128	0.269	0.136	0.282	0.121	0.260	0.124	0.261	0.121	0.256
Weather	0.647	0.569	0.657	0.574	0.648	0.572	0.656	0.574	0.624	0.558	0.714	0.590	0.843	0.704	0.695	0.605	0.653	0.580	0.606	0.551	0.650	0.571

Table 1. Univariate test MSE/MAE, averaged over four horizons at

L{=}96

Dataset	RFD		WPMixer		TimeMixer		PatchTST		iTransformer		DLinear		FEDformer		SparseTSF		FITS		TimeXer		TQNet
Dataset	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.455	0.448	0.442	0.434	0.463	0.449	0.464	0.458	0.463	0.451	0.457	0.454	0.443	0.462	0.712	0.572	0.590	0.518	0.459	0.451	0.463	0.443
ETTh2	0.371	0.391	0.388	0.412	0.378	0.404	0.392	0.414	0.393	0.411	0.556	0.514	0.443	0.455	0.442	0.447	0.419	0.428	0.373	0.400	0.375	0.401
ETTm1	0.382	0.388	0.385	0.395	0.392	0.402	0.393	0.407	0.426	0.419	0.403	0.407	0.441	0.455	0.463	0.438	0.416	0.409	0.398	0.407	0.397	0.402
ETTm2	0.274	0.317	0.278	0.323	0.279	0.326	0.289	0.335	0.290	0.333	0.352	0.400	0.307	0.352	0.303	0.341	0.288	0.329	0.283	0.327	0.284	0.327
Weather	0.319	0.330	0.316	0.331	0.318	0.333	0.326	0.336	0.327	0.336	0.332	0.373	0.414	0.442	0.367	0.370	0.348	0.353	0.311	0.330	0.319	0.333

Table 2. Multivariate test MSE/MAE, averaged over four horizons at

L{=}96

		$H{=}96$	$H{=}192$	$H{=}336$	$H{=}720$
ETTh1	RFD	.3737	.4125	.4472	.4613
	best baseline	.3749 (WPMixer)	.4209 (FEDformer)	.4666 (WPMixer)	.4968 (WPMixer)
ETTh2	RFD	.2815	.3408	.3635	.4021
	best baseline	.2912 (TimeMixer)	.3729 (TQNet)	.4155 (TimeXer)	.4020 (TimeXer)
ETTm1	RFD	.2871	.3208	.3602	.4179
	best baseline	.3172 (WPMixer)	.3648 (WPMixer)	.3980 (WPMixer)	.4597 (WPMixer)

Table 3. Long-context test MSE against the strongest baseline per cell from Tables 1 – 2 .

	Stage factorization ( $d_{\mathrm{ff}}{=}128$ )					Width ( $K{=}3$ )
Dataset	$K{=}1$	$K{=}2$	$K{=}3$	$K{=}4$	$K{=}6$	$d_{\mathrm{ff}}{=}16$	$32$	$64$	$128$
ETTm1	.3816	.3817	.3814	.3801	.3817	.3900	.3860	.3832	.3814
ETTm2	.1208	.1217	.1216	.1218	.1210	.1234	.1229	.1225	.1216
ETTh2	.2109	.2129	.2134	.2146	.2146	.2147	.2133	.2127	.2134
ETTh1	.6813	.6734	.6702	.6673	.6664	.7482	.6962	.6816	.6702
params (k)	164	152	148	146	144	5	13	42	148

Table 4. RQ2 stage-factorization and width sweeps on the four ETT datasets (three-seed validation MSE).

Figure 4. RQ2 sensitivity across three seeds; error bars show sample standard deviation. Eight line plots in a two-by-four layout. The top row varies stage count and the bottom row varies width for ETTh1, ETTh2, ETTm1, and ETTm2; points show three-seed mean validation MSE with standard-deviation error bars.

Dataset	MSE	MSE	$\Delta\%$	MSE	$\Delta\%$	MSE	$\Delta\%$
	A0	A1 (gate open)		A2 (no feedback)		A3 (plain MSE)
ETTm1	.3812	.3810	0.0	.9125	$+139.4$	.3918	$+2.8$
ETTm2	.1216	.1203	$-1.1$	.1552	$+27.6$	.1269	$+4.3$
ETTh2	.2134	.2137	$+0.1$	.2523	$+18.2$	.2144	$+0.5$
ETTh1	.6699	.6669	$-0.4$	1.2057	$+80.0$	.6631	$-1.0$

Table 5. RQ3 bundled stress tests on the four ETT datasets; A0 is the full RFD model (three-seed validation MSE).

Figure 5. RQ3 bundled stress tests; error bars show sample standard deviation. Four bar charts for ETTh1, ETTh2, ETTm1, and ETTm2. Each compares the full model, open-gate initialization, unconditioned no-feedback, and final MSE using three-seed mean validation MSE and standard-deviation error bars.

$C$	Model	params (k)	mem (MiB)	thr (smp/s)
$7$	RFD ( $d_{\mathrm{ff}}{=}16$ )	4.8	81.3	4788
	RFD ( $d_{\mathrm{ff}}{=}32$ )	12.9	82.6	4725
	RFD ( $d_{\mathrm{ff}}{=}64$ )	41.5	85.4	4827
	RFD ( $d_{\mathrm{ff}}{=}128$ )	147.9	91.7	4798
	ReNF (beta)	1101.8	86.4	10310
$862$	RFD ( $d_{\mathrm{ff}}{=}16$ )	86.8	2067.6	548

Table 6. RQ4 parameter and compute profile (

L{=}H{=}96

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	$L$	$d_{\mathrm{ff}}$ ( $d$ )	dropout	pe	$\alpha$	$\gamma$	batch	lr	budget
ETTh1	512	2048 (128)	0.7	on	0.00	0.3	128	$6{\times}10^{-4}$	10 (3)
ETTh2	512	128	0.1	off	0.00	0.3	16	$5{\times}10^{-4}$	5 (3)
ETTm1	512	2048 (128)	0.5	on	0.20	0.9	64	$5{\times}10^{-4}$	5 (2)
ETTm2	512	128	0.3	on	0.45	0.9	64	$2{\times}10^{-3}$	5 (2)
Weather	720	64	0.8	on	0.50	0.1	64	$4{\times}10^{-3}$	5 (10)
Traffic	512	2048 (128)	0.5	off	0.20	0.9	16	$8{\times}10^{-4}$	40 (15)

Appendix

Table 7. Frozen RFD recipe per dataset (common to all horizons). The model width

d

equals

\min(d_{\mathrm{ff}},128)

. pe: positional offsets on the rolling window (shared across stages).

\alpha

: frequency weight of the TFMAE objective.

\gamma

: StepLR decay factor (step size 2). Budget: maximum epochs (early-stopping patience). ETTh1 uses the global-standardization variant of the model; all other datasets use anchored normalization.

Dataset	F0: feedback	F1: detached	F2: stage+feedback	F3: stage only
ETTh1	.6699 $\pm$ .0020	.6701 $\pm$ .0019	.6695 $\pm$ .0019	1.2064 $\pm$ .0033
ETTh2	.2134 $\pm$ .0012	.2131 $\pm$ .0012	.2124 $\pm$ .0013	.2513 $\pm$ .0007
ETTm1	.3812 $\pm$ .0007	.3842 $\pm$ .0013	.3811 $\pm$ .0009	.3917 $\pm$ .0012
ETTm2	.1216 $\pm$ .0011	.1217 $\pm$ .0011	.1214 $\pm$ .0006	.1227 $\pm$ .0002

Appendix

Table 8. Feedback controls: validation MSE (mean

\pm

sample standard deviation).

Dataset	$H$	L0: final MSE	L1: prefix MSE	L2: prefix MAE	L3: prefix TFMAE
ETTh1	96	.6631 $\pm$ .0007	.6530 $\pm$ .0008	.6699 $\pm$ .0020	.6699 $\pm$ .0020
ETTh2	96	.2144 $\pm$ .0004	.2160 $\pm$ .0024	.2134 $\pm$ .0012	.2134 $\pm$ .0012
ETTm1	96	.3918 $\pm$ .0024	.3915 $\pm$ .0007	.3813 $\pm$ .0012	.3812 $\pm$ .0007
ETTm1	720	.9656 $\pm$ .0070	.9544 $\pm$ .0120	.9600 $\pm$ .0042	.9464 $\pm$ .0018
ETTm2	96	.1269 $\pm$ .0005	.1264 $\pm$ .0010	.1234 $\pm$ .0004	.1216 $\pm$ .0011
ETTm2	720	.2783 $\pm$ .0038	.2761 $\pm$ .0012	.2681 $\pm$ .0015	.2629 $\pm$ .0043

Appendix

Table 9. Loss controls: validation MSE (mean

\pm

sample standard deviation).

Dataset	$H$	width	1 repeat	2 repeats	4 repeats
ETTh1	96	32	.6962 $\pm$ .0020	.6952 $\pm$ .0019	.6935 $\pm$ .0015
ETTh1	96	64	.6816 $\pm$ .0010	.6804 $\pm$ .0003	.6767 $\pm$ .0026
ETTh1	96	128	.6702 $\pm$ .0028	.6689 $\pm$ .0025	.6675 $\pm$ .0018
ETTh1	336	32	1.1275 $\pm$ .0011	1.1274 $\pm$ .0012	1.1278 $\pm$ .0028
ETTh1	336	64	1.1158 $\pm$ .0050	1.1159 $\pm$ .0047	1.1150 $\pm$ .0035
ETTh1	336	128	1.1027 $\pm$ .0156	1.0991 $\pm$ .0171	1.0959 $\pm$ .0162

Appendix

Table 10. Fixed-factorization shared-compute control: validation MSE (mean

\pm

sample standard deviation).

Dataset	$H$	gated PT	ungated PT	patch linear	rolling MLP
ETTm1	96	.3551 $\pm$ .0017	.3544 $\pm$ .0030	.3763 $\pm$ .0016	.3832 $\pm$ .0020
ETTm1	336	.6156 $\pm$ .0024	.6171 $\pm$ .0054	.6504 $\pm$ .0012	.6468 $\pm$ .0044
ETTm2	96	.1148 $\pm$ .0008	.1156 $\pm$ .0007	.1135 $\pm$ .0005	.1129 $\pm$ .0010
ETTm2	336	.2009 $\pm$ .0016	.2020 $\pm$ .0024	.1962 $\pm$ .0010	.1965 $\pm$ .0037
Traffic	96	.3197 $\pm$ .0006	.3199 $\pm$ .0003	.3383 $\pm$ .0004	.3416 $\pm$ .0004
Traffic	336	.3393 $\pm$ .0005	.3376 $\pm$ .0004	.3504 $\pm$ .0001	.3934 $\pm$ .0005

Appendix

Table 11. Matched block-rollout operators: validation MSE (mean

\pm

sample standard deviation).

Dataset	pred_len	RFD		WPMixer		TimeMixer		PatchTST		iTransformer		DLinear		FEDformer		SparseTSF		FITS		TimeXer		TQNet
Dataset	pred_len	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.073	0.199	0.056	0.179	0.056	0.180	0.057	0.181	0.061	0.189	0.062	0.184	0.080	0.218	0.068	0.202	0.060	0.188	0.059	0.185	0.055	0.178
	192	0.092	0.227	0.073	0.205	0.073	0.206	0.078	0.216	0.074	0.206	0.086	0.219	0.101	0.241	0.084	0.226	0.080	0.219	0.079	0.217	0.072	0.203
	336	0.113	0.261	0.087	0.228	0.082	0.222	0.088	0.234	0.084	0.223	0.105	0.251	0.113	0.264	0.099	0.250	0.097	0.247	0.091	0.238	0.085	0.227
	720	0.210	0.384	0.097	0.244	0.083	0.229	0.101	0.250	0.088	0.235	0.184	0.353	0.123	0.275	0.111	0.263	0.116	0.269	0.080	0.225	0.090	0.235
ETTh2	96	0.131	0.277	0.136	0.284	0.136	0.286	0.144	0.294	0.140	0.292	0.131	0.277	0.130	0.273	0.213	0.364	0.148	0.299	0.139	0.288	0.131	0.278

Appendix

Table 12. Comprehensive per-horizon test results (univariate S mode, look-back 96; lower is better). RFD is the three-seed mean (2021/2022/2023), each seed evaluated once on the official test split with the validation-selected checkpoint; baseline figures come from official releases or faithful re-implementations. Best in bold, second best underlined.

Dataset	pred_len	RFD		WPMixer		TimeMixer		PatchTST		iTransformer		DLinear		FEDformer		SparseTSF		FITS		TimeXer		TQNet
Dataset	pred_len	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.390	0.400	0.375	0.394	0.402	0.413	0.393	0.413	0.386	0.404	0.389	0.404	0.380	0.422	0.749	0.577	0.469	0.453	0.397	0.412	0.383	0.398
	192	0.442	0.434	0.428	0.425	0.452	0.440	0.447	0.439	0.443	0.434	0.438	0.433	0.421	0.447	0.659	0.539	0.531	0.486	0.434	0.436	0.444	0.428
	336	0.482	0.457	0.467	0.442	0.497	0.461	0.491	0.473	0.492	0.463	0.488	0.467	0.471	0.475	0.736	0.583	0.615	0.530	0.490	0.461	0.491	0.452
	720	0.507	0.501	0.497	0.475	0.500	0.483	0.524	0.506	0.531	0.504	0.513	0.511	0.501	0.502	0.706	0.589	0.743	0.602	0.514	0.496	0.534	0.495
ETTh2	96	0.286	0.331	0.298	0.348	0.291	0.343	0.318	0.361	0.294	0.346	0.342	0.396	0.351	0.392	0.381	0.406	0.317	0.360	0.295	0.342	0.293	0.343

Appendix

Table 13. Comprehensive per-horizon test results (multivariate-to-multivariate M mode, look-back 96; lower is better). RFD is the three-seed mean (2021/2022/2023), each seed evaluated once on the official test split with the validation-selected checkpoint; baseline figures come from official releases or faithful re-implementations. Best in bold, second best underlined.

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