Amortized Predictability-aware Training Framework for Time Series Forecasting and Classification

Xu Zhang; Peng Wang; Yichen Li; Wei Wang

cs.LGFeb 18, 2026

Amortized Predictability-aware Training Framework for Time Series Forecasting and Classification

Authors: Xu Zhang, Peng Wang, Yichen Li, Wei Wang

Organizations: Shanghai Key Laboratory of Data Science, College of Computer Science and Artificial Intelligence, Fudan University, Shanghai, China · Department of Electrical and Computer Engineering, University of British Columbia (UBC), Vancouver, Canada

Abstract

Time series data are prone to noise in various domains, and training samples may contain low-predictability patterns that deviate from the normal data distribution, leading to training instability or convergence to poor local minima. Therefore, mitigating the adverse effects of low-predictability samples is crucial for time series analysis tasks such as time series forecasting (TSF) and time series classification (TSC). While many deep learning models have achieved promising performance, few consider how to identify and penalize low-predictability samples to improve model performance from the training perspective. To fill this gap, we propose a general Amortized Predictability-aware Training Framework (APTF) for both TSF and TSC. APTF introduces two key designs that enable the model to focus on high-predictability samples while still learning appropriately from low-predictability ones: (i) a Hierarchical Predictability-aware Loss (HPL) that dynamically identifies low-predictability samples and progressively expands their loss penalty as training evolves, and (ii) an amortization model that mitigates predictability estimation errors caused by model bias, further enhancing HPL's effectiveness. The code is available at https://github.com/Meteor-Stars/APTF.

Figures & tables

Figure 1. The proposed general training framework APTF for time series forecasting and classification.

Figure 2. Illustration of different bucketing strategies for computing the predictability-aware loss. (a) A fixed number of buckets is used throughout training, ignoring the predictability evolution issue. (b) The number of buckets decreases as training progresses, addressing predictability evolution but discarding previous bucket partitions. (c) Hierarchical buckets consider both predictability evolution and previous bucket partitioning strategies.

	TimeMixer		PatchTST		Autoformer		NSformer		Scaleformer		NLinear		N-HiTS		Informer
	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original
Fund1	87.914	89.963	86.549	88.811	103.286	110.295	95.112	97.198	100.027	107.31	102.536	106.636	86.856	91.69	102.912	106.952
Fund2	84.356	85.512	85.094	86.472	99.712	107.141	91.516	94.406	96.736	109.01	90.791	93.538	86.13	91.531	91.407	97.193
Fund3	83.659	85.14	84.821	86.593	98.838	107.063	90.364	92.646	95.235	107.405	90.542	93.652	84.719	90.332	90.476	96.589

Table 1. Short-term TSF results of APTF for forecasting 1, 5, 8, and 10 future steps, where the average WMAPE over all horizons is reported. APTF also shows improvements over baselines CATS, TimeXer, and ModernTCN, as shown in Appendix Table 9 .

	TimeMixer		PatchTST		Autoformer		NSformer		Scaleformer		NLinear		N-HiTS		Informer
	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original
ETTh1	0.492	0.517	0.40	0.43	0.492	0.504	0.511	0.523	0.543	0.55	0.453	0.469	0.491	0.516	1.038	1.058
ETTh2	0.395	0.415	0.366	0.37	0.398	0.434	0.407	0.434	0.409	0.441	0.38	0.383	0.546	0.599	1.577	2.357
ETTm1	0.398	0.407	0.357	0.369	0.506	0.556	0.553	0.599	0.506	0.518	0.375	0.377	0.385	0.399	0.675	0.74
ETTm2	0.3	0.304	0.278	0.29	0.351	0.36	0.337	0.356	0.354	0.414	0.286	0.291	0.311	0.334	1.55	2.092
weather	0.264	0.269	0.241	0.244	0.383	0.386	0.307	0.314	0.452	0.534	0.263	0.27	0.257	0.262	0.508	0.595

Table 2. Long-term forecasting results for horizons of 96, 192, 336, and 720, reporting averaged MSE and MAE. Due to space constraints, partial results for each horizon are provided in Appendix 13 . APTF also shows improvements over baselines CATS, TimeXer, and ModernTCN, as shown in Appendix Table 10 and Table 12 .

	Methods	Average accuracy
	Methods	Origin	w/ APTF	Win	Tie
128 UCR datasets	FCNet	79.82 $\pm$ 0.011	81.21 $\pm$ 0.014	82	22
	ResNet	79.93 $\pm$ 0.0141	81.04 $\pm$ 0.0142	86	25
	InceptionTime	85.90 $\pm$ 0.008	86.73 $\pm$ 0.008	74	34
	OSCNN	84.82 $\pm$ 0.008	85.37 $\pm$ 0.0081	65	40
	FormerTime	74.42 $\pm$ 0.0157	75.31 $\pm$ 0.0152	69	33

Table 3. TSC task. The average classification accuracy over 128 UCR datasets is reported.

		Autoformer-Our	w/o Amortize	w/o HPL	Scaleformer-Our	w/o amortize	w/o HPL	NSFormer-Our	w/o Amortize	w/o HPL	NLinear-Our	w/o Amortize	w/o HPL
Fund-123	1	87.15	88.22	88.29	90.63	92.7	93.19	84.99	86.05	86.22	89.44	91.15	92.55
	5	104.42	107.31	108.33	98.12	100.09	103.04	94.94	96.31	96.96	92.11	92.96	94.37
	8	105.33	106.13	107.03	100.67	102.26	103.16	94.65	96.84	95.92	97.14	97.67	99.09
	10	106.31	108.04	107.88	101.29	103.58	102.3	95.77	96.59	96.38	99.8	100.51	101.62

Table 4. Ablation study on fund sales datasets. The WMAPE of Fund1, Fund2, and Fund3 on each prediction step are averaged and reported. “w/o Amortize” denotes not using the amortization model and “w/o HPL” denotes removing hierarchical buckets in Figure 2 (c) and the design of Figure 2 (b) is adopted. The best results are in bold.

	TimeMixer	PatchTST	Autoformer	NSFormer	Scaleformer	NLinear	NHits
Electricity (w/o HPL)	0.306	0.252	0.354	0.338	0.356	0.364	0.261
Electricity (w/o HPL)	$\pm$ 4e-3	$\pm$ 3e-4	$\pm$ 0.01128	$\pm$ 0.01135	$\pm$ 0.017	$\pm$ 0.002	$\pm$ 2e-4
Electricity (w/ HPL)	0.288	0.249	0.339	0.325	0.346	0.319	0.254
Electricity (w/ HPL)	$\pm$ 3e-3	$\pm$ 2e-4	$\pm$ 0.010	$\pm$ 0.007	$\pm$ 0.014	$\pm$ 0.001	$\pm$ 2e-4
Traffic (w/o HPL)	0.655	0.512	0.654	0.66	0.657	0.794	0.510
Traffic (w/o HPL)	$\pm$ 0.013	$\pm$ 1e-3	$\pm$ 0.02	$\pm$ 0.012	$\pm$ 0.017	$\pm$ 3e-3	$\pm$ 6e-3

Table 5. Further demonstrating the effectiveness of hierarchical predictability-aware loss (HPL) on two large-scale datasets. We report the standard deviations and average of MSE and MAE across 96, 192, 336, and 720 prediction steps.

Figure 3. Loss landscapes of Autoformer with and without APTF on the Fund sales dataset for 10-step forecasting. A flatter minimum (larger blue area) indicates better generalization. Additional cases are provided in Appendix Figure 7 .

	ETTh1	ETTm1	weather	Fund1	Fund2	Fund3
TimeMixer-Original	0.517	0.404	0.269	89.963	85.512	85.14
-Wavebound	0.706	0.489	0.279	206.9	186.198	328.27
-Coteaching	0.508	0.402	0.268	100.23	95.716	94.893
-APTF (Ours)	0.492	0.396	0.264	87.91	84.36	83.659
PatchTST-Original	0.430	0.369	0.244	88.811	86.472	86.593
-Wavebound	0.434	0.364	0.245	191.085	197.374	201.886

Table 6. Comparison with WaveBound and Co-teaching. We report average performance across all prediction steps (average WMAPE for Fund and average MSE and MAE for other datasets). Hyperparameter search results for Co-teaching are provided in Appendix Table 11 .

Figure 4. Hyperparameter sensitivity analysis.

	TimeMixer	Autoformer	NSFormer	Scaleformer	NLinear	NHits
Original	0.306	0.344	0.330	0.35	0.355	0.262
+APTF	0.288	0.335	0.321	0.336	0.319	0.255
+ARL	0.307	0.357	0.329	0.347	0.367	0.273
+SPL	0.300	0.349	0.332	0.451	0.360	0.262

Table 7. Comparison with predictability-aware loss functions on the Electricity dataset. We report the average MSE and MAE over prediction horizons of 96, 192, 336, and 720 steps. ARL ( Barron, 2019 ) denotes adaptive robust loss, while SPL ( Kumar et al., 2010 ) denotes self-paced learning loss.

	ET.h1	ET.h2	ET.m1	ET.m2	Fund1	Fund2	Fund3
TimeMixer-Original	0.537	0.405	0.402	0.281	89.96	85.51	85.14
-Ours (w/o amortization)	0.512	0.385	0.396	0.279	88.5	84.62	84.02
PatchTST-Original	0.423	0.347	0.351	0.258	88.81	86.47	86.59
-Ours (w/o amortization)	0.416	0.345	0.345	0.250	87.2	85.7	85.45
NHiTS-Original	0.526	0.612	0.386	0.307	91.69	91.53	90.33
-Ours (w/o amortization)	0.499	0.582	0.373	0.291	87.9	87.1	85.8

Table 8. APTF can still improve baseline accuracy without introducing additional time complexity or memory overhead when the amortization model is not used. We report the average WMAPE for the Fund datasets and the average MSE for the other datasets.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5. Time series visualization of Fund sales dataset.

	CATS		ModernTCN		TimeXer
	+APTF	Original	+APTF	Original	+APTF	Original
Fund1	85.72	88.17	86.47	90.28	83.97	86.29
Fund2	82.92	84.79	83.83	85.07	82.1	83.55
Fund3	80.56	83.91	82.55	85.16	81.38	84.02

Appendix

Table 9. Additional baselines for short-term TSF with average WMAPE on 1, 5, 8, and 10 future steps. Results for each prediction length are omitted due to limited space.

	CATS		ModernTCN		TimeXer
	+APTF	Original	+APTF	Original	+APTF	Original
ETTh1	0.426	0.430	0.426	0.438	0.436	0.445
ETTh2	0.363	0.373	0.365	0.374	0.367	0.373
ETTm1	0.357	0.366	0.371	0.386	0.377	0.382
ETTm2	0.288	0.313	0.289	0.302	0.284	0.295
Weather	0.245	0.246	0.246	0.251	0.248	0.249

Appendix

Table 10. Additional baselines for long-term TSF with average MSE and MAE on future 96, 192, 336, and 720 steps. Table 12 shows results for each prediction length.

Forget rate $r$	ETTh1	ETTm1	Weather	Fund1	Fund2	Fund3
TimeMixer, r =0.1	0.509	0.403	0.268	100.23	95.716	94.893
TimeMixer, r =0.2	0.508	0.403	0.269	101.29	96.11	95.571
TimeMixer, r =0.3	0.508	0.402	0.27	102.278	96.846	96.057
TimeMixer, r =0.4	0.509	0.402	0.27	102.972	97.133	96.714
PatchTST, r =0.1	0.429	0.37	0.245	88.604	88.668	87.022
PatchTST, r =0.2	0.429	0.371	0.246	89.706	90.582	88.074

Appendix

Table 11. Hyperparameter search for Coteaching. We report the average of MSE and MAE on predicting all timesteps. The forget rate

r

determines the number of samples discarded during the training process.

Figure 6. Visualizations of forecasting future 10-time steps on Fund1 and Fund2 datasets with the original baseline NHiTS (left) and Autoformer (right), as well as the versions of applying APTF (ours), Coteaching, and Wavebound.

		CATS-APTF		CATS		ModernTCN-APTF		ModernTCN		TimeXer-APTF		TimeXer
		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.379	0.4	0.389	0.408	0.37	0.399	0.381	0.411	0.394	0.412	0.396	0.416
	192	0.4	0.417	0.409	0.424	0.403	0.422	0.42	0.439	0.421	0.431	0.423	0.438
	336	0.421	0.434	0.422	0.436	0.428	0.438	0.441	0.452	0.436	0.448	0.441	0.459
	720	0.482	0.473	0.482	0.471	0.47	0.477	0.476	0.481	0.462	0.485	0.488	0.498
ETTh2	96	0.278	0.345	0.28	0.35	0.269	0.335	0.279	0.347	0.278	0.346	0.279	0.347

Appendix

Table 12. Full results of CATS, ModernTCN, and TimeXer on long-term TSF on popular public datasets with average performance on four random seeds. Due to space limitations, the standard deviation is omitted, and we will show it in our full version paper.

Figure 7. Full results of loss landscapes on different baselines with the Fund1 dataset. The flatter the bottom (the larger the blue area) indicates the better the generalization of the trained models.

		TimeMixer		PatchTST		Autoformer		NSformer		Scaleformer		NLinear		N-HiTS		Informer
		+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original	+APTF	Original
ETTh1	96	0.434	0.45	0.377	0.38	0.465	0.492	0.512	0.544	0.506	0.507	0.423	0.442	0.432	0.448	0.9	0.948
	192	0.49	0.522	0.406	0.412	0.479	0.494	0.5	0.51	0.494	0.495	0.449	0.464	0.466	0.484	1.031	1.05
	336	0.512	0.535	0.427	0.436	0.476	0.484	0.49	0.496	0.566	0.579	0.466	0.478	0.496	0.516	1.079	1.056
	720	0.532	0.56	0.472	0.49	0.548	0.544	0.541	0.542	0.607	0.619	0.475	0.492	0.57	0.616	1.141	1.176
ETTh2	96	0.316	0.326	0.304	0.31	0.376	0.409	0.39	0.406	0.381	0.408	0.318	0.328	0.349	0.385	1.225	1.568

Appendix

Table 13. Full results of long-term forecasting on remaining baselines. with average performance on four random seeds. Due to space limitations, the standard deviation is omitted, and we will show it in our full version paper.

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Training time-series forecasting models poses unique challenges in loss function design. Most existing approaches adopt temporal mean squared error, but this study reveals two critical limitations: (1) it ignores the presence of label autocorrelation, which biases it from the true label sequence likelihood; (2) it involves excessive number of tasks, which complicates optimization, especially for long-term forecasting. To address these issues, we introduce Time-o1, a transform-enhanced loss function for time-series forecasting. The central idea is to transform the label sequence into decorrelated components with discriminated significance. Models are then trained to align the most significant components, thereby effectively mitigating label autocorrelation and reducing task amount. Experiments demonstrate that Time-o1 achieves state-of-the-art performance and is compatible with various forecast models. Code is available at https://github.com/Master-PLC/Time-o1.

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