Energy Time-Series Imputation with Differentially Private Diffusion Models via Clipping-Aware Objective Conditioning

Huizhen Huang; Yu Li; Tao Huang; Chen Hou

cs.LGSep 21, 2026

Energy Time-Series Imputation with Differentially Private Diffusion Models via Clipping-Aware Objective Conditioning

Authors: Huizhen Huang, Yu Li, Tao Huang, Chen Hou

Organizations: School of Computer and Big Data, Minjiang University, No. 200 Xiyuangong Road, Fuzhou, 350108, Fujian, China

Abstract

Reliable recovery of missing measurements is important for monitoring and analysis in energy time-series systems, where fine-grained measurements may contain sensitive temporal information. Diffusion models trained with differentially private stochastic gradient descent (DP-SGD) provide a promising framework for privacy-sensitive energy time-series imputation. Under cosine diffusion schedules, late timesteps correspond to low signal-to-noise ratio (SNR) conditions, where standard $\varepsilon$ -prediction can induce large pre-clipping gradients. Such gradients are more likely to be clipped, reducing the retained optimization signal. The artificial intelligence (AI) contribution lies in formulating this objective--clipping interaction as an objective optimization problem under fixed-threshold DP-SGD and developing timestep-aware objective conditioning for diffusion-based energy time-series imputation. The method adopts $v$ -prediction to mitigate late-timestep gradient amplification, uses static loss weighting as a uniform-scaling control, and introduces diffusion-schedule-aware dynamic weighting for stronger attenuation before clipping. For the engineering application, we evaluate the method on five real-world energy time-series datasets across random point missingness, contiguous block missingness, persistent outages, and multiple missing-data severities. Under matched DP-SGD settings, the proposed method consistently improves imputation utility over the $\varepsilon$ -prediction baseline. Gradient diagnostics reveal lower upper-tail pre-clipping gradient norms, reduced clipping fractions, and stronger attenuation at late low-SNR timesteps, supporting the effectiveness of clipping-aware objective conditioning for energy time-series imputation.

Figures & tables

Figure 1: Cosine schedule and late low-SNR region. The cumulative signal-retention coefficient

\bar{\alpha}_{t}

decreases at late diffusion timesteps, while the illustrative inverse-signal trend increases.

Dataset	Missingness Pattern	MSE		MAE		RMSE
Dataset	Missingness Pattern	$\varepsilon$ -pred.	$v$ -pred., dynamic	$\varepsilon$ -pred.	$v$ -pred., dynamic	$\varepsilon$ -pred.	$v$ -pred., dynamic
ETTh1	Random Point	1181.36	9.12	29.61	2.36	34.37	3.02
	Contiguous Block	867.77	27.91	22.71	4.47	29.46	5.28
	Persistent Outage	941.54	20.28	24.85	3.62	30.68	4.48
ETTm1	Random Point	1173.89	8.83	30.14	1.56	34.26	2.97
	Contiguous Block	1051.89	18.84	27.86	3.21	32.43	4.36

Table 1: Main interpolation results under

50\%

missingness. Here,

\varepsilon

-pred. denotes

\varepsilon

-prediction, and

v

-pred., dynamic denotes

v

-prediction with timestep-aware dynamic weighting. All methods use the same matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. Lower values are better. Results are averaged over three random seeds (

42

43

44

Dataset	Method	Random Point		Contiguous Block		Persistent Outage
Dataset	Method	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	$\varepsilon$ -pred.	1181.36	29.61	867.77	22.71	941.54	24.85
	$v$ -pred., unweighted	21.90	3.67	29.80	4.57	21.81	3.75
	$v$ -pred., static ( $\lambda=0.6$ )	19.86	3.69	49.30	6.13	47.03	5.83
	$v$ -pred., static ( $\lambda=0.9$ )	25.07	3.96	35.68	5.03	44.37	5.72
	$v$ -pred., dynamic	9.12	2.36	27.91	4.47	20.28	3.62

Table 2: Progressive objective ablation under

50\%

missingness. All variants use the same matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. Results are averaged over three random seeds (

42

43

44

). Lower MSE and MAE are better, and the best result for each dataset, missingness pattern, and metric is shown in bold.

Dataset	$\varepsilon$ -pred.	$\varepsilon$ -pred.	$v$ -pred., unweighted
	$\downarrow$	$\downarrow$	$\downarrow$
	$\varepsilon$ -pred., dynamic	$v$ -pred., unweighted	$v$ -pred., dynamic
	MSE / MAE	MSE / MAE	MSE / MAE
ETTh1	3.34 / 2.59	97.99 / 86.34	31.23 / 17.37
ETTm1	-6.88 / -5.25	97.17 / 84.18	50.09 / 34.02
Electricity	0.57 / 0.56	97.76 / 84.77	40.63 / 26.09

Table 3: Controlled target–weighting decomposition under the matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. Each entry reports the relative reduction (%) in mean MSE / MAE over 12 interpolation conditions comprising three missingness patterns and four missing ratios. Positive values indicate improvement, whereas negative values indicate degradation.

Figure 2: Illustrative timestep-wise gradient diagnostic across diffusion timesteps. Curves are computed from fixed checkpoints using a synthetic normalized batch to visualize objective-induced scaling trends. The dashed line is the clipping threshold

C=1.0

, and the shaded region marks late low-SNR diffusion timesteps. Training-time gradient diagnostics are reported in Table 4 .

Dataset	Method	Mean	Median	Grad p95	Grad p99	Overall Clipping Fraction
ETTh1	$\varepsilon$ -pred.	7.982	3.168	15.290	38.901	1.000
	$v$ -pred., unweighted	4.526	3.043	9.660	13.836	1.000
	$v$ -pred., dynamic	3.683	1.943	8.233	12.730	0.944
ETTm1	$\varepsilon$ -pred.	19.284	2.558	19.233	69.501	1.000
	$v$ -pred., unweighted	5.601	2.438	12.481	21.748	1.000
	$v$ -pred., dynamic	4.702	2.136	10.664	18.361	0.948

Table 4: Training-time gradient diagnostics under the matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. Mean, Median, Grad p95 , and Grad p99 are computed from full-parameter gradient norms for individual private examples before DP-SGD clipping during the final

20\%

of training iterations. Overall Clipping Fraction denotes the fraction of recorded gradients exceeding the fixed clipping threshold

C=1.0

. For the gradient-norm statistics, smaller values indicate lower pre-clipping gradient magnitudes; for Overall Clipping Fraction, smaller values indicate less frequent threshold exceedance. The lowest value within each dataset is shown in bold.

Dataset	Method	10%	20%	30%	50%
ETTh1	$\varepsilon$ -pred.	24.68	28.50	25.55	25.72
	$v$ -pred., unweighted	3.15	3.52	3.60	4.00
	$v$ -pred., dynamic	2.35	2.82	3.13	3.48
ETTm1	$\varepsilon$ -pred.	29.58	27.80	27.97	28.66
	$v$ -pred., unweighted	4.22	4.55	4.67	4.59
	$v$ -pred., dynamic	2.37	3.11	3.32	3.11

Table 5: Robustness to different missing severities under the same matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. MAE is averaged over random point missingness, contiguous block missingness, and persistent outage missingness. Results are averaged over three random seeds (

42

43

44

). Lower values indicate better interpolation performance, and the best result for each dataset and missing ratio is shown in bold.

Dataset	Method	MSE			MAE
Dataset	Method	$H=24$	$H=48$	$H=72$	$H=24$	$H=48$	$H=72$
ETTh1	$\varepsilon$ -pred.	785.81	981.47	885.18	22.12	25.65	23.41
	$v$ -pred., unweighted	26.02	26.28	45.55	4.17	4.20	5.97
	$v$ -pred., dynamic	17.37	22.80	26.58	3.77	3.91	4.39
ETTm1	$\varepsilon$ -pred.	1005.67	1033.41	1060.82	26.55	27.25	27.91
	$v$ -pred., unweighted	53.06	53.41	36.31	6.67	6.52	5.16

Table 6: Suffix-missing forecasting performance under different forecast horizons. All methods use the same matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. MSE and MAE are computed only over the forecast horizon. Results are averaged over three random seeds (

42

43

44

). Lower values indicate better forecasting performance, and the best result for each dataset, forecast horizon, and metric is shown in bold.

Metric	Model	$\epsilon=8$			$\epsilon=10$
Metric	Model	ETTh1	ETTm1	Electricity	ETTh1	ETTm1	Electricity
MSE	PatchTST	3349.51	1965.84	2514.88	3348.61	1963.74	2516.32
	TimesNet	2271.85	1128.46	3213.12	2273.45	1130.65	3214.54
	iTransformer	4344.20	2966.35	3458.10	4341.34	2968.12	3457.21
	TimeDiT ( $\varepsilon$ -pred.)	1216.15	958.36	2597.01	1181.36	1173.89	2415.27
	Ours ( $v$ -pred., dynamic)	16.08	36.58	17.99	9.12	8.83	32.88

Table 7: Comparison with representative time-series baselines under matched DP-SGD interpolation settings at

50\%

random point missingness. Results are reported under two matched DP-SGD configurations indexed by

\epsilon=8

and

\epsilon=10

. Within each matched DP-SGD configuration, all compared methods use the same preprocessing, normalization, masking, and evaluation protocol. Here,

\epsilon

indexes the matched DP-SGD configuration, TimeDiT (

\varepsilon

-pred.) denotes the diffusion baseline with

\varepsilon

-prediction, and Ours denotes the proposed method with

v

-prediction and timestep-aware dynamic weighting. Lower MSE and MAE are better. Results are averaged over three random seeds (

42

43

44

Dataset	Selected $C$	Selected Learning Rate	Best Validation Loss	Selected Checkpoint Step
ETTh1	1	$3\times 10^{-5}$	0.203965	87000
ETTm1	1	$1\times 10^{-4}$	0.217985	99000
Appliances	1	$3\times 10^{-5}$	0.180863	87000

Table 8: Validation tuning and evaluation of the

\varepsilon

-prediction baseline. (a) Validation-selected clipping threshold, learning rate, and checkpoint for each dataset. (b) Comparison of the tuned

\varepsilon

-prediction baseline with the original

\varepsilon

-prediction baseline and unweighted

v

-prediction using the same evaluation protocol as the main experiments. For each missingness pattern, results are averaged over missing ratios of

10\%

20\%

30\%

, and

50\%

; for suffix-missing forecasting, results are averaged over

H\in\{24,48,72\}

. Lower MAE and RMSE are better, and the best result for each dataset–condition and metric is shown in bold. Results in panel (b) are averaged over three random seeds (

42

43

44

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Configuration
GPU	NVIDIA A100 80GB PCIe
System Memory	512GB RAM
Python Version	3.12.4
Deep Learning Framework	PyTorch 2.9.0
CUDA Version	12.8

Appendix

Table 9: Hardware and software environment used for all experiments.

Component	Configuration
Backbone	DiT-style Transformer
Number of DiT Blocks	8
Hidden Dimension	256
Self-Attention	8 heads
Timestep Embedding	256-d sinusoidal embedding + MLP
Conditioning Mechanism	AdaLN-Zero

Appendix

Table 10: Diffusion-backbone configuration used in the core objective comparisons.

Hyperparameter	Value
Learning Rate	$10^{-4}$
Batch Size	32 (ETTh1, ETTm1, Household, Appliances) 16 (Electricity)
Training Steps	100,000
Optimizer	AdamW
Diffusion Steps ( $T$ )	1000
Clipping Threshold ( $C$ )	1.0

Appendix

Table 11: Training configuration used for the core objective comparisons. The main experiments use the matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. All core objective variants use the same sampling scheme, clipping threshold, noise multiplier, and number of optimization steps.

Dataset	Method	Metric	Random Point	Contiguous Block	Persistent Outage
ETTh1	$\varepsilon$ -pred.	MSE	1117.24–1235.33	826.55–917.09	886.24–976.69
	$\varepsilon$ -pred.	MAE	28.39–30.38	21.61–23.61	23.90–25.95
	$v$ -pred., unweighted	MSE	19.80–23.50	26.65–31.90	20.00–23.51
	$v$ -pred., unweighted	MAE	3.46–3.84	4.22–4.94	3.50–3.90
	$v$ -pred., static ( $\lambda=0.6$ )	MSE	18.40–21.46	44.50–53.00	43.50–50.13
	$v$ -pred., static ( $\lambda=0.6$ )	MAE	3.50–3.87	5.60–6.45	5.40–6.23

Appendix

Table 12: Seed variability for the progressive objective ablation under

50\%

missingness. The MSE and MAE rows report the corresponding min–max ranges across three random seeds (42, 43, and 44). The main-text ablation table reports the corresponding seed averages.

Dataset	Method	Random Point		Contiguous Block		Persistent Outage
Dataset	Method	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	$\varepsilon$ -pred.	1181.36	29.61	867.77	22.71	941.54	24.85
	$v$ -pred., unweighted	21.90	3.67	29.80	4.57	21.81	3.75
	$v$ -pred., static ( $\lambda=0.6$ )	19.86	3.69	49.30	6.13	47.03	5.83
	$v$ -pred., static ( $\lambda=0.9$ )	25.07	3.96	35.68	5.03	44.37	5.72
	$v$ -pred., dynamic	9.12	2.36	27.91	4.47	20.28	3.62

Appendix

Table 13: Full progressive objective ablation under

50\%

missingness across all five datasets. All variants use the main matched DP-SGD configuration indexed by

(\epsilon,\delta)=(10,10^{-5})

. MSE and MAE are reported for Random Point, Contiguous Block, and Persistent Outage missingness. Bold marks the best result among the evaluated progressive objective variants for each dataset, missingness pattern, and metric. Results are averaged over three random seeds (42, 43, and 44).

Figure 3: Validation MSE and gradient norm as functions of the static loss weighting coefficient

\lambda

Dataset	Missing Ratio	$\epsilon=6$		$\epsilon=8$		$\epsilon=10$
Dataset	Missing Ratio	$\varepsilon$ -pred.	$v$ -pred., dynamic	$\varepsilon$ -pred.	$v$ -pred., dynamic	$\varepsilon$ -pred.	$v$ -pred., dynamic
ETTh1	10%	24.98	5.25	26.28	3.13	24.68	2.35
	20%	28.57	6.24	27.81	4.38	28.50	2.82
	30%	26.89	7.66	29.51	4.38	25.55	3.13
	50%	28.07	8.30	29.68	4.92	25.72	3.48
ETTm1	10%	26.23	5.04	28.55	5.08	29.58	2.37

Appendix

Table 14: Interpolation MAE under matched DP-SGD configurations indexed by

\epsilon\in\{6,8,10\}

. Here,

\epsilon

indexes the matched DP-SGD configuration,

\varepsilon

-pred. denotes

\varepsilon

-prediction, and

v

-pred., dynamic denotes

v

-prediction with timestep-aware dynamic weighting

(\lambda_{t}=\bar{\alpha}_{t})

. For each dataset and missing ratio, MAE is averaged over random point missingness, contiguous block missingness, and persistent outage missingness. Lower values are better, and the lower value within each matched configuration is shown in bold. Results are averaged over three random seeds (

42

43

44

Dataset	Median Aggregate-Gradient Norm		95th-Percentile Aggregate-Gradient Norm		Gradient Cosine Similarity
Dataset	Unweighted	Dynamic	Unweighted	Dynamic	Gradient Cosine Similarity
ETTh1	4.748	2.123	14.099	14.066	0.774
ETTm1	4.417	2.516	11.427	11.189	0.822
Appliances	4.437	1.786	28.443	28.319	0.754

Appendix

Table 15: Mixed-timestep aggregate-gradient diagnostic. The checkpoint, inputs, masks, diffusion noise, and sampled timesteps are matched between unweighted

v

-prediction and

v

-prediction with timestep-aware dynamic weighting; only the weighting rule is changed.

Dataset	Timestep Interval	Median Aggregate-Gradient Norm		95th-Percentile Aggregate-Gradient Norm		Gradient Cosine Similarity
Dataset	Timestep Interval	Unweighted	Dynamic	Unweighted	Dynamic	Gradient Cosine Similarity
ETTh1	0–99	4.022	3.981	5.298	5.271	1.000
	100–199	4.047	3.833	9.552	9.467	1.000
	200–299	3.383	2.854	8.520	8.392	0.999
	300–399	3.489	2.440	9.362	9.094	0.997
	400–499	4.054	2.262	12.376	12.101	0.996

Appendix

Table 16: Timestep-stratified aggregate-gradient diagnostic for unweighted

v

-prediction and

v

-prediction with timestep-aware dynamic weighting. Each interval contains 30 controlled evaluations with matched model state and stochastic inputs between the two objective evaluations.

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