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When Denoising Hurts: Rethinking the Terminal Step of Diffusion Time Series Forecasters -- Extended Version

Dat Nguyen-Cong, Luong Tran, Tung Kieu

arXiv:2608.14067v1cs.LG

TL;DR

Diffusion time-series forecasters are insufficiently understood, particularly because reverse-diffusion steps may contribute unequally and late refinement can degrade forecasts. This paper introduces label-free early stopping and Bernoulli timestep sampling, achieving comparable or better forecasting performance while reducing sampling cost by 20–50%.

  • Problem

    Diffusion time-series forecasters remain insufficiently understood, including how unequal timestep contributions and low-noise refinement affect forecast quality.

  • Method

    The framework estimates a dataset-level stopping point from pilot trajectories and uses it to terminate sampling early and bias Bernoulli timestep sampling toward high-noise regions while retaining full-horizon coverage.

  • Results

    Sampling cost decreases by 20–50% while forecasting performance remains comparable to or better than the full reverse process.

  • Takeaways & Limitations

    Avoiding late-stage denoising can reduce inference cost without requiring future ground-truth observations at inference time.

Abstract

from arXiv · show

Diffusion models offer a natural way to model uncertainty in time series forecasting, yet their iterative sampling process is often treated as a uniformly beneficial refinement procedure. Our study challenges this view by examining how forecast quality evolves throughout reverse diffusion. We find that general temporal structure is often recovered at relatively high noise levels, whereas continued low-noise refinement can introduce statistical drift and degrade the final forecast. Our analysis further suggests that this behavior explains why prior methods often favor relatively narrow diffusion architecture and schedule design. Building on this observation, we propose a label-free global stopping criterion that detects the optimal termination point, eventually speeding up inference and improving predictive accuracy. Additionally, since early stopping terminates inference in high-noise regions, we propose a Bernoulli timestep sampler that concentrates training on this region while preserving coverage of the full diffusion process. Extensive experiments conducted across eight real-world datasets demonstrate the superior performance of our method compared to existing approaches.

1 Introduction

The paper finds that diffusion forecasting can recover key temporal structure before the final denoising steps, while low-noise refinement may introduce statistical drift and degrade forecasts. It therefore proposes label-free early stopping and Bernoulli timestep sampling to reduce unnecessary computation and emphasize high-noise denoising.

  • Motivation: Diffusion time-series forecasting remains insufficiently understood because existing methods largely overlook how temporal structure interacts with the denoising trajectory.Prior work mainly modifies temporal architectures, training objectives, or guidance mechanisms without addressing this interaction.
  • Key finding: The reverse process has two regimes: high-to-intermediate noise rapidly recovers forecast-relevant structure, whereas low-noise refinement can cause statistical drift and worse forecasts.Reducing reverse-sampling steps can match or outperform substantially more expensive configurations.
  • Key finding: Local denoising loss improvements do not guarantee better forecasts because late-stage denoising increasingly predicts intractable inherent noise, allowing residual errors to accumulate.This exposes a mismatch between local denoising accuracy and end-to-end forecasting performance.
  • Proposed method: A label-free early-stopping mechanism terminates reverse diffusion before late-stage drift, requires no future ground truth, and reduces sampling cost by 20–50%.The stopping point is determined from the evolution of generated-sample statistics and yields comparable or better forecasting performance.
  • Proposed method: Bernoulli timestep sampling shifts training emphasis toward high-noise prediction while retaining coverage of the full diffusion trajectory.The strategy preserves low-noise coverage and improves prediction on heavily corrupted inputs.

2 Related Work

Prior forecasting work models trend, seasonality, and multi-scale temporal structure through basis expansion, decomposition, frequency-domain methods, and Transformer architectures. Diffusion forecasting instead generates future trajectories through iterative denoising, with existing methods using conditional guidance and temporal decomposition or structured architectures.

  • Classical and structured forecasting: Basis-expansion, decomposition, and frequency-domain forecasting methods target trend, seasonality, and multi-scale temporal structure.N-BEATS uses interpretable trend and seasonal components, while N-HiTS extends basis expansion through hierarchical interpolation at multiple resolutions.
  • Transformer-based forecasting: Transformer-based forecasters capture long-range dependencies through specialized architectures and tokenization strategies.Informer, Autoformer, and FEDformer use sparse attention, decomposition, and frequency-domain representations; PatchTST uses temporal patching, and iTransformer models variables as tokens.
  • Diffusion forecasting: Diffusion models provide a probabilistic alternative by generating future trajectories through iterative denoising.Existing approaches improve temporal generation using conditional guidance, temporal decomposition, or structured architectures, including TSDiff, TimeDiT, TimeDiff, CDPM, and D3U.

3 Preliminaries

This section defines the diffusion time-series forecasting notation and formulates forecasting as conditional generation. It describes iterative denoising from Gaussian noise and explains why diffusion models represent full predictive distributions for uncertain, multimodal futures.

  • Notation: The framework denotes historical context x, future horizon y_0, and diffusion latent y_t, with Q, P, and D representing context length, prediction length, and variable count.The noise schedule uses β_t, α_t = 1 − β_t, and cumulative ᾱ_t, with ᾱ_0 = 1 and ᾱ_T ≈ 0.
  • Problem Formulation: Forecasting is formulated as generating the conditional distribution p(y_0|x) of future time series y_0 given historical context x.The diffusion model treats this conditional generation task through an iterative learnable denoising process.
  • Diffusion Process: Sampling starts from simple Gaussian noise p(y_T) = N(0, I) and iteratively applies a learnable denoising process.This establishes the reverse diffusion procedure used to generate future trajectories.
  • Reverse Transitions: Because p_θ(y_{t−1}|y_t, x) is generally intractable, it is approximated using a predefined posterior q(y_{t−1}|y_t, y_0) derived from an auxiliary noising process.The posterior expression combines y_t and y_0, so learning the reverse transition reduces to approximating the clean prediction y_0.
  • Predictive Distribution: Unlike deterministic predictors, diffusion models measure the full predictive distribution, supporting time series with irreducible stochasticity, measurement noise, and multiple plausible futures.These properties make distributional forecasting suitable for real-world time series.

4 Methodology

The methodology aligns diffusion training and inference with time-series denoising dynamics, identifying a transition from global structure recovery to unstable late refinement. It estimates a ground-truth-free stopping boundary and reallocates training toward high-noise timesteps while preserving full diffusion coverage.

  • Denoising Dynamics: Forecasts recover dominant temporal structure early, but low-noise refinement increasingly diverges and raises errors toward the final step.This behavior is especially noticeable under the quadratic schedule.
  • Denoising Dynamics: The method distinguishes a structure-recovery regime from a fine-grained regime where denoising focuses on unpredictable local residual variations.Strong corruption encourages use of historical context for dominant patterns such as trend and seasonality.
  • Global Breakpoint Estimation: A ground-truth-free score curve uses expected clean predictions and piecewise-linear regression to detect the transition breakpoint.The expectation is approximated with independently sampled conditional reverse trajectories, and averaging reduces sampling variability.
  • Global Breakpoint Estimation: The global stopping timestep is the median breakpoint across N pilot samples, estimated once before evaluation and applied to all test instances.Inference returns the clean prediction at the stopping boundary, skipping redundant reverse steps with negligible estimation overhead.
  • Bernoulli Timestep Sampling: A Bernoulli timestep sampler emphasizes highly corrupted inputs while its complementary branch retains low-noise supervision and full diffusion-process coverage.Unlike hard timestep clipping, the allocation adapts as the estimated drift boundary evolves during training.

5 Experiments

Experiments across eight real-world benchmarks show that early stopping and Bernoulli timestep sampling improve forecasting quality, robustness, and efficiency. The proposed method achieves strong deterministic performance while balancing probabilistic accuracy and calibration under truncated reverse diffusion.

  • Experimental setup: Experiments use eight real-world benchmarks spanning diverse domains, sampling frequencies, and temporal dynamics, with standard train–validation–test splits.The datasets are ETT{h1, h2, m1, m2}, Weather, Electricity, Exchange, and Traffic.
  • Deterministic forecasting: The method achieves average MSE 0.320 and MAE 0.330, improving 4.3% and 5.8% over TimeMixer and 16.4% and 12.1% over CNDiff, respectively.The largest gains over second-best results occur on Exchange and Traffic, with MSE and MAE improvements of 7.4% and 13.9%.
  • Probabilistic forecasting: Early-stopped probabilistic forecasting trades lower CRPS under full diffusion for improved predictive accuracy and distributional calibration by avoiding low-noise statistical drift.Full diffusion’s lower CRPS is attributed largely to increased sample dispersion rather than more accurate predictions.
  • Component analysis: Ablations show that early stopping provides the largest point-forecasting improvement, Bernoulli sampling strengthens probabilistic forecasting, and their combination achieves the strongest MSE and MAE.Bernoulli sampling alone yields the best CRPS, while performance is generally strongest for p ∈[0.8, 0.9].
  • Robustness across noise schedules: Early stopping consistently yields lower and more stable MSE across noise schedules, indicating that denoising weakly predictable residual variations drives instability rather than noise-schedule or diffusion-step selection.Full sampling exhibits large instability throughout training.

6 Conclusion

The conclusion challenges the assumption that every reverse-diffusion step improves time-series forecasts, finding that later generation can cause statistical drift and degrade final forecasts. It introduces a label-free stopping criterion and a Bernoulli timestep sampler motivated by this behavior.

  • Conclusion: Later reverse-diffusion steps can induce statistical drift and degrade the final time-series forecast.This challenges the assumption that every reverse-diffusion step improves forecasting quality.
  • Conclusion: A label-free global stopping criterion estimates the breakpoint from pilot trajectories and terminates sampling before harmful drift occurs.The criterion is motivated by the observed degradation during the later generation phase.
  • Conclusion: A Bernoulli timestep sampler emphasizes the more influential high- portion of the diffusion process.

Supplementary Material … B Interpretation Of Late-Stage Drift

The supplementary material defines the conditional diffusion forecasting framework and accelerated sampling procedures, then explains how approximately identity-like low-noise transitions can preserve or amplify accumulated errors. This mechanism clarifies why additional denoising steps may degrade end-to-end forecasting quality, while not implying universal degradation.

  • A Denoising Diffusion Probabilistic Models: DDPMs map trajectories to an approximately Gaussian terminal state through a fixed forward process and learn a reverse process to recover the original trajectory.For sufficiently long horizons, yT becomes approximately standard Gaussian.
  • A.1 Forward Diffusion Process: The forward process permits direct sampling of a noisy trajectory from the clean trajectory using yt = √¯αty0 + √1 −¯αtϵ.The noise variable satisfies ϵ ∼N(0, I), and ¯αT approaches zero over a sufficiently long diffusion horizon.
  • A.2 Conditional Reverse Process: For forecasting, generation conditions reverse diffusion on historical context and trains a denoiser to predict the clean future with L1 reconstruction and Fourier-transform losses.The weights for the reconstruction and Fourier objectives are set to wt = λt = 1 throughout the experiments.
  • A.3 Sampling with DDPM: DDPM sampling starts from yT ∼N(0, I), predicts the clean trajectory at each reverse step, and repeats transitions from t = T to t = 1.The final step uses z = 0, producing one sample from pθ(y0 | x).
  • A.4 Accelerated Sampling With DDIM: DDIM accelerates inference by transitioning across a subsequence of diffusion timesteps without changing the clean-trajectory prediction objective.Setting η = 0 gives a deterministic trajectory, whereas this work uses η = 1 to improve forecasting diversity.
  • B Interpretation Of Late-Stage Drift: Linearizing a learned reverse transition around an ideal trajectory yields recursive deviations governed by local approximation errors and transition Jacobians.Unrolling the recursion shows how errors accumulated across reverse steps contribute to later deviations.
  • B Interpretation Of Late-Stage Drift: When low-noise transitions become approximately identity-like, ρt approaches one, so extra reverse steps can preserve accumulated errors or amplify them when ρt > 1.With δT = 0, the initial-state term vanishes; therefore, small per-timestep denoising loss need not improve end-to-end forecasting quality. The mechanism is sufficient but does not imply universal low-noise degradation.

C Model Architecture And Training

The model combines patch-based, channel-independent denoising with historical-context conditioning and timestep-aware Transformer blocks. Training uses fixed diffusion and optimization settings, while DDIM with 20 sampling steps accelerates inference.

  • Architecture: The denoising network processes noisy future observations, historical context, and diffusion timesteps using channel-independent patch representations with RevIN.Future sequences are split into non-overlapping patches of length Lp and projected into dmodel-dimensional representations.
  • Architecture: The architecture uses DiT encoder-decoder blocks, adaptive layer normalization for timestep integration, and cross-attention between future patches and historical representations.The timestep is sinusoidally embedded and passed through a multilayer perceptron before modulating denoising blocks.
  • Architecture: Final patch representations pass through a two-layer MLP and are rearranged to produce clean-trajectory estimates in the original forecasting space.
  • Implementation Details: The model fixes Lp = 24, Sp = 24, hidden dimension 128, 4 attention heads, and 512-dimensional feed-forward layers, tuning encoder-decoder depth and dropout.Higher dropout probability is beneficial for small datasets.
  • Implementation Details: Training uses Adam with batch size 64, initial learning rate 0.0005, cosine decay, up to 30 epochs, early stopping, and a linear schedule with T = 1000 diffusion steps.The schedule sets β1 = 0.0001 and βT = 0.1.
  • Implementation Details: Inference uses DDIM with M = 20 sampling steps unless otherwise specified to accelerate sampling while maintaining comparable performance.

D Implementation of Breakpoint Estimation and Bernoulli Sampling … F.2 Noise schedule design

The paper specifies concrete breakpoint-estimation and Bernoulli-sampling procedures, evaluates forecasts under standardized multi-dataset protocols, and reports that truncating reverse diffusion preserves temporal structure while improving robustness across noise schedules.

  • D Implementation of Breakpoint Estimation and Bernoulli Sampling: Breakpoint estimation uses N = 128 validation samples, S = 100 candidate samples, and 16 candidate tδ values after excluding 10% of timesteps at both ends.The sampling set uses T = 1000 diffusion steps and M = 20, giving K = {t20 = 1000, t19 = 950, . . . , t1 = 50}.
  • D Implementation of Breakpoint Estimation and Bernoulli Sampling: The Bernoulli sampler uses p = 0.9 to favor the high-noise region, updates tδ after each training epoch, and adds negligible computational overhead.The overhead comparison is reported in Tab. 4 relative to overall training and inference costs.
  • E Datasets And Evaluation Protocol: Experiments cover ETTh1, ETTh2, ETTm1, ETTm2, Weather, Electricity, Exchange, and Traffic using dataset-specific train/validation/test splits and training-statistics standardization.The ETT datasets use 12/4/4-month splits, while the other datasets use 7:1:2 splits; variables are standardized to zero mean and unit variance.
  • F Additional Ablations And Analyses: Additional experiments analyze omitted ablations and analyses, including qualitative forecasts and noise-schedule effects.These experiments are presented because of limited space in the main paper.
  • F.1 Qualitative forecasts: Across four representative datasets, early-stopped sampling produces more concentrated uncertainty and better preserves trends, periodicity, peaks, and recurring temporal patterns than full sampling.Full sampling often yields increasingly dispersed predictive intervals and median trajectories that deviate from dominant temporal patterns, especially on Weather and ETTh1; similar behavior appears on Traffic and Electricity.
  • F.2 Noise schedule design: Under varied diffusion schedules and step counts, the method consistently improves MSE and MAE across datasets and configurations, whereas full sampling can degrade substantially as T increases, especially on Exchange.The comparison is reported in Tab. A4 and indicates that the method's effectiveness is not tied to a particular schedule or diffusion-step count.

F.3 Sensitivity to stopping-point estimation · F.4 Inference Time

The estimated stopping point lies in a stable, broadly near-optimal region, making performance relatively robust to small estimation errors. Early stopping removes substantial reverse-sampling computation and lowers inference time across datasets while remaining slightly slower than NsDiff in absolute runtime.

  • F.3 Sensitivity to stopping-point estimation: MSE curves remain relatively flat around the estimated stopping point, indicating robustness to small stopping-point estimation errors.The analysis identifies a stable termination region across ETTm1, Weather, and Exchange.
  • F.3 Sensitivity to stopping-point estimation: 0.330, 0.198, and 0.174 are the selected boundary’s MSE values on ETTm1, Weather, and Exchange, respectively.The selected boundary achieves near-optimal performance on all three evaluated datasets.
  • F.4 Inference Time: The inference-time evaluation generates one forecast sample with prediction length P = 192 using S = 100 candidates.Table A3 reports inference time in seconds and defines Save as the percentage of reverse-sampling computation removed by early stopping relative to the complete trajectory.
  • F.3 Sensitivity to stopping-point estimation: The sensitivity analysis evaluates stopping steps on ETTm1, Weather, and Exchange with noise scaled by T/M = 50.The estimated stopping point is marked by a circle in Figure A2.
  • F.4 Inference Time: 35–55% of reverse-sampling computation is removed across datasets by early stopping.This reduction is reported for the method evaluated in Table A3.
  • F.4 Inference Time: The method consistently achieves lower inference time than CSDI, TMDM, D3U, and CNDiff, especially on Electricity and Traffic.NsDiff is slightly faster in absolute runtime, although the passage ends before stating the remaining comparison.

G Full Results · H Reproducibility Statement

The full-results section points readers to comprehensive forecasting results across all forecasting lengths and datasets. The reproducibility statement records the implementation framework, hardware, and use of 3 random…

  • G Full Results: Forecasting performance is reported for every forecasting length setting in Tables A5 and A6.These tables cover the ETT benchmarks and remaining datasets.
  • H Reproducibility Statement: All models are implemented in PyTorch and trained on a single A100 GPU with 40 GB of VRAM.
  • H Reproducibility Statement: The training setup uses 3 random…The supplied passage ends after this phrase.
  • H Reproducibility Statement: Table A4 compares forecasting performance under different noise schedules and diffusion steps for prediction length P = 192.Bold values indicate better performance.
  • G Full Results: Table A5 presents full forecasting results on the ETT benchmarks, with best and second-best results highlighted.Rankings use the original unrounded values.
  • G Full Results: Table A6 presents full forecasting results on the remaining datasets, highlighting the best and second-best results.
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