SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Approximation of  $\log p_{\theta}(\mathbf{y})$ . To approximate the log-likelihood  $\log p_{\theta}(\mathbf{y})$  we can utilize the objective used to train diffusion models,
$$
\log p _ {\theta} (\mathbf {y}) \approx - \mathbb {E} _ {\boldsymbol {\epsilon}, t} \left[ \left\| \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} ^ {t}, t\right) - \boldsymbol {\epsilon} \right\| ^ {2} \right], \tag {18}
$$
which is a simplification of the ELBO [20]. However, a good approximation requires sampling several diffusion steps  $(t)$  incurring computational overhead and slowing down inference. To speed up inference, we propose to approximate Eq. (18) using only a single diffusion step. Instead of randomly sampling  $t$ , we use the representative step,  $\tau$ , which improved refinement stability in our experiments. The representative step corresponds to the diffusion step that best approximates Eq. (18), i.e.,
$$
\tau = \underset {\tilde {t}} {\arg \min } \left(\mathbb {E} _ {\boldsymbol {\epsilon}, t, \mathbf {y}} \left[ \| \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} ^ {t}, t\right) - \boldsymbol {\epsilon} \| ^ {2} \right] - \mathbb {E} _ {\boldsymbol {\epsilon}, \mathbf {y}} \left[ \| \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {x} ^ {\tilde {t}}, \tilde {t}\right) - \boldsymbol {\epsilon} \| ^ {2} \right]\right) ^ {2}. \tag {19}
$$
The representative step is computed only once per dataset. It can be efficiently computed post training by computing the loss at every diffusion step on a randomly-sampled batch of training datapoints and then finding the diffusion step closest to the average loss. Alternatively, we can keep a running average loss for each  $t$  and compute the representative step using these running averages. This can be done efficiently because the loss used to obtain  $\tau$  is the same as the training loss for the diffusion model.
In this section, we present empirical results on several real-world datasets. Our goal is to investigate whether unconditional time series diffusion models can be employed for downstream tasks that typically require conditional models. Concretely, we tested if (a) the self-guidance mechanism in TSDiff can generate probabilistic forecasts (also in the presence of missing values), (b) the implicit probability density learned by TSDiff can be leveraged to refine the predictions of base forecasters, and (c) the synthetic samples generated by TSDiff are adequate for training downstream forecasters.<sup>4</sup>
Datasets and Evaluation. We conducted experiments on eight univariate time series datasets from different domains, available in GluonTS [2] — Solar [29], Electricity [11], Traffic [11], Exchange [29], M4-Hourly [37], UberTLC-Hourly [13], KDDCup [16], and Wikipedia [14]. We evaluated the quality of probabilistic forecasts using the continuous ranked probability score (CRPS) [15]. We approximated the CRPS by the normalized average quantile loss using 100 sample paths, and report means and standard deviations over three independent runs (see App. B for details on the datasets and evaluation metric).
4Our code is available at: github.com/amazon-science/unconditional-time-series-diffusion
Table 1: Forecasting results on eight benchmark datasets. The best and second best models have been shown as bold and underlined, respectively.
Method	Solar	Electricity	Traffic	Exchange	M4-Hourly	UberTLC-Hourly	KDDcup	Wikipedia
Seasonal Naive	0.512±0.000	0.069±0.000	0.221±0.000	0.011±0.000	0.048±0.000	0.299±0.000	0.561±0.000	0.410±0.000
ARIMA	0.545±0.006	-	-	0.008±0.000	0.044±0.001	0.284±0.001	0.547±0.003	-
ETS	0.611±0.040	0.072±0.004	0.433±0.050	0.008±0.000	0.042±0.001	0.422±0.001	0.753±0.008	0.715±0.002
Linear	0.569±0.021	0.088±0.008	0.179±0.003	0.011±0.001	0.039±0.001	0.360±0.023	0.513±0.011	1.624±1.114
DeepAR	0.389±0.001	0.054±0.000	0.099±0.001	0.011±0.003	0.052±0.006	0.161±0.002	0.414±0.027	0.231±0.008
MQ-CNN	0.790±0.063	0.067±0.001	-	0.019±0.006	0.046±0.003	0.436±0.020	0.516±0.012	0.220±0.001
DeepState	0.379±0.002	0.075±0.004	0.146±0.018	0.011±0.001	0.041±0.002	0.288±0.087	-	0.318±0.019
Transformer	0.419±0.008	0.076±0.018	0.102±0.002	0.010±0.000	0.040±0.014	0.192±0.004	0.411±0.021	0.214±0.001
TFT	0.417±0.023	0.086±0.008	0.134±0.007	0.007±0.000	0.039±0.001	0.193±0.006	0.581±0.053	0.229±0.006
CSDI	0.352±0.005	0.054±0.000	0.159±0.002	0.033±0.014	0.040±0.003	0.206±0.002	0.318±0.002	0.289±0.017
TSDiff-Cond	0.338±0.014	0.050±0.002	0.094±0.003	0.013±0.002	0.039±0.006	0.172±0.008	0.754±0.007	0.218±0.010
TSDiff-MS	0.391±0.003	0.062±0.001	0.116±0.001	0.018±0.003	0.045±0.000	0.183±0.007	0.325±0.028	0.257±0.001
TSDiff-Q	0.358±0.020	0.049±0.000	0.098±0.002	0.011±0.001	0.036±0.001	0.172±0.005	0.311±0.026	0.221±0.001
4.1 Forecasting using Observation Self-Guidance
We tested the forecasting performance of the two proposed variants of observation self-guidance, mean square guidance (TSDiff-MS) and quantile guidance (TSDiff-Q), against several forecasting baselines. We included Seasonal Naive, ARIMA, ETS, and a Linear (ridge) regression model from the statistical literature [21]. Additionally, we compared against deep learning models that represent various architectural paradigms such as the RNN-based DeepAR [46], the CNN-based MQ-CNN [54], the state space model-based DeepState [42], and the self-attention-based Transformer [53] and TFT [32] models. We also compared against two conditional diffusion models, CSDI [52] and TSDiff-Cond, a conditional version of TSDiff closely related to SSSD [1] (see App. B.4 for a more in-depth discussion on the baselines). Note that we did not seek to obtain state-of-the-art forecasting results on the datasets studied but to demonstrate the efficacy of unconditional diffusion models against task-specific conditional models.
Table 1 shows that TSDiff-Q is competitive with state-of-the-art conditional models, but does not require task-specific training, achieving the lowest or second-lowest CRPS on 5/8 datasets. We also observe that the choice of guidance distribution is critical. While Gaussian (TSDiff-MS) yields reasonable results, using the asymmetric Laplace distribution (TSDiff-Q) lowers the CRPS further. We hypothesize that taking the different quantile levels into account during guidance improves the results on the quantile-based evaluation metric (CRPS). Fig. 3 shows example forecasts from TSDiff-Q. Note that as self-guidance only imposes a soft-constraint on the observed timesteps, the resultant diffused values are not guaranteed to match the observations for these timesteps. The alignment between predictions and observations can be controlled via the scale parameter,  $s$  (see Eq. 9). In
Figure 3: Example forecasts generated by TSDiff-Q for time series in Electricity, KDDCup, and Exchange — three datasets with different frequencies and/or prediction lengths.
practice we observed that the diffused values were close to the observations for our selected values of  $s$ , as shown in Fig. 5 (Appendix).
Forecasting with Missing Values. We evaluated the forecasting performance with missing values in the historical context during inference to demonstrate the flexibility of self-guidance. Specifically, we tested three scenarios: (a) random missing (RM), (b) blackout missing (i.e., a window with consecutive missing values) at the beginning of the context window (BM-B), and (c) blackout missing at the end of the context window (BM-E). We removed a segment equal to the context window from the end of the training time series to ensure that the model is not trained on this section. During inference, we masked  $50\%$  of the timesteps from this held-out context window for each scenario
while the forecast horizon remained unchanged. We trained the unconditional model on complete sequences only once per dataset, without making any modifications to the objective function. As a result, the three scenarios described above are inference only for TSDiff. For comparison, we trained conditional models (TSDiff-Cond) specifically on the target missingness scenarios. These models were trained to predict the missing values in the input time series, where some timesteps were masked according to the target missingness scenario (see App. B.3 for a detailed discussion on the missing values experiment setup).
Table 2: Forecasting with missing values results on six benchmark datasets.
Method	Solar	Electricity	Traffic	Exchange	UberTLC-Hourly	KDDCup
RM	TSDiff-Cond	0.357±0.023	0.052±0.001	0.097±0.003	0.012±0.004	0.180±0.015	0.757±0.024
TSDiff-Q	0.387±0.015	0.052±0.001	0.110±0.004	0.013±0.000	0.183±0.002	0.397±0.042
BM-B	TSDiff-Cond	0.377±0.017	0.049±0.001	0.094±0.005	0.009±0.000	0.181±0.009	0.699±0.009
TSDiff-Q	0.387±0.019	0.051±0.000	0.110±0.006	0.011±0.001	0.182±0.004	0.441±0.096
BM-E	TSDiff-Cond	0.376±0.036	0.065±0.003	0.123±0.023	0.035±0.021	0.179±0.013	0.819±0.033
TSDiff-Q	0.435±0.113	0.068±0.009	0.139±0.013	0.020±0.001	0.183±0.005	0.344±0.012
The results in Table 2 show that TSDiff-Q performs competitively against task-specific conditional models, demonstrating its robustness with respect to missing values during inference. This makes TSDiff-Q particularly useful in real-world applications where missing values are common but the missingness scenario is not known in advance. It is noteworthy that we did not tune the guidance scale for each missingness scenario but used the same value as in the previous standard forecasting setup. We expect the results to further improve if the scale is carefully updated based on the missingness ratio and scenario.
4.2 Refining Predictions of Base Forecasters
We evaluated the quality of the implicit probability density learned by TSDiff by refining predictions from base forecasters, as described in Sec. 3.2. We considered two point forecasters (Seasonal Naive and Linear) and two probabilistic forecasters (DeepAR and Transformer) as base models. We refine the forecasts generated from these base models for 20 steps, as described in Algorithm 2 (see App. B.3 for a discussion on the number of refinement steps). Similar to self-guidance, we experimented with the Gaussian and asymmetric Laplace negative log-likelihoods as regularizers,  $\mathcal{R}$ , for both energy (denoted by LMC) and maximum likelihood (denoted by ML)-based refinement.
The CRPS scores before (denoted as "Base") and after refinement are reported in Table 3. At least one refinement method reduces the CRPS of the base forecast on every dataset. Energy-based refinement generally performs better than ML-based refinement for point forecasters. We hypothesize that this is
Table 3: Refinement results on eight benchmark datasets. The best and second best settings have been shown as bold and underlined, respectively.
Setting	Solar	Electricity	Traffic	Exchange	M4-Hourly	UberTLC-Hourly	KDDcup	Wikipedia
Seasonal Naive Linear	Base	0.512±0.000	0.069±0.000	0.221±0.000	0.011±0.000	0.048±0.000	0.299±0.000	0.561±0.000	0.410±0.000
LMC-MS	0.480±0.009	0.059±0.004	0.126±0.001	0.013±0.001	0.040±0.002	0.186±0.005	0.505±0.027	0.339±0.001
LMC-Q	0.480±0.007	0.051±0.001	0.134±0.004	0.009±0.000	0.036±0.001	0.204±0.007	0.399±0.003	0.357±0.001
ML-MS	0.489±0.007	0.064±0.006	0.130±0.002	0.015±0.002	0.046±0.003	0.202±0.004	0.519±0.028	0.349±0.001
ML-Q	0.480±0.007	0.050±0.001	0.135±0.004	0.009±0.000	0.036±0.001	0.215±0.008	0.403±0.003	0.365±0.001
Base	0.569±0.021	0.088±0.008	0.179±0.003	0.011±0.001	0.039±0.001	0.360±0.023	0.513±0.011	1.624±1.114
LMC-MS	0.494±0.019	0.059±0.004	0.113±0.001	0.013±0.001	0.040±0.002	0.187±0.007	0.458±0.015	1.315±0.992
LMC-Q	0.516±0.020	0.055±0.003	0.119±0.002	0.009±0.000	0.034±0.001	0.228±0.010	0.346±0.010	1.329±1.002
ML-MS	0.503±0.016	0.063±0.005	0.117±0.002	0.015±0.002	0.045±0.003	0.203±0.007	0.472±0.015	1.327±0.993
ML-Q	0.523±0.021	0.056±0.003	0.121±0.003	0.010±0.001	0.032±0.001	0.240±0.010	0.350±0.011	1.335±1.002
DeepAR	Base	0.389±0.001	0.054±0.000	0.099±0.001	0.011±0.003	0.052±0.006	0.161±0.002	0.414±0.027	0.231±0.008
LMC-MS	0.398±0.004	0.059±0.004	0.111±0.001	0.012±0.001	0.040±0.002	0.184±0.005	0.469±0.034	0.227±0.002
LMC-Q	0.388±0.002	0.053±0.001	0.101±0.001	0.010±0.001	0.035±0.001	0.161±0.002	0.401±0.021	0.220±0.005
ML-MS	0.402±0.009	0.064±0.006	0.115±0.002	0.014±0.001	0.046±0.003	0.198±0.005	0.477±0.034	0.235±0.002
ML-Q	0.386±0.002	0.052±0.001	0.099±0.001	0.010±0.002	0.035±0.001	0.160±0.002	0.401±0.021	0.221±0.006
Transformer	Base	0.419±0.008	0.076±0.018	0.102±0.002	0.010±0.000	0.040±0.014	0.192±0.004	0.411±0.021	0.214±0.001
LMC-MS	0.415±0.009	0.059±0.004	0.111±0.001	0.013±0.001	0.040±0.002	0.185±0.005	0.462±0.014	0.229±0.003
LMC-Q	0.415±0.008	0.058±0.003	0.101±0.001	0.010±0.000	0.038±0.006	0.177±0.005	0.384±0.005	0.211±0.002
ML-MS	0.418±0.010	0.063±0.005	0.115±0.002	0.014±0.002	0.046±0.003	0.198±0.005	0.470±0.014	0.238±0.003
ML-Q	0.413±0.008	0.059±0.005	0.099±0.001	0.010±0.000	0.037±0.006	0.177±0.005	0.384±0.006	0.210±0.002
Table 4: Results of forecasters trained on synthetic samples from different generative models on eight benchmark datasets. Best scores are shown in bold.
Generator	Solar	Electricity	Traffic	Exchange	M4-Hourly	UberTLC-Hourly	KDDcup	Wikipedia
Linear (LPS)	Real	0.569±0.021	0.088±0.008	0.179±0.003	0.011±0.001	0.039±0.001	0.360±0.023	0.513±0.011	1.624±1.114
TimeVAE	0.933±0.147	0.128±0.005	0.236±0.010	0.024±0.004	0.074±0.003	0.354±0.020	1.020±0.179	0.643±0.068
TimeGAN	1.140±0.583	0.234±0.064	0.398±0.092	0.011±0.000	0.140±0.053	0.665±0.104	0.713±0.009	0.421±0.023
TSDiff	0.581±0.032	0.065±0.002	0.164±0.002	0.012±0.001	0.045±0.007	0.291±0.084	0.481±0.013	0.392±0.013
DeepAR	Real	0.389±0.001	0.054±0.000	0.099±0.001	0.011±0.003	0.052±0.006	0.161±0.002	0.414±0.027	0.231±0.008
TimeVAE	0.493±0.012	0.060±0.001	0.155±0.006	0.009±0.000	0.039±0.010	0.278±0.009	0.621±0.003	0.440±0.012
TimeGAN	0.976±0.739	0.183±0.036	0.419±0.122	0.008±0.001	0.121±0.035	0.594±0.125	0.690±0.091	0.322±0.048
TSDiff	0.478±0.007	0.058±0.001	0.129±0.003	0.017±0.009	0.042±0.024	0.191±0.018	0.378±0.012	0.222±0.005
Transf.	Real	0.419±0.008	0.076±0.018	0.102±0.002	0.010±0.000	0.040±0.014	0.192±0.004	0.411±0.021	0.214±0.001
TimeVAE	0.520±0.030	0.071±0.009	0.163±0.018	0.011±0.001	0.035±0.011	0.291±0.008	0.717±0.181	0.451±0.017
TimeGAN	0.972±0.687	0.182±0.008	0.413±0.204	0.009±0.001	0.114±0.052	0.685±0.448	0.632±0.016	0.314±0.045
TSDiff	0.457±0.008	0.056±0.001	0.143±0.020	0.030±0.021	0.030±0.008	0.225±0.055	0.356±0.030	0.239±0.010
because the additive noise in LMC incentivizes exploration of the energy landscape which improves the "spread" of the initial point forecast. In contrast, ML-Q refinement outperforms energy-based refinement on probabilistic base models suggesting that sampling might not be necessary for these models due to the probabilistic nature of the initial forecasts.
Refinement provides a trade-off; while it has a lower computational overhead than self-guidance, its performance strongly depends on the chosen base forecaster. It substantially improves the performance of simple point forecasters (e.g., Seasonal Naive) and yields improvements on stronger probabilistic forecasters (e.g., Transformer) as well.
4.3 Training Downstream Models using Synthetic Data
Finally, we evaluated the quality of generated samples through the forecasting performance of downstream models trained on these synthetic samples. Several metrics have previously been proposed to evaluate the quality of synthetic samples [12, 57, 26, 23]. Of these, we primarily focused on predictive metrics that involve training a downstream model on synthetic samples and evaluating its performance on real data (i.e., the train on synthetic-test on real setup). In the absence of a standard downstream model (such as the Inception network [51] in the case of images), prior works have proposed different network architectures for the downstream model. This makes the results sensitive to architecture choice, random initialization, and even the choice of deep learning library. Furthermore, training a downstream model introduces additional overhead per metric computation. We propose the Linear Predictive Score (LPS) to address these issues. We define the LPS as the test CRPS of a linear (ridge) regression model trained on the synthetic samples. The ridge regression model is a simple, standard model available in standard machine learning libraries (e.g., scikit-learn) that can effectively gauge the predictive quality of synthetic samples. Moreover, a ridge regression model is cheap to fit — it can be solved in closed-form, eliminating variance introduced by initialization and training.
We compared samples generated by TSDiff against those from TimeGAN [57] and TimeVAE [9], two time series generative models from alternative frameworks. In addition to the linear model, we trained two strong downstream forecasters (DeepAR and Transformer) on synthetic samples from each generative model. The test CRPS of these forecasters is presented in Table 4. TSDiff's samples significantly outperform TimeVAE and TimeGAN in terms of the LPS showcasing their excellent predictive quality with respect to a simple (linear) forecaster. On the stronger DeepAR and Transformer forecasters, TSDiff outperforms the baselines on most of the datasets. Moreover, the scores obtained by TSDiff are reasonable when compared to those attained by downstream forecasters trained on real samples. Notably, these forecasters trained on real data had the advantage of accessing time features and lags that extend far beyond the sequence length modeled by TSDiff. These results serve as evidence that TSDiff effectively captures crucial patterns within time series datasets and generates realistic samples (see App. C for a qualitative comparison between real time series and those generated by TSDiff).
Diffusion Models. Diffusion models were originally proposed for image synthesis [48, 20, 10], but have since been applied to other domains such as audio synthesis [28], protein modeling [3], and graph modeling [24]. Prior works have also studied image inpainting using diffusion models [35, 45, 25] which is a problem related to time series imputation. Similar to the maximum likelihood variant of our refinement approach, Graikos et al. [17] showed the utility of diffusion models as plug-and-play priors. The architecture of our proposed model, TSDiff, is based on a modification [1] of DiffWave [28] which was originally developed for audio synthesis.
Diffusion Models for Time Series. Conditional diffusion models have been applied to time series tasks such as imputation and forecasting. The first work is due to Rasul et al. [44], who proposed TimeGrad for multivariate time series forecasting featuring a conditional diffusion head. Biloš et al. [6] extended TimeGrad to continuous functions and made modifications to the architecture enabling simultaneous multi-horizon forecasts. CDSI [52] is a conditional diffusion model for imputation and forecasting which is trained by masking the observed time series with different strategies. SSSD [1] is a modification of CDSI that uses S4 layers [18] as the fundamental building block instead of transformers. The models discussed above [44, 52, 1] are all conditional models, i.e., they are trained on specific imputation or forecasting tasks. In contrast, TSDiff is trained unconditionally and conditioned during inference using diffusion guidance, making it applicable to various downstream tasks.
Diffusion Guidance. Dhariwal and Nichol [10] proposed classifier guidance to repurpose pretrained unconditional diffusion models for conditional image generation using auxiliary image classifiers. Ho and Salimans [19] eliminated the need for an additional classifier by jointly training a conditional and an unconditional diffusion model and combining their score functions for conditional generation. Diffusion guidance has also been employed for text-guided image generation and editing [39, 4] using CLIP embeddings [41]. In contrast to prior work, our proposed observation self-guidance does not require auxiliary networks or modifications to the training procedure. Furthermore, we apply diffusion guidance to the time series domain while previous works primarily focus on images.
In this work, we proposed TSDiff, an unconditional diffusion model for time series, and a self-guidance mechanism that enables conditioning TSDiff for probabilistic forecasting tasks during inference, without requiring auxiliary guidance networks or modifications to the training procedure. We demonstrated that our task-agnostic TSDiff, in combination with self-guidance, is competitive with strong task-specific baselines on multiple forecasting tasks. Additionally, we presented a refinement scheme to improve predictions of base forecasters by leveraging the implicit probability density learned by TSDiff. Finally, we validated its generative modeling capabilities by training multiple downstream forecasters on synthetic samples generated by TSDiff. Samples from TSDiff outperform alternative generative models in terms of their predictive quality. Our results indicate that TSDiff learns important characteristics of time series datasets, enabling conditioning during inference and high-quality sample generation, offering an attractive alternative to task-specific conditional models.
Limitations and Future Work. While observation self-guidance provides an alternative approach for state-of-the-art probabilistic time series forecasting, its key limitation is the high computational cost of the iterative denoising process. Faster diffusion solvers [49, 33] would provide a trade-off between predictive performance and computational overhead without altering the training procedure. Furthermore, solvers developed specifically for accelerated guided diffusion [34, 55] could be deployed to speed up sampling without sacrificing predictive performance. Forecast refinement could be further improved by better approximations of the probability density and the utilization of momentum-based MCMC techniques such as Hamiltonian Monte Carlo [38] and underdamped Langevin Monte Carlo [50]. In this work, we evaluated TSDiff in the context of probabilistic forecasting, however, it is neither limited nor tailored to it. It can be easily applied to any imputation task similar to our forecasting with missing values experiment where we only evaluated the forecasting performance. Moreover, the core idea behind observation self-guidance is not limited to time series and may be applicable to other domains. Tailoring the guidance distribution appropriately, enables the use of self-guidance to a variety of inverse problems.
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A.1 Asymmetric Laplace Distribution