Theoretical Guarantees of Learning Ensembling Strategies with Applications to Time Series Forecasting

Ensembling is among the most popular tools in machine learning (ML) due to its effectiveness in minimizing variance and thus improving generalization. Most ensembling methods for black-box base learners fall under the umbrella of "stacked generalization," namely training an ML algorithm that takes the inferences from the base learners as input. While stacking has been widely applied in practice, its theoretical properties are poorly understood. In this paper, we prove a novel result, showing that choosing the best stacked generalization from a (finite or finite-dimensional) family of stacked generalizations based on cross-validated performance does not perform "much worse" than the oracle best. Our result strengthens and significantly extends the results in Van der Laan et al. (2007). Inspired by the theoretical analysis, we further propose a particular family of stacked generalizations in the context of probabilistic forecasting, each one with a different sensitivity for how much the ensemble weights are allowed to vary across items, timestamps in the forecast horizon, and quantiles. Experimental results demonstrate the performance gain of the proposed method.

Testing Causality for High Dimensional Data

Determining causal relationship between high dimensional observations are among the most important tasks in scientific discoveries. In this paper, we revisited the \emph{linear trace method}, a technique proposed in~\citep{janzing2009telling,zscheischler2011testing} to infer the causal direction between two random variables of high dimensions. We strengthen the existing results significantly by providing an improved tail analysis in addition to extending the results to nonlinear trace functionals with sharper confidence bounds under certain distributional assumptions. We obtain our results by interpreting the trace estimator in the causal regime as a function over random orthogonal matrices, where the concentration of Lipschitz functions over such space could be applied. We additionally propose a novel ridge-regularized variant of the estimator in \cite{zscheischler2011testing}, and give provable bounds relating the ridge-estimated terms to their ground-truth counterparts. We support our theoretical results with encouraging experiments on synthetic datasets, more prominently, under high-dimension low sample size regime.

Adaptive Sampling for Probabilistic Forecasting under Distribution Shift

The world is not static: This causes real-world time series to change over time through external, and potentially disruptive, events such as macroeconomic cycles or the COVID-19 pandemic. We present an adaptive sampling strategy that selects the part of the time series history that is relevant for forecasting. We achieve this by learning a discrete distribution over relevant time steps by Bayesian optimization. We instantiate this idea with a two-step method that is pre-trained with uniform sampling and then training a lightweight adaptive architecture with adaptive sampling. We show with synthetic and real-world experiments that this method adapts to distribution shift and significantly reduces the forecasting error of the base model for three out of five datasets.

First De-Trend then Attend: Rethinking Attention for Time-Series Forecasting

Transformer-based models have gained large popularity and demonstrated promising results in long-term time-series forecasting in recent years. In addition to learning attention in time domain, recent works also explore learning attention in frequency domains (e.g., Fourier domain, wavelet domain), given that seasonal patterns can be better captured in these domains. In this work, we seek to understand the relationships between attention models in different time and frequency domains. Theoretically, we show that attention models in different domains are equivalent under linear conditions (i.e., linear kernel to attention scores). Empirically, we analyze how attention models of different domains show different behaviors through various synthetic experiments with seasonality, trend and noise, with emphasis on the role of softmax operation therein. Both these theoretical and empirical analyses motivate us to propose a new method: TDformer (Trend Decomposition Transformer), that first applies seasonal-trend decomposition, and then additively combines an MLP which predicts the trend component with Fourier attention which predicts the seasonal component to obtain the final prediction. Extensive experiments on benchmark time-series forecasting datasets demonstrate that TDformer achieves state-of-the-art performance against existing attention-based models.

Towards Robust Multivariate Time-Series Forecasting: Adversarial Attacks and Defense Mechanisms

As deep learning models have gradually become the main workhorse of time series forecasting, the potential vulnerability under adversarial attacks to forecasting and decision system accordingly has emerged as a main issue in recent years. Albeit such behaviors and defense mechanisms started to be investigated for the univariate time series forecasting, there are still few studies regarding the multivariate forecasting which is often preferred due to its capacity to encode correlations between different time series. In this work, we study and design adversarial attack on multivariate probabilistic forecasting models, taking into consideration attack budget constraints and the correlation architecture between multiple time series. Specifically, we investigate a sparse indirect attack that hurts the prediction of an item (time series) by only attacking the history of a small number of other items to save attacking cost. In order to combat these attacks, we also develop two defense strategies. First, we adopt randomized smoothing to multivariate time series scenario and verify its effectiveness via empirical experiments. Second, we leverage a sparse attacker to enable end-to-end adversarial training that delivers robust probabilistic forecasters. Extensive experiments on real dataset confirm that our attack schemes are powerful and our defend algorithms are more effective compared with other baseline defense mechanisms.

Robust Probabilistic Time Series Forecasting

Probabilistic time series forecasting has played critical role in decision-making processes due to its capability to quantify uncertainties. Deep forecasting models, however, could be prone to input perturbations, and the notion of such perturbations, together with that of robustness, has not even been completely established in the regime of probabilistic forecasting. In this work, we propose a framework for robust probabilistic time series forecasting. First, we generalize the concept of adversarial input perturbations, based on which we formulate the concept of robustness in terms of bounded Wasserstein deviation. Then we extend the randomized smoothing technique to attain robust probabilistic forecasters with theoretical robustness certificates against certain classes of adversarial perturbations. Lastly, extensive experiments demonstrate that our methods are empirically effective in enhancing the forecast quality under additive adversarial attacks and forecast consistency under supplement of noisy observations.

Multivariate Quantile Function Forecaster

We propose Multivariate Quantile Function Forecaster (MQF$^2$), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF$^2$ combines the benefits of both approaches, by directly making predictions in the form of a multivariate quantile function, defined as the gradient of a convex function which we parametrize using input-convex neural networks. By design, the quantile function is monotone with respect to the input quantile levels and hence avoids quantile crossing. We provide two options to train MQF$^2$: with energy score or with maximum likelihood. Experimental results on real-world and synthetic datasets show that our model has comparable performance with state-of-the-art methods in terms of single time step metrics while capturing the time dependency structure.

Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecasting

Quantile regression is an effective technique to quantify uncertainty, fit challenging underlying distributions, and often provide full probabilistic predictions through joint learnings over multiple quantile levels. A common drawback of these joint quantile regressions, however, is \textit{quantile crossing}, which violates the desirable monotone property of the conditional quantile function. In this work, we propose the Incremental (Spline) Quantile Functions I(S)QF, a flexible and efficient distribution-free quantile estimation framework that resolves quantile crossing with a simple neural network layer. Moreover, I(S)QF inter/extrapolate to predict arbitrary quantile levels that differ from the underlying training ones. Equipped with the analytical evaluation of the continuous ranked probability score of I(S)QF representations, we apply our methods to NN-based times series forecasting cases, where the savings of the expensive re-training costs for non-trained quantile levels is particularly significant. We also provide a generalization error analysis of our proposed approaches under the sequence-to-sequence setting. Lastly, extensive experiments demonstrate the improvement of consistency and accuracy errors over other baselines.

Variance Reduction in Training Forecasting Models with Subgroup Sampling

In real-world applications of large-scale time series, one often encounters the situation where the temporal patterns of time series, while drifting over time, differ from one another in the same dataset. In this paper, we provably show under such heterogeneity, training a forecasting model with commonly used stochastic optimizers (e.g. SGD) potentially suffers large gradient variance, and thus requires long time training. To alleviate this issue, we propose a sampling strategy named Subgroup Sampling, which mitigates the large variance via sampling over pre-grouped time series. We further introduce SCott, a variance reduced SGD-style optimizer that co-designs subgroup sampling with the control variate method. In theory, we provide the convergence guarantee of SCott on smooth non-convex objectives. Empirically, we evaluate SCott and other baseline optimizers on both synthetic and real-world time series forecasting problems, and show SCott converges faster with respect to both iterations and wall clock time. Additionally, we show two SCott variants that can speed up Adam and Adagrad without compromising generalization of forecasting models.

Attention-based Domain Adaptation for Time Series Forecasting

Recent years have witnessed deep neural networks gaining increasing popularity in the field of time series forecasting. A primary reason of their success is their ability to effectively capture complex temporal dynamics across multiple related time series. However, the advantages of these deep forecasters only start to emerge in the presence of a sufficient amount of data. This poses a challenge for typical forecasting problems in practice, where one either has a small number of time series, or limited observations per time series, or both. To cope with the issue of data scarcity, we propose a novel domain adaptation framework, Domain Adaptation Forecaster (DAF), that leverages the statistical strengths from another relevant domain with abundant data samples (source) to improve the performance on the domain of interest with limited data (target). In particular, we propose an attention-based shared module with a domain discriminator across domains as well as private modules for individual domains. This allows us to jointly train the source and target domains by generating domain-invariant latent features while retraining domain-specific features. Extensive experiments on various domains demonstrate that our proposed method outperforms state-of-the-art baselines on synthetic and real-world datasets.