Technical Summary: DeepLévy

Problem Statement

Deep probabilistic forecasting models, such as DeepAR and DeepVAR, have demonstrated success in modeling temporal dependencies. However, they predominantly rely on the assumption that noise follows light-tailed distributions (e.g., Gaussian or Student's t) with finite variance. This assumption fundamentally fails in heavy-tailed regimes (e.g., financial market crashes or epidemic surges) where variance may be infinite and extreme deviations decay polynomially rather than exponentially.

Existing methods struggle to capture "black swan" events because standard parametric families cannot express the infinite variance and extreme asymmetry inherent in such data. Consequently, Gaussian-based models systematically underestimate the probability of catastrophic events, leading to critical failures in tail risk assessment. While Lévy $\alpha$-stable distributions offer a rigorous framework for modeling these non-Gaussian behaviors (characterized by stability index $\alpha$, skewness $\beta$, scale $\gamma$, and location $\delta$), their direct application in deep learning is hindered by the lack of closed-form probability density functions (PDFs), making conventional maximum likelihood estimation (MLE) intractable.

Methodology: DeepLévy

The authors introduce DeepLévy, a neural framework designed to learn mixtures of Lévy stable distributions for multi-horizon probabilistic forecasting without requiring closed-form PDFs.

Core Architecture

DeepLévy employs an encoder-decoder structure:

1. Sequence Encoder: Maps historical observations $x_{1:T}$ to a context-aware latent representation $c$ using architectures like Transformers or LSTMs.
2. Autoregressive Decoder: Generates horizon-specific hidden states $h^{(h)}$ by incorporating the latent context and autoregressive feedback from previous time steps.
3. Constrained Projection Layer: Transforms latent representations into valid mixture parameters for $K$ components. To ensure mathematical validity and numerical stability, the network outputs are projected through specific constraints:
   • Mixing Weights ($\pi$): Enforced via Softmax to sum to 1.
   • Stability Index ($\alpha$): Projected to $[0.1, 1.95]$ to avoid singularities near 0 and 2.
   • Skewness ($\beta$): Projected to $(-0.98, 0.98)$ to avoid numerical fragility at boundaries.
   • Scale ($\gamma$): Projected to be strictly positive.
   • Location ($\delta$): Unconstrained real values.

Training via Characteristic Function Matching

Since the PDF is intractable, DeepLévy operates in the spectral domain by matching the Characteristic Function (CF).

• Objective: The model minimizes the weighted squared distance between the predicted mixture CF and the empirical CF of the target observations over a discrete frequency grid.
• Scale-Adaptive Frequency Weighting: To address the rapid decay of the signal-to-noise ratio in empirical CFs at high frequencies, the authors introduce a weighting scheme that dynamically adapts to the predicted distribution's decay rate. This prevents overfitting to high-frequency noise.
• Parameterization: The framework utilizes Nolan's $S_0$-parameterization, which ensures continuity of the characteristic function at $\alpha = 1$, resolving numerical instability issues common in standard parameterizations.
• Regularization: An entropy regularization term is added to the loss to prevent mode collapse and encourage the utilization of multiple mixture components.

Inference

During inference, the model generates predictive samples using the Chambers-Mallows-Stuck (CMS) algorithm, which allows for efficient sampling from the heavy-tailed predictive distribution defined by the learned parameters.

Key Contributions

1. Novel Framework: The paper proposes the first deep neural architecture that effectively integrates the full parameter space of stable distributions ($\alpha, \beta, \gamma, \delta$) with deep learning for time series forecasting, moving beyond finite-variance assumptions.
2. Robust Optimization Strategy: The authors address the numerical instability of stable distributions through:
   • A scale-adaptive frequency weighting scheme.
   • The use of continuous $S_0$-parameterization.
   • Constrained parameter projections to maintain mathematical validity.
3. Spectral-Domain Learning: By matching characteristic functions rather than maximizing intractable likelihoods, the method enables end-to-end training of heavy-tailed models via standard gradient descent.

Experimental Results

DeepLévy was evaluated on three datasets: Bitcoin Returns, COVID-19 Cases, and Synthetic Lévy sequences.

• Tail Risk Performance: DeepLévy significantly outperforms state-of-the-art baselines (including DeepAR with Gaussian, Laplace, Student's t, and mixture variants, as well as DeepVAR and ProbTransformer) in tail-risk metrics.
  • On Bitcoin Returns, DeepLévy achieved a Tail-CRPS of 0.441 compared to 0.468 for the best baseline (Mixture Student's t).
  • Coverage: At the extreme 0.995 quantile, DeepLévy achieved a coverage of 0.979 (deviation 0.016), whereas the best baseline reached 0.968.
• Ablation Studies:
  • Removing the scale-adaptive weighting resulted in a significant degradation in Tail-CRPS (e.g., 0.522 vs. 0.441 on Bitcoin), confirming the necessity of frequency re-weighting.
  • Using the discontinuous $S_1$ parameterization instead of $S_0$ led to worse performance, particularly in volatile regimes, highlighting the importance of numerical stability.
  • The full mixture model ($K>1$) outperformed the single-component variant, demonstrating the value of capturing multi-modal tail dynamics.
• Trade-off: The results indicate a "center-tail trade-off." While asymmetric/mixture Student's t variants often perform better on global metrics (like standard CRPS), DeepLévy provides superior accuracy in extreme-tail regions, which is critical for risk assessment.

Significance and Claims

The paper claims that DeepLévy demonstrates the feasibility of integrating Lévy stable distributions with deep neural networks for heavy-tailed uncertainty learning. By relaxing the finite-variance assumption, the framework enables more accurate quantification of tail risks in high-stakes domains like financial risk assessment and epidemiological crisis management.

The authors acknowledge limitations, including the reliance on a parametric stable-family assumption and the requirement for larger batch sizes compared to likelihood-based methods to stabilize the empirical CF estimation. However, they assert that the improvements in tail calibration justify these costs for applications where underestimating extreme events is unacceptable. The work suggests that future research could explore multivariate stable distributions and temporal dependence in tail parameters.