Abstract
Normalizing flows are a flexible class of probability distributions, expressed as transformations of a simple base distribution. A limitation of standard normalizing flows is representing distributions with heavy tails, which arise in applications to both density estimation and variational inference. A popular current solution to this problem is to use a heavy tailed base distribution. Examples include the tail adaptive flow (TAF) methods of Laszkiewicz et al (2022). We argue this can lead to poor performance due to the difficulty of optimising neural networks, such as normalizing flows, under heavy tailed input. This problem is demonstrated in our paper. We propose an alternative: use a Gaussian base distribution and a final transformation layer which can produce heavy tails. We call this approach tail transform flow (TTF). Experimental results show this approach outperforms current methods, especially when the target distribution has large dimension or tail weight.
Overview
-
The paper introduces Tail Transform Flow (TTF), a new normalizing flow technique designed to effectively model heavy-tailed distributions by using non-Lipschitz transformations.
-
TTF outperforms traditional approaches, which struggle with heavy tails due to their reliance on Gaussian base distributions and Lipschitz functions, by integrating a final transformation layer to handle the tails directly.
-
Experiments demonstrate TTF's superior performance on synthetic data, financial return data, and variational inference tasks, validating its robustness and applicability to real-world scenarios.
Flexible Tails for Normalizing Flows
The paper "Flexible Tails for Normalizing Flows" by Tennessee Hickling and Dennis Prangle addresses the challenge of modelling probability distributions with heavy tails using normalizing flows (NFs). Traditional NFs, using Lipschitz transformations of Gaussian base distributions, struggle to model heavy tails effectively. The authors propose a novel alternative called tail transform flow (TTF) to overcome this limitation.
Introduction
Normalizing flows represent complex probability distributions through a series of bijective transformations applied to samples from a base distribution, typically Gaussian. These flows are widely applied in density estimation and variational inference, and are optimized by stochastic gradient descent on an objective function. Despite their flexibility, standard NFs are not effective in modelling heavy-tailed distributions such as those encountered in climate modeling, finance, and epidemiology. This inadequacy arises because Gaussian tails cannot be transformed to heavy tails using Lipschitz functions as shown by \citet{jaini_2020}.
Existing Approaches
Current solutions often employ heavy-tailed base distributions. For instance, the tail adaptive flow (TAF) models introduce Student's T distributions as base distributions with degrees of freedom that are optimized along with the NF parameters. However, this method can degrade the performance of neural network optimization due to the stochastic gradients’ heavy-tailed nature \citep{zhang2020why}.
Proposed Method: Tail Transform Flow (TTF)
The proposed TTF approach uses a Gaussian base distribution in combination with a final non-Lipschitz transformation layer designed to produce heavy tails. This final layer, referred to as $R$, is mathematically defined to transform standard normal tails to generalized Pareto distribution (GPD) tails, with tunable parameters for tail heaviness. This avoids the problematic gradient issues associated with heavy-tailed input distributions.
Mathematical Foundation
The tail transform flow (TTF) transformation ( R ) is given by: [ R(z; \lambda+, \lambda-) = \mu + \sigma \frac{s}{\lambdas}[\erfc(|z| / \sqrt{2}){-\lambdas} - 1], ] where (\lambda+ > 0) and (\lambda- > 0) are parameters controlling the tail weights for positive and negative tails respectively. The transformation ensures that the output distribution can have heavy tails, allowing for more accurate modelling of heavy-tailed phenomena.
Experiments
The authors conducted several experiments to validate the effectiveness of the TTF method:
- Synthetic Data: Using a model with varying dimensions and tail weights, TTF significantly outperformed existing methods, particularly in high-dimensional settings with very heavy tails ((\nu < 2)).
- S&P 500 Data: TTF demonstrated superior performance on financial return data, showcasing its practical applicability to real-world data with heavy tails.
- Variational Inference (VI): In a proof-of-concept VI experiment with an artificial target distribution, TTF consistently provided more accurate approximations than methods based on heavy-tailed base distributions.
Conclusion
The TTF method advances the state of the art in normalizing flows by robustly handling heavy-tailed distributions without compromising optimization stability. This approach avoids the degradation of neural network optimization seen with heavy-tailed base distributions, ensuring more reliable modelling in high-dimensional and extreme-value contexts.
Implications and Future Developments
Practically, TTF can be used in a variety of applications requiring accurate tail modelling, such as financial risk management, climate extremes, and epidemiological forecasting. Theoretically, this work expands the capabilities of normalizing flows, encouraging future research to explore more sophisticated and potentially automated methods of tail parameter estimation and transformations that could handle multivariate dependencies in tails.
Future research may focus on improving initialization strategies, extending the tail modelling to capture tail dependencies, and integrating these approaches with simulation-based inference frameworks. Additionally, exploring the applicability of TTF in probabilistic programming and automated Bayesian inference represents an exciting avenue for expanding the practicality and robustness of these methodologies in diverse scientific and industrial applications.
This research lays the groundwork for next-generation methods in machine learning and statistics, fostering more robust and accurate probabilistic modelling tools.
For more details and access to the source code of this research, please visit the GitHub repository.
Create an account to read this summary for free: