Variational Inference with Normalizing Flows (1505.05770v6)

Abstract: The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference, focusing on mean-field or other simple structured approximations. This restriction has a significant impact on the quality of inferences made using variational methods. We introduce a new approach for specifying flexible, arbitrarily complex and scalable approximate posterior distributions. Our approximations are distributions constructed through a normalizing flow, whereby a simple initial density is transformed into a more complex one by applying a sequence of invertible transformations until a desired level of complexity is attained. We use this view of normalizing flows to develop categories of finite and infinitesimal flows and provide a unified view of approaches for constructing rich posterior approximations. We demonstrate that the theoretical advantages of having posteriors that better match the true posterior, combined with the scalability of amortized variational approaches, provides a clear improvement in performance and applicability of variational inference.

• The paper presents normalizing flows as a method to construct flexible, complex approximate posteriors that overcome the limitations of simple mean-field approximations.
• The paper leverages both finite and infinitesimal flow transformations to maintain computational tractability while capturing detailed posterior structures.
• The paper empirically validates its approach on benchmark datasets, demonstrating significant improvements over traditional variational inference methods.

Variational Inference with Normalizing Flows

The paper "Variational Inference with Normalizing Flows" by Danilo Jimenez Rezende and Shakir Mohamed presents an innovative approach to improve the flexibility and accuracy of variational inference (VI) by utilizing normalizing flows. The key contribution of this work lies in addressing the core problem of selecting an approximate posterior distribution in VI, which traditionally relies on simple families of distributions such as mean-field approximations. These simple approximations often fail to capture the true complexity of the posterior distribution, leading to suboptimal inference outcomes.

Contributions and Theoretical Foundations

The main contributions of the paper can be summarized as follows:

1. Normalizing Flows for Approximate Posteriors: The authors propose the use of normalizing flows to construct flexible, arbitrarily complex approximate posterior distributions. A normalizing flow transforms a simple initial density through a sequence of invertible mappings, thereby generating a more complex final density. This method allows for a better match with the true posterior distribution compared to traditional approaches.
2. Finite and Infinitesimal Flows: The paper differentiates between finite and infinitesimal flows. Finite flows involve a finite sequence of transformations, while infinitesimal flows represent the continuous-time limit of these transformations. The latter can recover the true posterior distribution in the asymptotic regime.
3. Efficient Inference Mechanism: The paper introduces a method for efficient VI using normalizing flows that maintains computational tractability. By leveraging invertible linear-time transformations, the authors ensure that the Jacobian determinants required for density transformations can be computed efficiently.
4. Experimental Validation: The efficacy of normalizing flows is empirically validated on several benchmark datasets. The results demonstrate that normalizing flows provide significant improvements over traditional posterior approximations and other contemporary methods such as Hamiltonian variational inference (HVI) and Non-linear Independent Components Estimation (NICE).

Practical Implications

Normalizing flows substantially enhance the performance and applicability of variational inference by accommodating more complex posterior distributions. Specifically, the ability to specify multi-modal and intricate dependencies in the approximate posterior is crucial for numerous real-world applications. This advancement makes VI more reliable for probabilistic modeling in domains such as natural language processing, computer vision, and scientific data analysis.

Key Findings:

• Quantitative Improvements: The paper reports a notable reduction in the negative log-probabilities on test datasets like MNIST and CIFAR-10 when using normalizing flows, with flow lengths systematically improving the bounds.
• Comparative Performance: In comparison to HVI and NICE, normalizing flows achieve superior results, highlighting their effectiveness in capturing the true posterior distribution.

Theoretical Implications

The introduction of normalizing flows extends the theoretical framework of variational inference, offering a rigorous mechanism to model complex target distributions. This approach also unifies various existing techniques under the broader umbrella of normalizing flows, providing a cohesive understanding of posterior approximation methods. This theoretical advancement paves the way for future research on the design and analysis of more sophisticated transformations within the normalizing flow framework.

Future Directions

The promising results of this paper suggest several avenues for further research:

• Design of New Transformations: Exploring alternative invertible transformations that can capture different characteristics of the posterior while maintaining computational efficiency.
• Scalability: Investigating methods to further enhance the scalability of normalizing flows for extremely high-dimensional data.
• Application-Specific Adaptations: Tailoring normalizing flow-based variational inference for specific applications in scientific research, where capturing complex dependencies is critical.

In conclusion, the paper "Variational Inference with Normalizing Flows" makes a significant contribution to the field of machine learning by proposing a robust and flexible framework for posterior approximation in variational inference. The use of normalizing flows addresses longstanding limitations of traditional VI methods and has the potential to substantially improve the reliability and accuracy of probabilistic modeling in various domains.