EigenVI: A Novel Approach to Black-Box Variational Inference Using Score Matching and Orthogonal Function Expansions

Conceitos Básicos

EigenVI is a new algorithm for black-box variational inference that leverages orthogonal function expansions to construct flexible variational approximations and utilizes score matching to derive a computationally efficient optimization procedure based on solving a minimum eigenvalue problem.

Resumo

• Bibliographic Information: Cai, D., Modi, C., Margossian, C. C., Gower, R. M., Blei, D. M., & Saul, L. K. (2024). EigenVI: score-based variational inference with orthogonal function expansions. Advances in Neural Information Processing Systems, 38.

• Research Objective: This paper introduces EigenVI, a novel method for black-box variational inference (BBVI) that aims to overcome limitations of traditional gradient-based BBVI methods by employing score matching and orthogonal function expansions.

• Methodology: EigenVI constructs variational approximations using orthogonal function expansions, where the lowest order term corresponds to a Gaussian distribution, and higher-order terms introduce non-Gaussianity. The algorithm minimizes the Fisher divergence between the variational approximation and the target distribution, which, due to the chosen structure, reduces to a minimum eigenvalue problem. This approach avoids iterative gradient-based optimization, making EigenVI potentially more robust and computationally efficient.

• Key Findings:

  • EigenVI demonstrates the ability to effectively approximate a variety of complex target distributions, including multimodal, asymmetric, and heavy-tailed distributions.
  • Empirical evaluations on synthetic targets show that EigenVI achieves more accurate approximations compared to Gaussian BBVI methods, particularly for distributions exhibiting significant non-Gaussian characteristics.
  • Experiments on real-world hierarchical Bayesian models from posteriordb benchmark EigenVI against established BBVI algorithms like ADVI, GSM, and BaM. EigenVI consistently yields more accurate posterior approximations, as measured by the empirical Fisher divergence.

• Main Conclusions: EigenVI presents a novel and effective approach to BBVI that leverages the properties of orthogonal function expansions and score matching. The method exhibits advantages in terms of accuracy and computational efficiency compared to existing Gaussian BBVI techniques, particularly for modeling complex, non-Gaussian target distributions.

• Significance: This research contributes to the field of variational inference by introducing a new class of variational families and a computationally efficient optimization method. EigenVI has the potential to impact various domains that rely on probabilistic modeling and inference, such as Bayesian statistics, machine learning, and data analysis.

• Limitations and Future Research:

  • The reliance on importance sampling in EigenVI may introduce challenges for high-dimensional problems. Exploring adaptive importance sampling techniques could address this limitation.
  • Investigating alternative orthogonal function expansions beyond the Hermite polynomial family could further enhance the flexibility and efficiency of EigenVI for specific types of target distributions.
  • Developing iterative versions of EigenVI that process subsets of data points could enable its application to large-scale Bayesian inference problems.

Estatísticas

The lowest-order basis function of the Hermite family, when squared, results in a standard multivariate Gaussian distribution.
EigenVI with a 16th-order Hermite polynomial expansion approximates an uncentered Gaussian distribution with comparable accuracy to a standardized Gaussian using the base distribution.
For a translated mixture distribution, EigenVI requires twice as many basis functions (K=14) compared to the untranslated version (K=7) to achieve a similar approximation quality.
In experiments with sinh-arcsinh normal distributions, increasing the number of importance samples generally improves the accuracy of EigenVI approximations, especially as the number of basis functions increases.
EigenVI consistently outperforms ADVI, GSM, and BaM in terms of achieving lower empirical Fisher divergence on a set of hierarchical Bayesian models from posteriordb.

Citações

"EigenVI effectively sidesteps the iterative gradient-based optimizations that are required for many other BBVI algorithms."
"On these distributions, we find that EigenVI is more accurate than existing methods for Gaussian BBVI."

Principais Insights Extraídos De

EigenVI: score-based variational inference with orthogonal function expansions

by Diana Cai, C... às arxiv.org 11-01-2024

https://arxiv.org/pdf/2410.24054.pdf

Perguntas Mais Profundas

How could EigenVI be extended to handle discrete latent variables or mixed discrete-continuous distributions?

Extending EigenVI to handle discrete latent variables or mixed discrete-continuous distributions presents a fascinating challenge and a promising research direction. Here's a breakdown of potential approaches and considerations:
1. Discrete Latent Variables:

Basis Function Adaptation: The core concept of EigenVI hinges on orthogonal function expansions. For discrete variables, we need to employ suitable basis functions.

One-Hot Encoding: A straightforward approach is to represent each discrete state with a one-hot vector. We could then construct multi-dimensional basis functions as tensor products of these one-hot vectors.
Other Basis Sets: Exploring alternative basis sets designed for discrete spaces, such as Walsh functions or Haar wavelets, could offer computational or representational advantages.

Score Function Estimation: The score function, central to EigenVI, needs careful handling for discrete variables.

Finite Differences: One option is to approximate the score function using finite differences. However, this can be noisy and might require smoothing techniques.
Score Function Reparameterization: Recent advances in score-based methods for discrete data (e.g., using Concrete distributions or Gumbel-Softmax tricks) could provide ways to define differentiable score functions for discrete variables.
2. Mixed Discrete-Continuous Distributions:

Hybrid Expansions: A natural extension is to combine basis functions suitable for both continuous and discrete variables. For instance, we could use Hermite polynomials for continuous dimensions and one-hot encodings for discrete ones, forming product basis functions.

Conditional Modeling: Another strategy is to factorize the variational approximation into continuous and discrete parts. We could model the continuous part conditioned on the discrete part (or vice-versa) using EigenVI with appropriate basis functions for each.
Challenges and Considerations:

Computational Complexity: The eigenvalue problem's size in EigenVI scales with the number of basis functions. With discrete variables, the basis function count can grow rapidly, potentially demanding efficient eigenvalue solvers or low-rank approximations.

Approximation Quality: The choice of basis functions and their order will significantly impact how well EigenVI can approximate the target distribution, especially in the presence of complex dependencies between discrete and continuous variables.

While EigenVI demonstrates advantages in accuracy, how does its computational cost compare to gradient-based methods for high-dimensional problems and large datasets?

EigenVI's computational cost relative to gradient-based methods for high-dimensional problems and large datasets involves a trade-off:
EigenVI's Strengths:

No Iterative Optimization: EigenVI's key advantage is its avoidance of iterative gradient descent. It solves a single eigenvalue problem, which can be parallelized efficiently. This contrasts with gradient-based methods (e.g., ADVI, BBVI) that require numerous gradient computations.

Large Batch Sizes: EigenVI can benefit from large batch sizes, as the eigenvalue problem becomes more accurate with more samples. Gradient-based methods often struggle with large batches due to memory constraints or slow convergence.
EigenVI's Challenges:

Eigenvalue Problem Size: The primary computational bottleneck in EigenVI is the size of the eigenvalue problem, which grows linearly with the number of basis functions (K). In high dimensions, using a sufficient number of basis functions to capture complex distributions can lead to a very large K.

Importance Sampling: The accuracy of EigenVI's score matching objective relies on the quality of importance sampling. In high dimensions, finding a good proposal distribution can be challenging, and poor importance sampling can lead to inaccurate approximations.
Gradient-Based Methods:

Scalability with Data Size: Gradient-based methods can handle large datasets efficiently using stochastic or mini-batch optimization.

Dimensionality Challenges: In high dimensions, gradient-based methods can suffer from slow convergence or get trapped in local optima. Techniques like control variates or natural gradients are often needed.
In Summary:

Low-to-Moderate Dimensions: EigenVI can be computationally competitive, especially if the target distribution can be well-approximated with a moderate number of basis functions.

High Dimensions: As dimensionality increases, the eigenvalue problem's size in EigenVI can become prohibitive. Gradient-based methods, despite their iterative nature, might be more practical, especially with techniques to handle high-dimensional optimization.

Large Datasets: EigenVI's ability to leverage large batch sizes can be advantageous. However, if the dataset is too large to fit in memory, gradient-based methods with mini-batch optimization would be more suitable.

Could the principles of orthogonal function expansions and score matching be applied to develop novel algorithms for other probabilistic inference tasks beyond variational inference?

Absolutely! The principles of orthogonal function expansions and score matching hold significant potential for innovation in probabilistic inference beyond variational inference. Here are some intriguing avenues:
1. Importance Sampling and Density Estimation:

Proposal Distribution Design: Orthogonal function expansions could be used to construct highly flexible proposal distributions for importance sampling. Score matching could then be employed to optimize the weights of these expansions to closely match the target distribution.

Non-Parametric Density Estimation: Score matching combined with orthogonal function expansions offers a route to non-parametric density estimation. By increasing the order of the expansion, we can approximate increasingly complex densities.
2. Markov Chain Monte Carlo (MCMC) Methods:

Efficient Proposal Mechanisms: Orthogonal function expansions could be incorporated into MCMC samplers to design more effective proposal mechanisms, especially in high-dimensional spaces where standard proposals struggle.

Target Distribution Approximation: Score matching could be used within MCMC to learn a flexible approximation of the target distribution, which can then guide the sampler towards regions of high probability more efficiently.
3. Probabilistic Programming and Implicit Models:

Inference in Intractable Models: For probabilistic models with intractable likelihoods or posteriors, score matching with orthogonal function expansions provides a way to perform approximate inference.

Generative Modeling: These techniques could be leveraged to train generative models that learn to represent complex data distributions using orthogonal function expansions, optimized via score matching.
4. Time Series and Sequential Data:

State-Space Models: Orthogonal function expansions could be employed to represent the temporal evolution of states in state-space models, with score matching used for parameter estimation.
Advantages and Considerations:

Flexibility: Orthogonal function expansions offer a highly flexible way to represent complex functions and distributions.

Score Matching's Simplicity: Score matching provides a potentially simpler alternative to likelihood-based methods, especially when dealing with intractable distributions.

Computational Challenges: As with EigenVI, computational cost, particularly in high dimensions, needs careful consideration. Efficient algorithms for score matching and handling large basis function expansions are crucial.