The Generalized Reparameterization Gradient (1610.02287v3)

Abstract: The reparameterization gradient has become a widely used method to obtain Monte Carlo gradients to optimize the variational objective. However, this technique does not easily apply to commonly used distributions such as beta or gamma without further approximations, and most practical applications of the reparameterization gradient fit Gaussian distributions. In this paper, we introduce the generalized reparameterization gradient, a method that extends the reparameterization gradient to a wider class of variational distributions. Generalized reparameterizations use invertible transformations of the latent variables which lead to transformed distributions that weakly depend on the variational parameters. This results in new Monte Carlo gradients that combine reparameterization gradients and score function gradients. We demonstrate our approach on variational inference for two complex probabilistic models. The generalized reparameterization is effective: even a single sample from the variational distribution is enough to obtain a low-variance gradient.

• The paper introduces the Generalized Reparameterization Gradient (G-REP), a novel technique that extends reparameterization methods to handle a broader set of variational distributions beyond Gaussian.
• The G-REP method leverages invertible transformations on latent variables and combines reparameterization and score function gradients to generate low-variance Monte Carlo estimates with fewer samples.
• Empirical results show G-REP achieves competitive or superior accuracy and computational efficiency compared to Black-Box Variational Inference (BBVI) and Automatic Differentiation Variational Inference (ADVI) for models requiring sparse posterior solutions, like those with gamma and beta distributions.
• The Generalized Reparameterization Gradient (G-REP) extends reparameterization methods to a wider range of variational distributions using invertible transformations.
• The paper introduces the Generalized Reparameterization Gradient (G-REP), extending reparameterization methods to handle a broader set of variational distributions.
• The method utilizes invertible transformations on latent variables and combines reparameterization and score function gradients to generate low-variance Monte Carlo estimates.
• Empirical results show G-REP achieves accurate inference with fewer samples and better computational efficiency compared to existing methods like ADVI and BBVI for complex models.

The Generalized Reparameterization Gradient: Expanding Variational Inference Capabilities

The paper "The Generalized Reparameterization Gradient" introduces a significant expansion of the reparameterization gradient, enabling its application to a broader set of variational distributions. Variational Inference (VI) remains a cornerstone in probabilistic modeling, allowing the approximation of intractable posterior distributions through optimization. Standard reparameterization gradients excel in reducing variance for Monte Carlo estimates but are predominantly used with Gaussian distributions, often requiring additional approximations for other distributions like gamma and beta. This work addresses these limitations through the Generalized Reparameterization Gradient (G-REP), presented as a novel technique that combines reparameterization and score function gradients.

Overview and Methodology

The core contribution of the paper lies in extending reparameterization techniques by using invertible transformations on latent variables, resulting in transformed distributions with weak parameter dependencies. This approach generates new Monte Carlo gradients that maintain low variance with fewer samples—sometimes even one. The authors leverage the mathematical framework of VI, focusing particularly on nonconjugate probabilistic models where traditional gradients pose challenges.

Two models serve as empirical demonstrations: a sparse gamma deep exponential family and a beta-gamma matrix factorization model. The results illustrate how a single sample from the variational distribution suffices to achieve low-variance gradients, asserting the efficacy of the G-REP approach.

Numerical Results and Claims

The experiments conducted show competitive results against Black-Box Variational Inference (BBVI) and Automatic Differentiation Variational Inference (ADVI). Notably, G-REP outperforms ADVI in accuracy with superior computational efficiency compared to BBVI. Given the complexity of handling gamma and beta distributions, these results highlight the practical benefits of G-REP, especially in terms of computational cost and fitting accuracy for models that necessitate sparse posterior solutions.

Theoretical and Practical Implications

Theoretically, the extension of reparameterization techniques suggests a paradigm shift in handling complex variational distributions, potentially enhancing model diversity and inference efficiency across the field. Practically, G-REP broadens the arsenal for probabilistic modeling, facilitating applications in specialized domains where traditional methods struggle due to high dimensionality or restrictive prior assumptions.

Speculation on Future Developments

Future research might explore the automatic selection and implementation of transformations tailored to the statistical properties of a given dataset, optimizing the trade-offs between variance and bias in gradient estimations. Additionally, the integration of G-REP into probabilistic programming frameworks would democratize access to its capabilities, promoting broader adoption and experimentation.

Conclusion

"The Generalized Reparameterization Gradient" represents a meaningful advancement in VI techniques by efficiently addressing the limitations of reparameterization methods for non-Gaussian distributions. This development not only provides immediate benefits in terms of inference speed and accuracy but also lays the groundwork for further innovations and applications in complex probabilistic modeling.