Batch, match, and patch: low-rank approximations for score-based variational inference (2410.22292v2)

Abstract: Black-box variational inference (BBVI) scales poorly to high-dimensional problems when it is used to estimate a multivariate Gaussian approximation with a full covariance matrix. In this paper, we extend the batch-and-match (BaM) framework for score-based BBVI to problems where it is prohibitively expensive to store such covariance matrices, let alone to estimate them. Unlike classical algorithms for BBVI, which use stochastic gradient descent to minimize the reverse Kullback-Leibler divergence, BaM uses more specialized updates to match the scores of the target density and its Gaussian approximation. We extend the updates for BaM by integrating them with a more compact parameterization of full covariance matrices. In particular, borrowing ideas from factor analysis, we add an extra step to each iteration of BaM--a patch--that projects each newly updated covariance matrix into a more efficiently parameterized family of diagonal plus low rank matrices. We evaluate this approach on a variety of synthetic target distributions and real-world problems in high-dimensional inference.

• The paper introduces an EM-inspired patch step within the Batch-and-Match algorithm to form a diagonal plus low-rank covariance structure that reduces complexity in high-dimensional inference.
• The methodology transforms dense covariance updates into linear-complexity operations, dramatically improving scalability and convergence compared to traditional BBVI.
• Empirical results on synthetic and real datasets demonstrate enhanced inference quality and stability for high-dimensional Gaussian targets and Cox process models.

Overview of Low-Rank Approximation Methods for Score-Based Variational Inference

The paper under discussion explores advancements in the field of variational inference, specifically targeting the limitations of Black-Box Variational Inference (BBVI) when applied to high-dimensional distributions with complex covariance structures. The authors propose an enhancement of the existing Batch-and-Match (BaM) algorithm, introducing what they term as a "patch" to maintain efficiency and scalability in processing high-dimensional data. The focus of their work is on improving the covariance estimation by integrating low-rank approximations, consequently enabling the effective application of BBVI in large-scale inference problems.

Conceptual Framework

The fundamental challenge addressed in this research is the inefficiency of BBVI in managing high-dimensional data, particularly when such data requires the computation and storage of dense covariance matrices. Traditional gradient descent methods used in BBVI to minimize the reverse Kullback-Leibler (KL) divergence become computationally prohibitive as dimensionality increases. To mitigate this challenge, the authors extend the BaM framework with a novel integration strategy that projects dense covariance matrices into a more manageable form, characterized by a diagonal plus low-rank structure. This structuring is analogous to factor analysis models, which facilitate the representation of large covariance matrices through a combination of diagonal and low-rank matrices. The introduced "patch" in the BaM updates aligns with the maximum likelihood estimation methods employed in factor analysis but is adapted to variational inference.

Methodological Advancements

The modified BaM algorithm, now incorporating a patch step, transforms the original covariance update into one that fits within a structured family of low-rank and diagonal matrices. This transformation is achieved through an Expectation-Maximization (EM) inspired algorithm, which ensures that each iterative update of the covariance matrix reduces computational complexity from quadratic to linear concerning dimensionality. Notably, the EM-based patch projection aims to minimize the KL divergence between the intermediate dense covariance and the updated structured covariance. The authors assert that this leads to improved scalability and convergence rates across diverse inference tasks.

Empirical Evaluation and Results

Empirical validation of the proposed method is conducted on a range of synthetic and real-world high-dimensional datasets. The researchers evaluate the performance of the patched BaM (pBaM) algorithm against traditional and structured BBVI methods, highlighting the superior convergence speed and stability of pBaM. Their experiments demonstrate notable enhancements in inference quality for high-dimensional Gaussian targets and logistic Gaussian Cox processes. Importantly, the pBaM method achieves this efficiency without compromising on the accuracy of variational approximation, showcasing its potential as a practical tool in scenarios where data dimensionality presents a significant computational burden.

Implications and Future Work

The implications of this work are manifold, influencing both theoretical advancements in variational inference and practical applications in fields dealing with large-scale data. The introduction of structured covariance estimation through low-rank approximations aids in circumventing the prohibitive memory and computation constraints posed by high dimensions. Further, the adaptability of the algorithm to varying rank structures of covariance suggests its utility in dynamic and iterative data environments.

Looking forward, the authors propose extending this research to encompass more complex and diverse structured variational families. Potential adaptations might include variations of the covariance structure to incorporate sparsity or block-diagonal patterns, thereby enhancing the versatility and application range of variational inference techniques. Additionally, the exploration of adaptive rank-increasing strategies could further optimize the approximation quality in dynamically evolving datasets.

In conclusion, this paper presents a significant step towards addressing the computational challenges inherent in score-based BBVI, offering a robust methodology through the incorporation of low-rank approximations. It opens avenues for further exploration in enhancing both the scalability and applicability of variational inference algorithms in high-dimensional settings.