Coresets for Scalable Bayesian Logistic Regression (1605.06423v3)

Abstract: The use of Bayesian methods in large-scale data settings is attractive because of the rich hierarchical models, uncertainty quantification, and prior specification they provide. Standard Bayesian inference algorithms are computationally expensive, however, making their direct application to large datasets difficult or infeasible. Recent work on scaling Bayesian inference has focused on modifying the underlying algorithms to, for example, use only a random data subsample at each iteration. We leverage the insight that data is often redundant to instead obtain a weighted subset of the data (called a coreset) that is much smaller than the original dataset. We can then use this small coreset in any number of existing posterior inference algorithms without modification. In this paper, we develop an efficient coreset construction algorithm for Bayesian logistic regression models. We provide theoretical guarantees on the size and approximation quality of the coreset -- both for fixed, known datasets, and in expectation for a wide class of data generative models. Crucially, the proposed approach also permits efficient construction of the coreset in both streaming and parallel settings, with minimal additional effort. We demonstrate the efficacy of our approach on a number of synthetic and real-world datasets, and find that, in practice, the size of the coreset is independent of the original dataset size. Furthermore, constructing the coreset takes a negligible amount of time compared to that required to run MCMC on it.

Insights on Coresets for Scalable Bayesian Logistic Regression

The paper presents a novel approach for facilitating scalable Bayesian logistic regression by leveraging the concept of coresets. This research addresses the computational burden typically associated with Bayesian methods in the context of large-scale datasets. Standard Bayesian inference methods struggle with the scalability required by modern data analysis tasks, which can involve tens or even hundreds of millions of data points. Traditional methods such as MCMC or variational inference, while robust, are computationally intensive and often impede real-time or large-scale applications.

Key Contributions

The primary contribution of this paper is the development of an efficient algorithm for constructing coresets for Bayesian logistic regression. A coreset is a weighted subset of the original data that, despite its reduced size, allows for the estimation of posterior distributions with a specified level of approximation fidelity. The utility of a coreset lies in its ability to represent the original dataset with much fewer data points, thereby reducing computational costs when performing posterior inference.

1. Coreset Construction Algorithm: The authors propose an algorithm that constructs a coreset by carefully sampling and weighting the data points. The algorithm offers theoretical guarantees on both the approximation quality and the coreset size for fixed datasets and data generative models. Notably, the size of the coreset is often independent of the original dataset size, a crucial factor for scalability.
2. Theoretical Underpinnings: The paper offers rigorous theoretical insights into the coreset construction process. It provides bounds on the sensitivity of data points, ensuring that the constructed coresets maintain the statistical properties necessary for reliable Bayesian inference.
3. Streaming and Parallel Implementation: The coreset construction allows for efficient implementations in streaming and parallel computing environments. This is particularly advantageous in distributed computing settings or when the dataset cannot be fully loaded into memory at once.
4. Empirical Validation: The authors validate their approach with experiments on synthetic and real-world datasets. Their results show significant improvements in computational efficiency, with negligible time spent on coreset construction relative to MCMC-based inference on these coresets. Furthermore, in many cases, the coreset-based methods provided superior posterior approximations compared to traditional subsampling methods.

Implications and Future Directions

The introduction of coresets into Bayesian logistic regression opens up significant possibilities for both theoretical exploration and practical application. On the practical side, this approach allows for enhanced scalability of Bayesian methods, making them more applicable to real-world problems where data is abundant and computational resources may be limited. Theoretically, it poses interesting questions regarding the generalization of coresets to other Bayesian models and their respective inference pipelines.

The authors hint at the potential applicability of their methods beyond logistic regression to other generative models, which could be a fertile avenue for future research. Extending coresets across different kinds of likelihood models could vastly enhance the flexibility and applicability of Bayesian methods in machine learning.

Moreover, the paper's findings encourage further exploration into efficient sampling mechanisms, especially in structured data environments like time series or graph-based data. The potential combination of data compression techniques and coresets could lead to even more efficient algorithms, particularly in machine learning applications where large volumes of redundant data are common.

In conclusion, this paper makes a substantial contribution to scalable Bayesian inference by demonstrating that coresets can dramatically improve the efficiency of logistic regression models while maintaining the integrity of inferential statistics. This work paves the way for further research into optimizing Bayesian methodologies for large-scale data analysis, a critical need in the era of big data.