Stochastic Gradient Descent for Gaussian Processes Done Right (2310.20581v2)

Abstract: As is well known, both sampling from the posterior and computing the mean of the posterior in Gaussian process regression reduces to solving a large linear system of equations. We study the use of stochastic gradient descent for solving this linear system, and show that when \emph{done right} -- by which we mean using specific insights from the optimisation and kernel communities -- stochastic gradient descent is highly effective. To that end, we introduce a particularly simple \emph{stochastic dual descent} algorithm, explain its design in an intuitive manner and illustrate the design choices through a series of ablation studies. Further experiments demonstrate that our new method is highly competitive. In particular, our evaluations on the UCI regression tasks and on Bayesian optimisation set our approach apart from preconditioned conjugate gradients and variational Gaussian process approximations. Moreover, our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.

• The paper introduces a tailored stochastic dual descent algorithm for Gaussian processes that achieves competitive performance with simple implementation.
• It integrates advanced techniques like Nesterov's momentum and random coordinate selection to ensure enhanced convergence speed and stability.
• Empirical evaluations on UCI benchmarks, Bayesian optimization, and molecular binding affinity tasks validate its superiority over conventional methods.

An Analysis of Stochastic Gradient Descent for Gaussian Processes

The paper under review, titled "Stochastic Gradient Descent for Gaussian Processes Done Right," presents an in-depth examination of optimizing Gaussian process regression using stochastic gradient descent (SGD) methods. Traditionally, Gaussian process regression problems, particularly those utilizing squared loss, have been solved through exact methods such as conjugate gradient descent. However, the paper proposes an alternative: a specifically fine-tuned stochastic dual gradient descent (SDD) algorithm that integrates select insights from the optimization and kernel communities, thereby indicating the high efficacy of this approach when implemented correctly.

The primary contribution of this research lies in presenting a specific stochastic dual descent methodology tailored for Gaussian processes. This method is not only straightforward to code—requiring merely a few lines in any deep learning framework—but also competitive against robust existing approaches like preconditioned conjugate gradients, variational Gaussian process approximations, and previous SGD methods tailored for Gaussian processes. The authors have rigorously elucidated their design decisions through ablation studies, highlighting the advantages of their SDD approach against alternatives.

Key Results and Comparisons

Through various experimental evaluations, the authors showcase the practical viability of their proposed method:

1. UCI Regression Benchmarks: The SDD demonstrates superior or equivalent performance compared to established Gaussian process baselines on several standard datasets. Notably, SDD shows improvement or parity with preconditioned conjugate gradients and sparse variational Gaussian processes in terms of predictive performance.
2. Bayesian Optimization: For large-scale Bayesian optimization tasks, the SDD surpasses prior stochastic gradient descent approaches and other baseline models, showcasing both improved iteration and time efficiency.
3. Molecular Binding Affinity Prediction: SDD attains performance levels comparable to state-of-the-art graph neural networks on tasks involving molecular binding affinity prediction, a domain where Gaussian process methods have traditionally lagged behind deep learning approaches. This is a noteworthy accomplishment, placing Gaussian processes on a competitive footing with advanced neural network models for these problems.

Methodological Innovations

The novel stochastic dual descent algorithm incorporates several distinctive elements:

• Dual Objective Function: Unlike traditional primal objectives used in Gaussian process regression, the dual objective provides better conditioning that fosters superior convergence properties.
• Randomization Techniques: By employing random coordinates instead of random features, the method benefits from multiplicative noise properties that offer enhanced stability and faster convergence rates.
• Advanced Optimization Techniques: The use of Nesterov's momentum and geometric iterate averaging further accelerates convergence and stability of the optimization process. This combination allows for larger step sizes and improved performance that outshine competing stochastic methods.

Implications and Future Directions

The implications of this research are multifaceted, touching on both theoretical and practical aspects within machine learning:

• Theoretical Insights: This work adds to the understanding of dual optimization, demonstrating structural advantages that can be exploited to achieve faster, more reliable convergence in large-scale regression tasks.
• Practical Applications: The demonstrated effectiveness of SDD on diverse tasks underscores its potential applicability across various domains that rely on Gaussian process models, including Bayesian optimization and drug discovery.

Future AI research may build upon these findings, exploring further enhancements to dual optimization frameworks and extending stochastic methodologies to other probabilistic models. Additionally, exploring adaptive schemes for hyperparameter configuration in various data regimes could further bolster the utility and robustness of the proposed approach.

In conclusion, the paper delivers a comprehensive and technically rigorous exploration of a refined stochastic gradient descent methodology, firmly establishing stochastic dual descent as a competitive and promising algorithm for Gaussian process regression in both research and applied settings.