Gradient-free optimization via integration (2408.00888v1)

Abstract: In this paper we propose a novel, general purpose, algorithm to optimize functions $l\colon \mathbb{R}^d \rightarrow \mathbb{R}$ not assumed to be convex or differentiable or even continuous. The main idea is to sequentially fit a sequence of parametric probability densities, possessing a concentration property, to $l$ using a Bayesian update followed by a reprojection back onto the chosen parametric sequence. Remarkably, with the sequence chosen to be from the exponential family, reprojection essentially boils down to the computation of expectations. Our algorithm therefore lends itself to Monte Carlo approximation, ranging from plain to Sequential Monte Carlo (SMC) methods. The algorithm is therefore particularly simple to implement and we illustrate performance on a challenging Machine Learning classification problem. Our methodology naturally extends to the scenario where only noisy measurements of $l$ are available and retains ease of implementation and performance. At a theoretical level we establish, in a fairly general scenario, that our framework can be viewed as implicitly implementing a time inhomogeneous gradient descent algorithm on a sequence of smoothed approximations of $l$. This opens the door to establishing convergence of the algorithm and provide theoretical guarantees. Along the way, we establish new results for inhomogeneous gradient descent algorithms of independent interest.

• The paper introduces a novel gradient-free optimization algorithm that uses iterative probabilistic modeling and Bayesian updating to find minimizers for complex functions without requiring differentiability or continuity.
• The core method involves fitting sequential parametric probability densities, such as the exponential family, enabling compatibility with standard Monte Carlo methods and handling noisy function observations.
• The algorithm theoretically mimics time-inhomogeneous gradient descent on smoothed function approximations and demonstrates competitive performance in experiments, even with low sample sizes.

An Overview of Gradient-Free Optimization via Integration

The paper under discussion presents a novel approach to optimization—particularly, gradient-free optimization of functions that are not assumed to be differentiable, convex, or even continuous. The authors propose an algorithm that utilizes probabilistic modeling to approximate function minimizers through iterative updates.

Core Methodology

Algorithm Framework

The central methodology involves fitting a sequence of parametric probability densities to the target function using a Bayesian update mechanism. This results in a reprojection onto a parametric family, such as the exponential family of distributions. In a scenario where the density is from the exponential family, reprojection simplifies to the computation of expected values, reducing the complexity of implementation. This method is broadly compatible with standard Monte Carlo and Sequential Monte Carlo (SMC) methods.

The proposed algorithm is versatile enough to handle scenarios where only noisy observations of the function are available. This adaptability stems from its built-in capability to process uncertainty inherent in the observations while maintaining ease of implementation.

Theoretical Insights

In theoretical derivations, the authors demonstrate that, in general, the proposed algorithm effectively mimics a form of time-inhomogeneous gradient descent. This gradient descent operates on a sequence of smooth approximations of the non-differentiable target function. This progression opens avenues for establishing convergence and theoretical guarantees, extending the results to broader classes of functions and optimization landscapes.

Addressing Key Scenarios

1. Noiseless scenario: The algorithm is conceptualized in an environment where the objective function can be evaluated exactly. Here, the authors illustrate the concentration property of the algorithm, drawing parallels to Bayesian posterior concentration.
2. Noisy scenario: The capability to operate under noisy conditions is emphasized. The adaptation involves constructing stochastic gradients where the randomness of the gradient updates originates from observational noise rather than from the function itself.

Experimental Validation and Comparative Performance

The efficacy of the approach is validated through experiments on classification problems, where the algorithm demonstrates competitive performance. Specifically, it manages to perform well even with less refined setup, such as low sample sizes for Monte Carlo methods, indicating robustness.

Implications and Future Directions

Theoretical Implications

The gradient-free nature of the approach introduces significant theoretical implications, particularly in fields where derivative information is inaccessible or prohibitively expensive to compute. The algorithm's performance in noisy environments suggests robust applications in real-world scenarios where data is imperfect or approximate.

Practical Considerations

Practically, the paper's findings may be transformational for optimization tasks involving black-box functions or complex landscapes characterized by numerous local minima or non-smooth features. The methods outlined could revolutionize optimization operations in machine learning, control systems, and other computational domains reliant on high-dimensional function minimization.

Speculations on Future Developments

Looking forward, the methodological framework could be extended by experimenting with different types of probability distributions and variational inference techniques, potentially broadening the algorithm's applicability and efficiency. Additional work could also focus on refining convergence rates and exploring real-time adaptive schemes to optimize initial parameter settings dynamically.

Conclusion

This paper makes a significant contribution to the optimization domain, particularly in handling non-standard, challenging function classes. The bridging of Bayesian updating with optimization opens fresh avenues of research while offering robust practical tools for diverse applications. This approach, standing at the intersection between stochastic modeling and optimization, is poised to influence future developments in AI and allied fields.