Efficient Zeroth-Order Bilevel Optimization via Gaussian Smoothing

Core Concepts

This paper proposes a fully zeroth-order stochastic approximation method for solving bilevel optimization problems, where neither the upper/lower objective values nor their unbiased gradient estimates are available. The authors use Gaussian smoothing to estimate the first- and second-order partial derivatives of the functions with two independent block of variables, and establish non-asymptotic convergence analysis and sample complexity bounds for the proposed algorithm.

Abstract

The paper focuses on solving stochastic bilevel optimization problems where neither the objective function values nor their gradients are available. The authors make the following key contributions:

They generalize the Gaussian convolution technique to functions with two block-variables and establish relationships between such functions and their smooth Gaussian approximations. This allows them to exploit zeroth-order derivative estimates over just one block.

They provide the first fully zeroth-order stochastic approximation method for solving bilevel optimization problems, without assuming the availability of unbiased first/second order derivatives for the upper or lower level objective functions.

They provide a detailed non-asymptotic convergence analysis of the proposed method and present sample complexity results, which are the first established for a fully zeroth-order method for solving stochastic bilevel optimization problems.

The paper first lays out the necessary assumptions and notations. It then focuses on developing the required analysis tools by applying Gaussian smoothing techniques to functions with two block-variables. This includes estimating the first and second-order partial derivatives and bounding the discrepancies with their true values.
The paper then shifts its focus to the zeroth-order approximation of the bilevel optimization problem. It addresses issues such as the approximation error and the efficient evaluation of the gradient of the upper-level objective function.
The proposed solution algorithm is presented in Section 4, utilizing the tools and results from the previous sections to analyze the inner and outer loops of the bilevel programming scheme. The authors provide sample complexity results pertaining to the overall algorithm performance.

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Key Insights Distilled From

Fully Zeroth-Order Bilevel Programming via Gaussian Smoothing

by Alireza Agha... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00158.pdf

Deeper Inquiries

How can the proposed zeroth-order bilevel optimization framework be extended to handle additional constraints or more complex problem structures

The proposed zeroth-order bilevel optimization framework can be extended to handle additional constraints or more complex problem structures by incorporating regularization terms or penalty functions. These additional constraints can be integrated into the objective function using Lagrange multipliers or penalty methods. By introducing these constraints, the optimization problem becomes more robust and can accommodate a wider range of real-world scenarios. Moreover, the framework can be extended to handle multi-objective optimization problems by considering multiple upper-level objectives and incorporating trade-off parameters to balance between them. This extension would enable the framework to address more complex decision-making scenarios where multiple conflicting objectives need to be optimized simultaneously.

What are the potential challenges and limitations of the Gaussian smoothing approach used in this work, and how could alternative smoothing techniques be explored

The Gaussian smoothing approach used in this work may face challenges and limitations in scenarios where the underlying functions have discontinuities or sharp changes. Gaussian smoothing tends to oversmooth functions with sharp peaks or valleys, leading to inaccuracies in the estimated derivatives. To address this limitation, alternative smoothing techniques such as kernel smoothing or spline interpolation could be explored. These techniques offer more flexibility in capturing the underlying function's characteristics without oversmoothing. Additionally, exploring adaptive smoothing techniques that adjust the smoothing parameters based on the local function properties could enhance the accuracy of the derivative estimates in regions with varying smoothness.

Given the broad applications of bilevel optimization in machine learning, how could the insights from this work be leveraged to advance specific problem domains such as meta-learning, neural architecture search, or adversarial training

The insights from this work on zeroth-order bilevel optimization can be leveraged to advance specific problem domains in machine learning such as meta-learning, neural architecture search, and adversarial training. In meta-learning, the zeroth-order optimization framework can be applied to efficiently adapt meta-parameters across different tasks without the need for explicit gradient information. This can lead to faster meta-learning convergence and improved generalization to new tasks. In neural architecture search, the zeroth-order optimization approach can be utilized to optimize the search space of neural network architectures without the computational overhead of calculating exact gradients. This can accelerate the architecture search process and enable the discovery of more efficient network structures. In adversarial training, the zeroth-order optimization framework can be used to enhance the robustness of machine learning models against adversarial attacks by efficiently optimizing the model parameters in the presence of adversarial perturbations. This can lead to more resilient models that are less susceptible to adversarial manipulation.