Efficient Safe Zeroth-Order Optimization Using Quadratic Local Approximations

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This paper proposes a novel safe zeroth-order optimization method that iteratively constructs quadratic approximations of the constraint functions, builds local feasible sets, and optimizes over them. The method guarantees that all samples are feasible and returns an η-KKT pair within a polynomial number of iterations and samples.

Samenvatting

The paper addresses smooth constrained optimization problems where the objective and constraint functions are unknown but can be queried. The main goal is to generate a sequence of feasible points converging towards a KKT primal-dual pair.
The key highlights are:

The authors propose a zeroth-order method that iteratively computes quadratic approximations of the constraint functions, constructs local feasible sets, and optimizes over them. This ensures that all samples are feasible.

The method is proven to return an η-KKT pair within O(d/η^2) iterations and O(d^2/η^2) samples, where d is the problem dimension.

Numerical experiments show that the proposed method can achieve faster convergence compared to state-of-the-art zeroth-order safe approaches, such as LB-SGD and SafeOptSwarm. The effectiveness is also illustrated on nonconvex optimization problems in optimal control and power system operation.

The authors extend their previous work on convex problems to handle non-convex objective and constraint functions. They prove that the accumulation points of the iterates are KKT pairs under mild assumptions.

The termination conditions of the algorithm are designed to ensure that the final output is an η-KKT pair. The complexity analysis shows that the number of iterations and samples required scales polynomially with the problem dimension and inversely with the desired accuracy η.

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Belangrijkste Inzichten Gedestilleerd Uit

Safe Zeroth-Order Optimization Using Quadratic Local Approximations

by Baiwei Guo,Y... om arxiv.org 04-25-2024

https://arxiv.org/pdf/2303.16659.pdf

Diepere vragen

How can the proposed method be extended to handle stochastic or noisy function evaluations

To extend the proposed method to handle stochastic or noisy function evaluations, we can incorporate techniques from stochastic optimization. One approach is to use stochastic gradient estimation methods, such as the Finite Difference Stochastic Approximation (FDSA) algorithm. In FDSA, instead of using deterministic gradients, we estimate the gradients using noisy function evaluations. This allows us to handle stochastic or noisy function evaluations in the optimization process. Additionally, we can utilize techniques like Bayesian optimization, which can handle noisy evaluations by modeling the objective function as a Gaussian process and incorporating uncertainty into the optimization process. By integrating these stochastic optimization techniques into the proposed method, we can effectively handle stochastic or noisy function evaluations.

Can the complexity bounds be further improved by exploiting the specific structure of the problem, such as convexity or sparsity

To improve the complexity bounds further, we can exploit the specific structure of the problem, such as convexity or sparsity. For convex problems, we can leverage the properties of convex optimization to design more efficient algorithms. For example, for convex objective and constraint functions, we can potentially derive tighter convergence bounds by utilizing the properties of convex optimization, such as strong duality and optimality conditions. Additionally, for sparse optimization problems, we can incorporate techniques like proximal algorithms or coordinate descent methods to exploit the sparsity of the problem and reduce the computational complexity. By tailoring the algorithm to the specific structure of the problem, we can potentially achieve better complexity bounds and improve the efficiency of the optimization process.

What are the potential applications of this safe zeroth-order optimization approach beyond optimal control and power system operation

The safe zeroth-order optimization approach proposed in the context of optimal control and power system operation has a wide range of potential applications beyond these domains. Some potential applications include:

Machine Learning: The safe zeroth-order optimization method can be applied to hyperparameter tuning in machine learning models. By optimizing hyperparameters while ensuring sample feasibility, the method can enhance the performance of machine learning algorithms.
Finance: In financial modeling and portfolio optimization, the safe zeroth-order optimization approach can be used to optimize investment strategies while considering constraints and uncertainties in the market.
Healthcare: In healthcare applications, the method can be utilized for optimizing treatment plans or resource allocation in hospitals while ensuring safety and feasibility constraints are met.
Supply Chain Management: The approach can be applied to optimize supply chain logistics, inventory management, and production planning while considering complex constraints and uncertainties in the supply chain network.
By adapting the safe zeroth-order optimization approach to these diverse applications, we can address a wide range of optimization problems in various fields while ensuring sample feasibility and convergence to optimal solutions.