Zeroth-Order Optimization (ZOO)

• Zeroth-Order Optimization (ZOO) is an approach that relies solely on function evaluations to solve minimization problems without explicit gradient information.
• It employs finite-difference methods using one-point or two-point estimators to approximate gradients in high-dimensional black-box settings.
• ZOO is widely applied in adversarial attacks, model-agnostic hyperparameter tuning, and reinforcement learning, demonstrating its practical significance across diverse domains.

Zeroth-Order Optimization (ZOO) encompasses a class of optimization algorithms where only function evaluations are available, with no access to explicit gradients. These methods are of central importance in large-scale black-box settings such as black-box adversarial attacks, model-agnostic hyperparameter tuning, reinforcement learning, and on-device neural network training, particularly when first-order (gradient-based) or second-order (Hessian-based) information is inaccessible or prohibitively expensive (Liu et al., 2020).

1. Problem Formulation and Fundamental Principles

Zeroth-Order Optimization targets minimization problems of the form

$\min_{x \in \mathbb{R}^d} f(x)$

where the function $f: \mathbb{R}^d \to \mathbb{R}$ (possibly stochastic or nonconvex) is accessible only through a black-box evaluation oracle. In the absence of explicit gradient information, ZOO algorithms estimate the gradient at each iterate $x_t$ via function queries, typically using finite-difference methods along random or coordinate directions (Ruan et al., 2019, Liu et al., 2020).

Canonical stochastic ZO estimators include:

• One-point estimator:

$$g_{\text{1pt}}(x) = \frac{\phi(d)}{\mu} f(x+\mu u) u$$

where $u$ is a random vector (e.g., Gaussian or uniform on the sphere), and $\phi(d)$ is a normalizing factor.

• Two-point estimator:

[ g_{\text{2pt}}(x) = \