Principle:Avhz RustQuant Gradient Descent Optimization

Knowledge Sources	Cauchy 1847 - Methode Generale pour la Resolution des Systemes Nocedal & Wright - Numerical Optimization
Domains	Optimization, Numerical_Methods, Model_Calibration
Last Updated	2026-02-07 20:00 GMT

Overview

A first-order iterative optimization algorithm that finds a local minimum of a differentiable function by repeatedly stepping in the direction of the negative gradient.

Description

Gradient Descent is the foundational algorithm for unconstrained optimization. Given an objective function f(x) and its gradient, the algorithm iteratively updates the parameter vector by moving in the direction of steepest descent:

x_{k+1} = x_k - alpha * gradient(f(x_k))

The step size (learning rate) alpha controls the magnitude of each update. The algorithm converges when the gradient norm falls below a tolerance threshold (stationarity condition).

In quantitative finance, gradient descent is used for:

• Model calibration: fitting model parameters to market data (e.g., calibrating volatility)
• Portfolio optimization: finding optimal asset weights
• Parameter estimation: fitting statistical distributions to data

When combined with automatic differentiation, the gradient is computed exactly without manual derivation.

Usage

Use gradient descent when you need to minimize a differentiable objective function with respect to multiple parameters. It is the primary optimization method for RustQuant's model calibration workflow, working in conjunction with the autodiff system.

Theoretical Basis

The gradient descent iteration:

$x_{k+1}=x_k-{\alpha }_k\mathrm{\nabla }f(x_k)$

Convergence: The sequence is monotonic:

$f(x_0)\ge f(x_1)\ge f(x_2)\ge \cdots \ge f(x^{*})$

Stationarity condition:

$\Vert \mathrm{\nabla }f(x_{k+1})\Vert \le ϵ$

Key hyperparameters:

• alpha (learning rate): Step size. Too large causes divergence, too small causes slow convergence.
• max_iterations: Upper bound on iterations to prevent infinite loops.
• tolerance: Threshold for gradient norm that signals convergence.

Limitations:

• Only finds local minima (not global)
• Sensitive to learning rate choice
• Slow convergence near saddle points
• No line search in basic implementation

Related Pages

Implemented By

• Implementation:Avhz_RustQuant_GradientDescent_optimize

Uses Heuristic

• Heuristic:Avhz_RustQuant_Learning_Rate_Tuning