Principle:Google deepmind Mujoco Finite Difference Derivatives

Knowledge Sources	Domains	Last Updated
Google DeepMind MuJoCo	Physics Simulation, Numerical Methods	2025-02-15

Overview

Description: Finite-difference derivative approximation computes numerical gradients of the dynamics by perturbing inputs and observing output changes. This serves as both a fallback when analytical derivatives are unavailable and a verification tool for analytical derivative correctness.

Context: MuJoCo provides finite-difference derivative computation for validating analytical derivatives and for use cases where the analytical path does not cover certain model features (e.g., custom plugins or user-defined callbacks).

Theoretical Basis

Finite-difference methods approximate derivatives using function evaluations at perturbed points:

• Forward differences: df/dx ≈ (f(x+h) - f(x)) / h, requiring n+1 evaluations for n parameters
• Central differences: df/dx ≈ (f(x+h) - f(x-h)) / 2h, achieving O(h^2) accuracy at the cost of 2n evaluations
• Step size selection: The perturbation h must balance truncation error (large h) against round-off error (small h); typically h ≈ sqrt(epsilon) for forward differences

Finite-difference derivatives scale linearly with the number of parameters, making them expensive for high-dimensional systems but invaluable for debugging and validation.

Related Pages

Implementations

• Implementation:Google_deepmind_Mujoco_Engine_Derivative_FD

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