Policy Iteration for Continuous Dynamics

This repository contains an implementation of Policy Iteration (PI) applied to environments with continuous dynamics. The Value Function (VF) is approximated using linear interpolation within each simplex of the discretized state space. The interpolation coefficients act like probabilities in a stochastic process, which helps in approximating the continuous dynamics using a discrete Markov Decision Process (MDP). This algorithm it was tested by the environments Cartpole and Mountain car provided by Gymnasium.

Normalize VF vs Number of iterations

The evolution of the value function for the mountain car as a function of the number of iterations of the algorithm:

Installation and Setting

• Create Conda environment with dependencies

   conda create --name DynamicProgramming python=3.11 ipykernel

• Activate environment

   conda activate DynamicProgramming

• Install the Policy Iteration class:

   pip install -e .

Files

• src/PolicyIteration.py: This is the implementation of the policy iteration algorithm, interpolating the value function with the vertices of a simplex.
• src/utils/utils.py: Contains auxiliary functions for testing the environments with the already calculated value function and functions for plotting it.
• testing.ipynb: This is a notebook for testing the algorithm in the cartpole and mountain car environments.

Simplex in This Project

A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. It is a fundamental building block in computational geometry and is used in various applications, including the interpolation methods used in this project. Here are the key concepts:

• Vertices: A simplex is defined by its vertices. For example, a 2-dimensional simplex (a triangle) has 3 vertices, a 3-dimensional simplex (a tetrahedron) has 4 vertices, and so on.

• Edges: The line segments connecting pairs of vertices are called edges. A d-dimensional simplex has $\frac{d(d+1)}{2}$ edges.

• Barycentric Coordinates: Any point inside a simplex can be represented as a weighted sum of its vertices, where the weights (barycentric coordinates) are non-negative and sum to one. These coordinates are useful for interpolation.

• Linear Interpolation: Within a simplex, the value of a function at any point can be linearly interpolated from its values at the vertices. This property is leveraged in the Policy Iteration algorithm to approximate the value function over a continuous state space.

In the context of this project, the state space is discretized into a mesh of simplexes (triangles in 2D, tetrahedra in 3D, and higher-dimensional analogs in higher dimensions). The value function is then approximated by linearly interpolating within these simplexes.

• State Space Discretization: The continuous state space is divided into smaller regions using simplexes. This allows for a piecewise linear approximation of the VF.
• Interpolation Process: For any state $\eta(\xi,u)$ within a simplex, its value is interpolated from the values at the vertices of the simplex. The barycentric coordinates determine the weights for this interpolation.
• Stochastic Approximation: This interpolation process can be seen as a probabilistic transition to the vertices of the simplex, allowing the continuous dynamics to be approximated by a discrete Markov Decision Process (MDP).

Considering this approach the dynamic programming equation used to compute the VF (eq. 5 in [2]) is given by: :

$$V(\xi) = \max_u \left[ \gamma^{\tau(\xi, u)} \sum_{i=0}^{d} p(\xi_i | \xi, u) V(\xi_i) + R(\xi, u) \right]$$

Here:

• $V(\xi)$ is the value function at state $\xi.$
• $\gamma^{\tau(\xi, u)}$ is the discount factor, $\tau(\xi, u)$ is the time taken to transition from state $\xi$ under control $u$.
• $p(\xi_i | \xi, u)$ is the probability of transitioning to state $\xi_i$ from $\xi$ under control $u$.
• $R(\xi, u)$ is the immediate reward received from state $\xi$ under control $u$.

References

• [1] R. Munos and A. Moore, "Barycentric interpolators for continuous space & time reinforcement learning" Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, 1998.

• [2] R. Munos and A. Moore, "Variable resolution discretization in optimal control" Machine Learning, vol. 49, pp. 291–323, 2002.

• [3] Rémis Munos, "A Study of Reinforcement Learning in the Continuous Case by the Means of Viscosity Solutions" Machine Learning, 40, 265–299, 2000.