Simple-SGLD-Implementation

Overview

This repository contains an implementation of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm. SGLD is a stochastic optimization method that bridges traditional optimization and Bayesian posterior sampling. By adding Gaussian noise to gradient updates, SGLD transitions seamlessly from optimization to posterior sampling, making it a powerful tool for Bayesian inference on large datasets.

This implementation is inspired by the paper "Bayesian Learning via Stochastic Gradient Langevin Dynamics" by Max Welling and Yee Whye Teh (2011).

Contents

• SGLD Logistic.py: Python implementation of SGLD and SGD algorithms, applied to a logistic regression example.
• WelTeh2011a.pdf: Reference paper for SGLD.
• README.md: This documentation.

Features

• Implements both SGLD and SGD algorithms.
• Simulates logistic regression data.
• Compares SGLD and SGD in terms of convergence and parameter estimation.
• Provides visualization of parameter traces and convergence metrics.

Key Methods

1. SGLD
   • Combines stochastic gradient descent with Langevin dynamics by injecting noise into gradient updates.
   • Approximates posterior sampling as step size anneals to zero.
2. SGD
   • Standard stochastic gradient descent for comparison.

Prerequisites

• Python 3.x
• Libraries: scipy, numpy, matplotlib, pandas, autograd, sklearn

Install the required libraries using the following command:

pip install -r requirements.txt

Usage

Running the Script

1. Clone the repository:

   git clone https://github.com/shibo769/Simple-SGLD-Implementation.git
   cd Simple-SGLD-Implementation

2. Run the Python script:

   python SGLD\ Logistic.py

Outputs

The script will:

• Simulate logistic regression data.
• Optimize parameters using SGLD and SGD.
• Plot parameter traces and convergence metrics.

Example Code Snippets

Below is a brief snippet illustrating how SGLD is implemented:

def SGLD(n_steps, burn_in, batch_size, theta_init, gamma, X, Y, thin=1):
    theta = theta_init.copy()
    thetas = []
    for i in range(n_steps):
        X_batch, Y_batch = batch(X, Y, len(X), batch_size)
        proposal_loc = 0.5 * gamma * (grad_prior(theta) + (len(X) / batch_size) * grad_lik(theta, X_batch, Y_batch))
        theta = theta + stats.multivariate_normal.rvs(mean=proposal_loc, cov=gamma)
        if i >= burn_in and i % thin == 0:
            thetas.append(theta.copy())
    return np.array(thetas)

Results

• Visualization: The script generates trace plots comparing SGLD and SGD parameter updates.
• Performance: SGLD provides posterior samples while SGD converges to the maximum a posteriori (MAP) estimate.

References

• Max Welling and Yee Whye Teh. "Bayesian Learning via Stochastic Gradient Langevin Dynamics." Proceedings of the 28th International Conference on Machine Learning (ICML), 2011.

Acknowledgments

This work is based on the foundational research by Max Welling and Yee Whye Teh. The implementation draws upon their theoretical insights and examples from their paper.

License

This project is licensed under the MIT License. See the LICENSE file for details.