This repository hosts a notebook that extends the research presented in "Training neural networks using Metropolis Monte Carlo and an adaptive variant" by Whitelam et al. (2022), accessible at arXiv:2205.07408. This notebook was developed by Giosuè Castellano, Alan Picucci and Fabio Pernisi (myself).

The referenced paper investigates the application of the zero-temperature Metropolis Monte Carlo method for neural network training. This approach aims to minimize a loss function through an algorithm that initially resembles Gradient Descent. In the original version, as described by Whitelam and colleagues, the algorithm updates each network weight $x_i$ by adding a Gaussian random number $\epsilon_i$, as follows:

$$ x_i \rightarrow x_i + \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(0, \sigma^2) $$

Here, $\sigma^2$ is a uniform variance across all weights. The update is accepted only if it results in a non-increasing loss, consistent with the zero-temperature Metropolis algorithm.

The paper introduces an adaptive version of the Monte Carlo algorithm (aMC), which proposes weight updates in a more nuanced manner:

$$ x_i \rightarrow x_i + \epsilon_i \mathcal{N}(\mu_i, \sigma_i^2) $$

In this adaptive approach, both the mean $\mu_i$ and variance $\sigma_i^2$ of the Gaussian distribution vary for each weight. The mean $\mu_i$ is adjusted after every accepted move:

$$ \mu_i \rightarrow \mu_i + \epsilon (\epsilon_i - \mu_i) $$

Here, $\epsilon$ is a model hyperparameter. The variance $\sigma_i$ is adapted using a simple learning-rate schedule: after every $n_S$ consecutive rejections (denoted n_reset in the code), the following updates occur:

$$ \sigma_i \rightarrow \sigma_i \times 0.95 \quad \text{and} \quad \mu_i = 0 $$

Initially, the step-size $\sigma$ is set to a constant value $\sigma_0$.

In this project, we delve into exploring a non-zero-temperature variant of the aMC algorithm. Our aim is to understand its implications and effectiveness in neural network training under different temperature settings.

Note: This README is intended to provide a brief overview of our work in this repository. For a comprehensive understanding, please refer to the original paper and the detailed documentation within the code.