# Dreaming machine learning: Lipschitz extensions for reinforcement learning on financial markets

## TL;DR

This paper introduces a novel reinforcement learning approach for financial markets that uses Lipschitz extensions to generate artificial 'dream' states, enhancing learning by interpolating real market data with added randomness.

## Contribution

The main novelty is the creation of artificial states, called 'dreams', through Lipschitz extensions and interpolation, to improve reinforcement learning in financial market modeling.

## Key findings

- Enhanced state space with artificially generated 'dreams' improves learning.
- Lipschitz extensions effectively interpolate real market states.
- Method shows promise for better investment decision strategies.

## Abstract

We consider a quasi-metric topological structure for the construction of a new reinforcement learning model in the framework of financial markets. It is based on a Lipschitz type extension of reward functions defined in metric spaces. Specifically, the McShane and Whitney extensions are considered for a reward function which is defined by the total evaluation of the benefits produced by the investment decision at a given time. We define the metric as a linear combination of a Euclidean distance and an angular metric component. All information about the evolution of the system from the beginning of the time interval is used to support the extension of the reward function, but in addition this data set is enriched by adding some artificially produced states. Thus, the main novelty of our method is the way we produce more states -- which we call "dreams" -- to enrich learning. Using some known states of the dynamical system that represents the evolution of the financial market, we use our technique to simulate new states by interpolating real states and introducing some random variables. These new states are used to feed a learning algorithm designed to improve the investment strategy by following a typical reinforcement learning scheme.