Deep Neural Network Approximations for Brownian Control Problems

In this article, we delve into the realm of deep reinforcement learning (DRL) and its application to Markov decision processes (MDPs). DRL is a subfield of machine learning that combines the power of deep neural networks with the reinforcement learning framework to learn complex behaviors from raw sensory input.
At its core, DRL involves optimizing an agent’s policy to maximize a cumulative reward signal. The agent interacts with an environment, taking actions based on its policy, and observing the consequences of those actions in terms of rewards. The goal is to learn a policy that maps states to actions that maximize the expected cumulative reward over time.
MDPs provide a mathematical framework for modeling complex environments with uncertainty. They consist of a set of states, actions, and transition probabilities, as well as a reward function that assigns rewards to each state-action pair. The objective of DRL in MDPs is to learn an optimal policy that maximizes the expected cumulative reward over an infinite horizon.
To tackle this challenge, we adopt a two-stage approach. First, we discretize time into discrete intervals and approximate the optimal policy using a reference policy. We then use dynamic programming to compute the value function of the reference policy, which serves as an upper bound on the optimal value function. Finally, we update our policy using backpropagation to minimize the difference between the value function and the optimal value function.
We explore various DRL algorithms for solving MDPs, including Q-learning, SARSA, and actor-critic methods. Each algorithm has its strengths and weaknesses, and we analyze their performance in terms of computational complexity, convergence rates, and sample efficiency.
One of the key challenges in DRL is handling large state spaces, which can lead to the curse of dimensionality. To address this issue, we propose a number of techniques, such as function approximation using neural networks, and use of off-policy learning methods that allow for exploration without requiring explicit exploration signals.
In conclusion, DRL for MDPs provides a powerful framework for solving complex decision-making problems in a wide range of domains. By combining the flexibility of deep neural networks with the reinforcement learning framework, we can learn optimal policies that maximize cumulative rewards in uncertain environments. While there are challenges to be addressed, the field of DRL holds great promise for tackling real-world decision-making problems and improving our understanding of complex systems.