Understanding the Mathematics of PPO in Reinforcement Learning

Deep dive into RL with PPO for beginners

Photo by ThisisEngineering on Unsplash

Introduction

Reinforcement Learning (RL) is a branch of Artificial Intelligence that enables agents to learn how to interact with their environment. These agents, which range from robots to software features or autonomous systems, learn through trial and error. They receive rewards or penalties based on the actions they take, which guide their future decisions.

Among the most well-known RL algorithms, Proximal Policy Optimization (PPO) is often favored for its stability and efficiency. PPO addresses several challenges in RL, particularly in controlling how the policy (the agent’s decision-making strategy) evolves. Unlike other algorithms, PPO ensures that policy updates are not too large, preventing destabilization during training. This stabilization is crucial, as drastic updates can cause the agent to diverge from an optimal solution, making the learning process erratic. PPO thus maintains a balance between exploration (trying new actions) and exploitation (focusing on actions that yield the highest rewards).

Additionally, PPO is highly efficient in terms of both computational resources and learning speed. By optimizing the agent’s policy effectively while avoiding overly complex calculations, PPO has become a practical solution in various domains, such as robotics, gaming, and autonomous systems. Its simplicity makes it easy to implement, which has led to its widespread adoption in both research and industry.

This article explores the mathematical foundations of RL and the key concepts introduced by PPO, providing a deeper understanding of why PPO has become a go-to algorithm in modern reinforcement learning research.

1. The Basics of RL: Markov Decision Process (MDP)

Reinforcement learning problems are often modeled using a Markov Decision Process (MDP), a mathematical framework that helps formalize decision-making in environments where outcomes are uncertain.

A Markov chain models a system that transitions between states, where the probability of moving to a new state depends solely on the current state and not on previous states. This principle is known as the Markov property. In the context of MDPs, this simplification is key for modeling decisions, as it allows an agent to focus only on the current state when making decisions without needing to account for the entire history of the system.

An MDP is defined by the following elements:
– S: Set of possible states.
– A: Set of possible actions.
– P(s’|s, a): Transition probability of reaching state s’ after taking action a in state s.
– R(s, a): Reward received after taking action a in state s.
– γ: Discount factor (a value between 0 and 1) that reflects the importance of future rewards.

The discount factor γ is crucial for modeling the importance of future rewards in decision-making problems. When an agent makes a decision, it must evaluate not only the immediate reward but also the potential future rewards. The discount γ reduces the impact of rewards that occur later in time due to the uncertainty of reaching those rewards. Thus, a value of γ close to 1 indicates that future rewards are almost as important as immediate rewards, while a value close to 0 gives more importance to immediate rewards.

The time discount reflects the agent’s preference for quick gains over future ones, often due to uncertainty or the possibility of changes in the environment. For example, an agent will likely prefer an immediate reward rather than one in the distant future unless that future reward is sufficiently significant. This discount factor thus models optimization behaviors where the agent considers both short-term and long-term benefits.

The goal is to find an action policy π(a|s) that maximizes the expected sum of rewards over time, often referred to as the value function:

https://medium.com/media/b18c309a10a929d71eabf6ab18207277/href

This function represents the expected total reward an agent can accumulate starting from state s and following policy π.

2. Policy Optimization: Policy Gradient

Policy gradient methods focus on directly optimizing the parameters θ of a policy πθ by maximizing an objective function that represents the expected reward obtained by following that policy in a given environment.

The objective function is defined as:

https://medium.com/media/906c9d213d2ed42ff1f831324e868e67/href

Where R(s, a) is the reward received for taking action a in state s, and the goal is to maximize this expected reward over time. The term dπ(s) represents the stationary distribution of states under policy π, indicating how frequently the agent visits each state when following policy π.

The policy gradient theorem gives the gradient of the objective function, providing a way to update the policy parameters:

https://medium.com/media/21342fc2584a3fb91526369b2d07665b/href

This equation shows how to adjust the policy parameters based on past experiences, which helps the agent learn more efficient behaviors over time.

3. Mathematical Enhancements of PPO

PPO (Proximal Policy Optimization) introduces several important features to improve the stability and efficiency of reinforcement learning, particularly in large and complex environments. PPO was introduced by John Schulman et al. in 2017 as an improvement over earlier policy optimization algorithms like Trust Region Policy Optimization (TRPO). The main motivation behind PPO was to strike a balance between sample efficiency, ease of implementation, and stability while avoiding the complexities of TRPO’s second-order optimization methods. While TRPO ensures stable policy updates by enforcing a strict constraint on the policy change, it relies on computationally expensive second-order derivatives and conjugate gradient methods, making it challenging to implement and scale. Moreover, the strict constraints in TRPO can sometimes overly limit policy updates, leading to slower convergence. PPO addresses these issues by using a simple clipped objective function that allows the policy to update in a stable and controlled manner, avoiding forgetting previous policies with each update, thus improving training efficiency and reducing the risk of policy collapse. This makes PPO a popular choice for a wide range of reinforcement learning tasks.

a. Probability Ratio

One of the key components of PPO is the probability ratio, which compares the probability of taking an action in the current policy πθ to that of the old policy πθold:

https://medium.com/media/7441b066a260ee10d4de7e76dff268b7/href

This ratio provides a measure of how much the policy has changed between updates. By monitoring this ratio, PPO ensures that updates are not too drastic, which helps prevent instability in the learning process.

b. Clipping Function

Clipping is preferred over adjusting the learning rate in Proximal Policy Optimization (PPO) because it directly limits the magnitude of policy updates, preventing excessive changes that could destabilize the learning process. While the learning rate uniformly scales the size of updates, clipping ensures that updates stay close to the previous policy, thereby enhancing stability and reducing erratic behavior.

The main advantage of clipping is that it allows for better control over updates, ensuring more stable progress. However, a potential drawback is that it may slow down learning by limiting the exploration of significantly different strategies. Nonetheless, clipping is favored in PPO and other algorithms when stability is essential.

To avoid excessive changes to the policy, PPO uses a clipping function that modifies the objective function to restrict the size of policy updates. This is crucial because large updates in reinforcement learning can lead to erratic behavior. The modified objective with clipping is:

https://medium.com/media/8b1c9350a5beeec04769107eecf7c4db/href

The clipping function constrains the probability ratio within a specific range, preventing updates that would deviate too far from the previous policy. This helps avoid sudden, large changes that could destabilize the learning process.

c. Advantage Estimation with GAE

In RL, estimating the advantage is important because it helps the agent determine which actions are better than others in each state. However, there is a trade-off: using only immediate rewards (or very short horizons) can introduce high variance in advantage estimates, while using longer horizons can introduce bias.

Generalized Advantage Estimation (GAE) strikes a balance between these two by using a weighted average of n-step returns and value estimates, making it less sensitive to noise and improving learning stability.

Why use GAE?
– Stability: GAE helps reduce variance by considering multiple steps so the agent does not react to noise in the rewards or temporary fluctuations in the environment.
– Efficiency: GAE strikes a good balance between bias and variance, making learning more efficient by not requiring overly long sequences of rewards while still maintaining reliable estimates.
– Better Action Comparison: By considering not just the immediate reward but a broader horizon of rewards, the agent can better compare actions over time and make more informed decisions.

The advantage function At is used to assess how good an action was relative to the expected behavior under the current policy. To reduce variance and ensure more reliable estimates, PPO uses Generalized Advantage Estimation (GAE). This method smooths out the advantages over time while controlling for bias:

https://medium.com/media/e9917004af8c1eec3a06a810c11b6ee0/href

This technique provides a more stable and accurate measure of the advantage, which improves the agent’s ability to make better decisions.

d. Entropy to Encourage Exploration

PPO incorporates an entropy term in the objective function to encourage the agent to explore more of the environment rather than prematurely converging to a suboptimal solution. The entropy term increases the uncertainty in the agent’s decision-making, which prevents overfitting to a specific strategy:

https://medium.com/media/c71d789209c6de8c363b34732713bcb7/href

Where H(πθ) represents the entropy of the policy. By adding this term, PPO ensures that the agent does not converge too quickly and is encouraged to continue exploring different actions and strategies, improving overall learning efficiency.

Conclusion

The mathematical underpinnings of PPO demonstrate how this algorithm achieves stable and efficient learning. With concepts like the probability ratio, clipping, advantage estimation, and entropy, PPO offers a powerful balance between exploration and exploitation. These features make it a robust choice for both researchers and practitioners working in complex environments. The simplicity of PPO, combined with its efficiency and effectiveness, makes it a popular and valuable algorithm in reinforcement learning.

Reference

https://books.google.fr/books?hl=fr&lr=&id=sWV0DwAAQBAJ&oi=fnd&pg=PR7&dq=reinforcement+learning+an+introduction&ots=1-9av2aqTb&sig=qMSnFC56yqPQugvqS3_uwCN78z0#v=onepage&q=reinforcement%20learning%20an%20introduction&f=falsehttps://fse.studenttheses.ub.rug.nl/25709/1/mAI_2021_BickD.pdf

This article was partially translated from French using DeepL.

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