What is an Accumulating Eligibility Trace?
An Accumulating Eligibility Trace is a type of eligibility trace, which is a method used in reinforcement learning to keep track of which actions and states are responsible for rewards or punishments. This trace is accumulative in nature, meaning it increments over time, and is used to update the value function of the agent.
Eligibility traces are used in reinforcement learning to keep track of the history of actions and states that led to a certain reward or punishment. An eligibility trace decays over time, so that older events will be less important than recent ones. Accumulating Eligibility Traces are a type of eligibility trace that use an additive update rule, so that past events will influence the value function more strongly the longer they remain in memory.
How do Accumulating Eligibility Traces work?
Accumulating Eligibility Traces are implemented using a memory vector $\textbf{e}\_{t}$, where t is the time step of the agent. The memory vector is initialized to zero at the beginning of each episode:
$$\mathbf{e\_{0}} = \textbf{0}$$
At each time step, the eligibility trace is updated using the gradient of the value function $\hat{v}$ with respect to the parameters $\mathbf{\theta}$, the discount factor $\gamma$, and the trace decay parameter $\lambda$:
$$\textbf{e}\_{t} = \nabla{\hat{v}}\left(S\_{t}, \mathbf{\theta}\_{t}\right) + \gamma\lambda\textbf{e}\_{t-1}$$
Where $\nabla{\hat{v}}\left(S\_{t}, \mathbf{\theta}\_{t}\right)$ is the gradient of the value function with respect to the parameters, evaluated at the current state $S_t$. The trace decay parameter $\lambda$ determines how fast the eligibility trace decays over time, and the discount factor $\gamma$ determines the importance of future rewards.
Using the accumulated eligibility trace, the value function can be updated using the following formula:
$$\mathbf{\theta}\_{t+1} = \mathbf{\theta}\_{t} + \alpha\left(R\_{t+1} + \gamma\hat{v}\left(S\_{t+1}, \mathbf{\theta}\_{t}\right) - \hat{v}\left(S\_{t}, \mathbf{\theta}\_{t}\right)\right)\textbf{e}\_{t}$$
Where $\alpha$ is the learning rate, $R_{t+1}$ is the reward received at the next time step, and $\hat{v}\left(S\_{t+1}, \mathbf{\theta}\_{t}\right)$ is the estimated value of the state $S_{t+1}$.
What are the advantages of using Accumulating Eligibility Traces?
Accumulating Eligibility Traces have several advantages over other types of eligibility traces:
- They allow the agent to remember past events more strongly over time, which may be useful in learning complex tasks that require long sequences of actions.
- They can be used with a wide variety of value function approximators, including neural networks, decision trees, and linear models.
- They are easy to implement and can be used in combination with other types of reinforcement learning algorithms, such as SARSA and Q-learning.
- They can be used in both on-policy and off-policy reinforcement learning algorithms.
Overall, Accumulating Eligibility Traces are a flexible and powerful tool for learning in reinforcement learning systems.
