Temporal Difference (TD) learning stands as a middle ground between Monte Carlo (MC) and Dynamic Programming (DP) methods, effectively combining the strengths of both. Like MC, TD is model-free and learns directly from sampling, meaning that complete knowledge of the environment is not necessary. Similar to DP, TD does not wait until the end of an episode to perform updates. Instead, it uses bootstrapping to estimate updates after each step.

**TD Update Rule:**

The update rule for TD is given by:

Vk+1(st)=Vk(st)+Î±[rt+1+Î³Vk(st+1)âˆ’Vk(st)]

Here, *Gt*â€‹, the return function in MC, is replaced by *rt*+1â€‹+*Î³Vk*â€‹(*st*+1â€‹), which is the immediate reward received plus the discounted estimate of the return from the subsequent state. This combination is known as the TD target. The TD target offers a more accurate or updated estimate for state *st*â€‹ based on new data.

**TD Error:**

The TD error, defined as the difference between the TD target and the current estimate of the stateâ€™s value, helps adjust the value functions:

*Î´t*â€‹=*rt*+1â€‹+*Î³Vk*â€‹(*st*+1â€‹)âˆ’*Vk*â€‹(*st*â€‹)

A positive TD error suggests that the actual outcome was better than expected, indicating that the value estimate *V*(*st*â€‹) might be too low and should be increased. Conversely, a negative TD error suggests that the outcome was worse than expected, and *V*(*st*â€‹) might be too high.

By continuously adjusting the value estimates based on the TD error, the learning algorithm incrementally improves its predictions about the value of each state. This adjustment process uses new experiences to correct past estimations, aligning them closer to the actual returns observed.

**Types of TD Learning:**

**One-Step TD (TD(0))**:

This is the simplest form of TD learning, where updates to the value function are made after every single step, using only the immediate reward and the value estimate of the next state.

**Multi-Step TD**:

This approach extends the idea of TD(0) by considering rewards and state values over multiple steps before making an update. It strikes a balance between the bias of TD(0) and the variance of MC, potentially involving two-step, three-step, up to n-step updates.

**TD(***Î»***)**:

A more general form of multi-step TD, which uses eligibility traces to combine information from all lengths of future paths, from one step up to full paths, weighted by a factor

*Î»*.

**Benefits of TD Learning:**

**Efficiency and Flexibility**: TD algorithms can learn after each step of agent-environment interaction, allowing them to update in an online manner. They do not need to wait until the end of the episode to perform an update, thus they can learn from incomplete sequences and apply to both episodic and non-episodic tasks.**Convergence**: Since each experience sample collected from the environment impacts the update step, TD algorithms generally converge faster than MC methods, with lower variance since the update step in TD learning is influenced by fewer factors (one action, reward, and next state).

**Drawbacks of TD Learning:**

**Sensitivity to Initial Conditions**: TD methods can be highly sensitive to the initial state, which might lead to longer convergence times.**Bias**: TD methods are biased due to their reliance on bootstrapping to update the estimated value function based on the current estimate of the value. In contrast, MC methods are unbiased, using the true return obtained from following the trajectory for the update step.

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