n-step state-value algorithms

Why Tree Backup Matters

The n-step tree-backup algorithm belongs to the off-policy material in Chapter 7. Its defining distinction is that it is off-policy without using importance sampling. That single property makes it important to separate tree backup from methods whose off-policy correction depends on importance-sampling ratios.

The Off-Policy Distinction

An off-policy method separates the policy generating experience from the policy being learned about. For n-step tree backup, the source makes a specific claim about this separation: the algorithm is off-policy and does not involve importance sampling. Therefore, the defining point is not merely that tree backup uses multiple steps. The defining point is the way it supports off-policy learning without importance-sampling ratios.

Tracing the Return Boundary

The supplied boundary rule is G^(n)_t = G_t whenever t + n ≥ T. Here, T is the terminal time named by the condition. The practical interpretation is direct: once the n-step index reaches or passes the terminal time, the n-step return is represented by G_t rather than by a separate unfinished n-step expression.

Original and Semi-Gradient Forms

The semi-gradient tree-backup algorithm is a semi-gradient version of the original tree-backup method. These are connected algorithms, but they should not be treated as unrelated names. The original method is the base tree-backup algorithm; the semi-gradient version applies semi-gradient methods to that original method.

Tree Backup and Q Sigma

n-step Q(σ) and semi-gradient tree backup play different roles. The source describes n-step Q(σ) as having a broader unifying role across action-value algorithms. By contrast, semi-gradient tree backup is specifically a semi-gradient version of the original tree-backup algorithm. Q(σ) is therefore the broader unifying idea in this comparison, while semi-gradient tree backup names a particular version of tree backup.

Common Misreadings

• Calling tree backup on-policy because it does not use importance sampling.

  The source explicitly identifies n-step tree backup as off-policy and separately emphasizes its lack of importance sampling.

  Fix: Remember both properties together: tree backup is off-policy and does not involve importance sampling.

• Treating the semi-gradient tree-backup algorithm as a completely different algorithm.

  The semi-gradient algorithm is described as a semi-gradient version of the original method.

  Fix: View the semi-gradient form as a version of the original tree-backup method.

• Reading t + n ≥ T as though the n-step return remains an unspecified expression.

  The supplied condition explicitly gives G^(n)_t = G_t in that case.

  Fix: Replace the n-step return representation with G_t whenever t + n ≥ T.

• Using n-step Q(σ) and semi-gradient tree backup as interchangeable labels.

  The source gives Q(σ) a broader unifying role across action-value algorithms, while semi-gradient tree backup is a version of tree backup.

  Fix: Match each term to its scope before explaining it.

Check Your Understanding

Key Takeaways

1. N-step tree backup is an off-policy method distinguished by its lack of importance sampling.
2. The semi-gradient tree-backup algorithm is a semi-gradient version of the original tree-backup method.
3. When t + n ≥ T, the supplied boundary rule is G^(n)_t = G_t.
4. N-step Q(σ) has a broader unifying role across action-value algorithms, so it should not be treated as a synonym for semi-gradient tree backup.
5. The target-policy and behavior-policy distinction must be kept separate from the question of whether importance sampling is used.