Fix Chapter 8 test: use truncated episodic λ-return to match backward-view sum

Author: srikanthbaride

ch8_td_lambda/tests/test_forward_backward_equiv.py

@@ -1,28 +1,30 @@
 import numpy as np
 
-def lambda_return(b, c, lam):
-    G1 = b
-    G2 = c
-    G3 = 1.0
-    return (1 - lam) * (G1 + lam * G2 + lam**2 * G3)
+def truncated_lambda_return(b, c, lam):
+    """
+    Finite-episode λ-return for a 3-step episode with rewards (0,0,1), γ=1.
+    G^(1)=b, G^(2)=c, G^(3)=1.
+    G^λ = (1-λ)(G1 + λ G2) + λ^2 G3
+    """
+    G1, G2, G3 = b, c, 1.0
+    return (1 - lam) * (G1 + lam * G2) + (lam ** 2) * G3
 
 def test_forward_backward_equivalence():
     a, b, c = 0.5, 0.3, 0.2
     lam = 0.5
 
-    # forward λ-return update for V(s0) = a
-    Glam = lambda_return(b, c, lam)
+    # Forward (truncated episodic λ-return)
+    Glam = truncated_lambda_return(b, c, lam)
     forward_update = Glam - a
 
-    # backward view TD error updates
+    # Backward: TD errors and eligibilities for s0 (γ=1)
     d0 = b - a
     d1 = c - b
     d2 = 1.0 - c
 
-    # eligibilities for s0 at each step
     e0 = 1.0
-    e1 = lam       # after one step
-    e2 = lam**2    # after two steps (γ=1 here)
+    e1 = lam
+    e2 = lam ** 2
 
     backward_update = d0 * e0 + d1 * e1 + d2 * e2