TD(λ) Asymptotic Error Bound Verification

Implement a function that computes and verifies the asymptotic error bound of TD(lambda) with linear function approximation for a given Markov Reward Process (MRP). In reinforcement learning, TD(lambda) with linear function approximation converges to a fixed point solution whose approximation error can be theoretically bounded relative to the best possible linear approximation error (the Monte Carlo projection error). Given an MRP defined by: A transition probability matrix P of shape (n, n) An expected reward vector r of shape (n,) A discount factor gamma in [0, 1) A feature matrix Phi of shape (n, k) for linear value function approximation V(s) = Phi @ w A stationary distribution d of shape (n,) (used as the weighting in the D norm) A list of lambda values to evaluate Your function should compute: 1. The true value function V of the MRP 2. The Monte Carlo (MC) projection of V onto the feature space under the D weighted norm, and its error 3. For each lambda, the TD(lambda) fixed point solution and its error under the D weighted norm 4. The error amplification ratio (TD error / MC error) for each lambda 5. The theoretical upper bound on the amplification ratio 6. Whether the bound holds for each lambda value The D weighted norm is defined as ||x|| D = sqrt(x^T D x) where D = diag(d). Return a dictionary with keys: 'true values', 'mc error', 'td errors', 'ratios', 'bound', 'bound holds'. All numeric values should be rounded to 4 decimal places. The 'bound holds' entry should be a list of booleans.