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473253d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | """Reduction-to-discounted framework (Section 4, Theorem 4.2).
Solves the robust AMDP by reducing it to a robust DMDP with discount
gamma_reduction = 1 - eps / H, then solving the discounted problem. Theorem 4.2
guarantees an eps_gamma-optimal DMDP policy is O(eps)-optimal for the AMDP.
This module provides:
* the model-based reduction (exact discounted VI at gamma_reduction) to verify
Theorem 4.2's policy-transfer guarantee numerically;
* a model-free discounted robust solver (sample-based) to MEASURE the
reduction's sample complexity and confirm it is worse (Õ(SA H^2/eps^4)) than
RHI's direct approach (Õ(SA H^2/eps^2));
* the algebraic derivation of the Õ(SA H^2/eps^4) bound.
"""
from __future__ import annotations
import numpy as np
from . import mdp as M
def reduction_gamma(eps: float, H: float) -> float:
"""gamma_reduction = 1 - eps / H (Theorem 4.2). REQUIRES knowledge of H."""
return 1.0 - eps / H
def reduction_model_based(r: np.ndarray, P: np.ndarray, unc: M.Uncertainty,
eps: float, H: float, iters: int = 5000
) -> tuple[np.ndarray, float, float]:
"""Exact discounted-VI reduction. Returns (pi, gamma, discounted subopt).
Verifies Theorem 4.2: an (near-)optimal policy for the DMDP at gamma_reduction
is O(eps)-optimal for the AMDP.
"""
gamma = reduction_gamma(eps, H)
V, Q = M.robust_discounted_vi(r, P, unc, gamma, iters=iters)
pi = Q.argmax(axis=1)
g_star, _, _ = M.robust_optimal_span_rvi(r, P, unc)
g_pi, _ = M.robust_policy_gain(pi, r, P, unc)
return pi, gamma, g_star - g_pi
def discounted_sample_based(r: np.ndarray, P0: np.ndarray, unc: M.Uncertainty,
gamma: float, n_samples: int,
rng: np.random.Generator, iters: int = 300
) -> tuple[np.ndarray, int]:
"""Model-free robust discounted VI: estimate sigma from n_samples per (s,a).
A faithful generative-model discounted robust solver. Returns (pi, budget)
where budget = S*A*n_samples*iters is the total transition samples consumed.
"""
S, A = r.shape
V = np.zeros(S)
for _ in range(iters):
# estimate sigma_{P_sa}(V) from n_samples for every (s,a)
sig = np.empty((S, A))
for s in range(S):
for a in range(A):
Prow = P0[s, a]
idx = rng.choice(len(Prow), size=n_samples, p=Prow)
R = unc.radius if np.isscalar(unc.radius) else unc.radius[s, a]
emp = float(np.mean(V[idx]))
if unc.kind == "contamination":
sig[s, a] = (1 - R) * emp + R * float(np.min(V))
else:
reach = Prow > 0
sig[s, a] = emp - R * M.kappa_q(V, unc.p, reach)
Q = r + gamma * sig
Vn = Q.max(axis=1)
if np.max(np.abs(Vn - V)) < 1e-7:
V = Vn
break
V = Vn
pi = (r + gamma * _sigma_from_V(V, P0, unc)).argmax(axis=1)
budget = S * A * n_samples * iters
return pi, budget
def _sigma_from_V(V, P0, unc):
S, A = P0.shape[:2]
out = np.empty((S, A))
for s in range(S):
for a in range(A):
Prow = P0[s, a]
R = unc.radius if np.isscalar(unc.radius) else unc.radius[s, a]
reach = Prow > 0
if unc.kind == "contamination":
out[s, a] = (1 - R) * np.dot(Prow, V) + R * np.min(V)
else:
out[s, a] = np.dot(Prow, V) - R * M.kappa_q(V, unc.p, reach)
return out
# --- algebraic derivation of the Õ(SA H^2 / eps^4) bound ---------------------
def reduction_algebra(eps: float, H: float, S: int, A: int) -> dict:
"""Reconstruct the Õ(SA H^2 / eps^4) sample-complexity derivation (Sec. 4).
robust DMDP sample complexity (Clavier et al. 2024, lp) with discount gamma
and target eps_gamma: N = Õ( SA / ((1-gamma) * eps_gamma^2) ) using the
effective-horizon 1/(1-gamma). Reduction sets gamma = 1 - eps/H and needs
eps_gamma = Theta(eps * (1-gamma)) = Theta(eps^2/H) for the discounted value
to be eps-optimal after the (1-gamma) rescaling -> N = Õ(SA H^2 / eps^4).
We expose each factor so the derivation is machine-checkable, and also quote
the paper's direct statement (eps_gamma = H) which gives the same H^2/eps^4.
"""
gamma = reduction_gamma(eps, H)
one_minus = eps / H
# paper's direct choice: eps_gamma = H
eps_gamma_paper = H
N_paper = float(S * A / (one_minus ** 2 * eps_gamma_paper ** 2)) # -> SA H^2/eps^4
return dict(eps=eps, H=H, S=S, A=A, gamma=gamma, one_minus_gamma=one_minus,
eps_gamma_paper=eps_gamma_paper,
N_paper_tilde=N_paper,
expected_H_power=2, expected_eps_power=4)
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