We study offline policy optimization in finite-horizon Markov decision processes where every deterministic Markov policy has an action-value function linear in known features, with coefficient norm at most $B$. From independent complete trajectories of one unknown Markov behavior policy, and under all-policy occupancy coverage, we give an algorithm with $\widetilde O(\varepsilon^{-2})$ sample dependence and computational complexity polynomial in $\log(2+B)$. The algorithm performs exactly one convex regression per layer. Its stopped-return targets are the values of a single consistent Markov witness policy. A shared reference design and bounded prediction constraints convert fast clipped-regression error into pointwise confidence for the raw linear predictions. Truncated leverage controls the performance loss without requiring empirical covariance approximation at scale $B^{-2}$. Rounding confidence radii upward preserves the exact witness and yields a polynomial-time bit implementation for rational numerical inputs. The guarantees allow any fixed initial-state distribution; realizability and coverage need only be checked for deterministic policies. We also establish a covariance-coverage variant with uniform approximate realizability and polynomial dependence on $B$. A two-action lower bound shows that the accuracy exponent $2$ is necessary.

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