Structure Matters: Dynamic Policy Gradient

In this work, we study γ-discounted infinite-horizon tabular Markov decision processes (MDPs) and introduce a framework called dynamic policy gradient (DynPG). The framework directly integrates dynamic programming with (any) policy gradient method, explicitly leveraging the Markovian property of the environment. DynPG dynamically adjusts the problem horizon during training, decomposing the original infinite-horizon MDP into a sequence of contextual bandit problems. By iteratively solving these contextual bandits, DynPG converges to the stationary optimal policy of the infinite-horizon MDP. To demonstrate the power of DynPG, we establish its non-asymptotic global convergence rate under the tabular softmax parametrization, focusing on the dependencies on salient but essential parameters of the MDP. By combining classical arguments from dynamic programming with more recent convergence arguments of policy gradient schemes, we prove that softmax DynPG scales polynomially in the effective horizon (1 -γ) -1 . Our findings contrast recent exponential lower bound examples for vanilla policy gradient.

When µ is a Dirac measure at s we let

(µ) to denote the value function of the stationary policy π being applied h times in a row. For h = ∞ the resulting infinite-horizon discounted MDP admits a stationary optimal policy [20]. We define V * ∞ (µ) := sup π∈Π V π ∞ (µ) and use π * to denote a stationary policy that achieves µ). In contrast, when h is finite, the finite-horizon MDP optimization problem needs non-stationary optimal policies; thus, we define

For any function V ∈ R |S| and stationary policy π, the Bellman expectation operator T π : R |S| → R |S| is defined for every s ∈ S by T π (V )(s) = a∈A π(a|s) r(s, a) + γ s ′ ∈S p(s ′ |s, a)V (s ′ ) .