Flow Model Using MDP Homomorphism
Mathematical derivation and definitions: Continuous MDP Homomorphisms
Step 1. Continuous Markov Decision Processes (MDPs)
Definition (Continuous MDP): A continuous MDP is defined as a tuple:
\[\mathcal{M} = (\mathcal{S}, \Sigma, \mathcal{A}, \{\tau_a\}_{a\in\mathcal{A}}, R, \gamma)\]where:
- $\mathcal{S}$ is a measurable state space (often $\mathbb{R}^n$).
- $\Sigma$ is the sigma-algebra on $\mathcal{S}$.
- $\mathcal{A}$ is a measurable action space (typically locally compact metric space).
-
$\tau_a: \mathcal{S}\times\Sigma\to [0,1]$ is the transition probability kernel such that for every state $s \in \mathcal{S}$ and action $a \in \mathcal{A}$, $\tau_a(\cdot s)$ is a probability measure on $\mathcal{S}$. - $R: \mathcal{S}\times\mathcal{A}\to\mathbb{R}$ is the reward function.
- $\gamma\in[0,1)$ is the discount factor.
A stochastic policy $\pi$ is a measurable mapping:
\[\pi : \mathcal{S} \to \Delta(\mathcal{A})\]The objective is to maximize expected return:
\[J(\pi) = \mathbb{E}_{s_0\sim\rho_0}\left[\sum_{t=0}^{\infty}\gamma^t\,R(s_t,a_t)\right]\]Step 2. Continuous MDP Homomorphisms
Definition (Continuous MDP Homomorphism): A measurable surjective map:
\[h = (f, g),\quad\text{where}\quad f:\mathcal{S}\to\tilde{\mathcal{S}}, \quad g:\mathcal{S}\times\mathcal{A}\to\tilde{\mathcal{A}}\]is an MDP homomorphism if the following two properties hold:
(1) Reward Invariance:
\[R(s,a) = \tilde{R}(f(s), g(s,a)) \quad \text{for all } (s,a)\in\mathcal{S}\times\mathcal{A}\](2) Transition Equivariance:
For any Borel set $B\subseteq\tilde{\mathcal{S}}$, we have:
\[\tau_a(f^{-1}(B)\mid s) = \tilde{\tau}_{g(s,a)}(B\mid f(s))\]This ensures transitions on the abstract space correctly represent the transitions in the original space after applying the homomorphism.
Step 3. Lifted (Push-forward) Policy
Definition (Lifted Policy): Given a stochastic policy defined in the abstract MDP:
\[\pi : \tilde{\mathcal{S}}\to\Delta(\tilde{\mathcal{A}})\]The lifted policy is defined by the push-forward measure via $g$:
\[\pi^\uparrow(g^{-1}(s,B)\mid s) = \pi(B\mid f(s)), \quad \text{for any measurable } B\subseteq\tilde{\mathcal{A}}\]where:
- $g^{-1}(s,B) = {a\in\mathcal{A}\mid g(s,a)\in B}$ is the set-valued preimage.
Step 4. Value Equivalence Theorem
Theorem (Value Equivalence): Let $h=(f,g)$ be a continuous MDP homomorphism. Then for all state-action pairs $(s,a)\in\mathcal{S}\times\mathcal{A}$:
\[Q^{\pi^\uparrow}(s,a)=Q^\pi(f(s),g(s,a))\]Proof (Sketch): Induction on Bellman recursion.
- Base case:
- Inductive step: Assume equality holds at step $n$. For step $n+1$:
Change variables under $h=(f,g)$ and apply the inductive assumption and push-forward measure condition to conclude equality at step $n+1$. $\square$
Step 5. Stochastic Homomorphic Policy Gradient (HPG) theorem
Theorem (Stochastic HPG): The gradient of the performance measure in the original MDP can be computed solely in the abstract space:
\[\nabla_\theta J(\pi_\theta^\uparrow)=\int_{\tilde{\mathcal{S}}}\rho^{\pi_\theta}(\tilde s)\int_{\tilde{\mathcal{A}}}Q^{\pi_\theta}(\tilde s,\tilde a)\nabla_\theta\pi_\theta(d\tilde a|\tilde s)\,d\tilde s\]- Here, $\rho^{\pi_\theta}$ is the stationary distribution in the abstract MDP, and $Q^{\pi_\theta}$ is the abstract action-value function.
- The theorem follows from Value Equivalence and the push-forward condition.
Step 6. Practical realization (Flow-Matching approach)
Use modern Flow Matching (FM) methods to instantiate $\pi_\theta$, $\tilde\tau$, and the lifting/decoding pipeline:
| Component | Implementation |
|---|---|
| $f_\phi$ | State encoder (NN) |
| $g_\eta$ | Action encoder (NN) |
| $\pi_\theta$ | Flow-matching actor (EFM/ReinFlow) |
| $\tilde\tau$ | Flow-matching transition dynamics |
| $h_\psi$ | Flexible non-invertible decoder |
All theoretical guarantees remain valid, provided training losses enforce the homomorphism properties.
Summary
This derivation gives a fully rigorous mathematical foundation for learning and exploiting compact latent MDPs via homomorphisms, backed by provable equivalence of policies and value functions, and supports modern flow-based methods for representation, control, and dynamics.
###