---
title: Q-learning with Adjoint Matching (QAM)
url: https://www.emergentmind.com/topics/q-learning-with-adjoint-matching-qam
type: topic
---

# Q-learning with Adjoint Matching (QAM)

Q-learning with Adjoint Matching (QAM) is a temporal-difference-based reinforcement learning method addressing the challenge of optimizing expressive diffusion or flow-matching policies with respect to a parameterized Q-function in continuous-action domains. QAM incorporates adjoint matching, a technique originally from generative modeling, to preserve first-order critic information for policy improvement while entirely circumventing the numerical instability associated with backpropagation through multi-step denoising processes inherent in flow or diffusion policy classes. In tandem with standard TD backup for critic learning, QAM yields provably unbiased, behavior-constrained optimal policies and demonstrates state-of-the-art empirical performance in offline and offline-to-online RL benchmarks [2601.14234].

## 1. Formal Problem Setup

QAM operates within a continuous-action Markov decision process (MDP) specified by $(\mathcal{S}, \mathcal{A}, P, \gamma, R, \mu)$, where $\mathcal{A} \subseteq \mathbb{R}^d$ and $\gamma \in [0,1)$ is the discount factor. The primary objective in offline RL is to learn a policy $\pi(a|s)$ maximizing the expected return, given a static dataset $\mathcal{D} = \{(s_i, a_i, r_i, s_i')\}$. 

Q-learning employs a parameterized critic $Q_\phi(s,a)$, trained to minimize the standard TD error:
\[
L(\phi) = \mathbb{E}_{(s, a, r, s') \sim \mathcal{D}} \bigl[(Q_\phi(s, a) - (r + \gamma Q_{\bar{\phi}}(s', a')))^2\bigr]
\]
where $a' \sim \pi_\theta(\cdot | s')$ and $\bar{\phi}$ tracks $\phi$ via a Polyak average.

Departing from conventional Gaussian policies, QAM utilizes a flow-matching policy $\pi_\theta(a|s)$ parameterized by a time-indexed velocity field $f_\theta: \mathcal{S} \times \mathcal{A} \times [0,1] \rightarrow \mathcal{A}$. This defines a memoryless SDE:
\[
da_t = \bigl(2 f_\theta(s, a_t, t) - \frac{a_t}{t}\bigr) dt + \sqrt{\frac{2(1-t)}{t}} dB_t,\quad a_1 \sim \text{data},\; a_0 \sim \mathcal{N}(0, I)
\]
Sampling the endpoint $a_0$ (via ODE integration) yields actions from $\pi_\theta(a|s)$. The base (behavior) policy $\pi_\beta$ is learned with standard flow matching:
\[
L_{FM}(\beta) = \mathbb{E}_{(s, a) \sim \mathcal{D},\, t \sim U[0,1],\, z \sim \mathcal{N}} [\| f_\beta(s, (1-t)z + ta, t) - (a - z) \|^2]
\]
QAM aims to extract the optimal policy constrained by behavior:
\[
\pi^*(a|s) \propto \pi_\beta(a|s) \exp[\tau Q_\phi(s,a)]
\]
by fine-tuning $f_\theta$ without unstable backpropagation through the SDE.

## 2. Derivation of the Adjoint-Matching Objective

### 2.1 Critic Gradients and BPTT Instability

Naive actor-critic approaches hill-climb the Q-function via the policy gradient $\nabla_a Q_\phi(s, a)$. For multi-step flow or diffusion policies, this requires backpropagation through all intermediate ODE/SDE steps and associated noise, leading to compounding of ill-conditioned Jacobians and significant numerical instability.

### 2.2 From Stochastic Optimal Control to Adjoint Matching

**Stochastic Optimal Control (SOC) Formulation:**  
Optimal flow-based policy extraction can be cast as minimizing the SOC loss:
\[
L_{SOC}(\theta) = \mathbb{E}_{s \sim \mathcal{D},\, a_{(\cdot)}} \Bigl[\int_0^1 \tfrac{1}{2} \|f_\theta(s, a_t, t) - f_\beta(s, a_t, t)\|^2 dt - \tau Q_\phi(s, a_1)\Bigr]
\]
where trajectories $a_t$ evolve under the memoryless SDE. Naive differentiation with respect to $\theta$ entails gradient backpropagation through the entire stochastic trajectory.

**Basic Adjoint Matching:**  
Define the adjoint (co-state) $g(s,a_t,t)$ via the backward ODE:
\[
\frac{dg}{dt} = -\bigl[\nabla_{a_t}(2 f_\theta - a_t/t)\bigr]g + \frac{2}{\sigma_t^2}\nabla_{a_t}\|f_\theta - f_\beta\|^2, \qquad g(s,a_1,1) = -\tau \nabla_a Q_\phi(s,a_1)
\]
Domingo-Enrich et al. (2025) established the equivalence $L_{SOC}(\theta) = L_{BAM}(\theta)$, with
\[
L_{BAM}(\theta) = \mathbb{E} \biggl[\int_0^1 \bigl\| 2(f_\theta - f_\beta)/\sigma_t + \sigma_t g \bigr\|^2 dt\biggr]
\]

**Lean Adjoint Matching (QAM):**  
At optimality ($f_\theta = f^*$), $\nabla_{a_t}f_\theta$ vanishes in the adjoint ODE, permitting a "lean" adjoint $\tilde{g}$ computed purely through the fixed base velocity field $f_\beta$:
\[
\frac{d\tilde{g}}{dt} = - \nabla_{a_t}[2 f_\beta(s,a_t,t) - a_t/t] \cdot \tilde{g},\qquad \tilde{g}(s,a_1,1) = -\tau \nabla_a Q_\phi(s,a_1)
\]
The resulting QAM objective circumvents unstable backpropagation through $f_\theta$:
\[
L_{AM}(\theta) = \mathbb{E}_{s, \{a_t\}} \left[\int_0^1 \|2(f_\theta(s, a_t, t) - f_\beta(s, a_t, t))/\sigma_t + \sigma_t \tilde{g}_t\|^2 dt\right]
\]
This construction exactly targets the desired optimal policy.

## 3. Algorithmic Workflow

QAM alternates between critic TD-updates and policy updates using adjoint matching.

- **Critic update:**  
  - Sample batch $(s, a, r, s') \sim \mathcal{D}$
  - Compute $a' \gets \text{ODE}(f_\theta(s',\cdot,\cdot), z \sim \mathcal{N})$
  - Calculate target $y \gets r + \gamma [\mū Q_\text{mean}(s', a') - \rho Q_\text{std}(s', a')]$
  - Update $Q_\phi$ to minimize squared Bellman error
  - Update target $\bar{\phi}$

- **Policy update:**  
  - Forward: Sample $z \sim \mathcal{N}$, rollout $a_t$ through the SDE under $f_\theta$
  - Adjoint: Initialize $\tilde{g}(1) = -\tau \nabla_a Q_\phi(s,a_1)$; backward step $\tilde{g}(t-h) = \tilde{g}(t) + h \cdot \mathrm{VJP}(\nabla_a [2 f_\beta - a/t], \tilde{g}(t))$
  - Calculate $L_{AM}$ as above
  - Update $\theta$ to minimize $L_{AM}$

Convergence is defined by $\partial L_{AM}/\partial f_\theta = 0$, ensuring that $f_\theta$ induces the policy $\pi^* \propto \pi_\beta \exp(\tau Q)$ without gradient bias and retaining full flow expressivity.

## 4. Theoretical Analysis

**Unique Optimality and Unbiasedness:**  
Proposition 4.1 (extension of Domingo-Enrich 2025) asserts that $L_{AM}$ admits a unique minimizer $f_\theta^*$ such that:
\[
\forall s \in \text{supp}(\mathcal{D}):\quad \pi^*_\theta(\cdot | s) \propto \pi_\beta(\cdot|s)\exp[\tau Q_\phi(s,\cdot)]
\]
and in the limit $L_{AM} \rightarrow 0$ (with sufficient function approximation capacity) QAM recovers the exact behavior-constrained optimum.

The "lean adjoint" $\tilde{g}$ propagates the Q-function’s boundary gradient according to the fixed $f_\beta$, rigorously linking the residual flow velocity to the SOC solution. The unbiasedness of the adjoint matching gradient for the SOC objective follows directly. Variance is controlled by the discretization steps $T$ of the SDE and the sampling noise in $\tilde{g}$; $T \approx 10$ is effective in practice.

## 5. Empirical Results and Comparative Evaluation

QAM was benchmarked on OGBench—ten long-horizon, sparse-reward environments (antmaze, humanoidmaze, cube, puzzle, scene)—against 17 baseline algorithms spanning Gaussian policies (ReBRAC), backprop-through-time flows (FBRAC, BAM), one-step distillations (FQL), advantage-weighted flows (FAWAC), gradient guidance methods (QSM, DAC, CGQL variants), and post-hoc editors (DSRL, FEdit, IFQL).

- **Offline RL (1M updates):**  
  QAM, with $\tau$ tuned per domain, attained $\approx$44/50 normalized score across tasks, outperforming the nearest baseline ($\approx$40). Discarding $\nabla_a Q$ (FAWAC) scored $\approx 8$, while backprop-based flows (BAM) scored $\approx 35$—demonstrating the performance and stability benefit of the lean adjoint technique. Reducing flows to one step (FQL) markedly degraded performance on multi-modal tasks. Guided and post-processed methods were consistently outperformed.

- **Offline-to-online fine-tuning (0.5M environment steps):**  
  The QAM-EDIT variant, introducing a minor relaxation beyond $\pi_\beta \exp(\tau Q)$, realized the fastest score improvements and highest asymptotic returns while being more sample-efficient than all considered baselines, including RLPD (behavior-unconstrained), QSM, and FQL.

- **Ablation and sensitivity analyses:**  
  Adjoint matching (BAM) versus lean adjoint (QAM) revealed higher stability and performance for QAM. The inverse-temperature $\tau$ is the most critical hyperparameter; results are robust to moderate numbers of flow steps ($T \approx 10$) and gradient clipping. In noise and data stitching robustness evaluations, QAM maintained near-optimality, unlike most baselines which significantly degraded.

## 6. Related Methodologies and Context

QAM subsumes and generalizes prior approaches for flow-based policy optimization in offline RL. Existing methods are limited either by their exclusion of first-order critic information, reliance on actor-critic updates susceptible to gradient bias or numerical instability (due to backpropagation through multiple ODE/SDE integration steps), or by constraining policy expressivity to simplify gradients. QAM, by leveraging adjoint matching, uniquely preserves full generative policy expressivity and unbiased policy gradients while entirely sidestepping backpropagation through noisy, multi-step diffusion processes. This positions QAM as a principled policy-extraction method for complex action distributions in RL, with guaranteed convergence to the optimal behavior-constrained distribution [2601.14234].

Source: https://www.emergentmind.com/topics/q-learning-with-adjoint-matching-qam