Q-Learning with Scalar Adjoint Matching

Yonghoon Dong, Minsung Yoon, Jaehyuk Kim, Jungwoo Park, Changyeon Kim +1

Flow policies capture rich and diverse action distributions, and fine-tuning them with off-policy RL to improve beyond the demonstrations has drawn growing interest. However, fine-tuning a flow policy against a learned value function is not trivial, because the policy generates its action over many flow steps. Adjoint matching offers a principled way to update the flow model itself by propagating value information from the final action back to each flow step, but it requires a vector--Jacobian product through the policy at every step, a cost that grows with the number of flow steps and the policy size. We observe that the batch-averaged velocity Jacobian of pretrained flow policies concentrates on its diagonal. Motivated by this finding, we derive a closed-form scalar adjoint that scales the value gradient at the final action by the flow time, eliminating the per-step vector--Jacobian products. We further find that controlling the critic's value at policy-generated actions is particularly important under the scalar adjoint. Based on these findings, we propose Q-learning with Scalar Adjoint Matching (SQAM), which combines the scalar adjoint with a value penalty at those actions.…