  Query-based object detectors directly decode image features into object
instances with a set of learnable queries. These query vectors are
progressively refined to stable meaningful representations through a sequence
of decoder layers, and then used to directly predict object locations and
categories with simple FFN heads. In this paper, we present a new query-based
object detector (DEQDet) by designing a deep equilibrium decoder. Our DEQ
decoder models the query vector refinement as the fixed point solving of an
{implicit} layer and is equivalent to applying {infinite} steps of refinement.
To be more specific to object decoding, we use a two-step unrolled equilibrium
equation to explicitly capture the query vector refinement. Accordingly, we are
able to incorporate refinement awareness into the DEQ training with the inexact
gradient back-propagation (RAG). In addition, to stabilize the training of our
DEQDet and improve its generalization ability, we devise the deep supervision
scheme on the optimization path of DEQ with refinement-aware
perturbation~(RAP). Our experiments demonstrate DEQDet converges faster,
consumes less memory, and achieves better results than the baseline counterpart
(AdaMixer). In particular, our DEQDet with ResNet50 backbone and 300 queries
achieves the $49.5$ mAP and $33.0$ AP$_s$ on the MS COCO benchmark under
$2\times$ training scheme (24 epochs).
