The lateral geniculate nucleus (LGN) relay cells act as a gateway for transmitting visual information from retina to the primary visual cortex (V1). The activities of thalamic relay cells are modulated by feedback connections emanating from layer 6 of V1. While the receptive field (RF) properties of these early parts of the visual system are relatively well understood, the function, computational role, and details of the feedback network from V1 to LGN are not. Computational models of efficient coding have been successful in deriving RF properties of retinal ganglion and V1 simple cells by optimizing the Shannon information. Further, previous experimental results have suggested that the feedback increases the Shannon information. Motivated by this earlier work, we try to understand the function of the feedback as optimizing the feedforward information to cortex. We build a model that learns feedback weights by maximizing the feedforward Shannon information on naturalistic stimuli. Our model predicts the strength and sign of feedback from a V1 cell to all ON- and OFF-center LGN relay cells that are within or surrounding the V1 cell RF. We find a highly specific pattern of influence on ON and OFF-center LGN overlapping the V1 RF depending on whether they overlapped the ON or OFF zone of the V1 RF. In addition, we find general inhibitory feedback in the further surround, which sharpens the RFs and increases surround suppression in LGN relay cells. This is consistent with results of recent experiments exploring the impact of feedback on stimuli integration.