Methods, systems, and apparatus, including computer programs encoded on a computer storage medium, for training a reinforcement learning system. In one aspect, a method of training an action selection policy neural network for use in selecting actions to be performed by an agent navigating through an environment to accomplish one or more goals comprises: receiving an observation image characterizing a current state of the environment; processing, using the action selection policy neural network, an input comprising the observation image to generate an action selection output; processing, using a geometry-prediction neural network, an intermediate output generated by the action selection policy neural network to predict a value of a feature of a geometry of the environment when in the current state; and backpropagating a gradient of a geometry-based auxiliary loss into the action selection policy neural network to determine a geometry-based auxiliary update for current values of the network parameters.