End-to-End Neuro-Symbolic Architecture for Image-to-Image Reasoning Tasks

Neural models and symbolic algorithms have recently been combined for tasks requiring both perception and reasoning. Neural models ground perceptual input into a conceptual vocabulary, on which a classical reasoning algorithm is applied to generate output. A key limitation is that such neural-to-symbolic models can only be trained end-to-end for tasks where the output space is symbolic. In this paper, we study neural-symbolic-neural models for reasoning tasks that require a conversion from an image input (e.g., a partially filled sudoku) to an image output (e.g., the image of the completed sudoku). While designing such a three-step hybrid architecture may be straightforward, the key technical challenge is end-to-end training -- how to backpropagate without intermediate supervision through the symbolic component. We propose NSNnet, an architecture that combines an image reconstruction loss with a novel output encoder to generate a supervisory signal, develops update algorithms that leverage policy gradient methods for supervision, and optimizes loss using a novel subsampling heuristic. We experiment on problem settings where symbolic algorithms are easily specified: a visual maze solving task and a visual Sudoku solver where the supervision is in image form. Experiments show high accuracy with significantly less data compared to purely neural approaches.