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Neural models and symbolic algorithms have recently been combined for tasksrequiring both perception and reasoning. Neural models ground perceptual inputinto a conceptual vocabulary, on which a classical reasoning algorithm isapplied to generate output. A key limitation is that such neural-to-symbolicmodels can only be trained end-to-end for tasks where the output space issymbolic. In this paper, we study neural-symbolic-neural models for reasoningtasks that require a conversion from an image input (e.g., a partially filledsudoku) to an image output (e.g., the image of the completed sudoku). Whiledesigning such a three-step hybrid architecture may be straightforward, the keytechnical challenge is end-to-end training -- how to backpropagate withoutintermediate supervision through the symbolic component. We propose NSNnet, anarchitecture that combines an image reconstruction loss with a novel outputencoder to generate a supervisory signal, develops update algorithms thatleverage policy gradient methods for supervision, and optimizes loss using anovel subsampling heuristic. We experiment on problem settings where symbolicalgorithms are easily specified: a visual maze solving task and a visual Sudokusolver where the supervision is in image form. Experiments show high accuracywith significantly less data compared to purely neural approaches.

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