NEURAL LEARNING OF ONE-OF-MANY SOLUTIONS FOR COMBINATORIAL PROBLEMS IN STRUCTURED OUTPUT SPACES

Abstract

Recent research has proposed neural architectures for solving combinatorial problems in structured output spaces. In many such problems, there may exist multiple solutions for a given input, e.g. a partially filled Sudoku puzzle may have many completions satisfying all constraints. Further, we are often interested in finding any one of the possible solutions, without any preference between them. Existing approaches completely ignore this solution multiplicity. In this paper, we argue that being oblivious to the presence of multiple solutions can severely hamper their training ability. Our contribution is two fold. First, we formally define the task of learning one-of-many solutions for combinatorial problems in structured output spaces, which is applicable for solving several problems of interest such as N-Queens, and Sudoku. Second, we present a generic learning framework that adapts an existing prediction network for a combinatorial problem to handle solution multiplicity. Our framework uses a selection module, whose goal is to dynamically determine, for every input, the solution that is most effective for training the network parameters in any given learning iteration. We propose an RL based approach to jointly train the selection module with the prediction network. Experiments on three different domains, and using two different prediction networks, demonstrate that our framework significantly improves the accuracy in our setting, obtaining up to 21 pt gain over the baselines.

1. INTRODUCTION

Neural networks have become the de-facto standard for solving perceptual tasks over low level representations, such as pixels in an image or audio signals. Recent research has also explored their application for solving symbolic reasoning tasks, requiring higher level inferences, such as neural theorem proving (Rocktäschel et al., 2015; Evans & Grefenstette, 2018; Minervini et al., 2020) , and playing blocks world (Dong et al., 2019) . The advantage of neural models for these tasks is that it will create a unified, end-to-end trainable representation for integrated AI systems that combine perceptual and high level reasoning. Our paper focuses on one such high level reasoning task -solving combinatorial problems in structured output spaces, e.g., solving a Sudoku or N-Queens puzzle. These can be thought of as Constraint Satisfaction problems (CSPs) where the underlying constraints are not explicitly available, and need to be learned from training data. We focus on learning such constraints by a non-autoregressive neural model where variables in the structured output space are decoded simultaneously (and therefore independently). Notably, most of the current state-of-the-art neural models for solving combinatorial problems, e.g., SATNET (Wang et al., 2019) , RRN (Palm et al., 2018) , NLM (Dong et al., 2019) , work with non autoregressive architectures because of their high efficiency of training and inference, since they do not have to decode the solution sequentially. One of the key characteristics of such problems is solution multiplicity -there could be many correct solutions for any given input, even though we may be interested in finding any one of these solutions. For example, in a game of Sudoku with only 16 digits filled, there are always multiple correct solutions (McGuire et al., 2012) , and obtaining any one of them suffices for solving Sudoku. Unfortunately, existing literature has completely ignored solution multiplicity, resulting in sub-optimally trained * Equal contribution. Work done while at IIT Delhi. Current email: deepanshu.jindal@alumni.iitd.ac.in 1

