PhD Proposal: Application of Homotopy Methods in Neural Networks and Transportation
In this research, we first study homotopy methods in the context of neural networks and use homotopy to interpret deep learning models. Homotopy methods have been studied by many scholars and proven to be effective in solving many hard optimization problems. However, despite their general effectiveness, these methods need to be specifically developed and tailored for individual problems, and as we will show here, neural networks can benefit from using these methods.In real-world applications, neural networks are usually trained for a specific task and then used as a tool to perform that task, for example to make decisions or predictions. These trained models are often described as black-boxes because they are so complex that one cannot interpret their output in terms of their inputs. This inexplainability becomes problematic in many ways, especially when the network is utilized in tasks consequential to human lives, such as in criminal justice, medicine and business.Despite the unprecedented success of neural networks in performing machine learning tasks, when they are used as black-box models, users cannot be sure how confident they can be in the correctness of each individual output. Furthermore, when a certain output is produced, it would be desirable to know the answer to questions such as, what changes in the input could have made the output different. A black-box cannot provide answers to such questions.There are many computational obstacles in pursuit of such interpretations. Some recent attempts to tackle this problem have either tried to avoid these computational difficulties by constructing alternative models that emulate a neural network, or are not mathematically rigorous and have considerable shortcomings. We take a mathematical/fundamental approach which can be described as a homotopy method that overcomes some of the computational obstacles and deals directly with the neural network. We further demonstrate that this homotopy facilitates the computations generally and can also be beneficial in the training process.In addition to our focus on the neural networks, we have also developed a homotopy method to optimize real-time dispatch decisions in freight transportation networks. We propose a real-time decision framework for intermodal freight dispatch through a system of hierarchical hubs, using a probabilistic model for transit times. Our homotopy algorithm (empirically) outperforms standard optimization algorithms on this problem by taking advantage of its structure.
Chair: Dr. Dianne O'Leary Dept. rep: Dr. Furong Huang Members: Dr. Howard Elman