- The paper introduces the Relax-and-Walk heuristic that uses LP relaxations to efficiently optimize over large-scale neural network models.
- The methodology iteratively refines initial global LP solutions through localized linear searches in ReLU-induced regions.
- Experiments show that Relax-and-Walk outperforms previous MILP-based heuristics in both speed and solution quality, especially for deeper networks.
Optimization Over Trained Neural Networks: A Heuristic Approach
This paper explores the computational challenges and potential solutions for mathematical optimization over trained neural networks. A significant hurdle in utilizing neural networks for optimization tasks arises from their non-linear nature and the dense constraints present within each layer. These challenges escalate with the expansion in network size. To address these, current techniques have leveraged methods like cutting plane algorithms, reformulations, and Mixed-Integer Linear Programming (MILP)-based heuristics.
The authors introduce a new heuristic, dubbed "Relax-and-Walk" (RW), which is tailored to optimize over neural networks more scalably by examining both global and local linear relaxations of the network model. This heuristic is designed to be competitive with state-of-the-art methodologies, including sophisticated MILP solvers and existing heuristics, while providing superior solutions for networks with increasing inputs, depths, and neurons.
Methodology
The RW heuristic fundamentally relies on solving Linear Programming (LP) models instead of MILP models at every search step. By focusing on LP relaxations, the heuristic can efficiently explore solution spaces while maintaining computational tractability. Here is how RW operates:
- It generates initial solutions using the global LP relaxation of the neural network models.
- These solutions are iteratively refined by locally searching within linear regions defined by Rectified Linear Units (ReLUs) and moving across these regions to identify the optimal solution.
- For each initialized solution, a directional search is applied to maximize the linear objectives derived from neural network outputs.
The algorithm entails walking from an initial point through adjacent linear regions by solving modified LP problems, contrasting with previous methods that repeatedly solve restricted MILPs, which scale poorly with network size.
Experimental Validation
The paper validates RW through rigorous experiments, including performance assessments against RW’s predecessor "Sample-and-MIP" (SM) and the commercial MILP solver Gurobi. The experiments were conducted on a range of neural networks with varying input sizes, depths, and widths. Significant findings are:
- RW performs favorably in terms of solution quality, especially notable when width and depth increase, where Gurobi and SM struggle due to longer computational times.
- RW is robust, regularly yielding better solutions than SM in over 64% of the test cases.
Furthermore, RW demonstrates marked efficiency in adversarial scenarios, such as optimizing adversarial inputs on MNIST-trained networks. Here, RW outperformed Gurobi in generating adversarial examples quicker and with higher adversarial efficacy in a majority of cases.
Implications and Future Directions
The development of RW extends the applicability of neural networks in optimization frameworks by offering a method that scales better than current alternatives. This is particularly impactful for models that demand high scalability and efficiency, such as large-scale neural networks in industrial applications. The heuristic's reliance on local linearity and its ability to manipulate high-dimensional geometry more fluidly opens avenues for its utilization in various settings, such as constraint learning, robust optimization, and dynamic system control.
Future work might focus on enhancing the RW approach by integrating it with other heuristic methods or exploring its effectiveness on other types of neural networks, such as those incorporating convolutional layers or more complex activations. Additionally, further theoretical work could explore analyzing the limits of applicability of the linear relaxation approach in extremely non-linear contexts encountered in deep architectures.
In summary, this paper provides a pertinent contribution to the literature by advancing the methodologies used for optimization over trained neural networks, presenting a heuristic that balances performance and scalability without a significant compromise on solution quality.