[Paper Review] Abstraction based Output Range Analysis for Neural Networks

Pavithra Prabhakar, Zahra Rahimi Afzal|arXiv (Cornell University)|Jul 18, 2020

Neural Networks and ApplicationsComputer Science22 citations

TL;DR

This paper proposes a novel abstraction technique that reduces the computational complexity of output range analysis for ReLU-based feed-forward neural networks by constructing an interval neural network (INN) with fewer neurons and interval weights. The method over-approximates the original network's output range and reduces the problem to mixed-integer linear programming (MILP), achieving a trade-off between computation time and precision, with significant speedups observed on ACAS Xu benchmarks when abstraction is applied.

ABSTRACT

In this paper, we consider the problem of output range analysis for feed-forward neural networks with ReLU activation functions. The existing approaches reduce the output range analysis problem to satisfiability and optimization solving, which are NP-hard problems, and whose computational complexity increases with the number of neurons in the network. To tackle the computational complexity, we present a novel abstraction technique that constructs a simpler neural network with fewer neurons, albeit with interval weights called interval neural network (INN), which over-approximates the output range of the given neural network. We reduce the output range analysis on the INNs to solving a mixed integer linear programming problem. Our experimental results highlight the trade-off between the computation time and the precision of the computed output range.

Motivation & Objective

• To address the high computational complexity of output range analysis for deep neural networks with ReLU activations.
• To develop a sound abstraction technique that over-approximates the output range while reducing the number of neurons.
• To enable efficient verification of neural network-controlled systems by simplifying the network structure using interval weights.
• To demonstrate a trade-off between computation time and precision in output range estimation.
• To provide a foundation for integrating abstraction with existing verification tools like MILP solvers and Reluplex.

Proposed method

• Construct an interval neural network (INN) by merging neurons in each layer, using interval weights derived from convex hulls scaled by the number of merged nodes.
• Ensure soundness of the abstraction by proving that the abstract network over-approximates the output range of the original network.
• Reduce output range computation on the INN to a mixed-integer linear programming (MILP) problem using an extended encoding from prior work.
• Use commercial MILP solvers like Gurobi to compute upper and lower bounds on the output range of the abstracted network.
• Apply random partitioning strategies to generate multiple abstractions for the same number of abstract nodes to evaluate precision variability.
• Measure and compare abstraction time, encoding time, and MILP solving time across different abstraction levels.

Experimental results

Research questions

• RQ1 Can abstraction of a deep neural network into an interval neural network (INN) significantly reduce the computational cost of output range analysis?
• RQ2 Does the proposed abstraction technique preserve soundness, ensuring that the INN over-approximates the output range of the original network?
• RQ3 How does the precision of the output range estimate vary with different numbers of abstract nodes and partitioning strategies?
• RQ4 What is the relative contribution of abstraction time, encoding time, and MILP solving time to the total computation cost?
• RQ5 Can the abstraction technique be used as a preprocessing step to make existing MILP-based or satisfiability-based verification methods scalable to larger networks?

Key findings

• The MILP solving time by Gurobi is the dominant cost in the overall computation pipeline, increasing drastically beyond a certain number of abstract nodes.
• Without abstraction, Gurobi failed to return a result on the full ACAS Xu benchmark, demonstrating the necessity of abstraction for scalability.
• Abstraction time and encoding time increase gradually with the number of abstract nodes, while MILP solving time grows sharply, indicating that abstraction is most effective in reducing the bottleneck.
• The upper bound of the output range decreases significantly with more abstract nodes, showing improved precision, but varies by up to a factor of 2 even for the same number of abstract nodes due to partitioning choice.
• The method achieves a favorable trade-off between computation time and output range precision, with substantial speedups observed when abstraction is applied.
• The precision of the output range is highly sensitive to the specific partitioning of neurons, suggesting that future work should focus on optimizing partitioning strategies.