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ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks

Yuezhu Xu, S. Sivaranjani

TL;DR

This paper provides a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks and considerably advances the scalability and efficiency of certifying neural network robustness, making it particularly attractive for online learning tasks.

Abstract

The Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Since calculating the exact Lipschitz constant is NP-hard, efforts have been made to obtain tight upper bounds on the Lipschitz constant. Typically, this involves solving a large matrix verification problem, the computational cost of which grows significantly for both deeper and wider networks. In this paper, we provide a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks. We first obtain an exact decomposition of the large matrix verification problem into smaller sub-problems. Then, leveraging the underlying cascade structure of the network, we develop two algorithms. The first algorithm explores the geometric features of the problem and enables us to provide Lipschitz estimates that are comparable to existing methods by solving small semidefinite programs (SDPs) that are only as large as the size of each layer. The second algorithm relaxes these sub-problems and provides a closed-form solution to each sub-problem for extremely fast estimation, altogether eliminating the need to solve SDPs. The two algorithms represent different levels of trade-offs between efficiency and accuracy. Finally, we demonstrate that our approach provides a steep reduction in computation time (as much as several thousand times faster, depending on the algorithm for deeper networks) while yielding Lipschitz bounds that are very close to or even better than those achieved by state-of-the-art approaches in a broad range of experiments. In summary, our approach considerably advances the scalability and efficiency of certifying neural network robustness, making it particularly attractive for online learning tasks.

ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks

TL;DR

This paper provides a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks and considerably advances the scalability and efficiency of certifying neural network robustness, making it particularly attractive for online learning tasks.

Abstract

The Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Since calculating the exact Lipschitz constant is NP-hard, efforts have been made to obtain tight upper bounds on the Lipschitz constant. Typically, this involves solving a large matrix verification problem, the computational cost of which grows significantly for both deeper and wider networks. In this paper, we provide a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks. We first obtain an exact decomposition of the large matrix verification problem into smaller sub-problems. Then, leveraging the underlying cascade structure of the network, we develop two algorithms. The first algorithm explores the geometric features of the problem and enables us to provide Lipschitz estimates that are comparable to existing methods by solving small semidefinite programs (SDPs) that are only as large as the size of each layer. The second algorithm relaxes these sub-problems and provides a closed-form solution to each sub-problem for extremely fast estimation, altogether eliminating the need to solve SDPs. The two algorithms represent different levels of trade-offs between efficiency and accuracy. Finally, we demonstrate that our approach provides a steep reduction in computation time (as much as several thousand times faster, depending on the algorithm for deeper networks) while yielding Lipschitz bounds that are very close to or even better than those achieved by state-of-the-art approaches in a broad range of experiments. In summary, our approach considerably advances the scalability and efficiency of certifying neural network robustness, making it particularly attractive for online learning tasks.
Paper Structure (18 sections, 9 theorems, 42 equations, 7 figures, 1 algorithm)

This paper contains 18 sections, 9 theorems, 42 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Background
  3. Problem Formulation
  4. Preliminaries
  5. Methodology
  6. Exact Decomposition
  7. Compositional Algorithms
  8. Theory
  9. Experiments
  10. Randomly Generated Neural Networks
  11. Neural Networks Trained on MNIST.
  12. Conclusion
  13. Technical Proofs
  14. Geometric Analysis for ECLipsE-Fast
  15. Computational Complexity Derivation
  16. ...and 3 more sections

Key Result

Theorem 1

For the FNN eqn: neural netowrk satisfying Assumption assm: slope_restrictedness, if there exists $F>0$ and positive diagonal matrices $\Lambda_i \in\mathbb{D}_{+}$, $i\in \mathbb{Z}_{l-1}$ such that with $p = \alpha\beta$ and $m=\frac{\alpha+\beta}{2}$, then $\left\|z_2^{(l)}-z_1^{(l)}\right\|_2\leq \sqrt{1/F}\left\|z_2^{(0)}-z_1^{(0)}\right\|_2$, which provides a sufficient condition for the Li

Figures (7)

  • Figure 1: Geometric Analysis of ECLipsE
  • Figure 2: Comparison between ECLipsE-Fast and ECLipsE with $c_i>1$
  • Figure 3: Performance of ECLipsE-Fast and ECLipsE, with respect to baselines for increasing network depth, with 80 neurons per layer. The red x markings indicate that the algorithm fails to provide an estimate within the computational cutoff time beyond this network size.
  • Figure 4: Performance of ECLipsE-Fast and ECLipsE with respect to baselines as network width increases, for a randomly generated network with 20 layers ((a) and (b)) and 50 layers ((c) and (d)). The Red x markings indicate that the algorithms fail to provide an estimate within the computational cutoff time of 15 min beyond this network size.
  • Figure 5: Computation time vs estimation accuracy for ECLipsE, ECLipsE-Fast and LipSDP splitting with different sub-network sizes.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1: LipSDP
  • Remark 1
  • Theorem 2: Restatement of Lemma 2 of agarwal2019sequential
  • Theorem 3
  • Remark 2
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • ...and 4 more