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ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks
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Proposes ECLipsE, a compositional layer-by-layer method that exactly decomposes the LipSDP matrix inequality via sequential block Cholesky, yielding two algorithms (SDP-based ECLipsE and closed-form ECLipsE-Fast) for fast Lipschitz upper bounds of feedforward ReLU networks.
Problem
Certifying Lipschitz constants of deep feed-forward networks via SDPs (LipSDP) is computationally prohibitive because the matrix verification problem grows with depth and width.
Approach
An exact sequential block Cholesky decomposition turns the large LipSDP matrix inequality into small per-layer sub-problems; ECLipsE solves a small SDP per layer, while ECLipsE-Fast replaces each with a closed-form choice of the multiplier lambda_i = 2/sigma_max(F_i).
Key results
- ECLipsE-Fast runs in about 0.14 s on a 100-layer, 80-neuron network versus 157 s for ECLipsE and over 15 min for several LipSDP variants.
- ECLipsE Lipschitz estimates are close to LipSDP-Neuron and ECLipsE-Fast estimates are close to LipSDP-Layer across the 40-configuration random sweep.
- LipDiff returns inadmissible estimates, up to 10^4-10^6 times the trivial bound on wide 50-layer networks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ECLipsE's compositional layer-wise SDP instead of LipSDP-Neuron | Random NN, 80 neurons · 20 layers | computation time (sec): 303.44 -> 28.45839 | Helped |
| Use ECLipsE-Fast's closed-form scalar identity-matrix relaxation instead of LipSDP-Layer (ablation) | Random NN, 80 neurons · 20 layers | computation time (sec): 125.05 -> 0.0515 | Helped |
| Use ECLipsE instead of LipSDP-Neuron | Random NN, 80 neurons · 20 layers | normalized Lipschitz estimate: 0.76481 -> 0.765184 | No clear effect |
| Use ECLipsE-Fast instead of LipSDP-Layer (ablation) | Random NN, 80 neurons · 20 layers | normalized Lipschitz estimate: 0.877786 -> 0.88079 | No clear effect |
| Use ECLipsE-Fast's closed-form scalar identity-matrix relaxation rather than ECLipsE's layer-wise SDP (ablation) | ECLipsE · Random NN, 80 neurons, 20 layers | computation time (sec): 28.45839 -> 0.0515 | Helped |
| Use ECLipsE-Fast's closed-form scalar identity-matrix relaxation rather than ECLipsE's layer-wise SDP (ablation) | ECLipsE · Random NN, 80 neurons, 20 layers | normalized Lipschitz estimate: 0.765184 -> 0.88079 | Hurt |
| Use ECLipsE instead of LipSDP-Layer | Random NN, 80 neurons · 30 layers | normalized Lipschitz estimate: 0.861134 -> 0.737564 | Helped |
| Use ECLipsE-Fast instead of LipSDP-Layer (ablation) | Random NN, 80 neurons · 30 layers | normalized Lipschitz estimate: 0.861134 -> 0.863682 | No clear effect |
| Use ECLipsE-Fast's closed-form scalar identity-matrix relaxation rather than ECLipsE's layer-wise SDP (ablation) | ECLipsE · Random NN, 80 neurons, 30 layers | computation time (sec): 43.25548 -> 0.05645 | Helped |
| Use ECLipsE-Fast's closed-form scalar identity-matrix relaxation rather than ECLipsE's layer-wise SDP (ablation) | ECLipsE · Random NN, 80 neurons, 30 layers | normalized Lipschitz estimate: 0.737564 -> 0.863682 | Hurt |
| Use ECLipsE's compositional layer-wise SDP rather than LipSDP-Neuron | Random NN, 80 neurons · 30 layers | computation time (sec) | Helped |
| Use ECLipsE-Fast's closed-form scalar identity-matrix relaxation rather than LipSDP-Layer (ablation) | Random NN, 80 neurons · 50 layers | computation time (sec): >15min -> 0.089058 | Helped |
Surprising findings
- The naive trivial bound is sometimes lower than the LipDiff estimate by orders of magnitude, so LipDiff gives inadmissible bounds.
- The paper notes that LipSDP-Network yields a counterexample where its estimate is not a strict upper bound.
Limitations
- Extension to convolutional and residual architectures is left to future work.
- Closed-form approximation in ECLipsE-Fast may lose tightness relative to SDP-based ECLipsE.
- Wide networks (width 300+) cause ECLipsE to exceed the 30 min cutoff.
Experimental setup
Datasets: MNIST
Benchmarks: randomly generated feedforward networks (2-100 layers, 20-100 neurons) MNIST-trained 3-layer networks
Metrics: normalized Lipschitz estimate (relative to trivial bound) computation time (seconds)
Baselines: LipSDP-Neuron LipSDP-Layer CPLip LipDiff naive product of induced norms
Builds on: LipSDP Chordal-LipSDP CPLip
Closest papers by meaning
- Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks (NeurIPS 2019 · similarity 0.95)
- Lipschitz regularity of deep neural networks: analysis and efficient estimation (NeurIPS 2018 · similarity 0.93)
- A Quantitative Geometric Approach to Neural-Network Smoothness (NeurIPS 2022 · similarity 0.92)
- Some Fundamental Aspects about Lipschitz Continuity of Neural Networks (ICLR 2024 · similarity 0.90)
- A Recipe for Improved Certifiable Robustness (ICLR 2024 · similarity 0.90)
- DP-SGD Without Clipping: The Lipschitz Neural Network Way (ICLR 2024 · similarity 0.90)
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