Enabling certification of verification-agnostic networks via memory-efficient semidefinite programming

Sumanth Dathathri
Sumanth Dathathri
Jonathan Uesato
Jonathan Uesato
Rudy Bunel
Rudy Bunel
Shreya Shankar
Shreya Shankar

NIPS 2020, 2020.

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Other Links: arxiv.org|dblp.uni-trier.de|academic.microsoft.com
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To visualize the improvements resulting from our solver, we include a comparison with guarantees based on interval arithmetic bound propagation, which we use to generate the bounds used in Algorithm 1

Abstract:

Convex relaxations have emerged as a promising approach for verifying desirable properties of neural networks like robustness to adversarial perturbations. Widely used Linear Programming (LP) relaxations only work well when networks are trained to facilitate verification. This precludes applications that involve verification-agnostic ne...More
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Introduction
  • Applications of neural networks to safety-critical domains requires ensuring that they behave as expected under all circumstances [31].
  • Several techniques that claimed to enhance neural network robustness were later shown to break under stronger attacks [60, 5].
  • This has motivated the search for verification algorithms that can provide provable guarantees on neural networks satisfying input-output specifications
Highlights
  • Applications of neural networks to safety-critical domains requires ensuring that they behave as expected under all circumstances [31]
  • One way to achieve this is to ensure that neural networks conform with a list of specifications, i.e., relationships between the inputs and outputs of a neural network that ought to be satisfied
  • We demonstrate the applicability of first-order semidefinite programming (SDP) techniques to neural network verification
  • We focus on SDP relaxations but develop customized solvers that can run on accelerators for deep learning (GPUs/TPUs) enabling their application to large convolutional neural network (CNN)
  • To visualize the improvements resulting from our solver, we include a comparison with guarantees based on interval arithmetic bound propagation (IBP) [22, 38], which we use to generate the bounds used in Algorithm 1
  • On all networks we study, we significantly improve on the baseline verified accuracies
  • Compared to IBP, SDP-FO can successfully verify at perturbation radii roughly 50x as large
Methods
  • The authors evaluate the SDP-FO verification algorithm on two specifications: robustness to adversarial perturbations for image classifiers (Sec. 6.1), and robustness to latent space perturbations for a generative model (Sec. 6.2).
  • In both cases, the authors focus on verification-agnostic networks.
  • The authors further compare to the SDP relaxation from [50] solved using MOSEK [3], a commercial interior point SDP (SDP-IP) solver, and the MIP approach from [58]
Results
  • The authors verify a VAE on MNIST with a convolutional decoder containing « 10K total activations.
  • While small-scale problems can be solved exactly with second-order interior point methods, these approaches have poor asymptotic scaling factors
  • Both the SDP primal and dual problems involve matrix variables with number of elements quadratic in the number of network activations N.
  • Using CROWN bounds in place of interval arithmetic bounds improves the overall verified accuracy from 83.0% to 91.2%
  • This closes most of the gap to the PGD upper bound of 93.4%.
Conclusion
  • The authors have developed a promising approach to scalable tight verification and demonstrated good performance on larger scale than was possible previously.
  • This solver is applicable to arbitrarily large networks, further innovations are necessary to get meaningful verified guarantees on larger networks
Summary
  • Introduction:

    Applications of neural networks to safety-critical domains requires ensuring that they behave as expected under all circumstances [31].
  • Several techniques that claimed to enhance neural network robustness were later shown to break under stronger attacks [60, 5].
  • This has motivated the search for verification algorithms that can provide provable guarantees on neural networks satisfying input-output specifications
  • Objectives:

    The authors' goal is to develop a custom solver for large-scale neural network verification with the following desiderata: (1) compute anytime upper bounds valid after each iteration, (2) rely on elementary computations with efficient implementations that can exploit hardware like GPUs and TPUs, and (3) have per-iteration memory and computational cost that scales linearly in the number of neurons.
  • The authors' aim is to certify robustness of the decoder to perturbations in the VAE latent space
  • Methods:

    The authors evaluate the SDP-FO verification algorithm on two specifications: robustness to adversarial perturbations for image classifiers (Sec. 6.1), and robustness to latent space perturbations for a generative model (Sec. 6.2).
  • In both cases, the authors focus on verification-agnostic networks.
  • The authors further compare to the SDP relaxation from [50] solved using MOSEK [3], a commercial interior point SDP (SDP-IP) solver, and the MIP approach from [58]
  • Results:

    The authors verify a VAE on MNIST with a convolutional decoder containing « 10K total activations.
  • While small-scale problems can be solved exactly with second-order interior point methods, these approaches have poor asymptotic scaling factors
  • Both the SDP primal and dual problems involve matrix variables with number of elements quadratic in the number of network activations N.
  • Using CROWN bounds in place of interval arithmetic bounds improves the overall verified accuracy from 83.0% to 91.2%
  • This closes most of the gap to the PGD upper bound of 93.4%.
  • Conclusion:

    The authors have developed a promising approach to scalable tight verification and demonstrated good performance on larger scale than was possible previously.
  • This solver is applicable to arbitrarily large networks, further innovations are necessary to get meaningful verified guarantees on larger networks
Tables
  • Table1: Comparison of verified accuracy across verification algorithms. Highlighted rows indicate models trained in a verification-agnostic manner. All numbers computed across the same 500 test set examples, except when using previously reported values. For all networks, SDP-FO outperforms previous approaches. The improvement is largest for verification-agnostic models
  • Table2: Architecture of CNN models used on MNIST and CIFAR-10. Each layer (except the last fully connected layer) is followed by ReLU activations. CONV T W×H+S corresponds to a convolutional layer with T filters of size W×H with stride of S in both dimensions. FC T corresponds to a fully connected layer with T output neurons
  • Table3: The VAE consists of an encoder and a decoder, and the architecture details for both the encoder and the decoder are provided here. CONV-T T W×H+S corresponds to a transpose convolutional layer with T filters of size W×H with stride of S in both dimensions
  • Table4: Comparison of verified accuracy across various networks and perturbation radii. All SDP-FO numbers computed on the first 100 test set examples, and numbers for LP on the first 1000 test set examples. The perturbations and training-modes considered here differ from those in Table 1. For all networks, SDP-FO outperforms the LP-relaxation baseline
Download tables as Excel
Related work
  • Neural network verification. There is a large literature on verification methods for neural networks. Broadly, the literature can be grouped into complete verification using mixed-integer programming [25, 17, 58, 10, 2], bound propagation [55, 67, 64, 20], convex relaxation [29, 16, 66, 50], and randomized smoothing [34, 11]. Verified training approaches when combined with convex relaxations have led to promising results [29, 50, 22, 6]. Randomized smoothing and verified training approaches requires special modifications to the predictor (smoothing the predictions by adding noise) and/or the training algorithm (training with additional noise or regularizers) and hence are not applicable to the verification-agnostic setting. Bound propagation approaches have been shown to be special instances of LP relaxations [36]. Hence we focus on describing the convex relaxations and complete solvers, as the areas most closely related to this paper.
Funding
  • This work was supported by NSF Award Grant no. 1805310
  • AR was supported by a Google PhD Fellowship and Open Philanthropy Project AI Fellowship
Study subjects and analysis
samples: 100
Enabling certification of verification-agnostic networks. For 100 random examples on MNIST and CIFAR-10, we plot the verified upper bound on φx against the adversarial lower bound (taking the worst-case over target labels for each). Recall, an example is verified when the verified upper bound φx ă 0. Our key result is that SDP-FO achieves tight verification across all examples, with all points lying close to the line y “ x. In contrast, LP or CROWN bounds produce much looser gaps between the lower and upper bounds. We note that many CROWN bounds exceed the plotted y-axis limits. Comparison of different approaches for verifying the robustness of the decoder of a VAE on MNIST, measured across 100 samples. The lower-bound on the robust accuracy computed with SDP-FO closely matches the upper bound based on a PGD adversarial attack upto perturbations of 0.1 σz, while the lower bound based on IBP begins to diverge from the PGD upper bound at much smaller perturbations. Two digits from the MNIST data set, and the corresponding images when perturbed with Gaussian noise, whose squared 2-norm is equal to the threshold (τ “ 40.97). τ corresponds to threshold on the reconstruction error used in equation (7)

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