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Training robust neural networks using Lipschitz bounds

Patricia Pauli, Anne Koch, Julian Berberich, Paul Kohler, Frank Allgöwer

arXiv:2005.02929v2cs.LGeess.SYmath.OCstat.ML

TL;DR

Neural networks’ vulnerability to adversarial perturbations and their limited rigorous guarantees hinder safety-critical use. The paper trains multi-layer networks with ADMM and semidefinite-programming-based Lipschitz regularization or explicit Lipschitz constraints. Across two examples, the framework increases robustness while trading off robustness and accuracy, and can preserve comparable accuracy to nominal networks.

  • Problem

    Neural networks can be deceived by imperceptible adversarial perturbations, while safety-critical applications require stronger robustness guarantees.

  • Method

    The framework uses ADMM and a semidefinite-programming-based upper-bound estimate to train networks with Lipschitz regularization or enforced Lipschitz bounds.

  • Results

    The two examples show that Lipschitz regularization effectively trains robust neural networks while trading off robustness and accuracy.

  • Takeaways & Limitations

    The framework can produce networks with low Lipschitz constants and comparable accuracy, while L2 regularization at comparable bounds can significantly compromise accuracy on MNIST.

Abstract

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Due to their susceptibility to adversarial perturbations, neural networks (NNs) are hardly used in safety-critical applications. One measure of robustness to such perturbations in the input is the Lipschitz constant of the input-output map defined by an NN. In this work, we propose a framework to train multi-layer NNs while at the same time encouraging robustness by keeping their Lipschitz constant small, thus addressing the robustness issue. More specifically, we design an optimization scheme based on the Alternating Direction Method of Multipliers that minimizes not only the training loss of an NN but also its Lipschitz constant resulting in a semidefinite programming based training procedure that promotes robustness. We design two versions of this training procedure. The first one includes a regularizer that penalizes an accurate upper bound on the Lipschitz constant. The second one allows to enforce a desired Lipschitz bound on the NN at all times during training. Finally, we provide two examples to show that the proposed framework successfully increases the robustness of NNs.

I. INTRODUCTION

The paper frames Lipschitz constants as a robustness measure for neural networks and proposes training procedures that directly encourage small or bounded Lipschitz constants. These procedures aim to address the lack of rigorous robustness guarantees that limits neural-network use in safety-critical applications.

  • Neural networks are broadly successful because backpropagation trains them straightforwardly and universal approximation enables complex nonlinear representations.
  • Adversarial attacks can deceive neural networks with imperceptible input perturbations, motivating provable robustness measures such as Lipschitz constants.
  • The proposed framework trains robust networks by adding either a regularizer or a constraint on the network’s Lipschitz constant.
  • The paper presents two Lipschitz-continuity-based training procedures using a tight upper bound and reports successful application in two examples.
  • A low Lipschitz constant indicates low output sensitivity to input changes and therefore higher robustness.
  • Unlike conventional weight regularization, the method penalizes a more accurate Lipschitz estimate as a direct robustness-oriented objective.

B. Lipschitz constant estimation

The paper estimates neural-network Lipschitz bounds with LipSDP, an SDP exploiting slope restrictions of nonlinear activations. Its valid formulation uses diagonal decision matrices and yields a global upper bound through an SDP optimization.

  • LipSDP estimates more accurate Lipschitz bounds than products of weight spectral norms by exploiting slope-restricted nonlinear activations.
  • Activation slope restrictions are expressed as incremental quadratic constraints and combined using diagonal weighting matrices for stacked activations.
  • A feed-forward network composes affine weight-and-bias maps with nonlinear activation functions across hidden layers.
  • The SDP formulates an upper bound on the Lipschitz constant using the network’s stacked structure and activation constraints.
  • Theorem 1 states that feasibility of the diagonal-matrix condition implies global Lipschitz continuity with bound L satisfying L ≥ L∗.
  • The smallest Lipschitz upper bound is obtained by minimizing L2 subject to the semidefinite constraint, with T and L2 as decision variables.
  • The paper reports that the larger decision-matrix parametrization from prior LipSDP work does not generally support the claimed result, so T is restricted to be diagonal.

C. Counterexample for LipSDP with coupling

The paper presents a counterexample showing that the most accurate LipSDP variant from can produce an arbitrarily small value that is not an upper bound on a network’s Lipschitz constant.

  • C. Counterexample for LipSDP with coupling: The authors challenge the most accurate LipSDP variant in with a minimal counterexample involving neuron coupling.The critique targets Theorems 1 and 2 and the assumptions underlying their decision-variable formulation.
  • C. Counterexample for LipSDP with coupling: A two-neuron tanh network fits the cosine function on x ∈[−π, π/2] with maximum output deviation 0.0843.
  • C. Counterexample for LipSDP with coupling: The cosine-like network has an approximately unit Lipschitz constant because the cosine’s maximum slope is ±1.
  • C. Counterexample for LipSDP with coupling: The LMI remains feasible for arbitrarily small L2, so the resulting L cannot upper-bound the network’s approximately unit Lipschitz constant.
  • C. Counterexample for LipSDP with coupling: The failure arises because the assumed inequality does not hold for every admissible coupling matrix T and slope-restricted activation function.

III. TRAINING ROBUST NNS

The proposed robust-NN training framework uses Lipschitz certificates to incorporate robustness directly into training through either regularization or an enforced Lipschitz bound.

  • III. TRAINING ROBUST NNS: The framework uses Lipschitz certificates to train robust neural networks.
  • III. TRAINING ROBUST NNS: One procedure minimizes an upper bound on the network’s Lipschitz constant through direct regularization.
  • III. TRAINING ROBUST NNS: The second procedure enforces a desired Lipschitz bound on the network during training.

A. Weights as decision variables

The training formulation fixes the activation lower slope at α = 0 to preserve convex LMI constraints, while fixing T introduces conservatism that requires careful selection.

  • A. Weights as decision variables: Using the Schur complement, the Lipschitz LMI is rearranged into an equivalent form linear in L2 and the weights W for fixed T ∈Dn.
  • A. Weights as decision variables: For α > 0, the constraint is nonconvex in W, so the formulation sets α = 0 and thereby uses LMI constraints.This choice is conservative for some activation functions but tight for ReLU, tanh, and sigmoid with α = 0 and β = 1.
  • A. Weights as decision variables: The single-layer case yields a simplified form of the matrix M1.
  • A. Weights as decision variables: Fixing T instead of optimizing over it introduces conservatism, so T should be selected to minimize that conservatism.The paper suggests determining T from Lipschitz estimation on a vanilla or L2-regularized network trained on the same problem.

B. Lipschitz regularization

The Lipschitz-regularization procedure separates loss minimization from Lipschitz-bound minimization and solves the resulting coupled problem with ADMM, requiring an SDP update at each iteration.

  • B. Lipschitz regularization: The training objective minimizes both the NN’s predefined loss and its Lipschitz constant.
  • B. Lipschitz regularization: ADMM splits the separable objectives into smaller subproblems connected by auxiliary weight variables and linear equality constraints.The NN loss depends on W, while the Lipschitz bound depends on W̄ through the LMI.
  • B. Lipschitz regularization: The objective weights the Lipschitz-bound term with µ > 0, which controls the trade-off between accuracy and robustness.
  • B. Lipschitz regularization: ADMM solves the formulation through iterative minimization steps on the augmented Lagrangian followed by a dual update.
  • B. Lipschitz regularization: Each training iteration uses backpropagation for the loss update and an SDP for the Lipschitz update, adding computation relative to vanilla training.
  • B. Lipschitz regularization: Optimizing L2, T, and W jointly can reduce conservatism but increases computation time through a multi-block ADMM scheme.

C. Enforcing Lipschitz bounds

The second training procedure enforces a chosen upper bound on the neural network’s Lipschitz constant throughout ADMM training. This provides robustness guarantees but introduces an accuracy trade-off as the bound decreases.

  • The procedure replaces Lipschitz-bound minimization with enforcement of a desired bound Ldes during training.Ldes is used instead of optimizing over L2 in the ADMM formulation.
  • Ldes remains an upper bound on the network’s Lipschitz constant at every training iteration.The guarantee is enforced during every Lipschitz update step through an LMI constraint on the weights.
  • The method allows users to select a Lipschitz bound and train networks with corresponding Lipschitz guarantees.The desired bound directly specifies the targeted degree of robustness.
  • Decreasing the Lipschitz-bound constraint generally worsens the fit, reflecting a trade-off between accuracy and robustness.Appropriate initialization may accelerate training and facilitate a better fit.

D. Convergence

The convergence discussion separates the ADMM subproblems: gradient descent almost surely reaches local minima for the non-convex loss step, while the Lipschitz update is a semidefinite program with a unique minimizer. A full convergence analysis of the overall scheme is beyond the paper’s scope.

  • A thorough convergence analysis is complicated because the neural-network loss is non-convex and lacks obvious structural properties.The authors state that analyzing the full ADMM scheme is beyond the scope of the work.
  • For subproblem (9a), gradient descent almost surely converges to local minima even though the problem is non-convex.
  • Subproblem (9b) is a semidefinite program with a unique minimizer.
  • Adding convex regularization and a convex-set indicator does not add non-convexity to the otherwise reliably convergent neural-network training optimization.

E. Computational tractability

The framework’s scalability is limited by the semidefinite program’s decision-variable count, making the Lipschitz update the computational bottleneck for larger networks and high-dimensional inputs. Pretraining and input downscaling are proposed to improve practical computation.

  • SDP solver complexity scales cubically with the number of decision variables, making the Lipschitz update more expensive than unconstrained gradient descent.
  • The Lipschitz update becomes the bottleneck as hidden-layer widths and depths increase the weight dimensions.
  • High-dimensional picture inputs can lead to high computation times or computational intractability.
  • Convolutional or pooling layers can downscale inputs and improve computation time on larger-scale problems.
  • The authors advise fully training a network first, using the result to initialize the matrix T and weights W, ¯W.This is intended to minimize the number of Lipschitz update steps and can accelerate subsequent refinement.

IV. SIMULATION RESULTS

The framework is evaluated on a 2D three-class problem and MNIST, comparing vanilla, L2-regularized, and Lipschitz-regularized neural networks. Across these examples, Lipschitz regularization produces lower Lipschitz bounds while preserving or improving accuracy and improves performance under selected noise perturbations.

  • 2D toy example: The 2D experiment uses a feed-forward network with two hidden layers of 10 tanh neurons each and compares vanilla, L2-, and Lipschitz-regularized networks.
  • 2D toy example: 242 is the nominal network’s Lipschitz bound, while Lipschitz regularization yields a lower bound and lower cross-entropy loss than L2 regularization.The Lipschitz-regularized network also achieves the highest accuracy in this example, despite a trade-off between accuracy and robustness.
  • 2D toy example: The Lipschitz-regularized network achieves the highest accuracy in the 2D example without compromising accuracy relative to the nominal network.The nominal network tends to overfit, whereas the L2-regularized network fails to provide a good fit in this example.
  • MNIST: On MNIST, the framework finds a network with a low Lipschitz constant and comparable accuracy to the nominal network, while L2 regularization significantly compromises accuracy at a comparably low bound.The MNIST evaluation adds Gaussian and uniform noise to normalized test data; Lipschitz-NN advantages appear for low-standard-deviation Gaussian noise and narrow uniform noise.
  • Overall findings: The experiments support using Lipschitz regularization to train robust neural networks while trading off robustness and accuracy.

V. CONCLUSION

The paper presents an ADMM-based framework that combines neural-network fitting with Lipschitz regularization or explicit Lipschitz-bound enforcement. It reports successful tests on two examples and identifies control applications, accelerated optimization, and broader robustness benchmarking as next steps.

  • Contributions: The framework trains multi-layer neural networks for robustness using Lipschitz regularization and explicit Lipschitz-bound enforcement.Its underlying semidefinite program estimates an upper bound using the slope-restricted structure of activation functions.
  • Optimization: ADMM solves the underlying optimization problem while incorporating the trade-off between accuracy and robustness.
  • Robustness guarantees: A framework variation allows neural networks to be trained with a desired Lipschitz bound enforced throughout training.
  • Validation: The method was successfully tested on two examples and benchmarked against L2 regularization.
  • Future work: Planned extensions include control applications with LMI-based verification, accelerated alternatives to ADMM, and comparisons with other robustness-oriented training procedures.
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