Reachability analysis of neural networks using mixed monotonicity
- URL: http://arxiv.org/abs/2111.07683v1
- Date: Mon, 15 Nov 2021 11:35:18 GMT
- Title: Reachability analysis of neural networks using mixed monotonicity
- Authors: Pierre-Jean Meyer
- Abstract summary: We present a new reachability analysis tool to compute an interval over-approximation of the output set of a feedforward neural network under given input uncertainty.
The proposed approach adapts to neural networks an existing mixed-monotonicity method for the reachability analysis of dynamical systems.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: This paper presents a new reachability analysis tool to compute an interval
over-approximation of the output set of a feedforward neural network under
given input uncertainty. The proposed approach adapts to neural networks an
existing mixed-monotonicity method for the reachability analysis of dynamical
systems and applies it to all possible partial networks within the given neural
network. This ensures that the intersection of the obtained results is the
tightest interval over-approximation of the output of each layer that can be
obtained using mixed-monotonicity. Unlike other tools in the literature that
focus on small classes of piecewise-affine or monotone activation functions,
the main strength of our approach is its generality in the sense that it can
handle neural networks with any Lipschitz-continuous activation function. In
addition, the simplicity of the proposed framework allows users to very easily
add unimplemented activation functions, by simply providing the function, its
derivative and the global extrema and corresponding arguments of the
derivative. Our algorithm is tested and compared to five other interval-based
tools on 1000 randomly generated neural networks for four activation functions
(ReLU, TanH, ELU, SiLU). We show that our tool always outperforms the Interval
Bound Propagation method and that we obtain tighter output bounds than ReluVal,
Neurify, VeriNet and CROWN (when they are applicable) in 15 to 60 percent of
cases.
Related papers
- Deep Learning without Global Optimization by Random Fourier Neural Networks [0.0]
We introduce a new training algorithm for variety of deep neural networks that utilize random complex exponential activation functions.
Our approach employs a Markov Chain Monte Carlo sampling procedure to iteratively train network layers.
It consistently attains the theoretical approximation rate for residual networks with complex exponential activation functions.
arXiv Detail & Related papers (2024-07-16T16:23:40Z) - Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis [6.9060054915724]
We propose a novel reachability analysis method tailored for neural networks with differentiable activations.
A key aspect of our method is loop transformation on the activation functions to exploit their monotonicity effectively.
The resulting end-to-end abstraction locally preserves the derivative information, yielding accurate bounds on small input sets.
arXiv Detail & Related papers (2024-06-06T20:02:49Z) - Globally Optimal Training of Neural Networks with Threshold Activation
Functions [63.03759813952481]
We study weight decay regularized training problems of deep neural networks with threshold activations.
We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
arXiv Detail & Related papers (2023-03-06T18:59:13Z) - Robust Training and Verification of Implicit Neural Networks: A
Non-Euclidean Contractive Approach [64.23331120621118]
This paper proposes a theoretical and computational framework for training and robustness verification of implicit neural networks.
We introduce a related embedded network and show that the embedded network can be used to provide an $ell_infty$-norm box over-approximation of the reachable sets of the original network.
We apply our algorithms to train implicit neural networks on the MNIST dataset and compare the robustness of our models with the models trained via existing approaches in the literature.
arXiv Detail & Related papers (2022-08-08T03:13:24Z) - Open- and Closed-Loop Neural Network Verification using Polynomial
Zonotopes [6.591194329459251]
We present a novel approach to efficiently compute tight non-contact activation functions.
In particular, we evaluate the input-output relation of each neuron by an approximation.
This results in a superior performance compared to other methods.
arXiv Detail & Related papers (2022-07-06T14:39:19Z) - Simultaneous approximation of a smooth function and its derivatives by
deep neural networks with piecewise-polynomial activations [2.15145758970292]
We derive the required depth, width, and sparsity of a deep neural network to approximate any H"older smooth function up to a given approximation error in H"older norms.
The latter feature is essential to control generalization errors in many statistical and machine learning applications.
arXiv Detail & Related papers (2022-06-20T01:18:29Z) - Comparative Analysis of Interval Reachability for Robust Implicit and
Feedforward Neural Networks [64.23331120621118]
We use interval reachability analysis to obtain robustness guarantees for implicit neural networks (INNs)
INNs are a class of implicit learning models that use implicit equations as layers.
We show that our approach performs at least as well as, and generally better than, applying state-of-the-art interval bound propagation methods to INNs.
arXiv Detail & Related papers (2022-04-01T03:31:27Z) - Training Certifiably Robust Neural Networks with Efficient Local
Lipschitz Bounds [99.23098204458336]
Certified robustness is a desirable property for deep neural networks in safety-critical applications.
We show that our method consistently outperforms state-of-the-art methods on MNIST and TinyNet datasets.
arXiv Detail & Related papers (2021-11-02T06:44:10Z) - PAC-Bayesian Learning of Aggregated Binary Activated Neural Networks
with Probabilities over Representations [2.047424180164312]
We study the expectation of a probabilistic neural network as a predictor by itself, focusing on the aggregation of binary activated neural networks with normal distributions over real-valued weights.
We show that the exact computation remains tractable for deep but narrow neural networks, thanks to a dynamic programming approach.
arXiv Detail & Related papers (2021-10-28T14:11:07Z) - The Connection Between Approximation, Depth Separation and Learnability
in Neural Networks [70.55686685872008]
We study the connection between learnability and approximation capacity.
We show that learnability with deep networks of a target function depends on the ability of simpler classes to approximate the target.
arXiv Detail & Related papers (2021-01-31T11:32:30Z) - Measuring Model Complexity of Neural Networks with Curve Activation
Functions [100.98319505253797]
We propose the linear approximation neural network (LANN) to approximate a given deep model with curve activation function.
We experimentally explore the training process of neural networks and detect overfitting.
We find that the $L1$ and $L2$ regularizations suppress the increase of model complexity.
arXiv Detail & Related papers (2020-06-16T07:38:06Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.