1,033 research outputs found
Plug-and-Play Methods Provably Converge with Properly Trained Denoisers
Plug-and-play (PnP) is a non-convex framework that integrates modern
denoising priors, such as BM3D or deep learning-based denoisers, into ADMM or
other proximal algorithms. An advantage of PnP is that one can use pre-trained
denoisers when there is not sufficient data for end-to-end training. Although
PnP has been recently studied extensively with great empirical success,
theoretical analysis addressing even the most basic question of convergence has
been insufficient. In this paper, we theoretically establish convergence of
PnP-FBS and PnP-ADMM, without using diminishing stepsizes, under a certain
Lipschitz condition on the denoisers. We then propose real spectral
normalization, a technique for training deep learning-based denoisers to
satisfy the proposed Lipschitz condition. Finally, we present experimental
results validating the theory.Comment: Published in the International Conference on Machine Learning, 201
System Identification Through Lipschitz Regularized Deep Neural Networks
In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs x˙(t)=f(t,x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks are universal approximators, we do not need any prior knowledge on the ODE system. Since the model is applied component wise, it can handle systems of any dimension, making it usable for real-world data
The Extended Regularized Dual Averaging Method for Composite Optimization
We present a new algorithm, extended regularized dual averaging (XRDA), for
solving composite optimization problems, which are a generalization of the
regularized dual averaging (RDA) method. The main novelty of the method is that
it allows more flexible control of the backward step size. For instance, the
backward step size for RDA grows without bound, while XRDA the backward step
size can be kept bounded
CLIP: Cheap Lipschitz Training of Neural Networks
Despite the large success of deep neural networks (DNN) in recent years, most
neural networks still lack mathematical guarantees in terms of stability. For
instance, DNNs are vulnerable to small or even imperceptible input
perturbations, so called adversarial examples, that can cause false
predictions. This instability can have severe consequences in applications
which influence the health and safety of humans, e.g., biomedical imaging or
autonomous driving. While bounding the Lipschitz constant of a neural network
improves stability, most methods rely on restricting the Lipschitz constants of
each layer which gives a poor bound for the actual Lipschitz constant.
In this paper we investigate a variational regularization method named CLIP
for controlling the Lipschitz constant of a neural network, which can easily be
integrated into the training procedure. We mathematically analyze the proposed
model, in particular discussing the impact of the chosen regularization
parameter on the output of the network. Finally, we numerically evaluate our
method on both a nonlinear regression problem and the MNIST and Fashion-MNIST
classification databases, and compare our results with a weight regularization
approach.Comment: 12 pages, 2 figures, accepted at SSVM 202
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