Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems trained with Gradient Descent

Advanced machine learning methods, and more prominently neural networks, have become standard to solve inverse problems over the last years. However, the theoretical recovery guarantees of such methods are still scarce and difficult to achieve. Only recently did unsupervised methods such as Deep Image Prior (DIP) get equipped with convergence and recovery guarantees for generic loss functions when trained through gradient flow with an appropriate initialization. In this paper, we extend these results by proving that these guarantees hold true when using gradient descent with an appropriately chosen step-size/learning rate. We also show that the discretization only affects the overparametrization bound for a two-layer DIP network by a constant and thus that the different guarantees found for the gradient flow will hold for gradient descent.

Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems

Neural networks have become a prominent approach to solve inverse problems in recent years. While a plethora of such methods was developed to solve inverse problems empirically, we are still lacking clear theoretical guarantees for these methods. On the other hand, many works proved convergence to optimal solutions of neural networks in a more general setting using overparametrization as a way to control the Neural Tangent Kernel. In this work we investigate how to bridge these two worlds and we provide deterministic convergence and recovery guarantees for the class of unsupervised feedforward multilayer neural networks trained to solve inverse problems. We also derive overparametrization bounds under which a two-layers Deep Inverse Prior network with smooth activation function will benefit from our guarantees.

Convergence Guarantees of Overparametrized Wide Deep Inverse Prior

Neural networks have become a prominent approach to solve inverse problems in recent years. Amongst the different existing methods, the Deep Image/Inverse Priors (DIPs) technique is an unsupervised approach that optimizes a highly overparametrized neural network to transform a random input into an object whose image under the forward model matches the observation. However, the level of overparametrization necessary for such methods remains an open problem. In this work, we aim to investigate this question for a two-layers neural network with a smooth activation function. We provide overparametrization bounds under which such network trained via continuous-time gradient descent will converge exponentially fast with high probability which allows to derive recovery prediction bounds. This work is thus a first step towards a theoretical understanding of overparametrized DIP networks, and more broadly it participates to the theoretical understanding of neural networks in inverse problem settings.

Maximizing Drift is Not Optimal for Solving OneMax

It seems very intuitive that for the maximization of the OneMax problem $f(x):=\sum_{i=1}^n{x_i}$ the best that an elitist unary unbiased search algorithm can do is to store a best so far solution, and to modify it with the operator that yields the best possible expected progress in function value. This assumption has been implicitly used in several empirical works. In [Doerr, Doerr, Yang: GECCO 2016] it was formally proven that this approach is indeed almost optimal. In this work we prove that drift maximization is \emph{not} optimal. More precisely, we show that for most fitness levels $n/2<\ell/2 < 2n/3$ the optimal mutation strengths are larger than the drift-maximizing ones. This implies that the optimal RLS is more risk-affine than the variant maximizing the step-wise expected progress. We show similar results for the mutation rates of the classic (1+1) Evolutionary Algorithm (EA) and its resampling variant, the (1+1) EA$_{>0}$. As a result of independent interest we show that the optimal mutation strengths, unlike the drift-maximizing ones, can be even.