Bilevel Learning with Inexact Stochastic Gradients

Bilevel learning has gained prominence in machine learning, inverse problems, and imaging applications, including hyperparameter optimization, learning data-adaptive regularizers, and optimizing forward operators. The large-scale nature of these problems has led to the development of inexact and computationally efficient methods. Existing adaptive methods predominantly rely on deterministic formulations, while stochastic approaches often adopt a doubly-stochastic framework with impractical variance assumptions, enforces a fixed number of lower-level iterations, and requires extensive tuning. In this work, we focus on bilevel learning with strongly convex lower-level problems and a nonconvex sum-of-functions in the upper-level. Stochasticity arises from data sampling in the upper-level which leads to inexact stochastic hypergradients. We establish their connection to state-of-the-art stochastic optimization theory for nonconvex objectives. Furthermore, we prove the convergence of inexact stochastic bilevel optimization under mild assumptions. Our empirical results highlight significant speed-ups and improved generalization in imaging tasks such as image denoising and deblurring in comparison with adaptive deterministic bilevel methods.

An Adaptively Inexact Method for Bilevel Learning Using Primal-Dual Style Differentiation

We consider a bilevel learning framework for learning linear operators. In this framework, the learnable parameters are optimized via a loss function that also depends on the minimizer of a convex optimization problem (denoted lower-level problem). We utilize an iterative algorithm called `piggyback' to compute the gradient of the loss and minimizer of the lower-level problem. Given that the lower-level problem is solved numerically, the loss function and thus its gradient can only be computed inexactly. To estimate the accuracy of the computed hypergradient, we derive an a-posteriori error bound, which provides guides for setting the tolerance for the lower-level problem, as well as the piggyback algorithm. To efficiently solve the upper-level optimization, we also propose an adaptive method for choosing a suitable step-size. To illustrate the proposed method, we consider a few learned regularizer problems, such as training an input-convex neural network.

A Primal-dual algorithm for image reconstruction with ICNNs

We address the optimization problem in a data-driven variational reconstruction framework, where the regularizer is parameterized by an input-convex neural network (ICNN). While gradient-based methods are commonly used to solve such problems, they struggle to effectively handle non-smoothness which often leads to slow convergence. Moreover, the nested structure of the neural network complicates the application of standard non-smooth optimization techniques, such as proximal algorithms. To overcome these challenges, we reformulate the problem and eliminate the network's nested structure. By relating this reformulation to epigraphical projections of the activation functions, we transform the problem into a convex optimization problem that can be efficiently solved using a primal-dual algorithm. We also prove that this reformulation is equivalent to the original variational problem. Through experiments on several imaging tasks, we demonstrate that the proposed approach outperforms subgradient methods in terms of both speed and stability.

Accelerated Convergent Motion Compensated Image Reconstruction

Motion correction aims to prevent motion artefacts which may be caused by respiration, heartbeat, or head movements for example. In a preliminary step, the measured data is divided in gates corresponding to motion states, and displacement maps from a reference state to each motion state are estimated. One common technique to perform motion correction is the motion compensated image reconstruction framework, where the displacement maps are integrated into the forward model corresponding to gated data. For standard algorithms, the computational cost per iteration increases linearly with the number of gates. In order to accelerate the reconstruction, we propose the use of a randomized and convergent algorithm whose per iteration computational cost scales constantly with the number of gates. We show improvement on theoretical rates of convergence and observe the predicted speed-up on two synthetic datasets corresponding to rigid and non-rigid motion.

A Guide to Stochastic Optimisation for Large-Scale Inverse Problems

Stochastic optimisation algorithms are the de facto standard for machine learning with large amounts of data. Handling only a subset of available data in each optimisation step dramatically reduces the per-iteration computational costs, while still ensuring significant progress towards the solution. Driven by the need to solve large-scale optimisation problems as efficiently as possible, the last decade has witnessed an explosion of research in this area. Leveraging the parallels between machine learning and inverse problems has allowed harnessing the power of this research wave for solving inverse problems. In this survey, we provide a comprehensive account of the state-of-the-art in stochastic optimisation from the viewpoint of inverse problems. We present algorithms with diverse modalities of problem randomisation and discuss the roles of variance reduction, acceleration, higher-order methods, and other algorithmic modifications, and compare theoretical results with practical behaviour. We focus on the potential and the challenges for stochastic optimisation that are unique to inverse imaging problems and are not commonly encountered in machine learning. We conclude the survey with illustrative examples from imaging problems to examine the advantages and disadvantages that this new generation of algorithms bring to the field of inverse problems.

Dynamic Bilevel Learning with Inexact Line Search

In various domains within imaging and data science, particularly when addressing tasks modeled utilizing the variational regularization approach, manually configuring regularization parameters presents a formidable challenge. The difficulty intensifies when employing regularizers involving a large number of hyperparameters. To overcome this challenge, bilevel learning is employed to learn suitable hyperparameters. However, due to the use of numerical solvers, the exact gradient with respect to the hyperparameters is unattainable, necessitating the use of methods relying on approximate gradients. State-of-the-art inexact methods a priori select a decreasing summable sequence of the required accuracy and only assure convergence given a sufficiently small fixed step size. Despite this, challenges persist in determining the Lipschitz constant of the hypergradient and identifying an appropriate fixed step size. Conversely, computing exact function values is not feasible, impeding the use of line search. In this work, we introduce a provably convergent inexact backtracking line search involving inexact function evaluations and hypergradients. We show convergence to a stationary point of the loss with respect to hyperparameters. Additionally, we propose an algorithm to determine the required accuracy dynamically. Our numerical experiments demonstrate the efficiency and feasibility of our approach for hyperparameter estimation in variational regularization problems, alongside its robustness in terms of the initial accuracy and step size choices.

Designing Stable Neural Networks using Convex Analysis and ODEs

Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially robust image classification, to image denoising, and finally to the inverse problem of deblurring.

On Optimal Regularization Parameters via Bilevel Learning

Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information, and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Existing strategies such as the discrepancy principle and L-curve can be used to determine a suitable parameter value, but in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters, and involves solving a nested optimisation problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still a developing field. One necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterises positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and large dimensional problems.

Analyzing Inexact Hypergradients for Bilevel Learning

Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that generalizes existing methods based on the implicit function theorem and automatic differentiation/backpropagation, showing that these two seemingly disparate approaches are actually tightly connected. Our framework is extremely flexible, allowing its subproblems to be solved with any suitable method, to any degree of accuracy. We derive a priori and computable a posteriori error bounds for all our methods, and numerically show that our a posteriori bounds are usually more accurate. Our numerical results also show that, surprisingly, for efficient bilevel optimization, the choice of hypergradient algorithm is at least as important as the choice of lower-level solver.

Compressed Sensing MRI Reconstruction Regularized by VAEs with Structured Image Covariance

Learned regularization for MRI reconstruction can provide complex data-driven priors to inverse problems while still retaining the control and insight of a variational regularization method. Moreover, unsupervised learning, without paired training data, allows the learned regularizer to remain flexible to changes in the forward problem such as noise level, sampling pattern or coil sensitivities. One such approach uses generative models, trained on ground-truth images, as priors for inverse problems, penalizing reconstructions far from images the generator can produce. In this work, we utilize variational autoencoders (VAEs) that generate not only an image but also a covariance uncertainty matrix for each image. The covariance can model changing uncertainty dependencies caused by structure in the image, such as edges or objects, and provides a new distance metric from the manifold of learned images. We demonstrate these novel generative regularizers on radially sub-sampled MRI knee measurements from the fastMRI dataset and compare them to other unlearned, unsupervised and supervised methods. Our results show that the proposed method is competitive with other state-of-the-art methods and behaves consistently with changing sampling patterns and noise levels.