Lindon Roberts

Matthias J. Ehrhardt, Lindon Roberts

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.

Mohammad Sadegh Salehi, Subhadip Mukherjee, Lindon Roberts et al.

Various tasks in data science are modeled utilizing the variational regularization approach, where manually selecting regularization parameters presents a challenge. The difficulty gets exacerbated when employing regularizers involving a large number of hyperparameters. To overcome this challenge, bilevel learning can be employed to learn such parameters from data. However, neither exact function values nor exact gradients with respect to the hyperparameters are attainable, necessitating methods that only rely on inexact evaluation of such quantities. State-of-the-art inexact gradient-based methods a priori select a sequence of the required accuracies and cannot identify an appropriate step size since the Lipschitz constant of the hypergradient is unknown. In this work, we propose an algorithm with backtracking line search that only relies on inexact function evaluations and hypergradients and show convergence to a stationary point. Furthermore, the proposed algorithm determines the required accuracy dynamically rather than manually selected before running it. Our numerical experiments demonstrate the efficiency and feasibility of our approach for hyperparameter estimation on a range of relevant problems in imaging and data science such as total variation and field of experts denoising and multinomial logistic regression. Particularly, the results show that the algorithm is robust to its own hyperparameters such as the initial accuracies and step size.

Albert S. Berahas, Lindon Roberts, Fred Roosta

The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given problem. In this work we introduce a local first-order smoothness oracle (LFSO) which generalizes the Lipschitz continuous gradients smoothness condition and is applicable to any twice-differentiable function. We show that this oracle can encode all relevant problem information for tuning stepsizes for a suitably modified gradient descent method and give global and local convergence results. We also show that LFSOs in this modified first-order method can yield global linear convergence rates for non-strongly convex problems with extremely flat minima, and thus improve over the lower bound on rates achievable by general (accelerated) first-order methods.

Mohammad Sadegh Salehi, Subhadip Mukherjee, Lindon Roberts et al.

Bilevel optimization is a central tool in machine learning for high-dimensional hyperparameter tuning. Its applications are vast; for instance, in imaging it can be used for learning data-adaptive regularizers and optimizing forward operators in variational regularization. These problems are large in many ways: a lot of data is usually available to train a large number of parameters, calling for stochastic gradient-based algorithms. However, exact gradients with respect to parameters (so-called hypergradients) are not available, and their precision is usually linearly related to computational cost. Hence, algorithms must solve the problem efficiently without unnecessary precision. The design of such methods is still not fully understood, especially regarding how accuracy requirements and step size schedules affect theoretical guarantees and practical performance. Existing approaches introduce stochasticity at both the upper level (e.g., in sampling or mini-batch estimates) and the lower level (e.g., in solving the inner problem) to improve generalization, but they typically fix the number of lower-level iterations, which conflicts with asymptotic convergence assumptions. In this work, we advance the theory of inexact stochastic bilevel optimization. We prove convergence and establish rates under decaying accuracy and step size schedules, showing that with optimal configurations convergence occurs at an $\mathcal{O}(k^{-1/4})$ rate in expectation. Experiments on image denoising and inpainting with convex ridge regularizers and input-convex networks confirm our analysis: decreasing step sizes improve stability, accuracy scheduling is more critical than step size strategy, and adaptive preconditioning (e.g., Adam) further boosts performance. These results bridge theory and practice, providing convergence guarantees and practical guidance for large-scale imaging problems.

Lindon Roberts, Edward Smyth

In distributed learning, a central server trains a model according to updates provided by nodes holding local data samples. In the presence of one or more malicious servers sending incorrect information (a Byzantine adversary), standard algorithms for model training such as stochastic gradient descent (SGD) fail to converge. In this paper, we present a simplified convergence theory for the generic Byzantine Resilient SGD method originally proposed by Blanchard et al. [NeurIPS 2017]. Compared to the existing analysis, we shown convergence to a stationary point in expectation under standard assumptions on the (possibly nonconvex) objective function and flexible assumptions on the stochastic gradients.

Yiwen Chen, Warren Hare, Lindon Roberts

We study three quadratic models in model-based derivative-free optimization: the minimum norm (MN), minimum Frobenius norm (MFN), and quadratic generalized simplex derivative (QS) models. Despite their widespread use, their approximation accuracy and relationships have not been systematically explored. We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption required in existing MN analyses and deriving the first such results for the QS model. We further analyze Hessian approximation accuracy via directional error bounds, showing that all three models achieve fully quadratic accuracy along sample directions under a mild condition on the sample set. This reveals a form of directional fully quadratic accuracy not captured by existing theory. Finally, we characterize the relationships among these models, identifying conditions under which they coincide and clarifying their structural connections.

Mohammad Sadegh Salehi, Subhadip Mukherjee, Lindon Roberts et al.

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.

Matthias J. Ehrhardt, Lindon Roberts

Many machine learning solutions are framed as optimization problems which rely on good hyperparameters. Algorithms for tuning these hyperparameters usually assume access to exact solutions to the underlying learning problem, which is typically not practical. Here, we apply a recent dynamic accuracy derivative-free optimization method to hyperparameter tuning, which allows inexact evaluations of the learning problem while retaining convergence guarantees. We test the method on the problem of learning elastic net weights for a logistic classifier, and demonstrate its robustness and efficiency compared to a fixed accuracy approach. This demonstrates a promising approach for hyperparameter tuning, with both convergence guarantees and practical performance.

Coralia Cartis, Tyler Ferguson, Lindon Roberts

Derivative-free - or zeroth-order - optimization (DFO) has gained recent attention for its ability to solve problems in a variety of application areas, including machine learning, particularly involving objectives which are stochastic and/or expensive to compute. In this work, we develop a novel model-based DFO method for solving nonlinear least-squares problems. We improve on state-of-the-art DFO by performing dimensionality reduction in the observational space using sketching methods, avoiding the construction of a full local model. Our approach has a per-iteration computational cost which is linear in problem dimension in a big data regime, and numerical evidence demonstrates that, compared to existing software, it has dramatically improved runtime performance on overdetermined least-squares problems.

Matthias J. Ehrhardt, Lindon Roberts

Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infeasible. In this work we propose to solve these problems using inexact derivative-free optimization algorithms which never require exact lower-level problem solutions, but instead assume access to approximate solutions with controllable accuracy, which is achievable in practice. We prove global convergence and a worstcase complexity bound for our approach. We test our proposed framework on ROFdenoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as highaccuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).