arXiv subjects

Sebastian Banert

Publications and source records attributed to Sebastian Banert.

18 recordsLinked to original sources

arXiv2025

Why the noise model matters: A performance gap in learned regularization

This article addresses the challenge of learning effective regularizers for linear inverse problems. We analyze and compare several types of learned variational regularization against the theoretical benchmark of the optimal affine reconstruction, i.e. the best possible affine linear map for minimizing the mean squared error. It is known that this optimal reconstruction can be achieved using Tikhonov regularization, but this requires precise knowledge of the noise covariance to properly weight the data fidelity term. However, in many practical applications, noise statistics are unknown. We therefore investigate the performance of regularization methods learned without access to this noise information, focusing on Tikhonov, Lavrentiev, and quadratic regularization. Our theoretical analysis and numerical experiments demonstrate that for non-white noise, a performance gap emerges between these methods and the optimal affine reconstruction. Furthermore, we show that these different types of regularization yield distinct results, highlighting that the choice of regularizer structure is critical when the noise model is not explicitly learned. Our findings underscore the significant value of accurately modeling or co-learning noise statistics in data-driven regularization.

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arXiv2025

The AutoLyap software suite for computer-assisted Lyapunov analyses of first-order methods

We introduce AutoLyap, a software suite that assists with Lyapunov analyses of a wide class of first-order methods for structured optimization and inclusion problems. Lyapunov analyses are structured proof patterns, with historical roots in the study of dynamical systems, commonly used to establish convergence results for first-order methods. Building on previous work, the core idea behind AutoLyap is to recast the verification of the existence of a Lyapunov analysis as a semidefinite program (SDP), which can then be solved numerically using standard SDP solvers. Users of the package specify (i) the class of optimization or inclusion problems, (ii) the first-order method in question, and (iii) the type of Lyapunov analysis they wish to test. Once these inputs are provided, AutoLyap handles the SDP modeling and proceeds to solve the SDP numerically. We use the package to numerically verify and extend several convergence results. AutoLyap is currently available in Python and Julia.

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arXiv2023

The Chambolle--Pock method converges weakly with $\theta>1/2$ and $\tau \sigma \|L\|^2<4/(1+2\theta)$

The Chambolle--Pock method is a versatile three-parameter algorithm designed to solve a broad class of composite convex optimization problems, which encompass two proper, lower semicontinuous, and convex functions, along with a linear operator $L$. The functions are accessed via their proximal operators, while the linear operator is evaluated in a forward manner. Among the three algorithm parameters $\tau $, $\sigma $, and $\theta$; $\tau,\sigma >0$ serve as step sizes for the proximal operators, and $\theta$ is an extrapolation step parameter. Previous convergence results have been based on the assumption that $\theta=1$. We demonstrate that weak convergence is achievable whenever $\theta> 1/2$ and $\tau \sigma \|L\|^2<4/(1+2\theta)$. Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.

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arXiv2023

Automated tight Lyapunov analysis for first-order methods

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, ii) first-order methods that can be written as a linear system in state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality-gap convergence in the Chambolle-Pock method when the linear operator is the identity mapping.

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arXiv2022

Circuit Analysis using Monotone+Skew Splitting

It is shown that the behavior of an $m$-port circuit of maximal monotone elements can be expressed as a zero of the sum of a maximal monotone operator containing the circuit elements, and a structured skew-symmetric linear operator representing the interconnection structure, together with a linear output transformation. The Condat-V\~u algorithm solves inclusion problems of this form, and may be used to solve for the periodic steady-state behavior, given a periodic excitation at each port, using an iteration in the space of periodic trajectories.

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arXiv2022

Incorporating History and Deviations in Forward--Backward Splitting

We propose a variation of the forward--backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward--backward splitting scheme and an accelerated O(1/n^2)-convergent forward--backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization.

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arXiv2022

Frugal Splitting Operators: Representation, Minimal Lifting, and Convergence

We investigate frugal splitting operators for finite sum monotone inclusion problems. These operators utilize exactly one direct or resolvent evaluation of each operator of the sum, and the splitting operator's output is dictated by linear combinations of these evaluations' inputs and outputs. To facilitate analysis, we introduce a novel representation of frugal splitting operators via a generalized primal-dual resolvent. The representation is characterized by an index and four matrices, and we provide conditions on these that ensure equivalence between the classes of frugal splitting operators and generalized primal-dual resolvents. Our representation paves the way for new results regarding lifting numbers and the development of a unified convergence analysis for frugal splitting operator methods, contingent on the directly evaluated operators being cocoercive. The minimal lifting number is $n-1-f$ where $n$ is the number of monotone operators and $f$ is the number of direct evaluations in the splitting. Notably, this lifting number is achievable only if the first and last operator evaluations are resolvent evaluations. These results generalize the minimal lifting results by Ryu and Malitsky--Tam that consider frugal resolvent splittings. Building on our representation, we delineate a constructive method to design frugal splitting operators, exemplified in the design of a novel, convergent, and parallelizable frugal splitting operator with minimal lifting.

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arXiv2022

DWIFOB: A Dynamically Weighted Inertial Forward-Backward Algorithm for Monotone Inclusions

We propose a novel dynamically weighted inertial forward-backward algorithm (DWIFOB) for solving structured monotone inclusion problems. The scheme exploits the globally convergent forward-backward algorithm with deviations in [26] as the basis and combines it with the extrapolation technique used in Anderson acceleration to improve local convergence. We also present a globally convergent primal-dual variant of DWIFOB and numerically compare its performance to the primal-dual method of Chambolle-Pock and a Tikhonov regularized version of Anderson acceleration applied to the same problem. In all our numerical evaluations, the primal-dual variant of DWIFOB outperforms the Chambolle-Pock algorithm. Moreover, our numerical experiments suggest that our proposed method is much more robust than the regularized Anderson acceleration, which can fail to converge and be sensitive to algorithm parameters. These numerical experiments highlight that our method performs very well while still being robust and reliable.

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arXiv2021

Nonlinear Forward-Backward Splitting with Momentum Correction

The nonlinear, or warped, resolvent recently explored by Giselsson and B\`ui-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflected-backward method of Malitsky-Tam, the primal-dual methods of V\~u-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-V\~u. A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding momentum to any special case of our nonlinear forward-backward method, in particular all the algorithms listed above.

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arXiv2021

Forward--Backward Splitting with Deviations for Monotone Inclusions

We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom. The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees. These guarantees include linear convergence of our method under a metric subregularity assumption without the need to adapt the algorithm parameters. Choosing suitable monotone operators allows for incorporating deviations into other algorithms, such as Chambolle--Pock and Krasnoselsky--Mann iterations. We propose a novel inertial primal--dual algorithm by selecting the deviations along a momentum direction and deciding their size using the norm condition. Numerical experiments demonstrate our convergence claims and show that even this simple choice of deviation vector can improve the performance, compared, e.g., to the standard Chambolle--Pock algorithm.

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arXiv2021

Accelerated Forward-Backward Optimization using Deep Learning

We propose several deep-learning accelerated optimization solvers with convergence guarantees. We use ideas from the analysis of accelerated forward-backward schemes like FISTA, but instead of the classical approach of proving convergence for a choice of parameters, such as a step-size, we show convergence whenever the update is chosen in a specific set. Rather than picking a point in this set using some predefined method, we train a deep neural network to pick the best update. Finally, we show that the method is applicable to several cases of smooth and non-smooth optimization and show superior results to established accelerated solvers.

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arXiv2018

A data-driven iteratively regularized Landweber iteration

We derive and analyse a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a nonlinear inverse problem for Schlieren tomography.

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arXiv2018

Data-driven nonsmooth optimization

In this work, we consider methods for solving large-scale optimization problems with a possibly nonsmooth objective function. The key idea is to first specify a class of optimization algorithms using a generic iterative scheme involving only linear operations and applications of proximal operators. This scheme contains many modern primal-dual first-order solvers like the Douglas-Rachford and hybrid gradient methods as special cases. Moreover, we show convergence to an optimal point for a new method which also belongs to this class. Next, we interpret the generic scheme as a neural network and use unsupervised training to learn the best set of parameters for a specific class of objective functions while imposing a fixed number of iterations. In contrast to other approaches of "learning to optimize", we present an approach which learns parameters only in the set of convergent schemes. As use cases, we consider optimization problems arising in tomographic reconstruction and image deconvolution, and in particular a family of total variation regularization problems.

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arXiv2016

Fixing and extending some recent results on the ADMM algorithm

We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration we allow the use of variable metrics, while the investigations are carried out in the setting of infinite dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties.

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arXiv2016

A general double-proximal gradient algorithm for d.c. programming

The possibilities of exploiting the special structure of d.c. programs, which consist of optimizing the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimization problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. Convergence of the iterates is shown if this objective function satisfies the Kurdyka--\L ojasiewicz property. In the last part, we apply the algorithm to an image processing model.

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arXiv2015

A forward-backward-forward differential equation and its asymptotic properties

In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant relaxation parameters and stepsizes of the resolvents. Besides proving existence and uniqueness of strong global solutions for the differential equation under consideration, we show weak convergence of the generated trajectories and, under strong monotonicity assumptions, strong convergence with exponential rate. In the particular setting of minimizing the sum of a proper, convex and lower semicontinuous function with a smooth convex one, we provide a rate for the convergence of the objective function along the ergodic trajectory to its minimum value.

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arXiv2014

Backward Penalty Schemes for Monotone Inclusion Problems

In this paper we are concerned with solving monotone inclusion problems expressed by the sum of a set-valued maximally monotone operator with a single-valued maximally monotone one and the normal cone to the nonempty set of zeros of another set-valued maximally monotone operator. Depending on the nature of the single-valued operator, we will propose two iterative penalty schemes, both addressing the set-valued operators via backward steps. The single-valued operator will be evaluated via a single forward step if it is cocoercive, and via two forward steps if it is monotone and Lipschitz continuous. The latter situation represents the starting point for dealing with complexly structured monotone inclusion problems from algorithmic point of view.

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arXiv2013

Backward-Backward Splitting in Hadamard Spaces

The backward-backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in Hilbert spaces. We generalize this setting to Hadamard spaces and prove the convergence of an error-tolerant version of the backward-backward method.

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