Peter Ochs

Michael Sucker, Peter Ochs

We apply the PAC-Bayes theory to the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-bounds) and explicit trade-off between a high probability of convergence and a high convergence speed. Even in the limit case, where convergence is guaranteed, our learned optimization algorithms provably outperform related algorithms based on a (deterministic) worst-case analysis. Our results rely on PAC-Bayes bounds for general, unbounded loss-functions based on exponential families. By generalizing existing ideas, we reformulate the learning procedure into a one-dimensional minimization problem and study the possibility to find a global minimum, which enables the algorithmic realization of the learning procedure. As a proof-of-concept, we learn hyperparameters of standard optimization algorithms to empirically underline our theory.

Sheheryar Mehmood, Peter Ochs

A large class of non-smooth practical optimization problems can be written as minimization of a sum of smooth and partly smooth functions. We examine such structured problems which also depend on a parameter vector and study the problem of differentiating its solution mapping with respect to the parameter which has far reaching applications in sensitivity analysis and parameter learning problems. Under partial smoothness and other mild assumptions, we apply Implicit (ID) and Automatic Differentiation (AD) to the fixed-point iterations of proximal splitting algorithms. We show that AD of the sequence generated by these algorithms converges (linearly under further assumptions) to the derivative of the solution mapping. For a variant of automatic differentiation, which we call Fixed-Point Automatic Differentiation (FPAD), we remedy the memory overhead problem of the Reverse Mode AD and moreover provide faster convergence theoretically. We numerically illustrate the convergence and convergence rates of AD and FPAD on Lasso and Group Lasso problems and demonstrate the working of FPAD on prototypical image denoising problems by learning the regularization term.

Severin Maier, Camille Castera, Peter Ochs

We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are almost exclusively achieved by non-autonomous methods via open-loop damping (e.g., Nesterov's algorithm), we show that our system, featuring a closed-loop damping, exhibits a rate arbitrarily close to the optimal one. We do so by coupling the damping and the speed of convergence of the system via a well-chosen Lyapunov function. By discretizing our system we then derive an algorithm and present numerical experiments supporting our theoretical findings.

Michael Sucker, Peter Ochs

We present a probabilistic model for stochastic iterative algorithms with the use case of optimization algorithms in mind. Based on this model, we present PAC-Bayesian generalization bounds for functions that are defined on the trajectory of the learned algorithm, for example, the expected (non-asymptotic) convergence rate and the expected time to reach the stopping criterion. Thus, not only does this model allow for learning stochastic algorithms based on their empirical performance, it also yields results about their actual convergence rate and their actual convergence time. We stress that, since the model is valid in a more general setting than learning-to-optimize, it is of interest for other fields of application, too. Finally, we conduct five practically relevant experiments, showing the validity of our claims.

Michael Sucker, Jalal Fadili, Peter Ochs

We use the PAC-Bayesian theory for the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-Bayesian bounds) and explicit trade-off between convergence guarantees and convergence speed, which contrasts with the typical worst-case analysis. Our learned optimization algorithms provably outperform related ones derived from a (deterministic) worst-case analysis. The results rely on PAC-Bayesian bounds for general, possibly unbounded loss-functions based on exponential families. Then, we reformulate the learning procedure into a one-dimensional minimization problem and study the possibility to find a global minimum. Furthermore, we provide a concrete algorithmic realization of the framework and new methodologies for learning-to-optimize, and we conduct four practically relevant experiments to support our theory. With this, we showcase that the provided learning framework yields optimization algorithms that provably outperform the state-of-the-art by orders of magnitude.

Sheheryar Mehmood, Peter Ochs

Numerous Optimization Algorithms have a time-varying update rule thanks to, for instance, a changing step size, momentum parameter or, Hessian approximation. In this paper, we apply unrolled or automatic differentiation to a time-varying iterative process and provide convergence (rate) guarantees for the resulting derivative iterates. We adapt these convergence results and apply them to proximal gradient descent with variable step size and FISTA when solving partly smooth problems. We confirm our findings numerically by solving $\ell_1$ and $\ell_2$-regularized linear and logisitc regression respectively. Our theoretical and numerical results show that the convergence rate of the algorithm is reflected in its derivative iterates.

Armin Beck, Peter Ochs

Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on a rotated MNIST dataset with a non-uniform rotation group, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.

Mahesh Chandra Mukkamala, Jalal Fadili, Peter Ochs

Lipschitz continuity of the gradient mapping of a continuously differentiable function plays a crucial role in designing various optimization algorithms. However, many functions arising in practical applications such as low rank matrix factorization or deep neural network problems do not have a Lipschitz continuous gradient. This led to the development of a generalized notion known as the $L$-smad property, which is based on generalized proximity measures called Bregman distances. However, the $L$-smad property cannot handle nonsmooth functions, for example, simple nonsmooth functions like $\abs{x^4-1}$ and also many practical composite problems are out of scope. We fix this issue by proposing the MAP property, which generalizes the $L$-smad property and is also valid for a large class of nonconvex nonsmooth composite problems. Based on the proposed MAP property, we propose a globally convergent algorithm called Model BPG, that unifies several existing algorithms. The convergence analysis is based on a new Lyapunov function. We also numerically illustrate the superior performance of Model BPG on standard phase retrieval problems, robust phase retrieval problems, and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.

Mahesh Chandra Mukkamala, Peter Ochs, Thomas Pock et al.

Backtracking line-search is an old yet powerful strategy for finding a better step sizes to be used in proximal gradient algorithms. The main principle is to locally find a simple convex upper bound of the objective function, which in turn controls the step size that is used. In case of inertial proximal gradient algorithms, the situation becomes much more difficult and usually leads to very restrictive rules on the extrapolation parameter. In this paper, we show that the extrapolation parameter can be controlled by locally finding also a simple concave lower bound of the objective function. This gives rise to a double convex-concave backtracking procedure which allows for an adaptive choice of both the step size and extrapolation parameters. We apply this procedure to the class of inertial Bregman proximal gradient methods, and prove that any sequence generated by these algorithms converges globally to a critical point of the function at hand. Numerical experiments on a number of challenging non-convex problems in image processing and machine learning were conducted and show the power of combining inertial step and double backtracking strategy in achieving improved performances.