Towards designing learned optimization algorithms that are usable beyond their training setting, we identify key principles that classical algorithms obey, but have up to now, not been used for Learning to Optimize (L2O). Following these principles, we provide a general design pipeline, taking into account data, architecture and learning strategy, and thereby enabling a synergy between classical optimization and L2O, resulting in a philosophy of Learning Optimization Algorithms. As a consequence our learned algorithms perform well far beyond problems from the training distribution. We demonstrate the success of these novel principles by designing a new learning-enhanced BFGS algorithm and provide numerical experiments evidencing its adaptation to many settings at test time.

Numerous optimization algorithms have a time-varying update rule thanks to, for instance, a changing step size, momentum parameter or, Hessian approximation. Often, such algorithms are used as solvers for the lower-level problem in bilevel optimization, and are unrolled when computing the gradient of the upper-level objective. 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 then adapt these convergence results and apply them to proximal gradient descent with variable step size and FISTA when solving partly-smooth problems. We test the convergence (rates) of these algorithms numerically through several experiments. Our theoretical and numerical results show that the convergence rate of the algorithm is reflected in its derivative iterates.

Learning-to-optimize leverages machine learning to accelerate optimization algorithms. While empirical results show tremendous improvements compared to classical optimization algorithms, theoretical guarantees are mostly lacking, such that the outcome cannot be reliably assured. Especially, convergence is hardly studied in learning-to-optimize, because conventional convergence guarantees in optimization are based on geometric arguments, which cannot be applied easily to learned algorithms. Thus, we develop a probabilistic framework that resembles classical optimization and allows for transferring geometric arguments into learning-to-optimize. Based on our new proof-strategy, our main theorem is a generalization result for parametric classes of potentially non-smooth, non-convex loss functions and establishes the convergence of learned optimization algorithms to critical points with high probability. This effectively generalizes the results of a worst-case analysis into a probabilistic framework, and frees the design of the learned algorithm from using safeguards.

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 worst-case analysis. The results rely on PAC-Bayesian bounds for general, possibly unbounded loss-functions based on exponential families. Further, we provide a concrete algorithmic realization of the framework and new methodologies for learning-to-optimize. Finally, 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.

Abstract. We introduce two quasi-Newton forward-backward splitting algorithms to solve a class of monotone inclusion problems. The bottleneck is the evaluation of the resolvent operator. Changing the metric makes its computation even harder, and this is even true for a simple operator whose resolvent is known for the standard metric. To fully exploit the advantage of adapting the metric, we develop a new efficient resolvent calculus for a low-rank perturbed standard metric, which accounts exactly for quasi-Newton metrics. Moreover, we prove the convergence of our algorithms, including linear convergence rates in case one of the two considered operators is strongly monotone. As a by-product of our general monotone inclusion framework, we introduce two variants of the quasi-Newton primal-dual hybrid gradient method (PDHG) for solving saddle point problems. The favorable performance of these two quasi-Newton PDHG methods is demonstrated on several numerical experiments in image processing.

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.

Abstract 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 $$\vert x^4-1 \vert $$ | 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 structured 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 and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.

We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in machine learning applications such as structured support vector machines, matrix factorization and min-min or minimax problems in general. Existing approaches for computing the derivative rely on strong assumptions of the parametric function. Therefore, in several scenarios there is no theoretical evidence that a given algorithmic differentiation strategy computes the true gradient information of the value function. We leverage a well known result from convex duality theory to relax the conditions and to derive convergence rates of the derivative approximation for several classes of parametric optimization problems in Machine Learning. We demonstrate the versatility of our approach in several experiments, including non-smooth parametric functions. Even in settings where other approaches are applicable, our duality based strategy shows a favorable performance.

We aim at computing the derivative of the solution to a parametric optimization problem with respect to the involved parameters. For a class broader than that of strongly convex functions, this can be achieved by automatic differentiation of iterative minimization algorithms. If the iterative algorithm converges pointwise, then we prove that the derivative sequence also converges pointwise to the derivative of the minimizer with respect to the parameters. Moreover, we provide convergence rates for both sequences. In particular, we prove that the accelerated convergence rate of the Heavy-ball method compared to Gradient Descent also accelerates the derivative computation. An experiment with L2-Regularized Logistic Regression validates the theoretical results.

The Conditional Gradient Method is generalized to a class of non-smooth non-convex optimization problems with many applications in machine learning. The proposed algorithm iterates by minimizing so-called model functions over the constraint set. Complemented with an Armijo line search procedure, we prove that subsequences converge to a stationary point. The abstract framework of model functions provides great flexibility in the design of concrete algorithms. As special cases, for example, we develop an algorithm for additive composite problems and an algorithm for non-linear composite problems which leads to a Gauss-Newton-type algorithm. Both instances are novel in non-smooth non-convex optimization and come with numerous applications in machine learning. We perform an experiment on a non-linear robust regression problem and discuss the flexibility of the proposed framework in several matrix factorization formulations.

Matrix Factorization is a popular non-convex optimization problem, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization variables. A remedy is non-alternating schemes. However, due to a lack of Lipschitz continuity of the gradient in matrix factorization problems, convergence cannot be guaranteed. A recently developed approach relies on the concept of Bregman distances, which generalizes the standard Euclidean distance. We exploit this theory by proposing a novel Bregman distance for matrix factorization problems, which, at the same time, allows for simple/closed form update steps. Therefore, for non-alternating schemes, such as the recently introduced Bregman Proximal Gradient (BPG) method and an inertial variant Convex--Concave Inertial BPG (CoCaIn BPG), convergence of the whole sequence to a stationary point is proved for Matrix Factorization. In several experiments, we observe a superior performance of our non-alternating schemes in terms of speed and objective value at the limit point.

Natural image statistics indicate that we should use nonconvex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge of optimization. Recently, iteratively reweighed $\ell_1$ minimization (IRL1) has been proposed as a way to tackle a class of nonconvex functions by solving a sequence of convex $\ell_2$-$\ell_1$ problems. We extend the problem class to the sum of a convex function and a (nonconvex) nondecreasing function applied to another convex function. The proposed algorithm sequentially optimizes suitably constructed convex majorizers. Convergence to a critical point is proved when the Kurdyka--Łojasiewicz property and additional mild restrictions hold for the objective function. The efficiency and practical importance of the algorithm are demonstrated in computer vision tasks such as image denoising and optical flow. Most applications seek smooth results with sharp discontinuities. These are achieved by combining nonconvexity with higher order regularization.

Motion is a strong cue for unsupervised object-level grouping. In this paper, we demonstrate that motion will be exploited most effectively, if it is regarded over larger time windows. Opposed to classical two-frame optical flow, point trajectories that span hundreds of frames are less susceptible to short-term variations that hinder separating different objects. As a positive side effect, the resulting groupings are temporally consistent over a whole video shot, a property that requires tedious post-processing in the vast majority of existing approaches. We suggest working with a paradigm that starts with semi-dense motion cues first and that fills up textureless areas afterwards based on color. This paper also contributes the Freiburg-Berkeley motion segmentation (FBMS) dataset, a large, heterogeneous benchmark with 59 sequences and pixel-accurate ground truth annotation of moving objects.

In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly nonconvex) and a convex (possibly nondifferentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a nonsmooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on nonconvex problems. The convergence result is obtained based on the Kurdyka--Łojasiewicz inequality. This is a very weak restriction, which was used to prove convergence for several other gradient methods. First, an abstract convergence theorem for a generic algorithm is proved, and then iPiano is shown to satisfy the requirements of this theorem. Furthermore, a convergence rate is established for the general problem class. We demonstrate iPiano on computer vision problems---image denoising with learned priors and diffusion based image compression.

Natural image statistics indicate that we should use non-convex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge to optimize them. Recently, iteratively reweighed ℓ1minimization has been proposed as a way to tackle a class of non-convex functions by solving a sequence of convex ℓ2- ℓ1problems. Here we extend the problem class to linearly constrained optimization of a Lipschitz continuous function, which is the sum of a convex function and a function being concave and increasing on the non-negative orthant (possibly non-convex and non-concave on the whole space). This allows to apply the algorithm to many computer vision tasks. We show the effect of non-convex regularizers on image denoising, deconvolution, optical flow, and depth map fusion. Non-convexity is particularly interesting in combination with total generalized variation and learned image priors. Efficient optimization is made possible by some important properties that are shown to hold.

Motion segmentation based on point trajectories can integrate information of a whole video shot to detect and separate moving objects. Commonly, similarities are defined between pairs of trajectories. However, pairwise similarities restrict the motion model to translations. Non-translational motion, such as rotation or scaling, is penalized in such an approach. We propose to define similarities on higher order tuples rather than pairs, which leads to hypergraphs. To apply spectral clustering, the hypergraph is transferred to an ordinary graph, an operation that can be interpreted as a projection. We propose a specific nonlinear projection via a regularized maximum operator, and show that it yields significant improvements both compared to pairwise similarities and alternative hypergraph projections.

We propose an algorithm for 3-D multiview deblurring using spatially variant point spread functions (PSFs). The algorithm is applied to multiview reconstruction of volumetric microscopy images. It includes registration and estimation of the PSFs using irregularly placed point markers (beads). We formulate multiview deblurring as an energy minimization problem subject to L1-regularization. Optimization is based on the regularized Lucy-Richardson algorithm, which we extend to deal with our more general model. The model parameters are chosen in a profound way by optimizing them on a realistic training set. We quantitatively and qualitatively compare with existing methods and show that our method provides better signal-to-noise ratio and increases the resolution of the reconstructed images.

Point trajectories have emerged as a powerful means to obtain high quality and fully unsupervised segmentation of objects in video shots. They can exploit the long term motion difference between objects, but they tend to be sparse due to computational reasons and the difficulty in estimating motion in homogeneous areas. In this paper we introduce a variational method to obtain dense segmentations from such sparse trajectory clusters. Information is propagated with a hierarchical, nonlinear diffusion process that runs in the continuous domain but takes superpixels into account. We show that this process raises the density from 3% to 100% and even increases the average precision of labels.