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Mathematical Optimization Group

Department of Mathematics, University of Tübingen, Germany

MOP Research Seminar

Research Seminar on Mathematical Optimization

Meeting: Thursdays 15.00 – 16.00 o'clock
Where: Virtual meeting via Zoom
Who: MOP Group and students or researchers with interest in Mathematical Optimization
Language: English
News Description Organization
News

Next Session: 01.12.2022

Talk by Sheheryar Mehmood about "Fixed-Point Automatic Differentiation of Forward--Backward Splitting Algorithms for Partly Smooth Functions".

Abstract: A large class of non-smooth practical optimization problems can be written as minimization of a sum of smooth and partly smooth functions. We consider 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 optmization problems. We show that under partial smoothness and other mild assumptions, Automatic Differentiation (AD) of the sequence generated by proximal splitting algorithms converges 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 practical image denoising problem by learning the regularization term.



Upcoming Sessions:

Date Topic Comment
01.12.2022 Fixed-Point Automatic Differentiation of Forward--Backward Splitting Algorithms for Partly Smooth Functions Talk by Sheheryar Mehmood
08.12.2022 SAM as an Optimal Relaxation of Bayes Talk by Thomas Möllenhoff
15.12.2022 PAC-Bayesian Learning of Optimization Algorithms Talk by Michael Sucker
22.12.2021 to 08.01.2022 Winter Break no session
12.01.2023 TBD reading session
19.01.2023 TBA Talk by Camille Castera


Past Sessions:

Date Topic Comment
24.11.2022 Inertial Quasi-Newton Methods for Monotone Inclusion: Efficient Resolvent Calculus and Primal-Dual Methods

We introduce an inertial quasi-Newton Forward-Backward Splitting Algorithm to solve a class of monotone inclusion problems. While the inertial step is computationally cheap, in general, the bottleneck is the evaluation of the resolvent operator. A change of the metric makes its computation hard even for (otherwise in the standard metric) simple operators. In order 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. Beyond the general monotone inclusion setup, we instantiate a novel inertial quasi-Newton Primal-Dual Hybrid Gradient Method for solving saddle point problems. The favourable performance of our inertial quasi-Newton PDHG method is demonstrated on several numerical experiments in image processing.

Talk by Shida Wang
17.11.2022 Bolte, Combettes, Pauwels: The Iterates of the Frank-Wolfe Algorithm May Not Converge, 2022. reading session
10.11.2022 Kunstner, Kumar, Schmidt: Homeomorphic-Invariance of EM: Non-Asymptotic Convergence in KL Divergence for Exponential Families via Mirror Descent, 2021. reading session
03.11.2022 Mei, Bai, Montanari: The Landscape of Empirical Risk for Non-convex Losses, 2017. reading session
27.10.2022 Aberdam, Beck: An Accelerated Coordinate Gradient Descent Algorithm for Non-separable Composite Optimization, 2021. reading session
20.10.2022 Zhang, Sra: First-order Methods for Geodesically Convex Optimization, 2016. reading session
13.10.2022 Du, Jin, Lee, Jordan: Gradient Descent Can Take Exponential Time to Escape Saddle Points, 2017. reading session
06.10.2022 Subgradient sampling for nonsmooth nonconvex minimization

We focus on a stochastic minimization problem in the nonsmooth and nonconvex setting which applies for instance to the training of deep learning models. A popular way in the machine learning community to deal with this problem is to use stochastic gradient descent (SGD). This method combines both subgradient sampling and backpropagation, which is an efficient implementation of the chain-rule formula on nonsmooth compositions. Due to the incompatibility of these operations in the nonsmooth world, the SGD can generate spurious critical points in the optimization landscape which does not guarantee the convergence of the iterates to the critical set. We will explain in this talk how the model of Conservative Gradients is compatible with subgradient sampling and backpropagation, allowing to obtain convergence results for nonsmooth SGD. By means of definable geometry, we will emphasize that functions used in machine learning are locally endown with geometric properties of piecewise affine functions. In this setting, chain-ruling nonsmooth functions, and sampling subgradients output conservative gradients, but also, spurious critical points are hardly attained when performing SGD in practice.

Talk by Tam Le
29.09.2022 Lyu, Li: Gradient Descent Maximizes the Margin of Homogeneous Neural Networks, 2020. reading session
22.07.2022 An SDE perspective on stochastic convex optimization

We analyze the global and local behavior of gradient-like flows under stochastic errors towards the aim of solving convex optimization problems with noisy gradient input. We first study the unconstrained differentiable convex case, using a stochastic differential equation where the drift term is minus the gradient of the objective function and the diffusion term is either bounded or square-integrable. In this context, under Lipschitz continuity of the gradient, our first main result shows almost sure convergence of the objective and the trajectory process towards a minimizer of the objective function. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and (local) Lojasiewicz case. The latter, which involves local analysis, is challenging and requires non-trivial arguments from measure theory. Then, we extend our study to the constrained case and more generally to certain nonsmooth situations. We show that several of our results have natural extensions obtained by replacing the gradient of the objective function by a cocoercive monotone operator. This makes it possible to obtain similar convergence results for optimization problems with an additively "smooth + non-smooth" convex structure. Finally, we consider another extension of our results to non-smooth optimization which is based on the Moreau envelope.

Talk by Rodrigo Maulen Soto
15.09.2022 Bianchi, Hachem, Schechtman: Convergence of constant step stochastic gradient descent for non-smooth non-convex functions, 2020. reading session
08.09.2022 Nonsmooth Implicit Differentiation for Machine-Learning and Optimization Talk by Tony Silveti-Falls
01.09.2022 Jin, Koppel, Rajawat, Mokhtari: Sharpened Quasi-Newton Methods: Faster Superlinear Rate and Larger Local Convergence Neighborhood, 2022. reading session
28.07. -- 25.08.2022 Summer Break no session
21.07.2022 Luo, Chen: From differential equation solvers to accelerated first-order methods for convex optimization, 2022. reading session
14.07.2022 Benaim, Hofbauer, Sorin: Stochastic Approximations and Differential Inclusions, 2005. reading session
07.07.2022 Davis, Drusvyatskiy, Kakade, Lee: Stochastic Subgradient Method Converges on Tame Functions, 2020. reading session
30.06.2022 Savarino, Schnörr: Continuous-Domain Assignment Flows, 2019. reading session
23.06.2022 Lewis, Malick: Alternating Projections on Manifolds, 2008. reading session
16.06.2022 Public Holiday no session
09.06.2022 Ahn, Sra: From Proximal Point Method to Nesterov's Acceleration, 2020. reading session
02.06.2022 Luo, Chen: From differential equation solvers to accelerated first-order methods for convex optimization, 2022. reading session
26.05.2022 Public Holiday no session
19.05.2022 Drori, Taylor: Efficient first-order methods for convex minimization: a constructive approach, 2020. reading session
12.05.2022 Li, Pong: Calculus of the exponent of Kurdyka-Lojasiewicz inequality and its applications to linear convergence of first-order methods, 2021. reading session
05.05.2022 Solving 0-1 ILPs with Binary Decision Diagrams

We present a Lagrange decomposition method for solving 0-1 integer linear programs occurring in structured prediction. We propose a sequential and a massively min-marginal averaging schemes for solving the Lagrangean dual and a perturbation technique for decoding primal solutions. For representing subproblems we use binary decision diagrams (BDDs), which support efficient computation of subproblem solutions and update steps. We present experimental results on combinatorial problems from MAP inference for Markov Random Fields, quadratic assignment and cell tracking for developmental biology. Our highly parallel GPU implementation improves, comes close to, or outperform some state-of-the-art specialized heuristics while being problem agnostic.

Talk by Paul Swoboda
28.04.2022 Bredies, Chenchene, Lorenz, Naldi: Degenerate Preconditioned Proximal Point algorithms, 2021. reading session
21.04.2022 Frecon, Salzo, Pontil, Gasso: Bregman Neural Networks, 2022. reading session
14.04.2022 Pedregosa, Scieur: Average-Case Acceleration Through Spectral Density Estimation, 2020. reading session
07.04.2022 Li, Voroninski: Sparse Signal Recovery from Quadratic Measurements via Convex Programming, 2012. reading session
31.03.2022 Master Thesis Defense Talk by Michael Sucker
24.03.2022 Lewis, Tian: The structure of conservative gradient fields, 2021. reading session
17.03.2022 Progress of current research Talk by Sheheryar Mehmood
10.03.2022 Progress of current research Talk by Rodrigo Maulen
03.03.2022 Progress of current research Talk by Jean-Jacques Godeme
24.02.2022 Raskutti, Mukherjee: The information geometry of mirror descent, 2013. Reading Session
17.02.2022 Progress of current research Talk by Oskar Adolfson
10.02.2022 Muehlebach, Jordan: Continuous-time Lower Bounds for Gradient-based Algorithms, 2020. reading session
03.02.2022 An Inertial Newton Algorithm for Deep Learning Talk by Camille Castera
27.01.2022 Alvarez, Attouch, Bolte, Redont: A second-order gradient-like dissipative dynamical system with Hessian-driven damping, 2002. Reading Session
20.01.2022 Progress of current research Talk by Shida Wang
13.01.2022 Apidopoulos, Ginatta, Villa: Convergence rates for the Heavy-Ball continuous dynamics for non-convex optimization, under Polyak-Ɓojasiewicz conditioning, 2021. Reading Session
23.12.2021 to 06.01.2022 Break no session
16.12.2021 Calatroni, Chambolle: Backtracking Strategies for Accelerated Descent Methods with Smooth Composite Objectives, 2019. reading session
09.12.2021 Sahin, Eftekhari, Alacaoglu, Latorre, Cevher: An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints, 2019. reading session
02.12.2021 Muehlebach, Jordan: On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems, 2020. reading session
25.11.2021 Andersen, Dahl, Vandenberghe: Logarithmic barriers for sparse matrix cones, 2013. reading session
18.11.2021 Nesterov: Implementable tensor methods in unconstrained convex optimization, 2021. reading session
11.11.2021 Wang, Fang, Liu: Stochastic Compositional Gradient Descent: Algorithms for Minimizing Nonlinear Functions of Expected Values, 2016. reading session
04.11.2021 Wilson, Recht, Jordan: A Lyapunov Analysis of Momentum Methods in Optimization, 2016. reading session
28.10.2021 Connor, Vandenberghe: On the equivalence of the primal-dual hybrid gradient method and Douglas-Rachford splitting, 2020. reading session
21.10.2021 Bilevel Learning for Inverse Problems

Variational regularization techniques are dominant in the field of inverse problems. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. This issue can be approached by machine learning where we estimate these parameters from data. This is known as "Bilevel Learning" and has been successfully applied to many tasks, some as small-dimensional as learning a regularization parameter, others as high-dimensional as learning a sampling pattern in MRI. While mathematically appealing this strategy leads to a nested optimization problem which is computationally difficult to handle. In this talk we discuss several applications of bilevel learning for imaging as well as new computational approaches. There are quite a few open problems in this relatively recent field of study, some of which I will highlight along the way.

Talk by Matthias Ehrhardt
14.10.2021 Fadili, Malick, Peyre: Sensitivity Analysis for Mirror-Stratifiable Convex Functions, 2017. reading session
07.10.2021 cancelled no session
30.09.2021 Gower, Schmidt, Bach, Richtarik: Variance-reduced methods for machine learning, 2020. reading session
23.09.2021 Sabach, Teboulle: Faster Lagrangian-Based Methods in Convex Optimization, 2020. reading session
16.09.2021 Proximal Algorithms for Scalar and Vector Optimization Problems

This presentation will go through several techniques for solving both scalar and vector optimization problems with various objective functions. On scalar version of optimization problems, we intend to discuss how third-order differential equations, or equations with an order higher than three, can contribute to development of fast optimization methods. The first problem to consider is to minimize the real-valued, convex, and continuously differentiable function defined on a Hilbert space. The problem will then be altered to minimize non- smooth and non-convex functions. We show that the convergence rate of the continuous dynamical system and the proximal algorithm obtained from discretization of it is the same. At the end of this route, we want to analyze the convergence of trajectories towards optimal solutions in both continuous and discrete versions. In the case of vector optimization problems, we aim to present a model of Alternating Direction Method of Multipliers (ADMM) which is a simple and powerful proximal algorithm, while the problem is minimization of sum of two vector-valued functions with a linear constraint respect to variables. This research is vital since there is a lack of literature on proximal techniques for tackling vector optimization problems.

Talk by Maede Ramazannejad
09.09.2021 Teboulle, Vaisbourd: Novel proximal gradient methods for nonnegative matrix factorization with sparsity constraints, SIIMS 13(1):381-421, 2020. reading session
08.2021 Break no session
29.07.2021 Reading: Chatterji, Diakonikolas, Jordan, Bartlett: Langevin Monte Carlo without smoothness, 2020. reading session
22.07.2021 Pokutta, Spiegel, Zimmer: Deep Neural Network Training with Frank-Wolfe, 2020. reading session
15.07.2021 Drori, Teboulle: Performance of first-order methods for smooth convex minimization: a novel approach, 2020. reading session
08.07.2021 Lin, Jin, Jordan: On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems, 2020. reading session
01.07.2021 Nonconvex Min-max Optimization in Machine Learning: Algorithms and Applications.

Abstract: Min-max Optimization has broad applications in machine learning, e.g., Adversarial Robustness, AUC Maximization, Generative Adversarial Networks, etc. However, modern machine learning models are intrinsically nonconvex (e.g., deep learning, matrix factorization), which brings tremendous challenge for min-max optimization. In this talk, I will talk about my work on provably efficient algorithms for min-max optimization in the presence of non-convexity. This talk will showcase a few results. First, I will talk about nonconvex-concave min-max optimization, with applications in deep AUC optimization. We designed polynomial-time algorithms to converge to stationary point, as well as maximal AUC value, under different assumptions. Second, I will talk about nonconvex-nonconcave min-max optimization, with applications in training Generative Adversarial Networks. The key result is that under the Minty Variational Inequality (MVI) condition, we design polynomial-time algorithms to find stationary points.

Talk by Mingrui Liu
24.06.2021 Stochastic trust-region method with adaptive sample sizes for finite-sum minimization problems.

In this talk, we present SIRTR (Stochastic Inexact Restoration Trust-Region method) for solving finite-sum minimization problems. At each iteration, SIRTR approximates both function and gradient by sampling. The function sample size is computed using a deterministic rule inspired by the inexact restoration method, whereas the gradient sample size can be smaller than the sample size employed in function approximations. Notably, our approach may allow the decrease of the sample sizes at some iterations. We show that SIRTR eventually reaches full precision in evaluating the objective function and we provide a worst-case complexity result on the number of iterations required to achieve full precision. Numerical results on nonconvex binary classification problems confirm that SIRTR is able to provide accurate approximations way before the maximum sample size is reached and without requiring a problem-dependent tuning of the parameters involved.

Talk by Simone Rebegoldi
17.06.2021 Jiang, Vandenberghe: "Bregman primal-dual first-order method and application to sparse semidefinite programming", 2020. reading session
10.06.2021 Salimans, Zhang, Radford, Metaxas: "Improving GANs Using Optimal Transport", ICLR 2018. reading session
03.06.2021 Public Holiday no session
27.05.2021 Cuturi, Teboul, Vert: "Differentiable Ranks and Sorting using Optimal Transport." NeurIPS, 2019. reading session
20.05.2021 Adaptive Acceleration for First-order Methods.

Abstract: First-order operator splitting methods are ubiquitous among many fields through science and engineering, such as inverse problems, image processing, statistics, data science and machine learning, to name a few. In this talk, through the fixed-point sequence, I will first discuss a geometry property of first-order methods when applying to solve non-smooth optimization problems. Then I will discuss the limitation of current widely used "inertial acceleration" technique, and propose a trajectory following adaptive acceleration algorithm. Global convergence is established for the proposed acceleration scheme based on the perturbation of fixed-point iteration. Locally, connections between the acceleration scheme and the well studied "vector extrapolation technique" in the field of numerical analysis will be discussed, followed by acceleration guarantees of the proposed acceleration scheme. Numeric experiments on various first-order methods are provided to demonstrate the advantage of the proposed adaptive acceleration scheme.

Talk by Jingwei Liang
13.05.2021 Public Holiday no session
06.05.2021 Rockafellar : "Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming." SIAM Journal on Control, 1974. reading session
29.04.2021 Attouch, Bolte, Svaiter: "Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods." Mathematical Programming, 2013. reading session
22.04.2021 Implicit differentiation for fast hyperparameter selection in non-smooth convex learning.

Abstract: Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian. Using implicit differentiation, we show it is possible to leverage the non-smoothness of the inner problem to speed up the computation. Finally, we provide a bound on the error made on the hypergradient when the inner optimization problem is solved approximately. Results on regression and classification problems reveal computational benefits for hyperparameter optimization, especially when multiple hyperparameters are required.

Talk by Quentin Bertrand
15.04.2021 Chen, Rubanova, Bettencourt, Duvenaud: "Neural ordinary differential equations." NeurIPS, 2018. reading session
08.04.2021 Bach: "Submodular functions: from discrete to continuous domains", Mathematical Programming, 2019. reading session
01.04.2021 Learning to optimize with unrolled algorithms

Abstract: When solving multiple optimization problems sharing the same underlying structure, using iterative algorithms designed for worst case scenario can be considered as inefficient. When one aim at having good solution in average, it is possible to improve the performances by learning the weights of a neural networked designed to mimic an unfolded optimization algorithm. However, the reason why learning the weights of such a network would accelerate the problem resolution is not always clear. In this talk, I will first review how one can design unrolled algorithms to solve the linear regression with l1 or TV regularization, with a particular focus on the choice of parametrization and loss. Then, I will discuss the reasons why such procedure can lead to better results compared to classical optimization, with a particular focus on the choice of step sizes.

Talk by Thomas Moreau
25.03.2021 Wright: "Coordinate Descent Algorithms", Mathematical Programming 151:3-34, 2015. reading session
18.03.2021 Dragomir, d'Aspremont, Bolte: "Quartic first-order methods for low rank minimization." Journal of Optimization Theory and Applications, 2021 reading session
11.03.2021 Monsaingeon, Vorotnikov: "The Schrödinger problem on the non-commutative Fisher-Rao space." Calculus of Variations and Partial Differential Equations, 2021. reading session
04.03.2021 Kolmogorov: "Recursive frank-wolfe algorithms", 2020. reading session
25.02.2021 Bolte, Pauwels: "A mathematical model for automatic differentiation in machine learning". NeurIPS, 2020. reading session
18.02.2021 Dragomir, Taylor, d'Aspremont, and Bolte: "Optimal complexity and certification of Bregman first-order methods". arXiv preprint, 2019. reading session
11.02.2021 Kolmogorov: "Convergent tree-reweighted message passing for energy minimization". IEEE transactions on pattern analysis and machine intelligence, 2006. reading session
04.02.2021 Dalalyan: "Theoretical guarantees for approximate sampling from smooth and log-concave densities". Journal of the Royal Statistical Society, 2017. reading session
28.01.2021 Djolonga, Krause: "Differentiable Learning of Submodular Models". Advances in Neural Information Processing Systems, 2017. reading session
21.01.2021 "A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization". Talk by Tony Silveti-Falls
14.01.2021 Van den Berg, Friedlander: "Sparse optimization with least-squares constraints". SIAM Journal on Optimization, 2011. reading session
17.12.2020 Stella, Themelis, Patrinos: "Forward-backward quasi-Newton methods for nonsmooth optimization problems". Computational Optimization and Applications 67:443-487, 2017. reading session
10.12.2020 "Lifting the Convex Conjugate in Lagrangian Relaxations". Talk by
Emanuel Laude
03.12.2020 Dlask, Werner: "A Class of Linear Programs Solvable by Coordinate-wise Minimization". 2020. reading session
02.12.2020 S. Wang: "Numerical scheme for Wasserstein distance on Manifold". Master's Thesis Mathematics, University of Bonn, 2020. Talk by
Shida Wang
26.11.2020 Attouch, Chbani, Fadili, Riahi: "First-order optimization algorithms via inertial systems with Hessian driven damping". Mathematical Programming, 2020. reading session
19.11.2020 Ablin, Peyré, Moreau: "Super-efficiency of automatic differentiation for functions defined as a minimum". ICML, 2020. reading session
12.11.2020 Mukkamala, Ochs: "Beyond Alternating Updates for Matrix Factorization with Inertial Bregman Proximal Gradient Algorithms". NeurIPS, 2019. Talk by
W. Bender
05.11.2020 Ghadimi, Lan: Stochastic first- and zeroth-order methods for nonconvex stochastic programming. 2013. reading session
29.10.2020 Swoboda, Kolmogorov: "MAP inference via Block-Coordinate Frank-Wolfe Algorithm". CVPR, 2018. reading session
22.10.2020 Ahookhosh, Hien, Gillis, Patrinos: A block inertial Bregman proximal algorithm for nonsmooth nonconvex problems, 2020. reading session
15.10.2020 Pilanci, Ergen: "Neural Networks are Convex Regularizers". 2020. reading session
Description
This is a research seminar about various topics in Mathematical Optimization that is organized by the MOP Group in form of reading sessions. We discuss important papers around mathematical optimization including (but not limited to) the following topics:
  • Non-smooth Analysis and Optimization
  • Convex or Non-convex Optimization
  • Parametric Optimization and Parameter Learning
  • Bilevel Optimization
  • Dynamic Programming and Discrete Optimization
  • Applications in Machine Learning, Computer Vision, Image Processing, Statistics, ...

Key for a convenient and beneficial reading session will be the active participation and discussion.
Organization

Format:

The reading session is (until further notice) organized via Zoom Meetings. Usually, one paper is discussed in a session. Everyone who wants to participate in the reading session should have read the paper. Before the meeting, a leader is assigned to the paper, who presents it. He or she is not supposed to be the expert of the topic.

Following are some guidelines:
  • The leader guides the reading by going through the paper step by step.
  • Everyone can add comments or start a discussion at any time.
  • Explicitly say ''I haven't understood this or that part.''
  • Discuss the 'big picture' of the paper and the context.
  • Discuss details of common interest and postpone specific details to individual discussions.
The goal is to answer the following questions:
  • What is the key contribution?
  • What do we learn from the paper?
  • What's the limitations of their approach?
  • What are the open questions?

Registration:

Please send an eMail to Peter Ochs, confirm your interest in this research seminar, and note whether you would like to join as guest or regular member (listed below). As a regular member, you should be more than willing to join the session regularly. You will have the permission to suggest papers directly in the spreadsheet and to vote for the paper of the next reading session.

Paper Collection:

The papers that are discussed in the reading session are organized in the following

google spreadsheet.


Access to the spreadsheet and the zoom link is provided by eMail after registration.

Paper Selection:

Every week, each regular members has in total three votes for papers in the section 'Upcoming Reading', which may be distributed to different papers. Deadline for voting is on Friday 2 pm for the upcoming week. (Votes cannot be accumulated.)


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