### October 8, 2018

The paper "Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications" (arXiv preprint here), coauthored with Antonin Chambolle , Matthias J. Ehrhardt, and Carola-Bibiane Schönlieb, was just published by the SIAM Journal on Optimization.

Here are related slides, poster, GitHub code and a YouTube talk.

### October 3, 2018

Filip Hanzely is now back from an internship at the Amazon Scalable Machine Learning group in Berlin. While there, he was working on Bayesian optimization for deep learning.

### September 25, 2018

New paper out: "Accelerated coordinate descent with arbitrary sampling and best rates for minibatches" - joint work with Filip Hanzely.

Abstract: Accelerated coordinate descent is a widely popular optimization algorithm due to its efficiency on large-dimensional problems. It achieves state-of-the-art complexity on an important class of empirical risk minimization problems. In this paper we design and analyze an accelerated coordinate descent (ACD) method which in each iteration updates a random subset of coordinates according to an arbitrary but fixed probability law, which is a parameter of the method. If all coordinates are updated in each iteration, our method reduces to the classical accelerated gradient descent method AGD of Nesterov. If a single coordinate is updated in each iteration, and we pick probabilities proportional to the square roots of the coordinate-wise Lipschitz constants, our method reduces to the currently fastest coordinate descent method NUACDM of Allen-Zhu, Qu, Richt\'{a}rik and Yuan. While mini-batch variants of ACD are more popular and relevant in practice, there is no importance sampling for ACD that outperforms the standard uniform mini-batch sampling. Through insights enabled by our general analysis, we design new importance sampling for mini-batch ACD which significantly outperforms previous state-of-the-art minibatch ACD in practice. We prove a rate that is at most O(√τ) times worse than the rate of minibatch ACD with uniform sampling, but can be O(n/τ) times better, where τ is the minibatch size. Since in modern supervised learning training systems it is standard practice to choose τ ≪ n, and often τ=O(1), our method can lead to dramatic speedups. Lastly, we obtain similar results for minibatch nonaccelerated CD as well, achieving improvements on previous best rates.

### September 23, 2018

I am at the Simons Institute, UC Berkeley, attending the workshop "Randomized Numerical Linear Algebra and Applications". This workshop is a part of the semester-long program "Foundations of Data Science".

My talk is on Tuesday, Sept 25, at 9:30am, PST. I will be talking about "Stochastic Quasi-Gradient Methods: Variance Reduction via Jacobian Sketching" (joint work with R.M. Gower and F. Bach). All talks are live-streamed and recorded, and will be uploaded onto YouTube.

Update (Sept 25): My talk is on YouTube.

### September 17, 2018

I have accepted an invite to serve as an Area Chair for The 36th International Conference on Machine Learning (ICML 2019). The event will be held in Long Beach, California, June 10-15, 2019.

### September 14, 2018

The paper "Importance sampling for minibatches", coauthored with Dominik Csiba, was just published by the Journal of Machine Learning Research.

### August 13, 2018

New paper out: "Nonconvex variance reduced optimization with arbitrary sampling" - joint work with Samuel Horváth.

Abstract: We provide the first importance sampling variants of variance reduced algorithms for empirical risk minimization with non-convex loss functions. In particular, we analyze non-convex versions of SVRG, SAGA and SARAH. Our methods have the capacity to speed up the training process by an order of magnitude compared to the state of the art on real datasets. Moreover, we also improve upon current mini-batch analysis of these methods by proposing importance sampling for minibatches in this setting. Surprisingly, our approach can in some regimes lead to superlinear speedup with respect to the minibatch size, which is not usually present in stochastic optimization. All the above results follow from a general analysis of the methods which works with arbitrary sampling, i.e., fully general randomized strategy for the selection of subsets of examples to be sampled in each iteration. Finally, we also perform a novel importance sampling analysis of SARAH in the convex setting.

A poster based on the results of this paper won a Best Poster Prize. Below I am recycling an earlier blog post I made about this in June (the paper was not available online at that time):

Best DS3 Poster Award for Samuel Horváth

Samuel Horváth attended the Data Science Summer School (DS3), which took place during June 25-29, 2018, at École Polytechnique in Paris, France. Based on the event website, the event gathered 500 participants from 34 countries and 6 continents, out of which 290 were MS and PhD students and postdocs, and 110 professionals. Selected guest/speaker names (out of 41): Cédric Villani, Nicoló Cesa-Bianchi, Mark Girolami, Yann Lecun, Suvrit Sra, Jean-Philippe Vert, Adrian Weller, Marco Cuturi, Arthur Gretton, and Andreas Krause.

The event also included a best-poster competition, with an impressive total of 170 posters. Samuel's poster won the Best DS3 Poster Award. The poster, entitled Nonconvex variance reduced optimization with arbitrary sampling, is based on a paper of the same title, joint work with me, and currently under review.

Here is the poster:

And here is the award:

This first prize carries a 500 EUR cash award.

Samuel: Congratulations!!!

### September 5, 2018

Three papers accepted to The Thirty-second Annual Conference on Neural Information Processing Systems (NIPS)

The long-awaited decisions arrived today! We've had three papers accepted:

"Stochastic spectral and conjugate descent methods" - joint work with Dmitry Kovalev, Eduard Gorbunov and Elnur Gasanov.

Abstract: The state-of-the-art methods for solving optimization problems in big dimensions are variants of randomized coordinate descent (RCD). In this paper we introduce a fundamentally new type of acceleration strategy for RCD based on the augmentation of the set of coordinate directions by a few spectral or conjugate directions. As we increase the number of extra directions to be sampled from, the rate of the method improves, and interpolates between the linear rate of RCD and a linear rate independent of the condition number. We develop and analyze also inexact variants of these methods where the spectral and conjugate directions are allowed to be approximate only. We motivate the above development by proving several negative results which highlight the limitations of RCD with importance sampling.

"Accelerated stochastic matrix inversion: general theory and speeding up BFGS rules for faster second-order optimization" - joint work with Robert M. Gower, Filip Hanzely and Sebastian Stich.

Abstract: We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the first accelerated (deterministic and stochastic) quasi-Newton updates. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.

"SEGA: Variance reduction via gradient sketching" - joint work with Filip Hanzely and Konstantin Mishchenko.

Abstract: We propose a novel randomized first order optimization method—SEGA (SkEtched GrAdient method)—which progressively throughout its iterations builds a variance-reduced estimate of the gradient from random linear measurements (sketches) of the gradient provided at each iteration by an oracle. In each iteration, SEGA updates the current estimate of the gradient through a sketch-and-project operation using the information provided by the latest sketch, and this is subsequently used to compute an unbiased estimate of the true gradient through a random relaxation procedure. This unbiased estimate is then used to perform a gradient step. Unlike standard subspace descent methods, such as coordinate descent, SEGA can be used for optimization problems with a non-separable proximal term. We provide a general convergence analysis and prove linear convergence for strongly convex objectives. In the special case of coordinate sketches, SEGA can be enhanced with various techniques such as importance sampling, mini-batching and acceleration, and its rate is up to a small constant factor identical to the best-known rate of coordinate descent.

As an added bonus, I got a free NIPS registration as one of the top-ranked reviewers this year. Thanks NIPS!

The conference will take place during December 3-8, 2018 in Montreal, Canada.

### September 3, 2018

Two new students just joined my group at KAUST:

- Dmitry Kovalev (from Moscow Institute of Physics and Technology)

- Elnur Gasanov (from Moscow Institute of Physics and Technology)

Welcome!

### August 29, 2018

Several people from my group are away on internships. Filip Hanzely has been with Amazon Scalable Machine Learning group in Berlin, Germany, since June and will stay until the end of September. Konstantin Mishchenko is with Amazon, Seattle, USA since August, and will stay there until the end of November. Nicolas Loizou is with FAIR at Facebook in Montreal, Canada since August and will stay there for four months.

### August 26, 2018

The Fall semester is starting at KAUST today. I am teaching CS390FF: "Selected Topics in Data Sciences" (Sundays and Tuesdays, 9-10:30am in Bldg 9: 4125).

### August 12, 2018

I am on my way to Bethlehem, Pennsylvania, to give a talk at the DIMACS Workshop on Optimization in Machine Learning, taking place at Lehigh University during August 13-15, 2018. The workshop is part of a larger event which also includes the MOPTA conference (Aug 15-17) and the TRIPODS Summer School for PhD students (Aug 10-12).

I am giving a talk on Tuesday entitled "Stochastic quasi-gradient methods: variance reduction via Jacobian sketching", joint work with Robert M. Gower and Francis Bach. Nicolas Loizou is attending as well; he is presenting a poster on Tuesday and giving a talk on Thursday, both on the same topic: "Revisiting randomized gossip algorithms", and based on these two papers: [GlobalSIP2016], [Allerton2018].

The speaker line-up is excellent. On the other hand, the weather in Bethlehem does not seem to be particularly welcoming:

Meanwhile, this is what we are supposed to have at KAUST during the same week:

I'd welcome a convex combination of the two instead ;-)

### August 10, 2018

New paper out: "Accelerated Bregman proximal gradient methods for relatively smooth convex optimization" - joint work with Filip Hanzely and Lin Xiao.

Abstract: We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. The relatively smooth condition is much weaker than the standard assumption of uniform Lipschitz continuity of the gradients, thus significantly increases the scope of potential applications. We present accelerated Bregman proximal gradient (ABPG) methods that employ the Bregman distance of the reference function as the proximity measure. These methods attain an O(1/k^γ) convergence rate in the relatively smooth setting, where γ ∈ [1,2] is determined by a triangle scaling property of the Bregman distance. We develop adaptive variants of the ABPG method that automatically ensure the best possible rate of convergence and argue that the O(1/k^2) rate is attainable in most cases. We present numerical experiments with three applications: D-optimal experiment design, Poisson linear inverse problem, and relative-entropy nonnegative regression. In all experiments, we obtain numerical certificates showing that these methods do converge with the O(1/k^2) rate.

### August 7, 2018

The paper "Stochastic primal-dual hybrid gradient algorithm with arbitrary sampling and imaging applications", joint work with Antonin Chambolle, Matthias J. Ehrhardt, and Carola-Bibiane Schönlieb, was accepted to SIAM Journal on Optimization.

Here is a YouTube video of a talk I gave on this topic. Here is the SPDHG GitHub code.

### August 4, 2018

The paper "Accelerated gossip via stochastic heavy ball method", joint work with Nicolas Loizou, was accepted to Allerton (56th Annual Allerton Conference on Communication, Control, and Computing, 2018).

### July 27, 2018

Working on NIPS author feedback... The deadline is on August 1st.

### July 23, 2018

The paper "The complexity of primal-dual fixed point methods for ridge regression", coauthored with Ademir Alves Ribeiro, just appeared online in Linear Algebra and its Applications.

### July 22, 2018

I am in Foz do Iguacu, Brazil, attending the conference Mathematics and it Applications and XII Brazilian Workshop on Continuous Optimization. I will give a plenary talk on Thursday.

### July 16, 2018

New paper out: "Matrix completion under interval uncertainty: highlights" - joint work with Jakub Mareček and Martin Takáč. To appear in ECML-PKDD 2018.

### July 10, 2018

Most-read paper in the journal Optimization Methods and Software in 2017

The paper Distributed Optimization with Arbitrary Local Solvers, joint work with Chenxin Ma, Jakub Konečný, Martin Jaggi, Virginia Smith, Michael I Jordan, and Martin Takáč, was the most-read paper in the OMS journal in year 2017.

This is how I know: I clicked on the "Read our most-read article of 2017 for free here" link available on this website, and got a nice surprise.

### July 9, 2018

New paper out: "Accelerated gossip via stochastic heavy ball method" - joint work with Nicolas Loizou.

Abstract: In this paper we show how the stochastic heavy ball method (SHB)—a popular method for solving stochastic convex and non-convex optimization problems—operates as a randomized gossip algorithm. In particular, we focus on two special cases of SHB: the Randomized Kaczmarz method with momentum and its block variant. Building upon a recent framework for the design and analysis of randomized gossip algorithms [19] we interpret the distributed nature of the proposed methods. We present novel protocols for solving the average consensus problem where in each step all nodes of the network update their values but only a subset of them exchange their private values. Numerical experiments on popular wireless sensor networks showing the benefits of our protocols are also presented.

### July 3, 2018

I have joined the Editorial Board of Optimization Methods and Software.

### July 1, 2018

I am on my way to Bordeaux, to attend ISMP (23rd International Symposium on Mathematical Programming). With Alexandre d’Aspremont, Olivier Beaumont, and Suvrit Sra, we have organized stream 4a: "Learning: Machine Learning, Big Data, Cloud Computing, and Huge-Scale Optimization". Here is the schedule of talks which are based on papers I am co-author of (highlighted in red):

Coordinate Descent and Randomized Direct Search Methods (Continuous Optimization)
RandomM - Mo 3:15pm-4:45pm, Format: 3x30 min
Room: Salle KC6 Building: K, Intermediate 1, Zone: 10
Invited Session 211

Organizer: Martin Takáč, Lehigh University, US

1 - When Cyclic Coordinate Descent Outperforms Randomized Coordinate Descent Speaker: Asu Ozdaglar, MIT, US, talk 1486
Co-Authors: Mert Gurbuzbalaban, Nuri Vanli, Pablo Parrilo

2 - Random direct search method for unconstrained smooth minimization Speaker: El houcine Bergou, KAUST-INRA, SA, talk 421 Co-Authors: Peter Richtárik, Eduard Gorbunov

3 - Active Metric Learning for Supervised Classification Speaker: Dimitri Papageorgiou, ExxonMobil, US, talk 1275 Co-Authors: Krishnan Kumaran, Martin Takáč

Machine learning and sparse optimisation (Continuous Optimization) NLP - Tu 8:30am-10:30am, Format: 4x30 min
Room: Salle 05 Building: Q, 1st floor, Zone: 11
Invited Session 109
Organizer: Coralia Cartis, University of Oxford, GB

1 - Condition numbers and weak average-case complexity in optimization
Speaker: Martin Lotz, The University of Manchester, GB, talk 957

Co-Authors: Dennis Amelunxen, Jake Walvin

2 - A Long (Random) Walk Solves All Your (Linear) Problems
Speaker: Armin Eftekhari, Alan Turing Institute, GB, talk 1199
Co-Authors: Martin Lotz, Peter Richtárik

3 - Manifold lifting: problems and methods
Speaker: Florentin Goyens, Oxford University, GB, talk 1182
Co-Authors: Armin Eftekhari, Coralia Cartis, Greg Ongie

4 - Sparse non-negative super-resolution: simplified and stabilized Speaker: Jared Tanner, University of Oxford, GB, talk 1462 Co-Authors: Armin Eftekhari, Andrew Thompson, Bogdan Toader, Hemant Tyagi

Recent advances in first-order algorithms for non-smooth optimization (Continuous Optimization) NonSmooth - We 8:30am-10:30am, Format: 4x30 min
Room: Salle LC4 Building: L, Intermediate 1, Zone: 9
Invited Session 198 Organizer: Thomas Pock, Graz University of Technology, AT

1 - Non-smooth Non-convex Bregman Minimization: Unification and New Algorithms Speaker: Peter Ochs, Saarland University, DE, talk 134 Co-Authors: Jalal Fadili

2 - Primal-dual algorithm for linearly constrained optimization problem Speaker: Yura Malitsky, Univeristy of Göttingen, DE, talk 155

3 - Stochastic PDHG with Arbitrary Sampling and Applications to Medical Imaging Speaker: Matthias Ehrhardt, University of Cambridge, GB, talk 127 Co-Authors: Carola Schoenlieb, Peter Richtárik, Antonin Chambolle

4 - Acceleration and global convergence of the NL- PDHGM Speaker: Stanislav Mazurenko, University of Liverpool, GB, talk 1655 Co-Authors: Tuomo Valkonen, C. Clason

Fast Converging Stochastic Optimization Algorithms (Continuous Optimization) RandomM - We 3:15pm-4:45pm, Format: 3x30 min
Room: Salle KC6 Building: K, Intermediate 1, Zone: 10
Invited Session 213 Organizer: Francis Bach, INRIA - ENS, FR

1 - Bridging the Gap between Constant Step Size SGD and Markov Chains Speaker: Aymeric Dieuleveut, EPFL, CH, talk 1102 Co-Authors: Alain Durmus, Francis Bach

2 - Stochastic Optimization for Large Scale Optimal Transport Speaker: Aude Genevay, ENS, FR, talk 888

3 - Variance Reduced Methods via Sketching  Speaker: Robert Gower, Telecom Paristech, FR, talk 859 Co-Authors: Peter Richtárik, Francis Bach

Non-Convex and Second-order Methods in Machine Learning (Continuous Optimization) RandomM - We 5:00pm-6:30pm, Format: 4x20 min
Room: Salle KC6 Building: K, Intermediate 1, Zone: 10
Invited Session 33 Organizer: Martin Takáč, Lehigh University, US

1 - Escaping Saddles with Stochastic Algorithms Speaker: Aurelien Lucchi, ETH Zurich, CH, talk 531

2 - Convergence Rate of Expectation-Maximization Speaker: Reza Babanezhad, UBC, CA, talk 1135 Co-Authors: Raunak Kumar, Mark Schmidt

3 - Parameter-free nonsmooth convex stochastic optimization through coin betting Speaker: Francesco Orabona, Stony Brook University, US, talk 1108

4 - SGD and Hogwild! Convergence Without the Bounded Gradients Assumption Speaker: Martin Takáč, Lehigh University, US, talk 1342 Co-Authors: Lam Nguyen, Phuong Nguyen, Marten van Dijk, Peter Richtárik, Katya Scheinberg

First-order methods for large-scale convex problems (Specific Models, Algorithms, and Software Learning) Th 8:30am-10:30am, Format: 4x30 min
Room: FABRE Building: J, Ground Floor, Zone: 8

Invited Session 316
Organizer: Stephen Vavasis, University of Waterloo, CA

1 - A single potential governing convergence of CG, AG and geometric descent Speaker: Stephen Vavasis, University of Waterloo, CA, talk 582 Co-Authors: Sahar Karimi

2 - Robust Accelerated Gradient Method Speaker: Mert Gurbuzbalaban, Rutgers University, US, talk 1106

3 - Randomized methods for convex feasibility problems and applications to ML Speaker: Peter Richtárik, KAUST, SA, talk 385 Co-Authors: Ion Necoara, Andrei Patrascu

4 - Bregman Divergence for Stochastic Variance Reduction Speaker: Yaoliang Yu, University of Waterloo, CA, talk 937 Co-Authors: Xinhua Zhang, Zhan Shi

Recent Advances in Coordinate Descent and Constrained Problems (Continuous Optimization)
RandomM - Fr 9:00am-10:30am, Format: 3x30 min Room: Salle KC6 Building: K, Intermediate 1, Zone: 10 Invited Session 208 Organizer: Ion Necoara, Univ. Politehnica Bucharest, RO

1 - Convergence Analysis of Inexact Randomized Iterative Methods Speaker: Nicolas Loizou, University of Edinburgh, GB, talk 835 Co-Authors: Peter Richtárik

2 - A Stochastic Penalty Model for Optimization with Many Convex Constraints Speaker: Konstantin Mishchenko, KAUST, SA, talk 1264 Co-Authors: Peter Richtárik, Ion Necoara

3 - Random coordinate descent methods for linearly constrained convex optimization Speaker: Ion Necoara, Univ. Politehnica Bucharest, RO, talk 809 Co-Authors: Martin Takáč

### June 29, 2018

Best DS3 Poster Award for Samuel Horváth

Samuel Horváth attended the Data Science Summer School (DS3), which took place during June 25-29, 2018, at École Polytechnique in Paris, France. Based on the event website, the event gathered 500 participants from 34 countries and 6 continents, out of which 290 were MS and PhD students and postdocs, and 110 professionals. Selected guest/speaker names (out of 41): Cédric Villani, Nicoló Cesa-Bianchi, Mark Girolami, Yann Lecun, Suvrit Sra, Jean-Philippe Vert, Adrian Weller, Marco Cuturi, Arthur Gretton, and Andreas Krause.

The event also included a best-poster competition, with an impressive total of 170 posters. Samuel's poster won the Best DS3 Poster Award. The poster, entitled Nonconvex variance reduced optimization with arbitrary sampling, is based on a paper of the same title, joint work with me, and currently under review.

Here is the poster:

And here is the award:

This first prize carries a 500 EUR cash award.

Samuel: Congratulations!!!

### June 18, 2018

I am now in Edinburgh for a week. On Tuesday, I am giving a talk in the ANC Seminar (School of Informatics), and on Wednesday I am giving the same talk in the ERGO Seminar (School of Mathematics).

### June 15, 2018

New paper out: "Improving SAGA via a probabilistic interpolation with gradient descent" - joint work with Adel Bibi, Alibek Sailanbayev, Bernard Ghanem and Robert Mansel Gower.

Abstract: We develop and analyze a new algorithm for empirical risk minimization, which is the key paradigm for training supervised machine learning models. Our method---SAGD---is based on a probabilistic interpolation of SAGA and gradient descent (GD). In particular, in each iteration we take a gradient step with probability $q$ and a SAGA step with probability $1−q$. We show that, surprisingly, the total expected complexity of the method (which is obtained by multiplying the number of iterations by the expected number of gradients computed in each iteration) is minimized for a non-trivial probability $q$. For example, for a well conditioned problem the choice $q=1/(n−1)^2$, where $n$ is the number of data samples, gives a method with an overall complexity which is better than both the complexity of GD and SAGA. We further generalize the results to a probabilistic interpolation of SAGA and minibatch SAGA, which allows us to compute both the optimal probability and the optimal minibatch size. While the theoretical improvement may not be large, the practical improvement is robustly present across all synthetic and real data we tested for, and can be substantial. Our theoretical results suggest that for this optimal minibatch size our method achieves linear speedup in minibatch size, which is of key practical importance as minibatch implementations are used to train machine learning models in practice. This is the first time linear speedup in minibatch size is obtained for a variance reduced gradient-type method by directly solving the primal empirical risk minimization problem.

### June 10, 2018

I am in Voronovo, Russia, attending the Traditional Youth School in "Control, Information and Optimization" organized by Boris Polyak  and Elena Gryazina. This is the 10th edition of the school. I will be teaching a 3h module on stochastic methods in optimization and machine learning.

Update 1: Slides from my two talks: TALK 1, TALK 2.

Update 2: Nikita Doikov won the Best Talk Award for the paper "Randomized Block Cubic Newton Method", to appear in ICML 2018.

### June 1, 2018

Jingwei Liang (Cambridge) is visiting me at KAUST.

### May 21, 2018

Adil Salim (Télécom ParisTech) is visiting me at KAUST this week.

### May 21, 2018

New paper out: "A nonconvex projection method for robust PCA" - joint work with Aritra Dutta and Filip Hanzely.

Abstract: Robust principal component analysis (RPCA) is a well-studied problem with the goal of decomposing a matrix into the sum of low-rank and sparse components. In this paper, we propose a nonconvex feasibility reformulation of RPCA problem and apply an alternating projection method to solve it. To the best of our knowledge, we are the first to propose a method that solves RPCA problem without considering any objective function, convex relaxation, or surrogate convex constraints. We demonstrate through extensive numerical experiments on a variety of applications, including shadow removal, background estimation, face detection, and galaxy evolution, that our approach matches and often significantly outperforms current state-of-the-art in various ways.

### May 19, 2018

The NIPS deadline is over now. Me and my group members will probably spend a few days sleeping...

### May 11, 2018

We have got two papers accepted to ICML 2018:

1) Randomized block cubic Newton method (with Nikita Doikov)

2) SGD and Hogwild! convergence without the bounded gradients assumption (with Lam M. Nguyen, Phuong Ha Nguyen, Marten van Dijk, Katya Scheinberg and Martin Takáč)

### May 1, 2018

New paper out: "Stochastic quasi-gradient methods: variance reduction via Jacobian sketching" - joint work with Robert Gower and Francis Bach.

Abstract: We develop a new family of variance reduced stochastic gradient descent methods for minimizing the average of a very large number of smooth functions. Our method---JacSketch---is motivated by novel developments in randomized numerical linear algebra, and operates by maintaining a stochastic estimate of a Jacobian matrix composed of the gradients of individual functions. In each iteration, JacSketch efficiently updates the Jacobian matrix by first obtaining a random linear measurement of the true Jacobian through (cheap) sketching, and then projecting the previous estimate onto the solution space of a linear matrix equation whose solutions are consistent with the measurement. The Jacobian estimate is then used to compute a variance-reduced unbiased estimator of the gradient, followed by a stochastic gradient descent step. Our strategy is analogous to the way quasi-Newton methods maintain an estimate of the Hessian, and hence our method can be seen as a stochastic quasi-gradient method. Indeed, quasi-Newton methods project the current Hessian estimate onto a solution space of a linear equation consistent with a certain linear (but non-random) measurement of the true Hessian. Our method can also be seen as stochastic gradient descent applied to a controlled stochastic optimization reformulation of the original problem, where the control comes from the Jacobian estimates.

We prove that for smooth and strongly convex functions, JacSketch converges linearly with a meaningful rate dictated by a single  convergence theorem which applies to general sketches. We also provide a refined convergence theorem which applies to a smaller class of sketches, featuring a novel proof technique based on a stochastic Lyapunov function. This enables us to obtain sharper complexity results for variants of JacSketch with importance sampling. By specializing our general approach to specific  sketching strategies, JacSketch reduces to the celebrated stochastic average gradient (SAGA) method, and its several existing and many new minibatch, reduced memory, and importance sampling variants. Our rate for SAGA with importance sampling is the current best-known rate for this method, resolving a conjecture by Schmidt et al (2015). The rates we obtain for minibatch SAGA are also superior to existing rates. Moreover, we obtain the first minibatch SAGA method with importance sampling.

### April 29, 2018

I am on my way to Birmingham, and then Coventry. I will be giving a talk at the DIMAP seminar (DIMAP = Centre for Discrete Mathematics and its Applications), University of Warwick, on "Stochastic Quasi-Gradient Methods: Variance Reduction via Jacobian Sketching". The talk is based on joint work with Robert M. Gower and Francis Bach.

### April 25, 2018

Today and tomorrow I am teaching a mini-course on "Optimization for Machine Learning" for students from various Saudi universities who were previously contestants in Saudi National Mathematical Olympiad and/or IMO. Several current contestants are attending as well.

This is a collaborative effort with Diogo Gomes, who is teaching a "Mathematica" mini-course.

### April 18, 2018

New paper out: "Weighted low-rank approximation of matrices and background modeling" - joint work with Aritra Dutta and Xin Li.

Abstract: We primarily study a special a weighted low-rank approximation of matrices and then apply it to solve the background modeling problem. We propose two algorithms for this purpose: one operates in the batch mode on the entire data and the other one operates in the batch-incremental mode on the data and naturally captures more background variations and computationally more effective. Moreover, we propose a robust technique that learns the background frame indices from the data and does not require any training frames. We demonstrate through extensive experiments that by inserting a simple weight in the Frobenius norm, it can be made robust to the outliers similar to the L1 norm. Our methods match or outperform several state-of-the-art online and batch background modeling methods in virtually all quantitative and qualitative measures.

### April 16, 2018

I am giving a talk today at the CS Graduate Seminar at KAUST. I will be talking about "Randomized Methods for Convex Feasibility Problems". This is joint work with Ion Necoara and Andrei Patrascu.

### April 5, 2018

My lab has openings for postdoc (straight after PhD, or a few years after PhD) and research scientist (several to many years after PhD; similar to a RS position at big data companies such as Google, Microsoft Research, Amazon, Baidu, Tencent, Facebook) positions.

Relevant areas: machine learning theory, optimization, algorithms, high performance computing, deep learning, randomized and stochastic algorithms, federated learning, computer vision, machine learning systems, data science, applied mathematics, theoretical computer science. Contact me by email if interested. Please send your CV (including publication record), a brief statement of interest, 3 reference letters (and PhD transcript for postdoc applicants).

Place of work: KAUST. Outstanding working conditions.

Starting date: Fall 2018 (flexible).

Contract duration: based on agreement (e.g., 1-3 years).

### April 2, 2018

Dominik Csiba's PhD thesis "Data sampling strategies in stochastic algorithms for empirical risk minimization" is online now.

### March 2, 2018

I am on vacation this week.

### March 23, 2018

Konstantin and Filip are attending the 2018 NFORMS Optimization Society Conference in Denver, Colorado.

### March 21, 2018

New paper out: "Fastest rates for stochastic mirror descent methods" - joint work with Filip Hanzely.

Abstract: Relative smoothness - a notion introduced by Birnbaum et al. (2011) and rediscovered by Bauschke et al. (2016) and Lu et al. (2016) - generalizes the standard notion of smoothness typically used in the analysis of gradient type methods. In this work we are taking ideas from well studied field of stochastic convex optimization and using them in order to obtain faster algorithms for minimizing relatively smooth functions. We propose and analyze two new algorithms: Relative Randomized Coordinate Descent (relRCD) and Relative Stochastic Gradient Descent (relSGD), both generalizing famous algorithms in the standard smooth setting. The methods we propose can be in fact seen as variants of stochastic mirror descent. One of them, relRCD is the first stochastic mirror descent algorithm with a linear convergence rate.

### March 18, 2018

Sarah Sachs, a master student from Technical University Munich (TUM), arrived at KAUST today. She will spend six months at KAUST (until early September) as a visiting student in my group, and will write her master's thesis under my supervision. In her thesis she is focusing on randomized optimization algorithms. Welcome!

Sarah's bachelor thesis at TUM focused on approximation of the infimal convolution for non-convex functions. She  previously worked on finding efficiently computable stopping criteria for ADMM and the Chambolle-Pock algorithm applied to LP relaxations of ILPs with integral extreme points. She is generally interested in optimization with applications to computer vision.

### March 4, 2018

Konstantin Mishchenko is visiting the Cambridge Image Analysis group of Carola-Bibiane Schönlieb at the University of Cambridge. During March 8-11 he is participating in VHacks, the first ever hackathon at the Vatican.

Aritra and El Houcine are also travelling.

Update (March 19): Konstantin, El Houcine and Aritra are back.

### February 25, 2018

I am on my way to Bratislava, Slovakia. Tomorrow, I am giving a statistics seminar talk at "Matfyz" - School of Mathematics, Physics and Informatics, Comenius University.

Title: On stochastic algorithms in linear algebra, optimization and machine learning
Place: FMFI UK, M/XII
Date: Monday, February 26, 2018
Time: 09:50am

If anyone is interested in MS / PhD / postdocs / research scientist positions at KAUST, I will be available to talk to you after the talk.

### February 20, 2018

Videos of the talks from the KAUST Research Workshop on Optimization and Big Data are now available. They can be found here.

Comment:
At the moment the videos are accessible to KAUST community only, they will soon be available globally.

### February 13, 2018

New paper out: "SGD and Hogwild! convergence without the bounded gradients assumption" - joint work with Lam M. Nguyen, Phuong Ha Nguyen, Marten van Dijk, Katya Scheinberg and Martin Takáč.

Abstract: Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical analysis of convergence of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al., 2016) a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds. Moreover, we propose an alternative convergence analysis of SGD with diminishing learning rate regime, which is results in more relaxed conditions that those in (Bottou et al., 2016). We then move on the asynchronous parallel setting, and prove convergence of the Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.

### February 12, 2018

New paper out: "Accelerated stochastic matrix inversion: general theory and speeding up BFGS rules for faster second-order optimization" - joint work with Robert M. Gower, Filip Hanzely and Sebastian Stich.

Abstract: We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the first accelerated (deterministic and stochastic) quasi-Newton updates. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.

### February 10, 2018

New paper out: "Randomized block cubic Newton method" - joint work with Nikita Doikov.

Abstract: We study the problem of minimizing the sum of three convex functions: a differentiable, twice-differentiable and a non-smooth term in a high dimensional setting. To this effect we propose and analyze  a randomized block  cubic Newton (RBCN) method, which in each iteration builds a model of the objective function formed as the sum of the {\em natural} models of its three components: a linear model with a quadratic regularizer for the differentiable term, a quadratic model with a cubic regularizer for the twice differentiable term, and perfect (proximal)  model for the nonsmooth term. Our method in each iteration minimizes the model over a random subset of  blocks of the search variable. RBCN is the first algorithm with these properties, generalizing several existing methods, matching the best known bounds in all special cases. We establish ${\cal O}(1/\epsilon)$, ${\cal O}(1/\sqrt{\epsilon})$ and ${\cal O}(\log (1/\epsilon))$ rates under different assumptions on the component functions. Lastly, we show numerically that our method outperforms the state-of-the-art on a variety of machine learning problems, including cubically regularized least-squares, logistic regression with constraints, and Poisson regression.

### February 10, 2018

New paper out: "Stochastic spectral and conjugate descent methods" - joint work with Dmitry Kovalev, Eduard Gorbunov and Elnur Gasanov.

Abstract: The state-of-the-art methods for solving optimization problems in big dimensions are variants of randomized coordinate descent (RCD). In this paper we introduce a fundamentally new type of acceleration strategy for RCD based on the augmentation of the set of coordinate directions by a few spectral or conjugate directions. As we increase the number of extra directions to be sampled from, the rate of the method improves, and interpolates between the linear rate of RCD and a linear rate independent of the condition number. We develop and analyze also inexact variants of these methods where the spectral and conjugate directions are allowed to be approximate only. We motivate the above development by proving several negative results which highlight the limitations of RCD with importance sampling.

### February 5, 2018

OBD 2018 is starting! The KAUST Workshop on Optimization and Big Data just started. We have 19 amazing speakers and 21 deluxe e-posters lined up.

Update (February 12): Thanks for all who participated in the workshop, thanks you to this was an excellent event! Group photos:

### February 4, 2018

KAUST Research Workshop on Optimization and Big Data is starting tomorrow! We have 19 amazing speakers, and 21 deluxe poster talks and ePoster presentations.

This year, Tamás Terlaky (Lehigh) is the keynote speaker.

Thanks to the KAUST Office for Sponsored Research, The Alan Turing Institute and KICP.

### February 1, 2018

Nicolas Loizou is back at KAUST on a research visit. Welcome!

### January 28, 2018

Aritra Dutta (postdoc), Alibek Sailanbayev (MS/PhD student) and Samuel Horvath (MS/PhD student) are attending Applied Machine Learning Days at EPFL, Lausanne, Switzerland.

### January 27, 2018

Let me welcome Dmitry Kovalev and Elnur Gasanov (master students visiting from MIPT, Moscow) and Slavomir Hanzely (undergraduate student at Comenius University), who arrived at KAUST about a week ago and are working with me as interns.They will be here for about a month.

### January 18, 2018

New paper out: "A randomized exchange algorithm for computing optimal approximate designs of experiments" - joint work with Radoslav Harman and Lenka Filová.

Abstract: We propose a class of subspace ascent methods for computing optimal approximate designs that covers both existing as well as new and more efficient algorithms. Within this class of methods, we construct a simple, randomized exchange algorithm (REX). Numerical comparisons suggest that the performance of REX is comparable or superior to the performance of state-of-the-art methods across a broad range of problem structures and sizes. We focus on the most commonly used criterion of D-optimality that also has applications beyond experimental design, such as the construction of the minimum volume ellipsoid containing a given set of datapoints. For D-optimality, we prove that the proposed algorithm converges to the optimum. We also provide formulas for the optimal exchange of weights in the case of the criterion of A-optimality. These formulas enable one to use REX for computing A-optimal and I-optimal designs.

### January 16, 2018

I was travelling and am back at KAUST now. Let me welcome Eduard Gorbunov (a master's student visiting from MIPT, Moscow; will be here untill Feb 8), Matthias Ehrhardt (visiting from Cambridge, UK, untill February 10) and Elhoucine Bergou (new postdoc in my group, starting today).

### January 15, 2018

New paper out: "Randomized projection methods for convex feasibility problems: conditioning and convergence rates" - joint work with Ion Necoara and Andrei Patrascu.

Abstract: Finding a point in the intersection of a collection of closed convex sets, that is the convex feasibility problem, represents the main modeling strategy for many computational problems. In this paper we analyze new stochastic reformulations of the convex feasibility problem in order to facilitate the development of new algorithmic schemes. We also analyze the conditioning problem parameters using certain (linear) regularity assumptions on the individual convex sets. Then, we introduce a general random projection algorithmic framework, which extends to the random settings many existing projection schemes, designed for the general convex feasibility problem. Our general random projection algorithm allows to project simultaneously on several sets, thus providing great flexibility in matching the implementation of the algorithm on the parallel architecture at hand. Based on the conditioning parameters, besides the asymptotic convergence results, we also derive explicit sublinear and linear convergence rates for this general algorithmic framework.

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