### January 14, 2019

ICML deadline is upon us (on Jan 23)... Everyone in my group is working hard towards the deadline.

### January 10, 2019

I've been asked to lead an Aritificial Intelligence Committee at KAUST whose role is to prepare a strategic plan for growing AI research and activities at KAUST over the next 5 years. This will be a substantial investment, and will involve a large number of new faculty, research scientist, postdoc and PhD and MS/PhD positions; investment into computing infrastructure and more. (The committee started its work in 2018; I am positing the news with some delay...)

Independently to this, Bernard Ghanem, Marco Canini, Panos Kalnis and me have established the Machine Learning Hub at KAUST, with the aim to advance ML research and training activities for the benefit of the entire KAUST community. The website is only visible from within the KAUST network at the moment.

### January 6, 2019

I am back at KAUST. El Houcine, Konstantin and Xun are here. Aritra is on his way to WACV 2019, Hawaii. Samuel and Filip will come back tomorrow. Alibek and Elnur are arriving soon, too.

I will have several interns/research visitors from my group at MIPT visiting me at KAUST during January-February:

- Egor Shulgin (Jan 6 - Feb 21)

- Dmitry Kamzolov (Jan 10 - Feb 18)

- Vladislav Elsukov (Jan 11 - Feb 15)

- Eduard Gorbunov (Jan 13 - Feb 24)

- Igor Sokolov (Jan 18 - Feb 25)

### January 3, 2019

I am visiting Radoslav Harman @ Comenius University, Slovakia.

### December 22, 2018

I am on vacation until the end of the year.

### December 22, 2018

The paper "Accelerated coordinate descent with arbitrary sampling and best rates for minibatches", coauthored with Filip Hanzely, was accepted to the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS 2019). The conference will take place in Naha, Okinawa, Japan, during April 16-18, 2019. The acceptance email said:

*"There were 1,111 submissions for AISTATS this year, of which the program committee accepted 360 for presentation at the conference; among these, 28 papers were accepted for oral presentation, and 332 for poster presentation."*

### December 22, 2018

I have accepted an invite to deliver half-a-day worth of summer school lectures on optimization in machine learning at the International Conference on Continuous Optimization (ICCOPT 2019). The Summer School and the main conference take place in Berlin in August 2019. The Summer School precedes the main event, and spans two days: August 3-4. The main conference runs from August 5 until August 8.

ICCOPT is the flagship conference series of the Mathematical Optimization Society (MOS) on continuous optimization, covering a wide range of topics in the field. The individual conferences are typically held once every three years. The last three editions of the conference took place in Tokyo, Japan (2016), Lisbon, Portugal (2013), and Santiago, Chile (2010). I attended all three.

There are two more key conferences in optimization that take place once in three years; each runs in a differenty year, so that one takes place every year. They are: ISMP (International Symposium on Mathematical Programming) and OP (SIAM Conference on Optimization). The last ISMP took place in Bordeaux in Summer 2018. The next OP conference will be in Hong Kong during May 26-29, 2020. I am a member of the organizing committee for OP2020 which is collectively responsible for the selection of invited plenary and tutorial speakers, summer school lecturers, and the organization of mini-symposia.

### December 14, 2018

Alibek and Samuel received their MS degrees today. Congratulations! Both will continue as PhD students in my group as of January 2019.

Earlier today, I had the great pleasure and honor to meet with Kai-Fu Lee (CEO of Sinovation Ventures; former president of Google China; founder & former managing director of Microsoft Research Asia) for a 2hr discussion about AI. I recommend that you watch some of his videos

TED Talk 2018: How AI Can Save Humanity

'AI Superpowers': A Conversation With Kai-Fu Lee

The Future of AI with Kai-Fu Lee: Udacity Talks

The Race for AI: Book Talk with Dr. Kai-Fu Lee

and read his most recent book:

AI Superpowers: China, Silicon Valley and the New World Order

### December 11, 2018

Konstantin and Filip are back (from Amazon internship / Microsoft Research visit, respectively). They stopped by NeurIPS on their way back.

### December 10, 2018

The final exam for CS 390FF course is today. Robert Gower arrived at KAUST for a research visit; he will stay until December 20.

### December 8, 2018

I am back at KAUST now.

### December 2, 2018

I have arrived in Montréal to attend the NeurIPS (formerly known as NIPS) conference. I was welcome with rain, which this is a good thing as far as I am concerned!. Tutorials are starting tomorrow; after that we have three days of the main conference and then two days of workshops. My group is presening three papers accepted to the main conference (paper SEGA, ASBFGS and SSCD) and one paper accepted to a workshop.

I am using the conference Whova app; feel free to get in touch! I am leaving on Thursday evening, so catch me before then... I've posted a few job openings we have at KAUST through the app: internships in my lab (apply by sending me your cv and transcript of university grades), postdoc and research scientist positions (apply by sending a cv + motivation letter), and machine learning faculty positions at all ranks (women and junior applicants are particularly encouraged to apply).

### November 30, 2018

New paper out: "New convergence aspects of stochastic gradient algorithms" - joint work with Lam M. Nguyen, Phuong Ha Nguyen, Katya Scheinberg, Martin Takáč, and Marten van Dijk.

Abstract:

*The classical convergence analysis 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 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. We show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime, which results in more relaxed conditions than those in Bottou et al. (2016). We then move on the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime in the case of diminished learning rate. It is well-known that SGD converges if a sequence of learning rates*

{ηt}

*satisfies*

∑∞t=0ηt→∞

*and*

∑∞t=0η2t<∞

*. We show the convergence of SGD for strongly convex objective function without using bounded gradient assumption when*

{ηt}

*is a diminishing sequence and*

∑∞t=0ηt→∞

*. In other words, we extend the current state-of-the-art class of learning rates satisfying the convergence of SGD.*

### November 28, 2018

Nicolas Loizou is on the job market; he will get is PhD in 2019. He is looking for research positions in academia (Assistant Prof / postdoc) and industry (Research Scientist). Nicolas will be at NeurIPS next week, presenting his work on privacy-preserving randomized gossip algorithms in the PPML workshop. At the moment, Nicolas is interning at Facebook AI Research (FAIR), where he has done some great work on decentralized training of deep learning models, and on accelerated decentralized gossip communication protocols.

### November 22, 2018

Here are the posters of our papers accepted to this year's NeurIPS:

[paper on arXiv]

[paper on arXiv]

[paper on arXiv]

The poster for our Privacy Preserving Machine Learning NeurIPS workshop paper was not finalized yet. I will include a link here once it is ready. Update (November 28): The poster is now ready:

[full-length paper on arXiv]

### November 18, 2018

Xun QIAN just joined my group at KAUST as a postdoc. He has a PhD in Mathematics (August 2017) from Hong Kong Baptist University. His PhD thesis is on "Continuous methods for convex programming and convex semidefinite programming" (pdf), supervised by Li-Zhi Liao.

Some of Xun's papers:

H. W. Yue, Li-Zhi Liao, and Xun Qian. Two interior point continuous trajectory models for convex quadratic programming with bound constraints, to appear in

*Pacific Journal on Optimization*

Xun Qian, Li-Zhi Liao, Jie Sun and Hong Zhu. The convergent generalized central paths for linearly constrained convex programming,

*SIAM Journal on Optimization*28(2):1183-1204, 2018

Xun Qian and Li-Zhi Liao. Analysis of the primal affine scaling continuous trajectory for convex programming,

*Pacific Journal on Optimization*14(2):261-272, 2018

Xun Qian and Li-Zhi Liao and Jie Sun. Analysis of some interior point continuous trajectories for convex programming,

*Optimization*66(4):589-608, 2017

### November 16, 2018

Nicolas Loizou is giving a talk today at Mila, University of Montréal. He is speaking about "Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods".

### November 13, 2018

Nicolas Loizou is giving a talk today in the Mathematics in Machine Learning Seminar at McGill University. He is speaking about "Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods", joint work with me.

### November 12, 2018

Today I am giving a talk at the Statistics and Data Science Workshop held here at KAUST. I am speaking about the JacSketch paper. Here is a YouTube video of the same talk, one I gave in September at the Simons Institute.

### November 4, 2018

A paper accepted to IEEE Winter Conference on Applications of Computer Vision (WACV 2019)

The paper "Online and batch incremental video background estimation", joint work with Aritra Dutta, has just been accepted to the WACV 2019. The conference will take place during January 7-January 11, 2019 in Honolulu, Hawaii.

### November 4, 2018

I am back from annual leave.

### November 3, 2018

A paper accepted to NIPS Workshop on Privacy-Preserving Machine Learning (PPML18)

The paper "A Privacy Preserving Randomized Gossip Algorithm via Controlled Noise Insertion", joint work with Nicolas Loizou, Filip Hanzely, Jakub Konečný and Dmitry Grishchenko, has been accepted to the NIPS Workshop on Privacy-Preserving Machine Learning. The full-length paper, which includes a number of additional algorithms and results, can be found on arXiv here.

The acceptance email said: "We received an astonishing number of high quality submissions to the Privacy Preserving Machine Learning workshop and we are delighted to inform you that your submission

*A Privacy Preserving Randomized Gossip Algorithm via Controlled Noise Insertion*(57) was accepted to be presented at the workshop."

### November 1, 2018

New paper out: "A stochastic penalty model for convex and nonconvex optimization with big constraints" - joint work with Konstantin Mishchenko.

Abstract:

*The last decade witnessed a rise in the importance of supervised learning applications involving*

__big data__and__big models__. Big data refers to situations where the amounts of training data available and needed causes difficulties in the training phase of the pipeline. Big model refers to situations where large dimensional and over-parameterized models are needed for the application at hand. Both of these phenomena lead to a dramatic increase in research activity aimed at taming the issues via the design of new sophisticated optimization algorithms. In this paper we turn attention to the__big constraints__scenario and argue that elaborate machine learning systems of the future will necessarily need to account for a large number of real-world constraints, which will need to be incorporated in the training process. This line of work is largely unexplored, and provides ample opportunities for future work and applications. To handle the big constraints regime, we propose a stochastic penalty formulation which reduces the problem to the well understood big data regime. Our formulation has many interesting properties which relate it to the original problem in various ways, with mathematical guarantees. We give a number of results specialized to nonconvex loss functions, smooth convex functions, strongly convex functions and convex constraints. We show through experiments that our approach can beat competing approaches by several orders of magnitude when a medium accuracy solution is required.### November 1, 2018

Aritra Dutta and El Houcine Bergou are on their way to Phoenix, Arizona, to give talks at the 2018 INFORMS Annual Meeting.

### October 31, 2018

New paper out: "Provably accelerated randomized gossip algorithms" - joint work with Nicolas Loizou and

Michael G. Rabbat.

Abstract:

*In this work we present novel provably accelerated gossip algorithms for solving the average consensus problem. The proposed protocols are inspired from the recently developed accelerated variants of the randomized Kaczmarz method - a popular method for solving linear systems. In each gossip iteration all nodes of the network update their values but only a pair of them exchange their private information. Numerical experiments on popular wireless sensor networks showing the benefits of our protocols are also presented.*

### October 31, 2018

A paper accepted to The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19)

The paper "A nonconvex projection method for robust PCA", joint work with Aritra Dutta and Filip Hanzely, has been accepted to AAAI-19. The conference will take place during January 27-February 1, 2019, in Honolulu, Hawaii, USA.

The acceptance email said: "We had a record number of over 7,700 submissions this year. Of those, 7,095 were reviewed, and due to space limitations we were only able to accept 1,150 papers, yielding an acceptance rate of 16.2%. There was especially stiff competition this year because of the number of submissions, and you should be proud of your success."

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.*

### October 30, 2018

A paper accepted to Journal of the American Statistical Association (JASA)

The paper "A randomized exchange algorithm for computing optimal approximate designs of experiments", joint work with Radoslav Harman and Lenka Filová, has been accepted to Journal of the American Statistical Association.

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 data-points. 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.*

### October 25, 2018

I am about to go on an annual leave to an island in the Indian ocean. I will likely have no functioning internet, and will not be reading my emails (maybe I'll read one or two *if* I get internet over there, but do not expect me to respond as the purpose of annual leave is to relax and recharge). I will be back at KAUST and operational on November 4, teaching at 9am.

### October 22, 2018

Sebastian Stich is visiting me at KAUST. He will stay here for three weeks, and will give a CS seminar talk on November 12.

### October 20, 2018

Filip Hanzely is visiting Lin Xiao at Microsoft Research in Redmond, Washington. He will be back roughly in a month. While in the US, he will also drop by Phoenix to give a talk at the 2018 INFORMS Annual Meeting.

### October 15, 2018

Congratulations to Filip Hanzely for receiving a NIPS Travel Award ($1,500). Filip is the coauthor of 2 papers accepted to NIPS this year:

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

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

### 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.

Submission deadline: January 23, 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 DS

^{3}Poster Award for Samuel Horváth

Samuel Horváth attended the Data Science Summer School (DS

^{3}), 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 DS

^{3}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!!!

Update (October 11, 2018): Here is a KAUST News article about this.

### 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

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

Organizer: Martin Takáč, Lehigh University, US

Co-Authors: Mert Gurbuzbalaban, Nuri Vanli, Pablo Parrilo

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

Speaker: Martin Lotz, The University of Manchester, GB, talk 957

Co-Authors: Dennis Amelunxen, Jake Walvin

Speaker: Armin Eftekhari, Alan Turing Institute, GB, talk 1199

Speaker: Florentin Goyens, Oxford University, GB, talk 1182

**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

Room: Salle LC4 Building: L, Intermediate 1, Zone: 9

**Invited Session 198**

**Organizer: Thomas Pock, Graz University of Technology, AT**

**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

Room: Salle KC6 Building: K, Intermediate 1, Zone: 10

**Invited Session 213**

**Organizer: Francis Bach, INRIA - ENS, FR**

**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

Room: Salle KC6 Building: K, Intermediate 1, Zone: 10

**Invited Session 33**

**Organizer: Martin Takáč, Lehigh University, US**

**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

Room: FABRE Building: J, Ground Floor, Zone: 8

Invited Session 316

Invited Session 316

**Organizer: Stephen Vavasis, University of Waterloo, CA**

**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

### June 29, 2018

Best DS

^{3}Poster Award for Samuel Horváth

Samuel Horváth attended the Data Science Summer School (DS

^{3}), 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 DS

^{3}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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