# Distributed Saddle-Point Problems: Lower Bounds, Near-Optimal and Robust Algorithms

@inproceedings{Beznosikov2020DistributedSP, title={Distributed Saddle-Point Problems: Lower Bounds, Near-Optimal and Robust Algorithms}, author={Aleksandr Beznosikov and Valentin Samokhin and Alexander V. Gasnikov}, year={2020} }

This paper focuses on the distributed optimization of stochastic saddle point problems. The first part of the paper is devoted to lower bounds for the cenralized and decentralized distributed methods for smooth (strongly) convex-(strongly) concave saddle-point problems as well as the near-optimal algorithms by which these bounds are achieved. Next, we present a new federated algorithm for cenralized distributed saddle point problems – Extra Step Local SGD. Theoretical analysis of the new method…

## 9 Citations

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## References

SHOWING 1-10 OF 57 REFERENCES

### Decentralized Distributed Optimization for Saddle Point Problems

- Computer Science, MathematicsArXiv
- 2021

The work of the proposed algorithm is illustrated on the prominent problem of computing Wasserstein barycenters (WB), where a non-euclidean proximal setup arises naturally in a bilinear saddle point reformulation of the WB problem.

### A Decentralized Proximal Point-type Method for Saddle Point Problems

- Computer Science, MathematicsArXiv
- 2019

This paper proposes a decentralized variant of the proximal point method that is the first decentralized algorithm with theoretical guarantees for solving a non-convexnon-concave decentralized saddle point problem and the numerical results for training a general adversarial network in a decentralized manner match the theoretical guarantees.

### A decentralized algorithm for large scale min-max problems

- Computer Science2020 59th IEEE Conference on Decision and Control (CDC)
- 2020

This work proposes a decentralized algorithm based on the Extragradient method, whose centralized implementation has been shown to achieve good performance on a wide range of min-max problems, and shows that the proposed method achieves linear convergence under suitable assumptions.

### Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems

- Computer Science, MathematicsMath. Program.
- 2021

It is proved that for strongly convex problems, O (1/t^2) is the best possible convergence rate, while it is known that gradient methods can have linear convergence on unconstrained problems.

### Multi-consensus Decentralized Accelerated Gradient Descent

- Computer ScienceArXiv
- 2020

A novel algorithm is proposed that can achieve near optimal communication complexity, matching the known lower bound up to a logarithmic factor of the condition number of the problem.

### Optimal Algorithms for Smooth and Strongly Convex Distributed Optimization in Networks

- Computer ScienceICML
- 2017

The efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression is verified.

### Lectures on modern convex optimization - analysis, algorithms, and engineering applications

- Computer ScienceMPS-SIAM series on optimization
- 2001

The authors present the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming as well as their numerous applications in engineering.

### Efficient Algorithms for Federated Saddle Point Optimization

- Computer ScienceArXiv
- 2021

This work designs an algorithm that can harness the benefit of similarity in the clients while recovering the Minibatch Mirror-prox performance under arbitrary heterogeneity (up to log factors) and gives the first federated minimax optimization algorithm that achieves this goal.

### Distributed Subgradient Methods for Multi-Agent Optimization

- Mathematics, Computer ScienceIEEE Transactions on Automatic Control
- 2009

The authors' convergence rate results explicitly characterize the tradeoff between a desired accuracy of the generated approximate optimal solutions and the number of iterations needed to achieve the accuracy.

### Solving variational inequalities with Stochastic Mirror-Prox algorithm

- Mathematics, Computer Science
- 2008

A novel Stochastic Mirror-Prox algorithm is developed for solving s.v.i. variational inequalities with monotone operators and it is shown that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters.