Original: Distributed Methods with Compressed Communication for Solving Variational Inequalities, with Theoretical Guarantees
Revised: Distributed Methods with Compressed Communication for Solving Variational Inequalities, with Theoretical Guarantees

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Variational inequalities in general and saddle point problems in particular are increasingly relevant in machine learning applications, including adversarial learning, GANs, transport and robust optimization.
Revised: Variational inequalities in general and saddle point problems in particular are increasingly relevant in machine learning applications, including adversarial learning, GANs, transport and robust optimization.

Original: With increasing data and problemsizes necessary to train high performing models across various applications, weneed to rely on parallel and distributed computing.
Revised: With increasing data and problemsizes necessary to train high performing models across various applications, weneed to rely on parallel and distributed computing.

Original: However, in distributed training, communication among the compute nodes is a key bottleneck during training, and this problem is exacerbated for high dimensional and over-parameterizedmodels.
Revised: However, in distributed training, communication among the compute nodes is a key bottleneck during training, and this problem is exacerbated for high dimensional and over-parameterizedmodels.

Original: Due to these considerations, it is important to equip existing methodswith strategies that would allow to reduce the volume of transmitted informationduring training while obtaining a model of comparable quality.
Revised: Due to these considerations, it is important to equip existing methodswith strategies that would allow to reduce the volume of transmitted informationduring training while obtaining a model of comparable quality.

Original: In this paper,we present the first theoretically grounded distributed methods for solving variational inequalities and saddle point problems using compressed communication:
Revised: In this paper,we present the ﬁrst theoretically grounded distributed methods for solving variational inequalities and saddle point problems using compressed communication:

Original: MASHA1 and MASHA2 .
Revised: MASHA1 and MASHA2 .

Original: Our theory and methods allow for the use of both unbiased (such as Rand k ; MASHA1 ) and contractive (such as Top k ; MASHA2 ) compressors.
Revised: Our theory and methods allow for the use of both unbiased (such as Rand k ; MASHA1 ) and contractive (such as Top k ; MASHA2 ) compressors.

Original: New algorithms support bidirectional compressions, and also can bemodified for stochastic setting with batches and for federated learning with partialparticipation of clients.
Revised: New algorithms support bidirectional compressions, and also can bemodiﬁed for stochastic setting with batches and for federated learning with partialparticipation of clients.

Original: We empirically validated our conclusions using two experimental setups: a standard bilinear min-max problem, and large-scale distributedadversarial training of transformers.
Revised: We empirically validated our conclusions using two experimental setups: a standard bilinear min-max problem and large-scale distributedadversarial training of transformers.

Original:  Introduction1.1 The expressive power of variational inequalities
Revised:  Introduction1.1 The expressive power of variational inequalities

Original: Due to their abstract mathematical nature and the associated flexibility they offer in modeling variouspractical problems of interests, variational inequalities (VI) have been an active area of research inapplied mathematics for more than half a century [64, 31, 22].
Revised: Due to their abstract mathematical nature and the associated ﬂexibility they offer in modeling variouspractical problems of interest, variational inequalities (VI) have been an active area of research inapplied mathematics for more than half a century [65, 31, 22].

Original: It is well known that VIs can be usedto formulate and study optimization problems, saddle point problems (SPPs) , games and fixed pointproblems, for example, in an elegant unifying mathematical framework [9].
Revised: It is well known that VIs can be usedto formulate and study optimization problems, saddle point problems (SPPs) , games and ﬁxed pointproblems in an elegant unifying mathematical framework [9]

Original: Recently, a series of works by various authors [15, 26, 57, 13, 48] built a bridge between VIs/SPPsand GANs [28].
Revised: Recently, a series of works by various authors [15, 26, 58, 13, 49] built a bridge between VIs/SPPsand GANs [28].

Original: This allows to successfully transfer established insights and well-known techniquesfrom the vast literature on VIs/SPPs, such as averaging and extrapolation to the study of GANs.
Revised: This allows transferring established insights and well-known techniques from thevast literature on VIs/SPPs, such as averaging and extrapolation, to the study of GANs.

Original: Be- sides their usefulness in studying GANs and alternative adversarial learning models [56], VIs/SPPs have recently attracted considerable attention of the machine learning community due to their ability to model other situations where the minimization of a single loss function does not suffice, such asauction theory [80], supervised learning with non-separable loss [39] or non-separable regularizer[7] and reinforcement learning [69, 65, 38]
Revised: Besidestheir usefulness in studying GANs and alternative adversarial learning models [57], VIs/SPPs haverecently attracted considerable attention of the machine learning community due to their abilityto model other situations where the minimization of a single loss function does not sufﬁce, such asauction theory [80], supervised learning with non-separable losses [39] or non-separable regularizers[7] and reinforcement learning [69, 66, 38]. to not

Original: In summary, VIs have recently become a potent tool enabling new advances in practical machinelearning situations reaching beyond supervised learning where optimization problems and techniques, which can be seen as special instances of VIs and methods for solving them, reign supreme.
Revised: In summary, variational inequalities have recently become a potent tool enabling new advances intheoretical and practical ML.

Original: 1.2 Training of supervised models via distributed optimization
Revised: 1.2 Training of supervised models via distributed optimization

Original: On the other hand, in the domain of classical, andhence also much better understood, supervisedmachine learning is characterized by the fact that standard optimization techniques apply and workwell, researchers and practitioners face other challenges that are currently beyond the reach of existing VI methods.
Revised: On the other hand, for classical and much better understood supervised machine learning or minimization problems, researchers and practitioners face other challenges, which, until recently, havebeen outside of research on VIs.

Original: Indeed,the trainingof modern supervised machine learning models in general, anddeep neural networks in particular, is still extremely challenging.
Revised: Indeed, training supervised machine learning models in general,and deep neural networks in particular, is still extremely challenging.

Original: Due to their desire to improve thegeneralization of deployed models, machine learning engineers need to rely on training datasets ofever increasing sizes and on elaborate over-parametrized models [5].
Revised: Due to their desire to improvethe generalization of deployed models, machine learning engineers need to rely on training datasetsof ever increasing sizes and on elaborate over-parametrized models [5].

Original: Supporting workloads of suchunprecedented magnitudes would be impossible without combining the latest advances in hardwareacceleration, distributed systems and distributed algorithm design [83].
Revised: Supporting workloads ofsuch unprecedented magnitudes would be impossible without combining the latest advances in hardware acceleration, distributed systems and distributed algorithm design [83].

Original: When training suchmodern supervised models in a distributed fashion,  communication cost is often the bottleneck of the training system, and for this reason, a lot of effort was recently targetedat the design of communication efficient distributed optimization methods [44, 76, 25, 29].
Revised: When training such models in a distributed fashion, the communication cost is often a bottleneckof the training system, and for this reason, a lot of effort was recently targeted at the design ofcommunication-efﬁcient distributed optimization methods [45, 76, 25, 29].

Original: A particularly successful technique for improving the communication efficiency of distributed first orderoptimization methods is communication compression.
Revised: A particularly successful technique for improving the communication efﬁciency of distributed ﬁrst-order optimizationmethods is communication compression.

Original: The idea behind this technique is rooted inthe observation that in practical implementations it  often advantageous to communicate messagescompressed via (often randomized) lossy compression techniques instead of communicating the fullmessages [75, 2].
Revised: The idea behind this technique is rooted in the observationthat in practical implementations, it is often advantageous to communicate messages compressedvia lossy compression techniques instead of communicating the full messages [75, 2].

Original: If the number of parallel workers is large enough, the noise introduced by compression is reduced, and training with compressed communicationwill often lead to the comparabletest error while reducing the amount of communicated bits, which results in faster training, both intheory and practice [58, 29].
Revised: If the numberof parallel workers is large enough, the noise introduced by compression is negligible, and trainingwith compressed communication often leads to a comparable test error while reducing the amountof communicated data, which results in faster training both in theory and practice [59, 29].

Original: 1.3 Two classes of compression operators
Revised: 1.3 Two classes of compression operators

Original: Since the paper focuses on compression methods for distributed VIs and SPPs.
Revised: This paper focuses on compression methods for distributed VIs and SPPs.

Original: Let us give the maindefinitions.
Revised: First, let us give the maindeﬁnitions used in the work.

Original: We say that a (possibly) stochastic mapping Q : R d → R d is an unbiased compressionoperator if there exists a constant q ≥ 1 such that
Revised: We say that a (possibly) stochastic mapping Q : R d ! R d is anunbiased compression operator if there exists a constant q  1 such that

Original: [Equation]
Revised: [Equation]

Original: Further, we say that a stochastic mapping C : R d → R d is a contractive compression operator ifthere exists a constant δ ≥ 1 such that
Revised: Furthermore, we say that a stochastic mapping C : R d ! R d is a contractive compression operatorif there exists a constant   1 such that

Original: [Equation]
Revised: [Equation]

Original: If b is the number of bits needed to represent a single float (e.g., b = 32 or b = 64 ), then the numberof bits needed to represent a generic vector z ∈ R d is ∥ z ∥ bits := bd .
Revised: If b is the number of bits needed to represent a single ﬂoating point number (e.g., b = 32 or b = 64 ),then the number of bits needed to represent a generic vector z 2 R d is k z k bits := bd .

Original: To describe how much acompression operator reduces its input vector on average, we define the notion of expected density,denoted via β − 1 := 1 bd E ∥ Q ( z ) ∥ bits , where ∥ Q ( z ) ∥ bits is the number of bits needed to represent thequantized vector Q ( z ) .
Revised: To describehow much a compression operator reduces its input vector on average, we deﬁne the notion ofexpected density, denoted via   1 := 1 bd E k Q ( z ) k bits , where k Q ( z ) k bits is the number of bits neededto represent the quantized vector Q ( z ) .

Original: Note that β ≥ 1 .
Revised: Note that   1 .

Original: For  the Rand k operator [3, 10] we have q = β =
Revised: For example, for the Rand k operator [3, 10]we have q =  =

Original: d/k .
Revised: d/k .

Original: 1.4 Towards communication-efficient distributed methods for VIs and SPPs
Revised: 1.4 Towards communication-efﬁcient distributed methods for VIs and SPPs

Original: Classical VI/SPP algorithms such as the Extra Gradient method  originally proposed by [45] and laterstudied by many authors [61, 40], including in  a distributed environment [77, 51, 60, 73].
Revised: Classical VI/SPP algorithms such as the Extra Gradient method were originally proposed by[46] and later studied by many authors [62, 41] in different setups, including a distributed one[77, 52, 61, 73].

Original: Amongthem, a number of works stand out trying to solve the communication bottleneck challenge usingvarious approaches such as local steps, data-similarity etc.[88, 34, 16, 11, 12].
Revised: Among them, a number of works stand out trying to solve the communication bottleneck challenge using various approaches such as local steps or data-similarity [88, 34, 16, 11, 12].

Original: But despite the factthat the use of compression is one of the most popular communication-efficient approaches for distributed minimization problems, no work has yet paid attention to the compression technique neitherfor distributed SPPs nor for VIs , with the exception of the work [88], which relies on rounding tothe nearest integer multiple of a certain quantity.
Revised: Despite the fact that compression is one of the most popular communication-efﬁcient approaches fordistributed minimization problems, no work has yet paid attention to the compression techniquesneither for distributed SPPs nor for VIs , with the exception of the work [88], which relies on rounding to the nearest integer multiple of a certain quantity.

Original: This compression mechanism does not offer theoretical benefits and does not even lead to convergence to the solution due to the errors introducedthrough rounding persist and prevent the method from solving the problem.
Revised: This compression mechanism does not offertheoretical beneﬁts and does not even lead to convergence to the solution, since the errors introducedthrough rounding persist and prevent the method from solving the problem.

Original:  Summary of Contributions
Revised:  Summary of Contributions

Original: In this paper, we investigate whether it is possible to design communication-efficient algorithmsfor solving distributed VI/SPP by borrowing generic communication compression techniques (1)and (2) from the optimization literature [75, 2, 58, 29, 72] and embedding them into established,efficient methods for solving VIs/SPPs [45, 61, 40, 1].
Revised: In this paper, we investigate whether it is possible to design communication-efﬁcient algorithms forsolving distributed VIs/SPPs by borrowing generic communication compression techniques (1) and (2) from the optimization literature [75, 2, 59, 29, 72] and embedding them into established, efﬁcientmethods for solving VIs/SPPs [46, 62, 41, 1].

Original: Whether or not this is possible is an openproblem.
Revised: Whether or not this is possible is an open problem.

Original: In summary,we design the first provably algorithms with compression for solving general distributed 
Revised: Insummary,we design the ﬁrst algorithms with compression for solving general distributed VI/SPP (see

Original: VI/SPP (see Section 3, Equation 3) in the deterministic (see (4) ), stochastic (see (42) ) andfederated (see (52) ) regimes, supporting both unbiased ( MASHA1 = Algorithms 1, 5, 7) andcontractive ( MASHA2 =
Revised: Section 3, Equation 3) in the deterministic (see (4) ), stochastic (see (45) ) and federated (see(55) ) regimes, supporting both unbiased ( MASHA1 = Algorithms 1, 5, 7) and contractive( MASHA2 =

Original: Algorithms 2, 6, 8) compressors.
Revised: Algorithms 2, 6, 8) compressors.

Original: Convergence of all our methodsare analyzed in strongly-monotone (strongly convex - strongly concave), monotone (convex- concave) non-monotone/minty (non-convex-non-concave) cases.
Revised: Convergence of all our methods is analyzedin strongly-monotone (strongly convex - strongly concave), monotone (convex - concave)non-monotone/ Minty (non-convex-non-concave) cases.

Original: 2.1 Two types of compressors
Revised: 2.1 Two types of compressors

Original: We develop two approaches for distributed VIs/SPPs depending on whether we use unbiased (1) orcontractive (2) compressors, since each type of  compressor demands a different algorithmic designand a different analysis.
Revised: We develop two approaches for distributed VIs/SPPs depending on whether we use unbiased (1) orcontractive (2) compressors, since each type of a compressor demands a different algorithmic designand a different analysis.

Original: In particular, contractive compressors are notoriously hard to analyze evenfor optimization problems [43, 72].
Revised: In particular, contractive compressors are notoriously hard to analyze evenfor optimization problems [44, 72].

Original: Our method based on unbiased compressors is called MASHA1(Algorithm 1), and our method based on contraction compressors is called MASHA2 (Algorithm 2).
Revised: Our method based on unbiased compressors is called MASHA1(Algorithm 1), and our method based on contraction compressors is called MASHA2 (Algorithm 2).

Original: 2.2 Theoretical complexity results
Revised: 2.2 Theoretical complexity results

Original: We establish a number of theoretical complexity results for our methods, which we summarizein Table 1.
Revised: We establish a number of theoretical complexity results for our methods, which we summarize in Table 1 (Appendix A).

Original: We consider the strongly monotone (strongly convex - strongly concave), monotone(convex - concave) regimes as well as the more general non-monotone/minty (non-convex-nonconcave) regime.
Revised: We consider the strongly monotone (strongly convex - strongly concave), monotone (convex - concave) regimes as well as the more general non-monotone/Minty (non-convex-nonconcave) regime.

Original: In the strongly monotone case we obtain linear convergence results ( O (log 1 /ϵ ) )in terms of the distance to solution, in the monotone  we obtain fast sublinear convergence results( O (1 /ϵ ) ) in terms of the gap function, and in the non-monotone case we have sublinear convergenceresults ( O (1 /ϵ 2 ) ) in terms of the second norm of the operator.
Revised: In the strongly monotone case, we obtain linear convergence results ( O (log 1 / ✏ ) )in terms of the distance to solution; in the monotone one, we obtain fast sublinear convergenceresults ( O (1 / ✏ ) ) in terms of the gap function; and in the non-monotone case, we have sublinear convergence results ( O (1 / ✏ 2 ) ) in terms of the Euclidean norm of the operator.

Original: To get an estimate for the numberof information transmitted, one need to multiply the estimates from Table 1 by 1 /β .
Revised: To get an estimate for thenumber of information transmitted, one needs to multiply the estimates from Table 1 by 1 /  .

Original: Then we getthat from the point of view ofthe transmitted information (and also time for communications), MASHA1 is better by a factor (cid:112) 1 /β + 1 /M
Revised: Thenwe obtain that in terms of transmitted information (and also time for communications), MASHA1is better by a factor p 1 /  + 1 /M

Original: ( M – number of workers) in comparison with theclassical Extra Gradient.
Revised: ( M is the number of workers) in comparison with the classical Extra Gradient.

Original: It means that we get an acceleration of min {√ β ; √ M } times.
Revised: It means that we get an acceleration of min { p  ; p M } times.

Original: For example,
Revised: For example,

Original: ADIANA from [47](the theoretical SOTA method with unbiased compressions for strongly convexminimization) has the same accelaration.
Revised: ADIANA from [48](the theoretical SOTA method with unbiased compressions for strongly convexminimization) has the same acceleration.

Original: The same situation is with MASHA2 .
Revised: The same situation is with MASHA2 .

Original: The method has thesame compression dependent multiplier as ECLK from [71] (the theoretical SOTA with contractivecompression for minimization).
Revised: The method has thesame compression dependent multiplier as ECLK from [71] (the theoretical SOTA with contractivecompression for minimization).

Original: Based on these facts, we hypothesize that MASHA1 and MASHA2have unimprovable estimates (see Appendix A).
Revised: Based on these facts, we hypothesize that MASHA1 and MASHA2have unimprovable estimates (see Appendix C).

Original: 2.3 Stochastic case and variance reduction
Revised: 2.3 Stochastic case and variance reduction

Original: MASHA1 and MASHA2 are designed to handle the deterministic setting.
Revised: MASHA1 and MASHA2 are designed to handle the deterministic setting.

Original: But often in practice, thecomputation of the full operators F m is  expensive, then we need to deal with stochastic realizations.
Revised: However, in practice, thecomputation of the full operators/gradients is often expensive, and thus we need to consider stochastic realizations.

Original: In particular, a popular case is when each operator F m has a finite-sum structure on its own, e.g. ,finite-sum of batches.
Revised: In particular, a popular case is when each operator/gradient has a ﬁnite-sum structureon its own, e.g., ﬁnite-sum of batches.

Original: For this issue, we consider two modifications: VR-MASHA1 (Algorithm 5)and VR-MASHA2 (Algorithm 6).
Revised: For this issue, we consider two modiﬁcations: VR-MASHA1(Algorithm 5) and VR-MASHA2 (Algorithm 6).

Original: Both are enhanced with bespoke variance-reduction techniques forbetter theoretical and practical performance.
Revised: Both are enhanced with bespoke variance-reductiontechniques for better theoretical and practical performance.

Original: These results can be interesting in the non-distributedcase.
Revised: These results can be interesting in thenon-distributed case.

Original: As far as we know, we are the first who consider variance reduction for non-monotone VIs.
Revised: As far as we know, we are the ﬁrst who consider variance reduction for nonmonotone VIs.

Original: Wefound only one paper on non-convex-concave saddle point problems [87] under the PL condition.
Revised: We found only one paper on non-convex-concave saddle point problems [87] underthe PL condition.

Original: See Appendix E fro details.
Revised: See Appendix G for details.

Original: 2.4 Federated learning and partial participation
Revised: 2.4 Federated learning and partial participation

Original: [Figure] Federated learning [44, 41] is an important and popular branch of distributed methods. Therefore,a good bonus for the algorithm is that it can be easily adapted for it.
Revised: Federated learning [45, 42] is an important and popular branch of distributed methods; therefore,a good bonus for the algorithm is that it can be easily adapted for it.

Original: In a federated setup wherethe computing devices are mobile phones, tablets, personal computers etc, the importance of thecommunication bottleneck is even higher.
Revised: In a federated setup, wherethe computing devices are mobile phones, tablets, personal computers etc, the importance of thecommunication bottleneck is even higher.

Original: Moreover, devices cannot only have a bad connection,in which one needs to compress data heavily, they can simply disconnect from the  learning process.
Revised: Also, participating devices can have weak and slow connections, or they can even disconnect from the network for a while. At such moments, it is notnecessary to interrupt the learning process, and only available devices can be used.

Original: [Table] Therefore, we introduce two modifications: PP-MASHA1 (Algorithm 7) and PP-MASHA2 (Algorithm 8), that support the mode of partial participation of devices in the learning process.
Revised: To handle thesecases, we introduce two modiﬁcations, PP-MASHA1 (Algorithm 7) and PP-MASHA2 (Algorithm 8),that support the mode of partial participation of devices in the learning process.

Original: For minimization problems, a combination of quantization and partial participation occurs in [33, 68, 29].
Revised: For minimization problems, a combination of quantization and partial participation occurs in [33, 68, 29].

Original: The results are contained in Appendix F.
Revised: The resultsare contained in Appendix H.

Original: 2.5 Bidirectional compression
Revised: 2.5 Bidirectional compression

Original: Most methods, especially  with contractive compressors, only use compression when transferringinformation from devices to the server.
Revised: Most methods, especially those with contractive compressors, only use compression when transferring information from devices to the server.

Original: Meanwhile, quite often in practical situations, the transfer of information from the server to the device is also expensive [32, 81, 68].
Revised: Meanwhile, quite often in practical situations, thetransfer of information from the server to the device is also expensive [32, 81, 68].

Original: In  the literatureon distributed heterogeneous minimization, we can highlighted only some works on bidirectionalunbiased [68] and contractive compressors [54, 23]. Also one can note some works [90, 81] foridentical/homogeneous data on workers.
Revised: In such situations it also makes sense to compress the information when sending it from the server to theagents. We can highlight several works on bidirectional unbiased [68] and contractive compressors[90, 81, 55, 23] for distributed minimization problems.

Original: But most of these methods have their small drawbacks,such as  only  deterministic case, homogeneity not suitable for federated learning etc.
Revised: Still, most of these methods have their smallshortcomings in theoretical analysis such as considering only the deterministic setting, or assumingthe homogeneity of local functions.

Original: All our methods MASHA1 , MASHA2 and their modifications support bidirectional compression.
Revised: All our methods MASHA1 , MASHA2 and their modiﬁcationssupport bidirectional compression.

Original: See Appendix C and D for details.
Revised: See Appendix E and F for details.

Original: 2.6 Experiments
Revised: 2.6 Experiments

Original: Toy experiments on bilinear problems show that methods with compression for minimization problems may not work (diverge) for SPPs.
Revised: Toy experiments on bilinear problems show that methods with compression for minimization problems may not work (diverge) for SPPs.

Original: Also we verify that MASHA1 and MASHA2 are much betterthan the classical Extra Gradient with added unbiased compression.
Revised: Also, we verify that MASHA1 and MASHA2 are much betterthan the classical Extra Gradient with added unbiased compression.

Original: Experiments on adversarialtraining of large-scale transformer  (ALBERT) show the practical importance of compression in distributed methods for large SPPs.
Revised: Experiments on adversarialtraining of large-scale transformer models (ALBERT) show the practical importance of compression in distributed methods for large SPPs.

Original:  Problem Formulation and Assumptions
Revised:  Problem Formulation and Assumptions

Original: 3.1 Problem formulation
Revised: 3.1 Problem formulation

Original: We study  distributed variational inequality (VI) problem 
Revised: We study a distributed variational inequality (VI) problem of the following form:

Original: Find z ∗ ∈ R d such that ⟨ F ( z ∗ ) , z − z ∗ ⟩ ≥ 0 , ∀ z ∈ R d , (3) where F : R d → R d is an operator with certain favorable properties (e.g., Lipschitzness andmonotonicity).
Revised: [Equation] [Equation] where F : R d ! R d is an operator with certain favorable properties (e.g., Lipschitzness andmonotonicity).

Original: We assume that the training data describing F is distributed across M workers/nodes/clients
Revised: We assume that the training data describing F is distributed across M workers/nodes/clients

Original: [Equation]
Revised: [Equation]

Original: where F m : R d → R d for all m ∈ { 1 , 2 , . .. , M } .
Revised: where F m : R d ! R d for all m 2 { 1 , 2 , . . , M } .

Revised: In Appendix B, we give several examples of VIsto show the breadth of this formalism.

Original: For the saddle point problem
Revised: Here we mention only that for the saddle point problem

Original: [Equation]
Revised: [Equation]

Original: we take z :=
Revised: we take z :=

Original: [ x, y ] and F m ( z ) :
Revised: [ x, y ] and F m ( z ) :

Original: = [ ∇ x g ( x, y ) , −∇ y g ( x, y )] .
Revised: = [ r x g ( x, y ) , r y g ( x, y )] .

Original: 3.2 Assumptions
Revised: 3.2 Assumptions

Original: Next, we list  two key assumptions - both  are standard in the literature on VIs.
Revised: Next, we list the two key assumptions, both of which are standard in the literature on VIs.

Original: Assumption 3.1 (Lipschitzness)
Revised: Assumption 3.1 (Lipschitzness)

Original: The operator F is L -Lipschitz continuous, i.e. for all z 1 , z 2 ∈ R dwe have ∥ F ( z 1 ) − F ( z 2 )
Revised: The operator F is L -Lipschitz continuous, i.e. for all z 1 , z 2 2 R dwe have k F ( z 1 )  F ( z 2 )

Original: ∥ ≤ L ∥ z 1 − z 2 ∥ .
Revised: k  L k z 1  z 2 k .

Original: Each operator F m is L m -Lipschitz continuous, i.e. for all z 1 , z 2 ∈ R d it holds ∥ F m ( z 1 ) −
Revised: Each operator F m is L m -Lipschitz continuous, i.e. for all z 1 , z 2 2 R d it holds k F m ( z 1 )  F m ( z 2 )

Original: [Equation]
Revised: k

Original:  z 2 ∥ .
Revised:  L m k z 1  z 2 k .

Original: Let us define new constant ˜ L as follows ˜
Revised: Let us deﬁne new constant ˜ L as follows:

Original:  L [Equation]
Revised: ˜ L 2 = 1 M M

Original: =
Revised: P

Original: [Equation]
Revised: [Equation]

Original: For saddle point problems, these properties are equivalent to smoothness.
Revised: For saddle point problems, these properties are equivalent to smoothness.

Original: Assumption 3.2 (Monotonicity) We need three cases of monotonicity(SM)
Revised: Assumption 3.2 (Monotonicity) We need three cases of monotonicity:(SM

Original: Strong monotonicity.
Revised: Strong monotonicity.

Original: The operator F is µ -strongly monotone, i.e. for all z 1 , z 2 ∈ R d we have⟨ F ( z 1 ) − F ( z 2 ) , z 1 − z 2 ⟩ ≥ µ ∥ z 1 − z 2 ∥ 2 .(M)
Revised: The operator F is µ -strongly monotone , i.e. for all z 1 , z 2 2 R d we haveh F ( z 1 )  F ( z 2 ) , z 1  z 2 i  µ k z 1  z 2 k 2 .

Original:  Monotonicity.
Revised: (M) Monotonicity.

Original: The operator F is monotone, i.e. for all z 1 , z2 ∈ R
Revised: The operator F is monotone, i.e. for all z 1 , z

Original:  d we have ⟨ F ( z 1 ) − F ( z 2 ) , z 1 − z 2 ⟩ ≥ 0 .(NM)
Revised: 2 2 R d we have h F ( z 1 )  170 F ( z 2 ) , z 1  z 2 i  0 .(NM)

Original: Non-monotonicity.
Revised: Non-monotonicity.

Original: The operator F is non-monotone (minty), if and only if there exists z ∗ ∈ 172{+⇤ 
+} R d such that for all z ∈ R d we have ⟨ F ( z ) , z − z ∗ ⟩ ≥ 0 .
Revised: The operator F is non-monotone (Minty), if and only if there exists z {+⇤ 
+} R d such that for all z 2 R d we have h F ( z ) , z  z ⇤ i  0 .

Original: The last assumption is called the minty or variational stability condition.
Revised: The last assumption is called the Minty or variational stability condition.

Original: It is not a general nonmonotonicity, but is already associated in the community with non-monotonicity [14, 37, 57, 52, 42,36, 19], particularly with the setup, which is somewhat appropriate for GANS [50, 51, 21, 8].
Revised: It does not representgeneral non-monotonicity, but is already associated in the community with non-monotonicity [14,37, 58, 53, 43, 36, 19], particularly with the setup that is somewhat appropriate for GANs [51, 52,21, 8].

Original:  MASHA
Revised:  MASHA

Original: In this Section we present new algorithms and their convergence.
Revised: In this section, we present new algorithms and their convergence.

Original: Section 4.1 is devoted to the algorithm ( MASHA1 ) with unbiased compression.
Revised: Section 4.1 is devoted to thealgorithm ( MASHA1 ) with unbiased compression.

Original: Section 4.2 – to  algorithm ( MASHA2 ) with contractivecompression.
Revised: Section 4.2 – to the algorithm ( MASHA2 ) withcontractive compression.

Original: Appendix gives modifications for the stochastic case – Section E, and for the  federated learning – Section F. Appendix A is devoted to the hypothesis about optimality of MASHA1and MASHA2 .
Revised: Appendix G describes modiﬁcations for the stochastic case, and Appendix H covers the case of federated learning. Appendix C is devoted to the hypothesis about the optimalityof MASHA1 and MASHA2 .

Original: 4.1 MASHA1 : Handling Unbiased Compressors
Revised: 4.1 MASHA1 : Handling Unbiased Compressors

Original: Before presenting our algorithm, let us discuss which approaches can be used to construct it.
Revised: Before presenting our algorithm, let us discuss which approaches can be used to construct it.

Original: Asdiscussed in Sections 1 and 2, compression methods play an important role in distributed minimization problems.
Revised: Asdiscussed in Sections 1 and 2, compression methods play an important role in distributed minimization problems.

Original: All these methods are modifications of the classical GD. For instance, the authorsof [2] compress stochastic gradients.
Revised: All these methods are modiﬁcations of the classical gradient descent (GD): forinstance, the authors of [2] compress stochastic gradients.

Original: Therefore, it is a natural idea to use GD-type methods for VIsas well.
Revised: Therefore, it is a natural idea to use GDtype methods for VIs as well.

Original: But it is a well-known fact that GD-type methods can give bad convergence estimates (see Section B.1 from [67]) or do not converge at all (see Section 7.2 and 8.2 from [27]) even on thesimplest SPPs and VIs.
Revised: However, it is a well-known fact that GD-type methods can give badconvergence estimates (see Section B.1 from [67]) or do not converge at all (see Section 7.2 and 8.2from [27]) even on the simplest SPPs and VIs.

Original: From a practical point of view, this approach can also fail (see QSGD and EF in Section 5.1).
Revised: From a practical point of view, this approach canalso fail (see QSGD and EF in Section 5.1).

Original: In the non-distributed case, this problem has long been solved and the Extra Gradient method [45, 61, 40] is used instead of GD: [Equation]
Revised: In the non-distributed case, this problem has long beensolved and the Extra Gradient method [46, 62, 41] is used instead of the gradient descent:

Original: [Equation]
Revised: [Equation]

Original: This method is optimal for both VIs and SPPs and has an estimate of convergence
Revised: This method is optimal for both VIs and SPPs and has an estimate of convergence

Original: ˜ O
Revised: ˜ O

Original: ( L / µ ) in thestrongly monotone case.
Revised: ( L / µ ) in thestrongly monotone case.

Original: Therefore, the second idea for the compressed method is to add compression operators to the method (6), e.g. use Q k ( F ( z k )) and
Revised: Therefore, the second idea for the compressed method is to add compression operators to the method (6), e.g. use Q k ( F ( z k )) and

Original: Q k +1 / 2 ( F ( z k +1 / 2 )) instead of F ( z k ) and F ( z k +1 / 2 ) .
Revised: Q k +1 / 2 ( F ( z k +1 / 2 )) instead of F ( z k )and F ( z k +1 / 2 ) .

Original: In Section G we analyse this method, but it gives an estimate ˜ O(cid:0) 1 + q/M ) ·
Revised: In Section I we analyse this method, but it gives an estimate ˜ O

Original: L 2 / µ 2 (cid:1) ,which is considerably worse in terms of L / µ than the original Extra Gradient method.
Revised: [Equation] which is considerably worse in terms of L / µ than the original Extra Gradient method.

Original: The keyproblem is that in the analysis one has to deal with ∥ Q k ( F ( z k +1 / 2 ))
Revised: The keyproblem is that in the analysis one has to deal with k Q k ( F ( z k +1 / 2 ))

Original: − Q k +1 / 2 ( F ( z k ))
Revised:  Q k +1 / 2 ( F ( z k ))

Original: ∥ 2 .
Revised: k 2 .

Original: Without compression operators, such difference is easily evaluated using Assumption 3.1. But when thecompression operators are different (in fact the same, but have different randomness) we cannotmake a good estimate for this term.
Revised: Withoutcompression operators, such difference is easily evaluated using Assumption 3.1, but when the compression operators are different (in fact the same, but have different randomness) we cannot make agood estimate for this term.

Original: The idea arises to use the same randomness in both steps of themethod (6), namely to substitute Q k ( F ( z k )) and Q k ( F ( z k +1 / 2 )) .
Revised: One possible solution is to use the same randomness in both steps of themethod (6), namely to substitute Q k ( F ( z k )) and Q k ( F ( z k +1 / 2 )) .

Original: But then z k +1 / 2 depends on therandomness Q k , and hence Q k ( F ( z k +1 / 2 )) is biased, which further complicates the analysis.
Revised: However, then z k +1 / 2 dependson the randomness Q k , and hence Q k ( F ( z k +1 / 2 )) is biased, which further complicates the analysis.

Original: Forexactly the same reasons, the various optimistic/single call modifications [70, 26, 35, 59] of the Extra Gradient method did not work for us either.
Revised: For exactly the same reasons, the various optimistic/single call modiﬁcations [70, 26, 35, 60] of the Extra Gradient method did not work for us either.

Original: We have also test the method (6) with compressionsin practice (see CEG in Section 5.1), and it turns out to be worse than the method we will presentbelow.
Revised: We have also tested the method (6) with compression techniques in practice (see CEG in Section 5.1), and it turns out to be worse than the method wewill present below.

Original: In the end, the use of variance reduction and negative momentum techniques [1] is key increating our algorithm.
Revised: In the end, the use of variance reduction and negative momentum techniques [1]is key in creating our algorithm.

Original: These tricks are not in themselves relevant to distributed problems, but, inour case, they help in creating MASHA1 and MASHA2 .
Revised: These tricks are not in themselves relevant to distributed problems,but, in our case, they help in creating MASHA1 and MASHA2 .

Original: Parameters: Stepsize γ > 0 , parameter τ ∈ (0; 1) , number of iterations K .
Revised: At the beginning of each MASHA1 iteration, all devices know the value of F ( w k ) , hence they cancalculate the value of z k +1 / 2 locally without communications. Furthermore, each device sends thecompressed version of the difference F m ( z k +1 / 2 )  F m ( w k ) to the server. The compression ofthese transfers is done by their local operators { Q dev m } .

Original: Initialization: Choose z 0 = w 0 ∈ Z . 
Revised: The server aggregates the information fromdevices, averages it, compresses by the Q serv operator and makes a broadcast to all devices. As aresult, an unbiased estimate of F ( z k +1 / 2 )  F ( w k ) appears at each node. Also, the nodes receiveone bit of information b k . This bit is generated randomly on the server and is equal to 1 withprobability 1  ⌧ , where 1  ⌧ is small.

Original: Devices send F m ( w 0 ) to server and get F ( w 0 ) for k = 0 , 1 , 2  , . . , K [-− 1 do for each device m in parallel do z k +1 / 
-]
Revised: Note that b k can be generated locally, it is enough to use the Algorithm Parameters: Stepsize  > 0 , parameter ⌧ 2 (0; 1) , number of iterations K [-− 1 do for each device m in parallel do z k +1 / 
-].

Revised: Initialization: Choose z

Original: =
Revised: =

Original: τz k
Revised: w 0 2 Z .

Original: + (1 − τ ) w k − γF ( w k ) 
Revised: Devices send F m ( w 0 ) to server and get F ( w 0 ) for k = 0 , 1 , 2 , . . , K  1 do for each device m in parallel do [Equation]

Original: Sends g km =
Revised: Send g km =

Original: Q dev m ( F m ( z k +1 / 2 ) − F m ( w k )) to server end for for server do
Revised: Q dev m [Equation]  F m ( w [Equation]

Original: Sends to devices g k = Q serv (cid:104) 1 M (cid:80)
Revised: to server end for for server do Send to devices g k = Q serv h

Original: Mm =1 g km (cid:105)
Revised: [Equation]

Original: Sends to devices one bit b k : 1 with probability 1 − τ , 0 withwith probability τ end for for each device m in parallel doz k
Revised: i Send to devices one bit b k : 1 with probability 1  ⌧ , 0 with probability ⌧ end for for each device m in parallel do

Original: [-+
 = z k +1 / 2 − γg k-]
Revised: [-+
 = z k +1 / 2 − γg k-][Equation]

Original: If b k = 1 then w k +1 =
Revised: If b k = 1 then w k +1 =

Original: z k , sends F m ( w k +1 ) to server and gets F ( w k +1 ) else w k [-+
-]
Revised: z k , send F m ( w k +1 ) to server and get F ( w k +1 ) else w k [-+
-]+1 =

Original: = w k end for end for 209 At the beginning of each iteration of MASHA1 , each device knows the value of F ( w k ) , hence itcan be calculated ¯ z k and z k +1 / 2 locally. Then it sends the compressed version of the difference F m ( z k +1 / 2 ) − F m ( w k ) to the server, and  the server does a reverse compressed broadcast.
Revised: w k end for end for same random generator and set the same seed on all devices.

Original: As aresult, an unbiased estimate of F ( z k +1 / 2 ) − F ( w k ) appears on each node. Also, the nodes receive abit of information b k . This bit is generated randomly on the server and is equal to 1 with probability1 − τ (where 1 − τ is small). Note that b k can be generated locally, it is enough to use the samerandom generator and set the same seed on all devices.
Revised: Next, the devices make a ﬁnal updateon z k +1 locally.

Original: Then, all devices make a final update onz k +1  , and also update the w k +1 = z k +1point if the b k = 1 or save it from the previous iterationw k +1{++
+} =
Revised: The ﬁnal step is an update of w k +1 : if b k = 1 , then w k +1 = z k +1 or otherwisew k {++
+} =

Original: w k .
Revised: w k .

Original: In the case whenthe point w k +

Revised: In the case when w k +

Original: =
Revised: =

Original: z k +1 , we need to exchange the full values of
Revised: z k +1 , we need to exchange the uncompressed values of

Original: F m ( w k +1 ) in order that at the beginning of the next iteration the value of F ( w k +1 ) is known to allnodes.
Revised: F m ( w k +1 ) to ensure that at the beginning of the next iteration, the value of F ( w k +1 ) is known toall agents.

Original: We use a possibly difference compressor on each device andalso on the server.
Revised: We use a possibly different compressors on the devices and on the server.

Original: To distinguishbetween them, we denote the following notation: Q dev m , q dev m , β dev m and Q serv , q serv , q serv .
Revised: To distinguishbetween them, we denote the following notation: Q dev m , q dev m ,  dev m and Q serv , q serv ,  serv .

Original: Theorem 4.1 Let Assumption 3.1 and one case of Assumption 3.2 are satisfied.
Revised: Theorem 4.1 Let Assumption 3.1 and one case of Assumption 3.2 be satisﬁed.

Original: Then for some stepγ the following estimates on MASHA1 number of iterations  to achieve ε -solution holds
Revised: Then for some stepsize  , the following estimates on the number of iterations of MASHA1 to achieve " -solution hold:

Original: • in strongly-monotone case: E [ ∥ z K − z∗ ∥ 2 + ∥ w K − z ∗ ∥ 2 ] 
Revised: • in the strongly monotone case (in terms of E [ k z K  z ⇤ k 2 ] ⇠ " ): O ([ 11  ⌧ +

Original: =
Revised: [Equation]

Original: O ([ 11 − τ + C q µ √ 1 − τ ] log 1 ε );
Revised: ] log 1 " );

Original: • in  monotone case: E max z ∈C [ ⟨ F ( u ) , ( 1 K[-(cid:80) K − 1 k =
-]
Revised: • in the monotone case ( E max z 2 C [ h F ( u ) , ( 1 K[-(cid:80) K − 1 k =
-]

Original: z k +1 / 2 ) − u ⟩ ] = O ( C q ∥ z 0 − z ∗ ∥ 2 ε √ 1 − τ );
Revised: [Equation]

Original: • in  non-monotone case: E [ 1 K[-(cid:80) K − 1 k =
-]
Revised: • in the non-monotone case (in terms of E [ 1 K[-(cid:80) K − 1 k =
-]

Original: ∥ F ( w k ) ∥ 2 ] =
Revised: [Equation] where

Original: O ( C 2 q ∥ z 0 − z ∗ ∥ 2 ε 2 (1 − τ ) ); where C 2 q = q serv M 2 (cid:80) Mm =1 ( q dev m L 2 m + ( M − 1)˜ L 2 ) .
Revised: [Equation]

Original: A full listing of the algorithm, as well as a full statement of the theorem with  proof, can be found in Appendix C.
Revised: The full description of the algorithm, as well as the full statement of the theorem with its proof, canbe found in Appendix E.

Original: The bounds in Theorem 4.1 are related to τ .
Revised: The bounds in Theorem 4.1 are related to ⌧ .

Original: Let us find optimal way to choose it.
Revised: Let us ﬁnd an optimal way to choose it.

Original: Note that (inaverage) once per 1 / (1 − τ ) iterations (when b k = 1 ), we send uncompressed information.
Revised: Note that (inaverage) once per 1 / (1  ⌧ ) iterations (when b k = 1 ), we send uncompressed information.

Original: Based onthis observation, we can find the best option for τ .
Revised: Based onthis observation, we can ﬁnd the best option for ⌧ .

Original: Let us analyze the case of compressions only onthe devices’ side ( q serv = 1 ).
Revised: Let us analyze the case of compression only onthe devices’ side ( q serv = 1 ).

Original: For simplicity, we put Q dev m =
Revised: For simplicity, we put Q dev m =

Original: Q with q dev m =
Revised: Q with q dev m =

Original: q and β dev m = β , also
Revised: q and  dev m =  , also

Original: L m =
Revised: L m =

Original: ˜ L = L .
Revised: ˜ L = L .

Original: Since compression is done only on devices, we assume that the server’s broadcast ischeap and we only care about devices.
Revised: Since compression is done only on devices, we assume that the server’s broadcast ischeap and we only care about devices.

Original: Then at each iteration the device sends O ( 1 / β + 1 − τ ) bits– each time information compressed by β times 
Revised: Then at each iteration the device sends O ( 1 /  + 1  ⌧ ) bits– each time information compressed by  times and with probability 1  ⌧ we send the full package.

Original: and with probability

Original: 1 − τ

Original: we send the

Original: full

Original: package.

Original: From where we immediately get the optimal choice for τ :
Revised: From this, we immediately get the optimal choice for ⌧ :

Original: Corollary 4.2 Let Assumption 3.1 and one case of Assumption 3.2 are satisfied.
Revised: Corollary 4.2 Let Assumption 3.1 and one case of Assumption 3.2 be satisﬁed.

Original: Then for some stepγ and 1 − τ = 1 / β the following estimates on MASHA1 number of iterations  to achieve ε -solutionholds
Revised: Then, for some stepsize  and 1  ⌧ = 1 /  , the following estimates on the number of iterations of MASHA1 to achieve" -solution hold

Original: • in str-monotone case: O ([ β + (cid:113) qβ M + β · Lµ ] log 1 ε );• in monotone case: O ( (cid:113) qβM + β · L ∥
Revised: • in the strongly monotone case: O ([  + [Equation]  [Equation]

Revised: [Equation]

Original: z 0 − z ∗ ∥ 2 ε )
Revised: ] log [Equation] • in the monotone case: O ( [Equation]

Original: ;• in  non-monotone case: O ([ qβM + β ] L 2 ∥{+k z 
+}
Revised:  [Equation] [Equation] • in the non-monotone case: O ([ q  M +  ] L 2 {+k z 
+}

Original:  z 0 − z ∗ ∥ 2 ε) .
Revised:  z ⇤ k 2 ") .

Original: We can see that MASHA1 can outperform the uncompressed Extra Gradient method.
Revised: We can see that MASHA1 can outperform the uncompressed Extra Gradient method.

Original: Let us comparethem in the strongly monotone case.
Revised: Let us comparethem in the strongly monotone case.

Original: The communication complexity of the Extra Gradient method is˜ O
Revised: The communication complexity of the Extra Gradient method is ˜ O

Original: ( L / µ ) .
Revised: ( L / µ ) .

Original: MASHA1 has communication complexity ˜
Revised: MASHA1 has communication complexity ˜

Original: O
Revised: O

Original: ( (cid:112) q / βM + 1 / β · L / µ ) .
Revised: ( p q /  M + 1 /  · L / µ ) .

Original: For practical compressors[10], β ≥ q .
Revised: For practical compressors[10],   q .

Original: Then once can note that  MASHA1 O ( (cid:112) 1 / M + 1 / β · L / µ ) better than the uncompressedextragradient method.
Revised: Then, one can note that the communication complexity of MASHA1 differs fromthe complexity of the uncompressed method by an additional factor ( p 1 / M + 1 /  ) .

Revised: It is easy tosee that even for a small number of devices M and expected density  , this factor is less than 1 ;hence, MASHA1 outperforms the uncompressed method.

Original: We think that the factorO ( (cid:112) 1 / M + 1 / β · L / µ ) is unimprovable and optimal –see Section A.
Revised: We believe that this factor ( p 1 / M + 1 /  )is theoretically unimprovable and optimal – see Section C for details.

Original: One can also consider the case of bidirectional compression ( q serv ̸ = 1 ).
Revised: One can also consider the case of bidirectional compression ( q serv 6 = 1 ).

Original: Table 1 (line 3) shows theresult for q serv =
Revised: Table 1 (line 3) shows theresult for q serv =

Original: q dev m = q , β serv = β dev m = β and 1 − τ = 1 / β .
Revised: q dev m = q ,  serv =  dev m =  and 1  ⌧ = 1 /  .

Original: 4.2 MASHA2 : Handling Contractive Compressors
Revised: 4.2 MASHA2 : Handling Contractive Compressors

Original: The use of contractive compressions is a more complex issue.
Revised: The use of contractive compressions is a more complex issue.

Original: In particular, it is known that if onesimply put a contractive cospressor instead of an unbiased one, the method may diverge even forquadratic problems [10].
Revised: In particular, it is known that if onesimply puts a contractive compressor instead of an unbiased one, the method may diverge even forquadratic problems [10].

Original: To fix this, an error compensation technique [78, 43, 79] is used.
Revised: To ﬁx this, an error compensation technique [78, 44, 79] is used.

Original: The pointis to keep untransmitted information and add  to  new information at the next iteration.
Revised: Thepoint of this approach is to keep untransmitted information and add it to a new package at thenext iteration.

Original: This is thekey difference between MASHA2 and MASHA1 .
Revised: This is the main difference between MASHA2 and MASHA1 .

Original: MASHA2 introduces additional sequences e k , e kmfor the server’s and devices’ error.
Revised: MASHA2 introducesadditional sequences e k , e km for the server’s and devices’ errors.

Original: To define contractive operators on devices and on the server, weintroduce the following notation: C dev m , δ dev , β dev and C serv m , δ serv , β serv .
Revised: To deﬁne contractive operators ondevices and on the server, we introduce the following notation: C dev m ,  dev ,  dev and C serv m ,  serv ,  serv .

Original: In the case of MASHA1 , an important theoretical question was how to design the algorithm to obtain convergence estimates as good as those for non-distributed methods. MASHA2 raises anotherproblem for theoretical analysis, how to combine MASHA1 and the error feedback. The analysis ofmethods with error compensation techniques for minimization problem min x f ( x ) is entirely tied tothe existence of the function f In particular, the differences f ( x k ) − f ( x ∗ ) appear inthe whole analysis and key in the technical lemmas.
Revised: Algorithm

Original: As a result f ( x ) − f ( x ∗ ) is used as a convergence criterion even in the strongly convex case. But in the case of VIs there is no function f , onlythe operator F (the existence of g ( x, y ) in SPP setup does not save the situation). This problem issolved in the proof of Theorem 4.3 by using an additional sequence ∥ z k +1 / 2 − w k ∥ .
Revised: Parameters: Stepsize  > 0 , parameter ⌧ , number of iterations K .

Original: Theorem 4.3 Let Assumption 3.1 and one case of Assumption 3.2 are satisfied.

Original: Then for some stepγ the following estimates on MASHA2 number of iterations to achieve ε -solution holds

Original: • in strongly-monotone case: E [ ∥ ˆ zK − z ∗ ∥ 2 + ∥ w K − z ∗ ∥ 2 ] = O ([ 11 − τ
Revised: Initialization: Choose z

Original: [-+ δ dev δ serv ˜ L µ √ 1 − τ ] log 1 ε );• in monotone case: E max z ∈C [ ⟨ F ( z ) , 1 K (cid:80) K − 1 k =
-]
Revised: [-+ δ dev δ serv ˜ L µ √ 1 − τ ] log 1 ε );• in monotone case: E max z ∈C [ ⟨ F ( z ) , 1 K (cid:80) K − 1 k =
-]

Original: z k +1 / 2 ) − z ⟩ ]

Original: =
Revised: =

Original: O ( δ dev δ serv ˜

Original: L ∥

Original: z 0 − z ∗ ∥ 2 ε √ 1 − τ ); Algorithm Parameters: Stepsize γ > 0 , parameter τ , number of iterations K . Initialization: Choose z 0 = w 0 ∈ Z , e 0 m = 0 , e 0 = 0 .
Revised: w 0 2 Z , e 0 m = 0 , e 0 = 0 .

Original: Devices send F m ( w 0 ) to server and get F ( w 0 ) for k = 0 , 1 , 2 , . .
Revised: Devices send F m ( w 0 ) to server and get F ( w 0 ) for k = 0 , 1 , 2 , . .

Original: , K − 1 do for each device m in parallel do z k +1 / [-
-] =
Revised: , K  1 do for each device m in parallel do z k +1 / [-
-]2 =

Original: τz k
Revised: ⌧ z

Original:  + (1 − τ ) w k − γF ( w k )
Revised: [Equation] + (1  ⌧ ) w k   F ( w k )

Original: Sends g km =
Revised: Send g km =

Original: C dev m ( γF m ( z k +1 / 2 ) − γF m ( w k ) + e km ) to server e k +1 m =
Revised: C dev m (  F m ( z k +1 / 2 )   F m ( w k ) + e km ) to server e k +1 m =

Original: e km + γF m ( z k +1 / 2 ) − γF m ( w k ) − g km end for for server do
Revised: e km +  F m ( z k +1 / 2 )   F m ( w k )  g km end for for server do

Original: Sends to devices g k = C serv (cid:104) 1 M (cid:80)
Revised: Send to devices g k = C serv h 1 M P

Original: Mm =
Revised: Mm =

Revised: [Equation]

Original: 1 g km + e k 
Revised: g km + e k i e k

Original: (cid:105){++
+}
Revised: {++
+}

Original:  e k +1 =
Revised: = e k + 1 M

Original: e k + 1 M (cid:80)
Revised: P

Original: Mm = 1 g km − g k Sends to devices one bit b  : 1 with probability 1 − with probability τ end for for each device m in parallel do zk
Revised: Mm =1 g km  g k Send to devices one bit b k : 1 with probability ⌧ , 0 with probability ⌧ end for for each device m in parallel do z

Original: [-+
 =-]
Revised: [-+
 =-][Equation]

Original: z k +1 / 2 − γg k If b k 
Revised: g k If b k = 1 then w k +1 =

Original: 1 then w k + k , sends to server and gets ( w k +1 ) else w k +1 =
Revised: z k , send F m ( w k +1 ) to server and get F ( w k +1 ) else w k +1 =

Original: w k end for end for
Revised: w k end for end for

Revised: In the case of MASHA1 , the key theoretical issue was the choice of a basic method (we discussedthis at the beginning of Section 4.1).

Revised: MASHA2 raises another problem for theoretical analysis,namely how to combine MASHA1 and the error feedback technique.

Revised: The analysis of methods witherror compensation for the minimization problem min x f ( x ) is entirely tied to the existence of thefunction f

Revised: In particular, the differences ( f ( · )  f ( x ⇤ )) appear in the entire analysis andare of key importance in the technical lemmas.

Original: • in non-monotone case: E [ 1 K (cid:80) K − 1 k [-=
-]
Revised: As a result, ( f ( · )  f ( x ⇤ )) is used as a convergencecriterion even in the strongly convex case. However, for VIs, there is no function f , only the operator F (the existence of g ( x, y ) in SPP setup does not save the situation). This problem is solved in theproof of Theorem 4.3 by using an additional sequence k z k +1 / 2  w k k [-=
-].

Revised: Theorem 4.3 Let Assumption 3.1 and one case of Assumption 3.2 be satisﬁed.

Original: ∥ F
Revised: Then for some step the following estimates on MASHA2 number of iterations to achieve " -solution hold: • in the strongly monotone case (in terms of E [ k ˆ z K  z ⇤ k 2 ] ⇠ " ): O ([ [Equation] ] log

Original: w k ) ∥ 2 ]
Revised: [Equation] [Equation] • in the monotone case ( E max z 2 C [ h F ( u ) , ( 1 K

Original: = O (
Revised: [Equation]

Original: δ dev δ
Revised: • in the non-monotone case (in terms of E [ 1 K [Equation]

Original: A full listing of the algorithm, as well as a full statement of the theorem with  proof, can be found in Appendix D.
Revised: The full listing of the algorithm, as well as the full statement of the theorem with its proof, can befound in Appendix F.

Original: The same way as in Section 4.1 we can consider only devices’ and bidirectional compression.
Revised: Similarly to Section 4.1, we can consider only devices’ or bidirectional compression.

Original: Inparticular, in the line 2 of Table 1 we put  results for δ serv = 1 , δ dev =
Revised: In particular,we put the results for  serv = 1 ,  dev =

Original: δ , L m =
Revised:  , L m =

Original: ˜ L = L and 1 − τ = β .
Revised: ˜ L = L and 1  ⌧ =  in the line 2 of Table 1.

Original: In the line 4 of Table 1 there are results for δ serv =
Revised: Inthe line 4 of Table 1, there are results for  serv =

Original: δ dev =
Revised:  dev =

Original: δ , L m =
Revised:  , L m =

Original: ˜ L = L and 1 − τ = β .
Revised: ˜ L = L and 1  ⌧ =  .

Original:  Experiments
Revised:  Experiments

Original: 5.1 Bilinear Saddle Point Problem
Revised: 5.1 Bilinear Saddle Point Problem

Original: We start our experiments with a distributed bilinear problem, i.e. the problem (5) with
Revised: We start our experiments with a distributed bilinear problem, i.e. the problem (5) with

Original: [Equation]
Revised: [Equation]

Original: where A m ∈ R d × d , a m , b m ∈ R d .
Revised: where A m 2 R d ⇥ d , a m , b m 2 R d .

Original: This problem is λ -strongly convex–strongly-concave and,moreover, all functions g m are ∥ A m ∥ 2 -smooth.
Revised: This problem is  -strongly convex–strongly-concave and,moreover, all functions g m are k A m k 2 -smooth.

Original: Therefore, such a distributed problem is well suitedfor the primary comparison of our methods.
Revised: Therefore, such a distributed problem is well suitedfor the primary comparison of our methods.

Original: We take d = 100 and generate positive definite matrices
Revised: We take d = 100 and generate positive deﬁnite matrices

Original: A m and vectors a m , b m randomly, λ is chosen as max m ∥
Revised: A m and vectors a m , b m randomly,  is chosen as max m k

Original: A m ∥ 2 / 10 5 .
Revised: A m k 2 / 10 5 .

Original: The purpose of the experiment is to understand whether the MASHA1 and MASHA2 methods aresuperior to those in the literature.
Revised: The purpose of the experiment is to understand whether the MASHA1 and MASHA2 methods aresuperior to those in the literature.

Original: As a comparison, we take QGD [2] with Random 30%, classical
Revised: As a comparison, we take QGD [2] with Random 30%, classical

Original: Error Feedback [78] with Top 30% compression, as well as CEG (Section G) – Compressed Extra 
Revised: Error Feedback [78] with Top 30% compression, as well as CEG (Compressed Extra Gradient,

Original: Gradient, each step of which we use Random 30%.
Revised: Section I), in each step of which we use Random 30%.

Original: In MASHA1 (Algorithm 1) we also used Random30%, in MASHA2 (Algorithm 2) –
Revised: For MASHA1 (Algorithm 1), we also used Random 30%, for MASHA2 (Algorithm 2) –

Original: Top 30%.
Revised: Top 30%.

Original: See Figure 1.
Revised: See Figure 1 for the results of this experiment.

Original: The stepsizes of all methods are chosenfor best convergence.
Revised: The stepsizes of all methods are chosen for best convergence.

Original: [Figure] || z N z * ||/|| z z * ||
Revised: ) in iterations and in Mbytes for (7).

Original: QSGDEFCEGAlg 1 Alg We see that methods based on gradient descent (QSGD and EF) diverge.
Revised: We see that methods basedon the gradient descent(QSGD and EF) convergeslowly.

Original: This confirmsthat one needs to use method specifically designed for saddle point problems (for example, the extragradient method), and not classical optimization methods.
Revised: This conﬁrms thatone needs to use methodsspeciﬁcally designed forsaddle point problems (forexample, the extragradientmethod), and not classicaloptimization methods.

Original: The much slower convergence of CEG shows the efficiency of our approach in which we compress the differences F m ( z k +1 / 2 ) − F ( w k ) .
Revised: Themuch slower convergenceof CEG demonstrates theefﬁciency of our approach, in which we compress the differences F m ( z k +1 / 2 )  F ( w k ) .

Original:  MASHA2 wins MASHA1 . This shows that in practice a contractive compressor can perform better than an unbiased one with the same parameters.
Revised: Also,it can be seen that MASHA2 outperforms MASHA1 , which shows that in practice a contractivecompressor can perform better than an unbiased one with the same parameters.

Original: 5.2 Adversarial Training of Transformers
Revised: 5.2 Adversarial Training of Transformers

Original: We now evaluate how compression performs for variational inequalities (and for saddle point problems, as a special case) in a more practically motivated scenario.
Revised: We now evaluate how compression performs for variational inequalities (and for saddle point problems, as a special case) in a more practically motivated scenario.

Original: Indeed, saddle point problems(special case of variational inequalities) have sample applications in machine learning, includingadversarial training .
Revised: Indeed, saddle point problems (aspecial case of VIs) have numerous applications in machine learning, including adversarial training .

Original: We consider one of these tasks and train a transformer-based masked language model [82, 18, 55] using a fleet of 16 low-cost preemptible workers with T4 GPU and lowbandwidth interconnect.
Revised: We consider one of these tasks and train a transformer-based masked language model [82, 18, 56]using a ﬂeet of 16 low-cost preemptible workers with T4 GPU and low-bandwidth interconnect.

Original: For this task, we use the compute-efficient adversarial training regimenproposed for transformers by [91, 53].
Revised: For this task, we use the compute-efﬁcient adversarial training task proposed for transformersby [91, 54].

Original: Formally, the adversarial formulation of the problem is themin-max problem
Revised: Formally, the adversarial formulation of the problem is the min-max problem

Original: [Equation]
Revised: [Equation]

Original: where w are the weights of the model, { ( x n , y n ) } N n =1 are pairs ofthe training data, ρ is the socalled adversarial noise which introduces a perturbation in the data, and λ are the regularizationparameters.
Revised: where w are the weights of the model, { ( x n , y n ) } Nn =1 are pairs of training data, ⇢ is the so-calledadversarial noise which introduces a perturbation in the data, and  is the regularization hyperparameter.

Original: To make our setup more realistic, we train ALBERT-large with layer sharing [46],which was recently shown to be much more communication-efficient during training [74, 20].
Revised: To make our setup more realistic, we train ALBERT-large with layer sharing [47], whichwas recently shown to be much more communication-efﬁcient during training [74, 20].

Original: Wetrain our model on a combination of Bookcorpus and Wikipedia datasets with the same optimizer(LAMB) and parameters as in the original paper [46], use the adversarial training configurationof [91], and follow system design considerations for preemptible instances [74].
Revised: We train ourmodel on a combination of BookCorpus and Wikipedia datasets with the same optimizer (LAMB) and parameters as in the original paper [47], use the adversarial training conﬁguration of [91], andfollow system design considerations for preemptible instances [74].

Original: In terms of communication, we consider 4 different setups for gradient compression: the “baseline”strategy with uncompressed gradients, full 8-bit quantization [17, 49], mixed 8-bit quantization, and
Revised: In terms of communication, we consider 4 different setups for gradient compression: the “baseline”strategy with uncompressed gradients, full 8-bit quantization [17, 50], mixed 8-bit quantization, and

Original: Power compression [84] with rank r =8 .
Revised: PowerSGD compression [84] with rank r =8 .

Original: For mixed 8-bit quantization and Power we only applycompression to gradient tensors with more than 2 16 elements, sending smaller ones uncompressed.
Revised: For mixed 8-bit quantization and PowerSGD, weonly apply compression to gradient tensors with more than 2 16 elements, leaving the smaller onesuncompressed.

Original: These small tensors represent layer biases and LayerNorm scales [6] that collectively amount to≤ 1% of the total gradient, but can be more difficult to compress than regular weight tensors.
Revised: These small tensors represent layer biases and LayerNorm scales [6] that collectivelyamount to  1% of the total gradient, but can be more difﬁcult to compress than regular weighttensors.

Original: Finally,since Power is a biased compression algorithm, we use error feedback [43, 72] with a modifiedformulation proposed by [84].
Revised: Finally, since PowerSGD is a biased compression algorithm, we use error feedback [44, 72]with a modiﬁed formulation proposed by [84].

Original: For all experimental setups, we report learning curves in terms of themodel training objective, similarly to [24, 74].
Revised: For all experimental setups, we report learningcurves in terms of the model training objective, similarly to [24, 74].

Original: To quantify the differences in training loss better, wealso evaluate the downstream performance for each model on several popular tasks from [85] aftereach model was trained on approximately 80 billion tokens.
Revised: To quantify the differencesin training quality better, we also evaluate the downstream performance for each model on severalpopular tasks from [85] after each model was trained on approximately 80 billion tokens.

Original: Finally, we measure the communicationefficiency of each proposed strategy by measuring the average wall time per communication roundwhen all 16 workers are active.
Revised: Finally,we measure the communication efﬁciency of each proposed strategy by measuring the average walltime per communication round when all 16 workers are active.

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: The learning curves in Figure 2(upper) follow a predictable pattern, with more extreme compression techniques demonstratingslower per-iteration convergence.
Revised: The learning curves in Figure 2 (upper) followa predictable pattern,with more extremecompression techniquesdemonstrating slowerper-iteration convergence.

Original: One curious exception to that is full8-bit quantization, which was unable to achieve competitive training loss.
Revised: One curiousexception to that is full8-bit quantization, whichwas unable to achievea competitive trainingloss.

Original: The remaining three setups converge to similar loss values below 2.
Revised: The remaining threesetups converge to similarloss values below 2.

Original: Both the baseline andmixed 8-bit compression show similar values in terms of downstreamperformance, with Power compression showing mild degradation.
Revised: Boththe baseline and mixed8-bit compression showsimilar results in termsof downstream performance, with PowerSGDshowing a mild degradation.

Original: Butin terms of information transfer time, methods using compression (especially Power) are significantly superior to the method without compression.
Revised: However, in terms of information transfer time, methods usingcompression (especially PowerSGD) are signiﬁcantly superior to the method without compression.

Original: This makes it possible to use such techniquesto increase the training time without sacrificing quality.
Revised: This makes it possible to use such techniques to reduce the training time without sacriﬁcing quality.

Revised:  Conclusion

Revised: In this paper, we present algorithms with unbiased and contractive compression for solving distributed VIs and SPPs.

Revised: Our algorithms are presented in deterministic, stochastic and federated versions.

Revised: All basic algorithms and their modiﬁcations support bidirectional compression.

Revised: Experimentsconﬁrm the efﬁciency of both our algorithms and the use of compression for solving large-scale VIsin general.

Revised: In future work, we believe that it is important to address the necessity of sending uncompressed information on some iterations.

Revised: Although full packages are rarely transmitted, this is a slight limitationof our approach.

Revised: Lower bounds for compression methods are also an interesting area of research.

Revised: Atthe moment, there are no such results neither for VIs and SPPs nor for minimization.

Revised: In Appendix

Revised: C, we only hypothesize the optimality of our methods and back it up with analogies; provable lowerestimates could complete the story with compressed methods.