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 [65, 31, 22].
Revised: 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].

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, for example, in an elegant unifying mathematical framework [9].

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

Original: Besides their usefulness in studying GANs and alternative adversarial learning models [57], VIs/SPPshave recently 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 loss [39] or non-separable regularizer[7] and reinforcement learning [69, 66, 38].
Revised: 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]

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, 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.

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, for classical and  much better understood supervised machine learning /minimization problems, researchers and practitioners face other challenges, which, until recently, have been outside of VI’s research.
Revised: On the other hand, in the domain of classical, and hence 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.

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

Original: Due to their desire to improve the generalization of deployed models, machine learning engineersneed to rely on training datasets of ever increasing sizes and on elaborate over-parametrized models[5].
Revised: 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].

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

Original: When training such modern 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 [45, 76, 25, 29].
Revised: When training such modern 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].

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 orderoptimization methods is communication compression.

Original: The idea behind this technique is rooted inthe observation that in practical implementations itis 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 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].

Original: If the number of parallel workers is large enough, the noise introduced by compression is reduced, and training with compressed communication will often lead to  comparabletest error while reducing the amount of communicated bits, which results in faster training, both intheory and practice [59, 29].
Revised: If the number of parallel workers is large enough, the noise introduced by compression is reduced, and training with compressed communication will often lead to the comparabletest error while reducing the amount of communicated bits, which results in faster training, both intheory and practice [58, 29].

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

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

Original: Let us give the maindeﬁnitions.
Revised: Let us give the maindefinitions.

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 an unbiased compressionoperator 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   1 such that
Revised: 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

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 2 R d is k z k bits := bd .
Revised: 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 .

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

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

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

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 [46] andlater studied by many authors [62, 41], including in a distributed environment [77, 52, 61, 73].
Revised: 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].

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

Original: But despitethe fact that the use of compression is one of the most popular communication-efﬁcient approachesfor distributed minimization problems, no work has yet paid attention to the compression techniqueneither 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.
Revised: 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.

Original: 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.
Revised: 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.

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, 59, 29, 72] and embedding them into established,efﬁcient methods for solving VIs/SPPs [46, 62, 41, 1].
Revised: 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].

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

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

Original:  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 =
Revised: 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 =

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

Original: Convergence of all our methods are analyzedin strongly-monotone (strongly convex - strongly concave), monotone (convex - concave)non-monotone/minty (non-convex-non-concave) cases.
Revised: Convergence of all our methodsare analyzed in 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 compressor demands a different algorithmic designand a different analysis.

Original: In particular, contractive compressors are notoriously hard to analyze evenfor optimization problems [44, 72].
Revised: In particular, contractive compressors are notoriously hard to analyze evenfor optimization problems [43, 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 summarize in Table 1 (Appendix A).
Revised: We establish a number of theoretical complexity results for our methods, which we summarizein Table 1.

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

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

Original: Then weget that from the point of view of the transmitted information (and also time for communications),
Revised: Then we getthat from the point of view of the transmitted information (and also time for communications),

Original: MASHA1 is better by a factor p 1 /  + 1 /M
Revised: MASHA1 is better by a factor (cid:112) 1 /β + 1 /M

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

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

Original: For example,
Revised: For example,

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

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

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,the computation of the full operators/gradients is expensive, then we need to deal with stochasticrealizations.
Revised: But often in practice, thecomputation of the full operators F m is expensive, then we need to deal with stochastic realizations.

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

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

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

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

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

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

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

Revised: [Figure]

Original: Federated learning [45, 42] is an important and popular branch of distributed methods.
Revised: Federated learning [44, 41] is an important and popular branch of distributed methods.

Original: Therefore,a good bonus for the algorithm is that it can be easily adapted for it.
Revised: 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: In such circumstances, devices can  have weak and slowconnections, or they can even disconnect for a while. At such moments, it is not necessary to interrupt the learning process, and only available devices can be used.
Revised: Moreover, devices can not only have a bad connection,in which one needs to compress data heavily, they can simply disconnect from the learning process.

Original:  Therefore, we introduce twomodiﬁcations: PP-MASHA1 (Algorithm 7) and PP-MASHA2 (Algorithm 8), that support the mode ofpartial participation of devices in the learning process.
Revised: [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.

Original: For minimization problems, a combinationof 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 H.
Revised: The results are contained in Appendix F.

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 with contractive compressors, only use compression when transferringinformation from devices to the server.

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

Original: Insuch situations it alsomakes sense to compress the information when sending it from the server to the agents. We canhighlight some works on bidirectional unbiased [68] and contractive compressors  [90, 81, 55, 23]for distributed minimization problems.
Revised: 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.

Original: But most of these methods have their small shortcomingsin theoretical analysis such as  deterministic setting only, homogeneity of local functions, etc.
Revised: But most of these methods have their small drawbacks,such as only deterministic case, homogeneity not suitable for federated learning etc.

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

Original: See Appendix E and F for details.
Revised: See Appendix C and D 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 (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) problemFind z ⇤ [Equation]
Revised: We study distributed variational inequality (VI) problem

Original:  R d such that [Equation] where F : R d ! R d is an operator with certain favorable properties (e.g., Lipschitzness andmonotonicity).
Revised: 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).

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

Original: In Appendix B, we give main examples of VIs toshow the breadth of this formalism.

Original: Here we mention only that for the saddle point problem
Revised: 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: = [ r x g ( x, y ) , r y g ( x, y )] .
Revised: = [ ∇ x g ( x, y ) , −∇ 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 two key assumptions - both 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 2 R dwe have k F ( z 1 )  F ( z 2 )
Revised: The operator F is L -Lipschitz continuous, i.e. for all z 1 , z 2 ∈ R dwe have ∥ F ( z 1 ) − F ( z 2 )

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

Original: 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 )
Revised: 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 ) −

Original: k
Revised: [Equation]

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

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

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

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 2 R d we haveh F ( z 1 )  F ( z 2 ) , z 1  z 2 i  µ k z 1  z 2 k 2 .(M)
Revised: 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)

Original: Monotonicity.
Revised: Monotonicity.

Original: The operator F is monotone, i.e. for all z 1 , z 
Revised: The operator F is monotone, i.e. for all z 1 , z 2 ∈ R

Original: 2 2 R d we have h F ( z 1 )  172 F ( z 2 ) , z 1  z 2 i  0 .(NM)
Revised: d we have ⟨ F ( z 1 ) − F ( z 2 ) , z 1 − z 2 ⟩ ≥ 0 .(NM)

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

Original: 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 .
Revised: 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 .

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, 58, 53, 43,36, 19], particularly with the setup, which is somewhat appropriate for GANS [51, 52, 21, 8].
Revised: 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].

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 the algorithm ( MASHA1 ) with unbiased compression.

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

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

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.
Revised: All these methods are modifications of the classical GD.

Original: For instance, the authorsof [2] compress stochastic gradients.
Revised: For instance, the authorsof [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 GD-type methods for VIsas well.

Original: But it is a well-known fact that GD-type methods can give bad convergence estimates (see
Revised: But it is a well-known fact that GD-type methods can give bad convergence estimates (see

Original: 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: 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.

Original: From a practical point of view, this approach can also fail (see QSGD and
Revised: From a practical point of view, this approach can also fail (see QSGD and

Original: EF in Section 5.1).
Revised: EF in Section 5.1).

Original: In the non-distributed case, this problem has long been solved and the Extra
Revised: In the non-distributed case, this problem has long been solved and the Extra

Original: Gradient method [46, 62, 41] is used instead of GD:
Revised: Gradient method [45, 61, 40] is used instead of GD:

Revised: [Equation]

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

Original: [Equation] which is considerably worse in terms of L / µ than the original Extra Gradient method.
Revised: L 2 / µ 2 (cid:1) ,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 k Q k ( F ( z k +1 / 2 ))
Revised: The keyproblem is that in the analysis one has to deal with ∥ Q k ( F ( z k +1 / 2 ))

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

Original: k 2 .
Revised: ∥ 2 .

Original: Without compression operators, such difference is easily evaluated using Assumption 3.1.
Revised: Without compression operators, such difference is easily evaluated using Assumption 3.1.

Original: But when thecompression operators are different (in fact the same, but have different randomness) we cannotmake a good estimate for this term.
Revised: But when thecompression operators are different (in fact the same, but have different randomness) we cannotmake a good 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: 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 )) .

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: 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.

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

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 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.

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 increating 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, inour case, they help in creating MASHA1 and MASHA2 .

Original: 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. .

Original: Further, each device sends thecompressed version of the difference F m ( z k +1 / 2 )

Original:  F m ( w k ) to the server.

Original: The compression onthese transfers is done by their local { Q dev m } operators.

Original: The server aggregates the information fromdevices, averages it, compresses by Q serv operator and makes a broadcast to all devices.

Original: As a result,

Original: Algorithm 1 MASHA

Original: Parameters: Stepsize  > 0 , parameter ⌧ 2 (0; 1) , number of iterations K .
Revised: Parameters: Stepsize γ > 0 , parameter τ ∈ (0; 1) , number of iterations K .

Original: Initialization: Choose z  = w 0 2 Z .
Revised: Initialization: Choose z 0 = w 0 ∈ Z .

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 / 
+}

Original: [Equation]
Revised: = τz k + (1 − τ ) w k − γF ( w k )

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

Original: Q dev m ( F m

Original: [Equation]

Original: z

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

Original: Sends to devices g k = Q serv h [Equation] P
Revised: Sends to devices g k = Q serv (cid:104) 1 M (cid:80)

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

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

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

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 )
Revised: z k , sends F m ( w k +1 ) to server and gets F ( w k +1 )

Original: else w k +

Revised: else w k +

Original: =
Revised: =

Revised: 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.

Revised: Then it sends the compressed version of the difference

Revised: F m ( z k +1 / 2 ) − F m ( w k ) to the server, and the server does a reverse compressed broadcast.

Original: w k end for end for an unbiased estimate of F ( z k +1 / 2 )  F ( w k ) appears at each node.
Revised: As aresult, an unbiased estimate of F ( z k +1 / 2 ) − F ( w k ) appears on each node.

Original: Also, the nodes receive one bitof information b k .
Revised: Also, the nodes receive abit of information b k .

Original: This bit is generated randomly on the server and is equal to 1 with probability1  ⌧ (where 1  ⌧ is small).
Revised: This bit is generated randomly on the server and is equal to 1 with probability1 − τ (where 1 − τ is small).

Original: 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: Note that b k can be generated locally, it is enough to use the samerandom generator and set the same seed on all devices.

Original: Next, the deviceslocally make a ﬁnal updateon z k +1 .
Revised: Then, all devices make a final update onz k +1 , and also update the w k +1 =

Original: The ﬁnal step is an update of w k +1 : if  b k = 1, then w k +1 = z k or otherwise w k [-+
-] =
Revised: z k +1 point if the b k = 1 or save it from the previous iterationw k [-+
-]+1 =

Original: w k .
Revised: w k .

Original: In the case when  w k +

Revised: In the case when the point w k +

Original: =
Revised: =

Original: z k  , we need to exchange the uncompressed values of F m ( w k +1 ) in orderto ensure that at the beginning of the next iteration the value of F ( w k +1 ) is known to all agents.
Revised: z k +1 , we need to exchange the full values of 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.

Original: Weuse a possibly difference compressor on each device and also on the server.
Revised: We use a possibly difference compressor on each device and also on the server.

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

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

Original: • in strongly monotone case (in terms of E [ k z K  z ⇤ k 2 ] ⇠ " ): O ([ 11  ⌧ + C q µ p 1  ⌧ ] log 1 " );• in monotone case (in terms of E max z 2 C [ h F ( u ) , ( 1 K
Revised: • in strongly-monotone case: E [ ∥ z K − z ∗ ∥ 2 + ∥ w K − z ∗ ∥ 2 ] = O ([ 11 − τ + C q µ √ 1 − τ ] log 1 ε );

Original: [Equation] K {+− 1 k =
+}
Revised: • in monotone case: E max z ∈C [ ⟨ F ( u ) , ( 1 K (cid:80) K {+− 1 k =
+}

Original: [Equation] 1 k 
Revised: z k +1 / 2 ) − u ⟩ ]

Original: [Equation]
Revised: =

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

Original: [Equation] ⇤ k 2 " p 1  ⌧ ); • in non-monotone case (in terms of E [ 1 K {+(cid:80) K − 1 k =
 ∥ F ( w k ) ∥ 2 ]+}
Revised: • in non-monotone case: E [ 1 K {+(cid:80) K − 1 k =
 ∥ F ( w k ) ∥ 2 ]+}

Original: [Equation]
Revised: = O ( C 2 q ∥ z 0 − z ∗ ∥ 2 ε 2 (1 − τ ) );

Original: where C 2 q = q serv M [Equation]
Revised: where C 2 q = q serv M 2 (cid:80) Mm =1 ( q dev m L 2 m + ( M − 1)˜ L 2 ) .

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

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

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

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

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

Original: Let us analyze the case of compressions only onthe devices’ side ( q serv = 1 ).
Revised: Let us analyze the case of compressions 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  dev m =  , also
Revised: q and β dev m = β , 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 /  + 1  ⌧ ) bits
Revised: Then at each iteration the device sends O ( 1 / β + 1 − τ ) bits– each time

Original: – each time information 
Revised: information compressed by β times

Original: compressed by  times and with probability 1  ⌧ we send the fullpackage.
Revised: and with probability 1 − τ we send the full

Revised: package.

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

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 are satisfied.

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

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

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

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

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

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

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

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

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

Original: One can also consider the case of bidirectional compression ( q serv 6 = 1 ).
Revised: One can also consider the case of bidirectional compression ( q serv ̸ = 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 ,  serv =  dev m =  and 1  ⌧ = 1 /  .
Revised: q dev m = q , β serv = β dev m = β and 1 − τ = 1 / β .

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 put a contractive cospressor instead of an unbiased one, the method may diverge even forquadratic problems [10].

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

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

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

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

Original: 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 .
Revised: 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: 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.

Original: Parameters: Stepsize  > 0 , parameter ⌧ , number of iterations K .
Revised: 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. As a result f ( x ) − f ( x ∗ ) is used as a convergence criterion even in the strongly convex case.

Original: Initialization: Choose z 
Revised: 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 ∥ . Theorem 4.3 Let Assumption 3.1 and one case of Assumption 3.2 are satisfied.

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

Revised: • in strongly-monotone case: E [ ∥ ˆ z K − z ∗ ∥ 2 + ∥

Revised: w K − z ∗ ∥ 2 ]

Original: =
Revised: =

Revised: O ([ 11 − τ

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 =
+}

Revised: z k +1 / 2 ) − z ⟩ ]

Original: w 0 2 Z , e 0 m =  0 , e 0 = 0 .
Revised: = O ( δ dev δ serv ˜ L ∥ z 0 − z ∗ ∥ 2 ε √ 1 − τ );

Revised: Algorithm

Revised: Parameters: Stepsize γ > 0 , parameter τ , number of iterations K .

Revised: Initialization: Choose z 0 = w 0 ∈ 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 / 2{+
+} =
Revised: , K − 1 do for each device m in parallel do z k +1 / {+
+} =

Original: ⌧ z
Revised: τz k

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

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

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

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

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

Original: Mm =1 g km + e k
Revised: Mm = 1 g km + e k

Original: i e k 
Revised: (cid:105) e k +1 = e k + 1 M (cid:80)

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

Original: e k + 1 M P
Revised: 1 g km − g k

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

Original: k +1{++
+} =
Revised: {++
+} =

Original: z k +1 / 2   g k
Revised: z k +1 / 2 − γg k

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

Original: z k , sendsF m ( w k +1 ) to server and getsF ( w k +1 ) 
Revised: 1 then w k + k , sends to server and gets ( w k +1 ) else

Original: else w k [-+
-] =
Revised: w k [-+
-]+1 =

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

Original: 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). MASHA2 raises another problem for theoretical analysis, howto combine MASHA1 and the error feedback technique. The analysis of methods with error compensation for the minimization problem min x f ( x ) is entirely tied to the existence of the functionf [79, 71, 72]. In particular, the differences ( f ( · )  f ( x ⇤ )) appear in the whole analysis and iskey in the technical lemmas.{+non-monotone case: E [ 1 K (cid:80) K − 1 k =
+}
Revised: • in {+non-monotone case: E [ 1 K (cid:80) K − 1 k =
+}

Original: As a result ( f ( · )  f ( x ⇤ )) is used as a convergence criterion even inthe strongly convex case. But for VIs there is no function f , only the operator F(the existence ofg ( x, y ) in SPP setup does not save the situation). This problem is solved in the proof of Theorem4.3 by using an additional sequence k z k +1 / 2  w k k . Theorem 4.3 Let Assumption 3.1 and one case of Assumption 3.2 are satisﬁed.
Revised: ∥ F

Original: Then for some step the following estimates on MASHA2 number of iterations to achieve " -solution holds • in strongly monotone case (in terms of E [ k ˆ z K  z ⇤ k2 ] ⇠ " ): O ([ 11  ⌧ +  dev  serv ˜ L µ p 1  ⌧ ] log 1 " ); • in monotone case ( E max z 2 C [ h F ( u ) [-, ( 1 K P K  1 k =
 z k +1 /-] 2)  u i ]⇠ " ):
Revised: w k ) [-, ( 1 K P K  1 k =
 z k +1 /-]∥ 2 ]

Original:  O ( dev  serv ˜
Revised: = O (

Original: [-L k z 
  z ⇤ k 2 " p ⌧ ); • in non-monotone case (in terms of E [ 1 K [Equation]-]
Revised: [-L k z 
  z ⇤ k 2 " p ⌧ ); • in non-monotone case (in terms of E [ 1 K [Equation]-]δ dev δ

Original: A full listing of the algorithm, as well as a full statement of the theorem with proof, can be found in
Revised: A full listing of the algorithm, as well as a full statement of the theorem with proof, can be found in

Original: Appendix F.
Revised: Appendix D.

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

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

Original:  , L m =
Revised: δ , L m =

Original: ˜ L = L and1  ⌧ =  .
Revised: ˜ L = L and 1 − τ = β .

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

Original:  dev =
Revised: δ dev =

Original:  , L m =
Revised: δ , L m =

Original: ˜ L = L and1  ⌧ =  .
Revised: ˜ L = L and 1 − τ = β .

Original:  Experiments5.1 Bilinear Saddle Point Problem
Revised:  Experiments

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

Original: This problem is  -strongly convex–strongly-concave and,moreover, all functions g m are k A m k 2 -smooth.
Revised: This problem is λ -strongly convex–strongly-concave and,moreover, all functions g m are ∥ A m ∥ 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,  is chosen as max m k
Revised: A m and vectors a m , b m randomly, λ is chosen as max m ∥

Original: A m k 2 / 10 5 .
Revised: A m ∥ 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 I) –
Revised: Error Feedback [78] with Top 30% compression, as well as CEG (Section G) –

Original: Compressed Extra
Revised: Compressed Extra

Original: Gradient, each step of which we use Random 30%.
Revised: Gradient, each step of which we use Random 30%.

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

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

Original: See Figure 1.
Revised: See Figure 1.

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

Revised: [Figure]

Revised: ||

Original: The much slower convergence of CEG shows the efﬁciency of our approach in which we compressthe differences F m ( z k +1 / 2 )  F ( w k ) .
Revised: z N z * ||/|| z z * || QSGDEFCEGAlg 1 Alg We see that methods based on gradient descent (QSGD and EF) diverge.

Revised: This confirmsthat one needs to use method specifically designed for saddle point problems (for example, the extragradient method), and not classical optimization methods.

Revised: The much slower convergence of CEG shows the efficiency of our approach in which we compress the differences

Revised: F m ( z k +1 / 2 ) − F ( w k ) .

Original: MASHA2 wins MASHA1 .
Revised: MASHA2 wins MASHA1 .

Original: This shows that in practice acontractive compressor can perform better than an unbiased one with the same parameters.
Revised: This shows that in practice a contractive compressor 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(special case of variational inequalities) have sample applications in machine learning, includingadversarial training .

Original: 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 lowbandwidth interconnect.
Revised: 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.

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

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

Original: [Equation]
Revised: [Equation]

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

Original: To make our setup more realistic, we train ALBERT-large with layer sharing [47],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 [46],which was 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 [47], use the adversarial training conﬁgurationof [91], and follow system design considerations for preemptible instances [74].
Revised: 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].

Original: 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
Revised: 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

Original: Power compression [84] with rank r =8 .
Revised: Power 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 Power we only applycompression to gradient tensors with more than 2 16 elements, sending smaller ones uncompressed.

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 collectively amount to≤ 1% of the total gradient, but can be more difficult to compress than regular weight tensors.

Original: Finally,since Power is a biased compression algorithm, we use error feedback [44, 72] with a modiﬁedformulation proposed by [84].
Revised: Finally,since Power is a biased compression algorithm, we use error feedback [43, 72] with a modifiedformulation 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 learning curves in terms of themodel 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 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.

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 communicationefficiency of each proposed strategy by measuring the average wall time per communication roundwhen all 16 workers are active. [Figure]

Original: [Table]
Revised: [Table]

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

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

Original: The remainingthree setups converge tosimilar loss values below2.
Revised: The remaining three setups converge to similar loss values below 2.

Original: Both the baseline andmixed 8-bit compressionshow similar values interms of downstreamperformance, with Powercompression showing mild degradation.
Revised: Both the baseline andmixed 8-bit compression show similar values in terms of downstreamperformance, with Power compression showing mild degradation.

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

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

Original: Conclusion

Original: In this paper we present algorithms with unbiased and contractive compressions for solving distributed VIs and SPPs.

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

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

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

Original: In future works it is important to address the issue of the necessity to forward uncompressed information in some iterations.

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

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

Original: At the moment there are neither such results for VIs and SPPs, nor for minimizations.

Original: In Appendix

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