Original: On Convergence of FedProx: Local Dissimilarity Invariant Bounds, Non-smoothness and Beyond
Revised: On Convergence of FedProx: Local Dissimilarity Invariant Bounds, Non-smoothness and Beyond

Original: Anonymous
Revised: Anonymous

Original: Abstract
Revised: Abstract

Original: The FedProx algorithm is a simple yet powerful distributed proximal point optimization method widely used for federated learning (FL) over heterogeneous data.
Revised: The FedProx algorithm is a simple yet powerful distributed proximal point optimization method widely used for federated learning (FL) over heterogeneous data.

Original: Despite its popularity and remarkable success witnessed in practice, the theoreticalunderstanding of FedProx is largely underinvestigated: the appealing convergencebehavior of FedProx is so far characterized under certain non-standard and unrealistic dissimilarity assumptions of local functions, and the results are limited tosmooth optimization problems.
Revised: Despite its popularity and remarkable success witnessed in practice, the theoreticalunderstanding of FedProx is largely underinvestigated: the appealing convergencebehavior of FedProx is so far characterized under certain non-standard and unrealistic dissimilarity assumptions of local functions, and the results are limited tosmooth optimization problems.

Original: In order to remedy these deﬁciencies, we developa novel local dissimilarity invariant convergence theory for FedProx and its minibatch stochastic extension through the lens of algorithmic stability.
Revised: In order to remedy these deﬁciencies, we developa novel local dissimilarity invariant convergence theory for FedProx and its minibatch stochastic extension through the lens of algorithmic stability.

Original: As a result, wecontribute to derive several new and deeper insights into FedProx for non-convexfederated optimization including: 1) convergence guarantees independent on local dissimilarity type conditions; 2) convergence guarantees for non-smooth FLproblems; and 3) linear speedup with respect to size of minibatch and number ofsampled devices.
Revised: As a result, wecontribute to derive several new and deeper insights into FedProx for non-convexfederated optimization including: 1) convergence guarantees independent on local dissimilarity type conditions; 2) convergence guarantees for non-smooth FLproblems; and 3) linear speedup with respect to size of minibatch and number ofsampled devices.

Original: Our theory for the ﬁrst time reveals that local dissimilarity andsmoothness are not must-have for FedProx to get favorable complexity bounds.
Revised: Our theory for the ﬁrst time reveals that local dissimilarity andsmoothness are not must-have for FedProx to get favorable complexity bounds.

Original:  Introduction
Revised:  Introduction

Original: Federated Learning (FL) has recently emerged as a promising paradigm for communication-efﬁcientdistributed learning on remote devices, such as smartphones, internet of things or agents [Koneˇcn ` yet al., 2016, Yang et al., 2019].
Revised: Federated Learning (FL) has recently emerged as a promising paradigm for communication-efﬁcientdistributed learning on remote devices, such as smartphones, internet of things or agents [Koneˇcn ` yet al., 2016, Yang et al., 2019].

Original: The goal of FL is to collaboratively train a shared model that worksfavorably for all the local data but without requiring the learners to transmit raw data across thenetwork.
Revised: The goal of FL is to collaboratively train a shared model that worksfavorably for all the local data but without requiring the learners to transmit raw data across thenetwork.

Original: The principle of optimizing a global model while keeping data localized can be beneﬁcialfor both computational efﬁciency and data privacy [Bhowmick et al., 2018].
Revised: The principle of optimizing a global model while keeping data localized can be beneﬁcialfor both computational efﬁciency and data privacy [Bhowmick et al., 2018].

Original: While resemblingthe classic distributed learning regimes, there are two most distinct features associated with FL: 1)large statistical heterogeneity of local data mainly due to the non-iid manner of data generalizationand collection across the devices [Hard et al., 2020]; and 2) partial participation of devices in thenetwork mainly due to the massive number of devices.
Revised: While resemblingthe classic distributed learning regimes, there are two most distinct features associated with FL: 1)large statistical heterogeneity of local data mainly due to the non-iid manner of data generalizationand collection across the devices [Hard et al., 2020]; and 2) partial participants of devices in thenetwork mainly due to the massive number of devices.

Original: These fundamental challenges make FL highlydemanding to tackle, both in terms of optimization algorithm design and in terms of theoreticalunderstanding of convergence behavior [Li et al., 2020a].
Revised: These fundamental challenges make FL highlydemanding to tackle, both in terms of optimization algorithm design and in terms of theoreticalunderstanding of convergence behavior [Li et al., 2020a].

Original: FL is most conventionally formulated as the following problem of global population risk minimizationaveraged over a set of M devices:
Revised: FL is most conventionally formulated as the following problem of global population risk minimizationaveraged over a set of M devices:

Original: [Equation]
Revised: [Equation]

Original: where R ( m ) is the local population risk on device m , (cid:96) ( m ) :
Revised: where R ( m ) is the local population risk on device m , (cid:96) ( m ) :

Original: R p × Z ( m ) (cid:55)→ R + is a non-negativeloss function whose value (cid:96) ( w ; Z ( m ) ) measures the loss over a random data point Z
Revised: R p × Z ( m ) (cid:55)→ R + is a non-negativeloss function whose value (cid:96) ( w ; Z ( m ) ) measures the loss over a random data point Z

Original: ( m ) ∈
Revised: ( m ) ∈

Original: Z ( m ) with parameter w , D ( m ) represents an underlying random data distribution over Z ( m ) .
Revised: Z ( m ) with parameter w , D ( m ) represents an underlying random data distribution over Z ( m ) .

Original: Since the datadistribution is typically unknown, the following empirical risk minimization (ERM) version of (1) isoften considered alternatively:
Revised: Since the datadistribution is typically unknown, the following empirical risk minimization (ERM) version of (1) isoften considered alternatively:

Original: [Equation]
Revised: [Equation]

Original: where R ( m ) erm is the local empirical risk over the training sample D ( m )
Revised: where R ( m ) erm is the local empirical risk over the training sample D ( m )

Original: = { z ( m ) i } N m i
Revised: = { z ( m ) i } N m i

Original: =1 on device m .
Revised: =1 on device m .

Original: Thesample size N m may vary signiﬁcantly across devices, which can be regarded as another source ofdata heterogeneity.
Revised: Thesample size N m may vary signiﬁcantly across devices, which can be regarded as another source ofdata heterogeneity.

Original: Federated optimization algorithms for solving (1) or (2) have attracted signiﬁcantresearch interest from both academia and industry, with a rich body of efﬁcient solutions developedthat can ﬂexibly adapt to the communication-computation tradeoffs and data/system heterogeneity.
Revised: Federated optimization algorithms for solving (1) or (2) have attracted signiﬁcantresearch interest from both academia and industry, with a rich body of efﬁcient solutions developedthat can ﬂexibly adapt to the communication-computation tradeoffs and data/system heterogeneity.

Original: Several popularly used FL algorithms for this setting include FedAvg [McMahan et al., 2017],
Revised: Several popularly used FL algorithms for this setting include FedAvg [McMahan et al., 2017],

Original: FedProx [Li et al., 2020b], SCAFFOLD [Karimireddy et al., 2020], and FedPD [Zhang et al., 2020], toname a few.
Revised: FedProx [Li et al., 2020b], SCAFFOLD [Karimireddy et al., 2020], and FedPD [Zhang et al., 2020], toname a few.

Original: A consensus among these methods on communication-efﬁcient implementation is tryingto extensively update the local models (e.g., with plenty epochs of local optimization) over subsetsof devices so as to quickly ﬁnd an optimal global model using a minimal number of inter-devicecommunication rounds for model aggregation.
Revised: A consensus among these methods on communication-efﬁcient implementation is tryingto extensively update the local models (e.g., with plenty epochs of local optimization) over subsetsof devices so as to quickly ﬁnd an optimal global model using a minimal number of inter-devicecommunication rounds for model aggregation.

Original: In this paper, we revisit the FedProx algorithm which is one of the most prominent frameworks forheterogeneous federated optimization.
Revised: In this paper, we revisit the FedProx algorithm which is one of the most prominent frameworks forheterogeneous federated optimization.

Original: Reasons for the interests of FedProx include implementationsimplicity, low communication cost, promise in dealing with data heterogeneity and tolerance topartial participation of devices [Li et al., 2020b].
Revised: Reasons for the interests of FedProx include implementationsimplicity, low communication cost, promise in dealing with data heterogeneity and tolerance topartial participants of devices [Li et al., 2020b].

Original: We analyze its convergence behavior, exposeproblems, and propose alternatives more suitable for scaling up and generalization.
Revised: We analyze its convergence behavior, exposeproblems, and propose alternatives more suitable for scaling up and generalization.

Original: We contributeto derive some new and deeper theoretical insights into the algorithm from a novel perspective ofalgorithmic stability theory.
Revised: We contributeto derive some new and deeper theoretical insights into the algorithm from a novel perspective ofalgorithmic stability theory.

Original: 1.1 Review of FedProx
Revised: 1.1 Review of FedProx

Original: For solving FL problems in the presence of data heterogeneity, methods such as FedAvg based onlocal stochastic gradient descent (SGD) can fail to converge in practice when the selected devicesperform too many local updates [Li et al., 2020b].
Revised: For solving FL problems in the presence of data heterogeneity, methods such as FedAvg based onlocal stochastic gradient descent (SGD) can fail to converge in practice when the selected devicesperform too many local updates [Li et al., 2020b].

Original: To mitigate this issue, FedProx [Li et al., 2020b]was recently proposed for solving the empirical FL problem (2) using the (inexact) proximal pointupdate for local optimization.
Revised: To mitigate this issue, FedProx [Li et al., 2020b]was recently proposed for solving the empirical FL problem (2) using the (inexact) proximal pointupdate for local optimization.

Original: The beneﬁts of FedProx include: 1) it provides more stable localupdates by explicitly enforcing the local optimization in the vicinity of the global model to date; 2)the method comes with convergence guarantees for both convex and non-convex functions, evenunder partial participation and very dissimilar amounts of local updates [Li et al., 2020a].
Revised: The beneﬁts of FedProx include: 1) it provides more stable localupdates by explicitly enforcing the local optimization in the vicinity of the global model to date; 2)the method comes with convergence guarantees for both convex and non-convex functions, evenunder partial participation and very dissimilar amounts of local updates [Li et al., 2020a].

Original: Morespeciﬁcally, at each time instance t , FedProx uniformly randomly selects a subset I t ⊆
Revised: Morespeciﬁcally, at each time instance t , FedProx uniformly randomly selects a subset I t ⊆

Original: [ M ] ofdevices and introduces for each device
Revised: [ M ] ofdevices and introduces for each device

Original: ξ ∈
Revised: ξ ∈

Original: I t the following proximal point ERM sub-problem forlocal update around the previous global model w t − 1 :
Revised: I t the following proximal point ERM sub-problem forlocal update around the previous global model w t − 1 :

Original: [Equation]
Revised: [Equation]

Original: where η t > 0 is the learning rate that controls the impact of the proximal term.
Revised: where η t > 0 is the learning rate that controls the impact of the proximal term.

Original: Then the global modelis updated by uniformly aggregating those local updates from I t as
Revised: Then the global modelis updated by uniformly aggregating those local updates from I t as

Original: [Equation]
Revised: [Equation]

Original: In the extreme case of allowing η t → + ∞ in (3), FedProx reduces to the regime of FedAvg if using
Revised: In the extreme case of allowing η t → + ∞ in (3), FedProx reduces to the regime of FedAvg if using

Original: SGD for local optimization.
Revised: SGD for local optimization.

Original: Since its inception, FedProx and its variants have received signiﬁcantinterests in research [Pathak and Wainwright, 2020, Nguyen et al., 2020, Li et al., 2019a] and becomean algorithm of choice in application areas such as automatous driving [Donevski et al., 2021] andcomputer vision [He et al., 2021].
Revised: Since its inception, FedProx and its variants have received signiﬁcantinterests in research [Pathak and Wainwright, 2020, Nguyen et al., 2020, Li et al., 2019a] and becomean algorithm of choice in application areas such as automatous driving [Donevski et al., 2021] andcomputer vision [He et al., 2021].

Original: Theoretically, FedProx comes with convergence guaranteesunder the following bounded local gradient dissimilarity assumption that captures the statisticalheterogeneity of local objectives across the network:
Revised: Theoretically, FedProx comes with convergence guaranteesunder the following bounded local gradient dissimilarity assumption that captures the statisticalheterogeneity of local objectives across the network:

Original: Deﬁnition 1 ( ( B, H ) -LGD) .
Revised: Deﬁnition 1 ( ( B, H ) -LGD) .

Original: We say the local functions R ( m ) have ( B, H ) -local gradient dissimilarity (LGD) if the following holds for all w ∈ R p :
Revised: We say the local functions R ( m ) have ( B, H ) -local gradient dissimilarity (LGD) if the following holds for all w ∈ R p :

Original: [Equation]
Revised: [Equation]

Original: The deﬁnition naturally extends to the local empirical risks { R ( m ) erm } Mm =1 .
Revised: The deﬁnition naturally extends to the local empirical risks { R ( m ) erm } Mm =1 .

Original: Specially in the homogenous setting where R ( m ) ≡ ¯ R , ∀ m ∈ [ M ] , we have B = 1 and H = 0 .
Revised: Specially in the homogenous setting where R ( m ) ≡ ¯ R , ∀ m ∈ [ M ] , we have B = 1 and H = 0 .

Original: Under( B, 0) -LGD and some regularization condition on the modulus B , it was shown that FedProx fornon-convex problems requires T = O (cid:0) 1 (cid:15) (cid:1) rounds of inter-device communication to reach an (cid:15) -stationary solution, i.e., 1 T (cid:80)
Revised: Under( B, 0) -LGD and some regularization condition on the modulus B , it was shown that FedProx fornon-convex problems requires T = O (cid:0) 1 (cid:15) (cid:1) rounds of inter-device communication to reach an (cid:15) -stationary solution, i.e., 1 T (cid:80)

Original: Tt =

Revised: Tt =

Original: (cid:107)∇ ¯ R erm ( w t ) (cid:107) 2 ≤ (cid:15)
Revised: (cid:107)∇ ¯ R erm ( w t ) (cid:107) 2 ≤ (cid:15)

Original: [Li et al., 2020b].
Revised: [Li et al., 2020b].

Original: Similar guarantees have alsobeen established for a variant of FedProx with non-uniform model aggregation [Nguyen et al., 2020].
Revised: Similar guarantees have alsobeen established for a variant of FedProx with non-uniform model aggregation [Nguyen et al., 2020].

Original: Open issues and motivation.
Revised: Open issues and motivation.

Original: In spite of the remarkable success achieved by FedProx and its variants,there are still a number of important theoretical issues regarding the unrealistic assumptions, restrictiveproblem regimes and expensive local oracle cost that remain open for exploration, as speciﬁed below.
Revised: In spite of the remarkable success achieved by FedProx and its variants,there are still a number of important theoretical issues regarding the unrealistic assumptions, restrictiveproblem regimes and expensive local oracle cost that remain open for exploration, as speciﬁed below.

Original: • Local dissimilarity .
Revised: • Local dissimilarity .

Original: The appealing convergence behavior of FedProx is so far characterized undera key but non-standard ( B, H ) -LGD (cf.
Revised: The appealing convergence behavior of FedProx is so far characterized undera key but non-standard ( B, H ) -LGD (cf.

Original: Deﬁnition 1) condition with B > 0 and H = 0 .
Revised: Deﬁnition 1) condition with B > 0 and H = 0 .

Original: Such acondition is obviously unrealistic in practice: it essentially requires the local objectives share thesame stationary point as the global objective since (cid:107)∇ ¯ R erm ( w ) (cid:107) = 

Revised: Such acondition is obviously unrealistic in practice: it essentially requires the local objectives share thesame stationary point as the global objective since (cid:107)∇ ¯ R erm ( w ) (cid:107) =

Original: implies (cid:107)∇ R ( m ) erm ( w ) (cid:107) =
Revised: implies (cid:107)∇ R ( m ) erm ( w ) (cid:107) =

Original: 0 forall m ∈
Revised: 0 forall m ∈

Original: [ M ] .
Revised: [ M ] .

Original: However, if the optima of R ( m ) erm are exactly (or even approximately) the same, therewould be little point in distributing data across devices for federated learning.
Revised: However, if the optima of R ( m ) erm are exactly (or even approximately) the same, therewould be little point in distributing data across devices for federated learning.

Original: It is thus desirable tounderstand the convergence behavior of FedProx for heterogeneous FL without imposing stringentlocal dissimilarity conditions like ( B, 0) -LGD with B > 0 .•
Revised: It is thus desirable tounderstand the convergence behavior of FedProx for heterogeneous FL without imposing stringentlocal dissimilarity conditions like ( B, 0) -LGD with B > 0 .•

Original: Non-smooth optimization .
Revised: Non-smooth optimization .

Original: The existing convergence guarantees of FedProx are only availablefor FL with smooth losses.
Revised: The existing convergence guarantees of FedProx are only availablefor FL with smooth losses.

Original: More often than not, however, FL applications involve non-smoothobjectives due to the popularity of non-smooth losses (e
Revised: More often than not, however, FL applications involve non-smoothobjectives due to the popularity of non-smooth losses (e

Original: g., hinge loss and absolute loss) in machinelearning, and training deep neural networks with non-smooth activation like ReLU.
Revised: g., hinge loss and absolute loss) in machinelearning, and training deep neural networks with non-smooth activation like ReLU.

Original: Therefore, it isdesirable to understand the convergence behavior of FedProx in non-smooth problem regimes.• Local oracle complexity .
Revised: Therefore, it isdesirable to understand the convergence behavior of FedProx in non-smooth problem regimes.• Local oracle complexity .

Original: Unlike the (stochastic) ﬁrst-order oracles such as SGD used by FedAvg ,the proximal point oracle (3) for local update is by itself a full-batch ERM problem which tends tobe expensive to solve even approximately per-iteration.
Revised: Unlike the (stochastic) ﬁrst-order oracles such as SGD used by FedAvg ,the proximal point oracle (3) for local update is by itself a full-batch ERM problem which tends tobe expensive to solve even approximately per-iteration.

Original: Plus, due to the potentially imbalanceddata distribution over devices, the computational overload of the proximal point oracle couldvary signiﬁcantly across the network.
Revised: Plus, due to the potentially imbalanceddata distribution over devices, the computational overload of the proximal point oracle couldvary signiﬁcantly across the network.

Original: Therefore, it is important to investigate whether using thestochastic approximation to the proximal point oracle (3) can provably improve the computationalefﬁciency of FedProx .
Revised: Therefore, it is important to investigate whether using thestochastic approximation to the proximal point oracle (3) can provably improve the computationalefﬁciency of FedProx .

Original: Last but not least, existing convergence analysis of FedProx mainly focuses on the empirical FLproblem (2).
Revised: Last but not least, existing convergence analysis of FedProx mainly focuses on the empirical FLproblem (2).

Original: The optimality in terms of the population FL problem (1) is not yet clear for FedProx .
Revised: The optimality in terms of the population FL problem (1) is not yet clear for FedProx .

Original: The primary goal of this work is to remedy these theoretical issues simultaneously, so as to lay amore solid theoretical foundation for the popularly applied FedProx algorithm.
Revised: The primary goal of this work is to remedy these theoretical issues simultaneously, so as to lay amore solid theoretical foundation for the popularly applied FedProx algorithm.

Original: 1.2 Our Contributions
Revised: 1.2 Our Contributions

Original: In this paper, we make progress towards understanding the convergence behavior of FedProx fornon-convex heterogenous FL under weaker and more realistic conditions.
Revised: In this paper, we make progress towards understanding the convergence behavior of FedProx fornon-convex heterogenous FL under weaker and more realistic conditions.

Original: The main results are a setof local dissimilarity invariant bounds for smooth or non-smooth problems.
Revised: The main results are a setof local dissimilarity invariant bounds for smooth or non-smooth problems.

Original: Main results for the vanilla FedProx .
Revised: Main results for the vanilla FedProx .

Original: As a starting point to address the restrictiveness of localdissimilarity assumption, we provide a novel convergence analysis for the vanilla FedProx algorithmindependent of local dissimilarity type conditions.
Revised: As a starting point to address the restrictiveness of localdissimilarity assumption, we provide a novel convergence analysis for the vanilla FedProx algorithmindependent of local dissimilarity type conditions.

Original: For smooth and non-convex optimization problems,our result in Theorem 1 shows that the rate of convergence to a stationary point is upper bounded by
Revised: For smooth and non-convex optimization problems,our result in Theorem 1 shows that the rate of convergence to a stationary point is upper bounded by

Original: [Equation]
Revised: [Equation]

Original: where I is the number devices randomly selected for local update at each iteration.
Revised: where I is the number devices randomly selected for local update at each iteration.

Original: If all the devicesparticipate in the local updates for every round, i.e. I t =
Revised: If all the devicesparticipate in the local updates for every round, i.e. I t =

Original: [ M ] , the rate of convergence can be improvedto O ( 1 T 2 / 3 ) .
Revised: [ M ] , the rate of convergence can be improvedto O ( 1 T 2 / 3 ) .

Original: For T < I 3 , the rate in (4) is dominated by O ( 1 T 2 / 3 ) which gives the communicationcomplexity 1 (cid:15) 3 / 2 to achieve an (cid:15) -stationary solution.
Revised: For T < I 3 , the rate in (4) is dominated by O ( 1 T 2 / 3 ) which gives the communicationcomplexity 1 (cid:15) 3 / 2 to achieve an (cid:15) -stationary solution.

Original: On the other hand when T ≥ I 3 , the rate isdominated by O ( 1 √ TI ) which gives the communication complexity 1 I(cid:15) 2 .
Revised: On the other hand when T ≥ I 3 , the rate isdominated by O ( 1 √ TI ) which gives the communication complexity 1 I(cid:15) 2 .

Original: Compared to the alreadyknown O ( 1 (cid:15) ) complexity bound of FedProx under the unrealistic ( B, 0)
Revised: Compared to the alreadyknown O ( 1 (cid:15) ) complexity bound of FedProx under the unrealistic ( B, 0)

Original: -LGD condition [Li et al.,
Revised: -LGD condition [Li et al.,

Original: 2020b], our rate in (4) is slower but it holds without needing to impose stringent regularity conditionson the dissimilarity of local functions, and it reveals the effect of device sampling for acceleratingconvergence.
Revised: 2020b], our rate in (4) is slower but it holds without needing to impose stringent regularity conditionson the dissimilarity of local functions, and it reveals the effect of device sampling for acceleratingconvergence.

Original: Further for non-smooth and non-convex problems, we establish in Theorem 2 thefollowing rate of convergence
Revised: Further for non-smooth and non-convex problems, we establish in Theorem 2 thefollowing rate of convergence

Original: [Equation]
Revised: [Equation]

Original: which is invariant to the number of selected devices in each round.
Revised: which is invariant to the number of selected devices in each round.

Original: In the case of I = O (1) ,the bounds in (4) and (5) are comparable, which indicates that smoothness is not must-have for
Revised: In the case of I = O (1) ,the bounds in (4) and (5) are comparable, which indicates that smoothness is not must-have for

Original: FedProx to get sharper convergence bound especially with low participation ratio.
Revised: FedProx to get sharper convergence bound especially with low participation ratio.

Original: On the other endwhen I =
Revised: On the other endwhen I =

Original: O ( M ) , the bound (5) for non-smooth problems is slower than the bound (4) for smoothfunctions in large-scale networks.
Revised: O ( M ) , the bound (5) for non-smooth problems is slower than the bound (4) for smoothfunctions in large-scale networks.

Original: Main results for minibatch stochastic FedProx .
Revised: Main results for minibatch stochastic FedProx .

Original: Then as the chief contribution of the present work,we propose a minibatch stochastic extension of FedProx along with its population optimizationperformance analysis from a novel perspective of algorithmic stability theory.
Revised: Then as the chief contribution of the present work,we propose a minibatch stochastic extension of FedProx along with its population optimizationperformance analysis from a novel perspective of algorithmic stability theory.

Original: Inspired by the recentsuccess of minibatch stochastic proximal point methods (MSPP)
Revised: Inspired by the recentsuccess of minibatch stochastic proximal point methods (MSPP)

Original: [Li et al., 2014, Wang et al., 2017,
Revised: [Li et al., 2014, Wang et al., 2017,

Original: Asi et al., 2020, Deng and Gao, 2021], we propose to implement FedProx using MSPP as the localupdate oracle.
Revised: Asi et al., 2020, Deng and Gao, 2021], we propose to implement FedProx using MSPP as the localupdate oracle.

Original: The resulting method, which is referred to as FedMSPP , is expected to attain improvedtrade-off between computation, communication and memory efﬁciency for large-scale FL.
Revised: The resulting method, which is referred to as FedMSPP , is expected to attain improvedtrade-off between computation, communication and memory efﬁciency for large-scale FL.

Original: In the caseof imbalanced data distribution, minibatching is also beneﬁcial for making the local computationmore balanced across the devices.
Revised: In the caseof imbalanced data distribution, minibatching is also beneﬁcial for making the local computationmore balanced across the devices.

Original: Based on some extended uniform stability arguments for gradients,we show in Theorem 3 the following local dissimilarity invariant rate of convergence for FedMSPP interms of population optimality:
Revised: Based on some extended uniform stability arguments for gradients,we show in Theorem 3 the following local dissimilarity invariant rate of convergence for FedMSPP interms of population optimality:

Original: [Equation]
Revised: [Equation]

Original: where b is the minibatch size of local update.
Revised: where b is the minibatch size of local update.

Original: For empirical FL, identical bound holds under samplingaccording to empirical distribution.
Revised: For empirical FL, identical bound holds under samplingaccording to empirical distribution.

Original: For T < ( bI ) 3 , the rate in (6) is dominated by O ( 1 T 2 / 3 ) whichgives the communication complexity 1 (cid:15) 3 / 2 , and it matches that of the vanilla FedProx .
Revised: For T < ( bI ) 3 , the rate in (6) is dominated by O ( 1 T 2 / 3 ) whichgives the communication complexity 1 (cid:15) 3 / 2 , and it matches that of the vanilla FedProx .

Original: For sufﬁcientlylarge T ≥ ( bI ) 3 , the rate is dominated by O ( 1 √ TbI ) which gives the communication complexity 1 bI(cid:15) 2 .
Revised: For sufﬁcientlylarge T ≥ ( bI ) 3 , the rate is dominated by O ( 1 √ TbI ) which gives the communication complexity 1 bI(cid:15) 2 .

Original: This shows that local minibatching and device sampling are both beneﬁcial for linearly speedingup communication.
Revised: This shows that local minibatching and device sampling are both beneﬁcial for linearly speedingup communication.

Original: Further, when applied to non-smooth problems, we can similarly show that
Revised: Further, when applied to non-smooth problems, we can similarly show that

Original: FedMSPP converges at the rate of
Revised: FedMSPP converges at the rate of

Original: [Equation]
Revised: [Equation]

Original: which is comparable to that of (6) when b =
Revised: which is comparable to that of (6) when b =

Original: O (1) and I =
Revised: O (1) and I =

Original: O (1) , but without showing the effect oflinear speedup with respect to b and I .
Revised: O (1) , but without showing the effect oflinear speedup with respect to b and I .

Original: Comparison with prior results.
Revised: Comparison with prior results.

Original: In Table 1, we summarize our communication complexity boundsfor FedProx ( FedMSPP ) and compare them with several related heterogeneous FL algorithms interms of the dependency on local dissimilarity, applicability to non-smooth problems and toleranceto partial participation.
Revised: In Table 1, we summarize our communication complexity boundsfor FedProx ( FedMSPP ) and compare them with several related heterogeneous FL algorithms interms of the dependency on local dissimilarity, applicability to non-smooth problems and toleranceto partial participation.

Original: A few observations are in order.
Revised: A few observations are in order.

Original: First , regarding the requirement of localdissimilarity, all of our O ( 1 (cid:15) 2 ) bounds are independent of local dissimilarity conditions, and they arecomparable to those of SCAFFOLD and FCO (for convex problems) which are also invariant to localdissimilarity.
Revised: First , regarding the requirement of localdissimilarity, all of our O ( 1 (cid:15) 2 ) bounds are independent of local dissimilarity conditions, and they arecomparable to those of SCAFFOLD and FCO (for convex problems) which are also invariant to localdissimilarity.

Original: Second , with regard to the applicability to non-smooth optimization, our convergenceguarantees in Theorem 2 and Theorem 4 are established for non-smooth and weakly convex functions.
Revised: Second , with regard to the applicability to non-smooth optimization, our convergenceguarantees in Theorem 2 and Theorem 4 are established for non-smooth and weakly convex functions.

Original: While FCO is the only one in the other considered algorithms that can be applied to non-smoothproblems, it is customized for federated convex composite optimization with potentially non-smoothregularizers [Yuan et al., 2021].
Revised: While FCO is the only one in the other considered algorithms that can be applied to non-smoothproblems, it is customized for federated convex composite optimization with potentially non-smoothregularizers [Yuan et al., 2021].

Original: Third , in terms of tolerance to partial participation, all of our resultsare robust to device sampling, and the O ( 1 bI(cid:15) 2 ) bound in Theorem 3 for FedMSPP is comparable tothe best known results under partial participation as achieved by FedAvg and SCAFFOLD .
Revised: Third , in terms of tolerance to partial participation, all of our resultsare robust to device sampling, and the O ( 1 bI(cid:15) 2 ) bound in Theorem 3 for FedMSPP is comparable tothe best known results under partial participation as achieved by FedAvg and SCAFFOLD .

Original: If assumingthat all the devices participate in local update for each communication round and under certain localdissimilarity conditions, substantially faster O ( 1 (cid:15) ) bounds are possible for STEM and FedPD , while the
Revised: If assumingthat all the devices participate in local update for each communication round and under certain localdissimilarity conditions, substantially faster O ( 1 (cid:15) ) bounds are possible for STEM and FedPD , while the

Original: O ( 1 (cid:15) 3 / 2 ) bounds can be achieved by FedAvg [Khanduri et al., 2021].
Revised: O ( 1 (cid:15) 3 / 2 ) bounds can be achieved by FedAvg [Khanduri et al., 2021].

Original: To summarize the comparison,
Revised: To summarize the comparison,

Original: Method
Revised: Method

Original: Work
Revised: Work

Original: Commun.
Revised: Commun.

Original: Complex.
Revised: Complex.

Original: LD Independ.
Revised: LD Independ.

Original: NS
Revised: NS

Original: PP
Revised: PP

Original: [Table]
Revised: [Table]

Original: [Li
Revised: [Li

Original: [Table]
Revised: [Table]

Original: al., 2020b]
Revised: al., 2020b]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Khanduri
Revised: [Khanduri

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Karimireddy
Revised: [Karimireddy

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Zhang et al., 2020]
Revised: [Zhang et al., 2020]

Original: [Table] [Table][Table]
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: our local dissimilarity invariant convergence bounds for FedProx ( FedMSPP ) are comparable to thebest-known rates in the identical setting, while covering the generic non-smooth and non-convexcases which to our knowledge so far has not been possible for other FL algorithms.
Revised: our local dissimilarity invariant convergence bounds for FedProx ( FedMSPP ) are comparable to thebest-known rates in the identical setting, while covering the generic non-smooth and non-convexcases which to our knowledge so far has not been possible for other FL algorithms.

Original: Highlight of contributions.
Revised: Highlight of contributions.

Original: The theoretical contributions of this work are highlighted as follows:
Revised: The theoretical contributions of this work are highlighted as follows:

Original: • From the perspective of algorithmic stability theory, we provide a set of novel local dissimilarityinvariant convergence guarantees for the widely used FedProx algorithm for non-convex heteroge-
Revised: • From the perspective of algorithmic stability theory, we provide a set of novel local dissimilarityinvariant convergence guarantees for the widely used FedProx algorithm for non-convex heteroge-

Original: neous FL, with smooth or non-smooth local functions.
Revised: neous FL, with smooth or non-smooth local functions.

Original: Our theory for the ﬁrst time reveals thatlocal dissimilarity and smoothness are not necessary to guarantee the convergence of FedProx with reasonable rates.
Revised: Our theory for the ﬁrst time reveals thatlocal dissimilarity and smoothness are not necessary to guarantee the convergence of FedProx withreasonable rates.

Original: • We present FedMSPP as a minibatch stochastic extension of FedProx and analyze its convergence
Revised: • We present FedMSPP as a minibatch stochastic extension of FedProx and analyze its convergence

Original: behavior in terms of population optimality, again without assuming any type of local dissimilarity conditions.
Revised: behavior in terms of population optimality, again without assuming any type of local dissimilarityconditions.

Original: The main result provably
Revised: The main result provably

Original: shows that FedMSPP enjoys linear speedup in terms ofminibatching size and partial participation ratio.
Revised: shows that FedMSPP enjoys linear speedup in terms ofminibatching size and partial participation ratio.

Original: Paper organization.
Revised: Paper organization.

Original: In Section 2 we present our local dissimilarity invariant convergence analysis forthe vanilla FedProx with smooth or non-smooth loss functions.
Revised: In Section 2 we present our local dissimilarity invariant convergence analysis forthe vanilla FedProx with smooth or non-smooth loss functions.

Original: In Section 3 we propose FedMSPP asa minibatch stochastic extension of FedProx and analyze its convergence behavior through the lens ofalgorithmic stability theory.
Revised: In Section 3 we propose FedMSPP asa minibatch stochastic extension of FedProx and analyze its convergence behavior through the lens ofalgorithmic stability theory.

Original: The concluding remarks are made in Section 4.
Revised: The concluding remarks are made in Section 4.

Original: Finally, all the technicalproofs and some additional related work are relegated to the appendix sections.
Revised: Finally, all the technicalproofs and some additional related work are relegated to the appendix sections.

Original:  Convergence of FedProx
Revised:  Convergence of FedProx

Original: We begin by providing an improved analysis for the vanilla FedProx independent of the localdissimilarity type conditions.
Revised: We begin by providing an improved analysis for the vanilla FedProx independent of the localdissimilarity type conditions.

Original: We ﬁrst introduce notations that will be used in the analysis to follow.
Revised: We ﬁrst introduce notations that will be used in the analysis to follow.

Original: Notations.
Revised: Notations.

Original: Throughout the paper, we use [ n ] to denote the set { 1 , ..., n } , (cid:107) · (cid:107) to denote the Euclideannorm and (cid:104)· , ·(cid:105) to denote the Euclidean inner product.
Revised: Throughout the paper, we use [ n ] to denote the set { 1 , ..., n } , (cid:107) · (cid:107) to denote the Euclideannorm and (cid:104)· , ·(cid:105) to denote the Euclidean inner product.

Original: We say a function f is G -Lipschitz continuousif | f ( w ) − f ( w (cid:48) )
Revised: We say a function f is G -Lipschitz continuousif | f ( w ) − f ( w (cid:48) )

Original: | ≤ G (cid:107) w − w (cid:48) (cid:107) for all w, w (cid:48)
Revised: | ≤ G (cid:107) w − w (cid:48) (cid:107) for all w, w (cid:48)

Original: ∈ R p , and it is L -smooth if |∇ f ( w ) − ∇ f ( w (cid:48) )
Revised: ∈ R p , and it is L -smooth if |∇ f ( w ) − ∇ f ( w (cid:48) )

Original: | ≤
Revised: | ≤

Original: L (cid:107) w − w (cid:48) (cid:107) for all w, w (cid:48)
Revised: L (cid:107) w − w (cid:48) (cid:107) for all w, w (cid:48)

Original: ∈ R p .
Revised: ∈ R p .

Original: Moreover, we say f is ν -weakly convex
Revised: Moreover, we say f is ν -weakly convex

Original: if for any w, w (cid:48) ∈ R p ,
Revised: if for any w, w (cid:48) ∈ R p ,

Original: [Equation]
Revised: [Equation]

Original: where ∂f ( w (cid:48) ) represents a subgradient of f evaluated at w (cid:48) .
Revised: where ∂f ( w (cid:48) ) represents a subgradient of f evaluated at w (cid:48) .

Original: We denote by
Revised: We denote by

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: min u
Revised: min u

Original: [Equation]
Revised: [Equation]

Original: the η -Moreau-envelope of f , and by
Revised: the η -Moreau-envelope of f , and by

Original: [Equation]
Revised: [Equation]

Original: the proximal mapping associated with f .
Revised: the proximal mapping associated with f .

Original: We also need to access the following deﬁnition of inexactlocal update oracle for FedProx .
Revised: We also need to access the following deﬁnition of inexactlocal update oracle for FedProx .

Original: Deﬁnition 2 (Local inexact oracle of FedProx ) .
Revised: Deﬁnition 2 (Local inexact oracle of FedProx ) .

Original: Suppose that the local proximal point regularizedobjective Q ( m ) erm ( w ; w t − 1 ) (cf. (3) ) admits a global minimizer.
Revised: Suppose that the local proximal point regularizedobjective Q ( m ) erm ( w ; w t − 1 ) (cf. (3) ) admits a global minimizer.

Original: For each time instance t , we say thatthe local update oracle of FedProx is ε t -inexactly solved with sub-optimality ε t ≥ 

Revised: For each time instance t , we say thatthe local update oracle of FedProx is ε t -inexactly solved with sub-optimality ε t ≥

Original: if
Revised: if

Original: [Equation]
Revised: [Equation]

Original: We assume that the objective value gap ¯∆ erm :
Revised: We assume that the objective value gap ¯∆ erm :

Original: = ¯ R erm ( w
Revised: = ¯ R erm ( w

Original: [Equation]
Revised: [Equation]

Original: − min w ∈ R [Equation] R
Revised: − min w ∈ R p ¯ R

Original: [Equation]
Revised: [Equation]

Original: ) is bounded.
Revised: ) is bounded.

Original: 2.1 Results for Smooth Problems
Revised: 2.1 Results for Smooth Problems

Original: The following theorem is our main result on the convergence rate of FedProx for smooth andnon-convex federated optimization problems.
Revised: The following theorem is our main result on the convergence rate of FedProx for smooth andnon-convex federated optimization problems.

Original: Theorem 1.
Revised: Theorem 1.

Original: Assume that for each m ∈
Revised: Assume that for each m ∈

Original: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz
Revised: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz

Original: and L -smoothwith respect to its ﬁrst argument.
Revised: and L -smoothwith respect to its ﬁrst argument.

Original: Set | I t | ≡ I and η t ≡ 13 L min (cid:110) 1 T 1 / 3 , (cid:113) IT (cid:111) .
Revised: Set | I t | ≡ I and η t ≡ 13 L min (cid:110) 1 T 1 / 3 , (cid:113) IT (cid:111) .

Original: Suppose that the localupdate oracle of FedProx is ε t -inexactly solved with ε t ≤ min (cid:110)
Revised: Suppose that the localupdate oracle of FedProx is ε t -inexactly solved with ε t ≤ min (cid:110)

Original: 2 L 2 G 2 η 3 t I 2 ( Lη t +1) ,

Original: G 2 η t 2 I Lη t +1) (cid:111) .
Revised: G 2 L √ I , Gη t I (cid:111) .

Original: Let t ∗ bean index uniformly randomly chosen in { 0 , 1 , ..., T − 1 } .
Revised: Let t ∗ be an indexuniformly randomly chosen in { 0 , 1 , ..., T − 1 } .

Original: Then it holds that
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: (cid:46) (cid:0) L ¯∆ erm + G 2 (cid:1) max (cid:26) 1 T 2 / 3 , 1 √

Original: TI [Table]
Revised: TI (cid:27) .

Original: Proof.
Revised: Proof.

Original: A proof of this result is deferred to Appendix B.1.
Revised: A proof of this result is deferred to Appendix B.1.

Original: A few remarks are in order.
Revised: A few remarks are in order.

Original: Remark 1.
Revised: Remark 1.

Original: Compared to the O ( 1 T ) bound from Li et al.
Revised: Compared to the O ( 1 T ) bound from Li et al.

Original: [2020b], our rate established in Theorem 1is slower
Revised: [2020b], our rate established in Theorem 1is slower

Original: but it is valid without assuming the unrealistic ( B, 0) -LGD
Revised: but it is valid without assuming the unrealistic ( B, 0) -LGD

Original: conditions and imposing strongregularization conditions on I [see, e.g., Li et al., 2020b, Remark 5].
Revised: conditions and imposing strongregularization conditions on I [see, e.g., Li et al., 2020b, Remark 5].

Original: Moreover, the dominant term1 √ TI in our bound reveals the beneﬁt of device sampling for linear speedup which is not clear in theoriginal analysis of Li et al.
Revised: Moreover, the dominant term1 √ TI in our bound reveals the beneﬁt of device sampling for linear speedup which is not clear in theoriginal analysis of Li et al.

Original: [2020b].
Revised: [2020b].

Original: Remark 2.
Revised: Remark 2.

Original: In the extreme case of full device participation, i.e., I t ≡ [ M ] , the terms related to I in
Revised: In the extreme case of full device participation, i.e., I t ≡ [ M ] , the terms related to I in

Original: Theorem 1 can be removed and thus the convergence rate becomes 1 T 2 / 3 under η t
Revised: Theorem 1 can be removed and thus the convergence rate becomes 1 T 2 / 3 under η t

Original: = O ( 1 LT 1 / 3 ) .
Revised: = O ( 1 LT 1 / 3 ) .

Original: Inthis same setting, we comment that the rate can also be improved to O ( 1 T ) using our proof augmentsif ( B, 0) -LGD is additionally assumed.
Revised: Inthis same setting, we comment that the rate can also be improved to O ( 1 T ) using our proof augmentsif ( B, 0) -LGD is additionally assumed.

Original: Remark 3.
Revised: Remark 3.

Original: The G -Lipschitz-loss assumption in Theorem 1 can be alternatively replaced by thebounded gradient condition as commonly used in the analysis of FL algorithms [Li et al., 2020b,
Revised: The G -Lipschitz-loss assumption in Theorem 1 can be alternatively replaced by thebounded gradient condition as commonly used in the analysis of FL algorithms [Li et al., 2020b,

Original: Zhang et al., 2020].
Revised: Zhang et al., 2020].

Original: Despite that our analysis does not explicitly access to any local dissimilarityconditions, the assumed G -Lipschitz (or bounded gradient) condition actually implies that the localobjective gradients are not too dissimilar, which shares a close spirit to the typically assumed (0 , H ) -
Revised: Despite that our analysis does not explicitly access to any local dissimilarityconditions, the assumed G -Lipschitz (or bounded gradient) condition actually implies that the localobjective gradients are not too dissimilar, which shares a close spirit to the typically assumed (0 , H ) -

Original: LGD condition [Karimireddy et al., 2020] and inter-client-variance condition [Khanduri et al., 2021].
Revised: LGD condition [Karimireddy et al., 2020] and inter-client-variance condition [Khanduri et al., 2021].

Original: It is noteworthy that these mentioned client heterogeneity conditions are substantially milder than the( B, 0)
Revised: It is noteworthy that these mentioned client heterogeneity conditions are substantially milder than the( B, 0)

Original: -LGD condition as required in the original analysis of FedProx .
Revised: -LGD condition as required in the original analysis of FedProx .

Original: 2.2 Results for Non-smooth Problems
Revised: 2.2 Results for Non-smooth Problems

Original: Now we turn to study the convergence of FedProx for weakly convex but not necessarily smoothproblems.
Revised: Now we turn to study the convergence of FedProx for weakly convex but not necessarily smoothproblems.

Original: For the sake of presentation clarity, we work on the exact FedProx in which the localupdate oracle is assumed to be exactly solved, i.e. ε t ≡ 0 .
Revised: For the sake of presentation clarity, we work on the exact FedProx in which the localupdate oracle is assumed to be exactly solved, i.e. ε t ≡ 0 .

Original: Extension to the inexact case is more orless straightforward, though with somewhat more involved perturbation treatments.
Revised: Extension to the inexact case is more orless straightforward, though with somewhat more involved perturbation treatment.

Original: We assume thatthe objective value gap ¯∆ erm ,ρ :=
Revised: We assume thatthe objective value gap ¯∆ erm ,ρ :=

Original: ¯ R erm ,ρ ( w 0 ) − min w ¯ R erm ,ρ ( w ) associated with ρ -Moreau-envelopeof ¯ R erm is bounded.
Revised: ¯ R erm ,ρ ( w 0 ) − min w ¯ R erm ,ρ ( w ) associated with ρ -Moreau-envelopeof ¯ R erm is bounded.

Original: The following is our main result on the convergence of FedProx for non-smoothand weakly convex problems.
Revised: The following is our main result on the convergence of FedProx for non-smoothand weakly convex problems.

Original: Theorem 2.
Revised: Theorem 2.

Original: Assume that for each m ∈
Revised: Assume that for each m ∈

Original: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz
Revised: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz

Original: and ν -weaklyconvex with respect to its ﬁrst argument.
Revised: and ν -weaklyconvex with respect to its ﬁrst argument.

Original: Set η t ≡ ρ √ T for arbitrary ρ < 12 ν .
Revised: Set η t ≡ ρ √ T for arbitrary ρ < 12 ν .

Original: Suppose that the localupdate oracle of FedProx is exactly solved with ε t ≡ 0 .
Revised: Suppose that the localupdate oracle of FedProx is exactly solved with ε t ≡ 0 .

Original: Let t ∗ be an index uniformly randomlychosen in { 0 , 1 , ..., T − 1 } .
Revised: Let t ∗ be an index uniformly randomlychosen in { 0 , 1 , ..., T − 1 } .

Original: Then it holds that
Revised: Then it holds that

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: The proof technique is inspired by the arguments from Davis and Drusvyatskiy [2019] developed for analyzing stochastic model-based algorithms, with several new elements along developedfor handling the challenges introduced by the model averaging and partial participation mechanismsassociated with FedProx .
Revised: The proof technique is inspired by the arguments from Davis and Drusvyatskiy [2019] developed for analyzing stochastic model-based algorithms, with several new elements along developedfor handling the challenges introduced by the model averaging and partial participation mechanismsassociated with FedProx .

Original: A particular crux here is that due to the random subset model aggregationof w t = 1 | I t | (cid:80) ξ ∈
Revised: A particular crux here is that due to the random subset model aggregationof w t = 1 | I t | (cid:80) ξ ∈

Original: I t w ( ξ ) t , the local function values R ( ξ ) erm ( w t ) are no longer independent of each otherthough ξ is uniformly random.
Revised: I t w ( ξ ) t , the local function values R ( ξ ) erm ( w t ) are no longer independent to each otherthough ξ is uniformly random.

Original: As a consequence, 1 | I t | (cid:80) ξ ∈
Revised: As a consequence, 1 | I t | (cid:80) ξ ∈

Original: I t R ( ξ ) erm ( w t ) is not an unbiased estimationof ¯ R erm ( w t ) .
Revised: I t R ( ξ ) erm ( w t ) is not an unbiased estimationof ¯ R erm ( w t ) .

Original: To overcome this technical obstacle, we make use of a key observation that w ( m ) twill be almost surely close enough to w t − 1 if the learning rate η t is small enough (which is thecase in our choice of η t ), and thus we can replace the former with the latter whenever beneﬁcialbut without introducing too much approximation error.
Revised: To overcome this technical obstacle, we make use of a key observation that w ( m ) twill be almost surely close enough to w t − 1 if the learning rate η t is small enough (which is thecase in our choice of η t ), and thus we can replace the former with the latter whenever beneﬁcialbut without introducing too much approximation error.

Original: A full proof of this result can be found in
Revised: A full proof of this result can be found in

Original: Appendix B.2.
Revised: Appendix B.2.

Original: A few comments are in order.
Revised: A few comments are in order.

Original: Remark 4.
Revised: Remark 4.

Original: To our best knowledge, Theorem 2 is the ﬁrst convergence guarantee for FL algorithmsapplicable to generic non-smooth and weakly convex problems.
Revised: To our best knowledge, Theorem 2 is the ﬁrst convergence guarantee for FL algorithmsapplicable to generic non-smooth and weakly convex problems.

Original: This is in sharp contrast with
Revised: This is in sharp contrast with

Original: FCO [Yuan et al., 2021] which focuses on composite convex and non-smooth problems such as (cid:96) 1estimation, or Fed-HT
Revised: FCO [Yuan et al., 2021] which focuses on composite convex and non-smooth problems such as (cid:96) 1estimation, or Fed-HT

Original: [Tong et al., 2020] which is specially customized for cardinality-constrainedsparse learning problems where the non-convexity essentially arises from the (cid:96) 0 -constraint.
Revised: [Tong et al., 2020] which is specially customized for cardinality-constrainedsparse learning problems where the non-convexity essentially arises from the (cid:96) 0 -constraint.

Original: Remark 5.
Revised: Remark 5.

Original: Let us consider ¯ w t ∗ := prox ρ ¯ R erm ( w t ∗ ) , the proximal mapping of w t ∗ associated with¯ R erm .
Revised: Let us consider ¯ w t ∗ := prox ρ ¯ R erm ( w t ∗ ) , the proximal mapping of w t ∗ associated with¯ R erm .

Original: In view of a feature of Moreau envelope to characterize stationarity [Davis and Drusvyatskiy,2019], if w t ∗ has small gradient norm (cid:13)(cid:13) ∇ ¯ R erm ,ρ ( w t ∗ ) (cid:13)(cid:13) , then ¯ w t ∗ must be a near-stationary solutionand w t ∗ stays in the proximity of ¯ w t ∗ due to the identity (cid:107) w t ∗ − ¯ w t ∗ (cid:107) =
Revised: In view of a feature of Moreau envelope to characterize stationarity [Davis and Drusvyatskiy,2019], if w t ∗ has small gradient norm (cid:13)(cid:13) ∇ ¯ R erm ,ρ ( w t ∗ ) (cid:13)(cid:13) , then ¯ w t ∗ must be a near-stationary solutionand w t ∗ stays in the proximity of ¯ w t ∗ due to the identity (cid:107) w t ∗ − ¯ w t ∗ (cid:107) =

Original: ρ (cid:13)(cid:13) ∇ ¯ R erm ,ρ ( w t ∗ ) (cid:13)(cid:13) .
Revised: ρ (cid:13)(cid:13) ∇ ¯ R erm ,ρ ( w t ∗ ) (cid:13)(cid:13) .

Original: Therefore,the bound in Theorem 2 suggests that in expectation ¯ w t ∗ converges to a stationary solution and w t ∗converges to ¯ w t ∗ , both at the rate of O ( 1 √ T ) .
Revised: Therefore,the bound in Theorem 2 suggests that in expectation ¯ w t ∗ converges to a stationary solution and w t ∗converges to ¯ w t ∗ , both at the rate of O ( 1 √ T ) .

Original: Remark 6.
Revised: Remark 6.

Original: We comment that the bound in Theorem 2 is not dependent on I , the number of selecteddevices.
Revised: We comment that the bound in Theorem 2 is not dependent on I , the number of selecteddevices.

Original: For I
Revised: For I

Original: = O (1) and sufﬁciently large T >
Revised: = O (1) and sufﬁciently large T >

Original: O
Revised: O

Original: ( I 3 ) , the bounds Theorem 1 and Theorem 2 arecomparable to each other, which demonstrates that the smoothness is not must-have for FedProx toget sharper convergence bound with small device sampling rate.
Revised: ( I 3 ) , the bounds Theorem 1 and Theorem 2 arecomparable to each other, which demonstrates that the smoothness is not must-have for FedProx toget sharper convergence bound with small device sampling rate.

Original: However, in the near-full participation setting where I =
Revised: However, in the near-full participation setting where I =

Original: O
Revised: O

Original: ( M ) , the bound in Theorem 2 for non-smooth problems will be slower when
Revised: ( M ) , the bound in Theorem 2 for non-smooth problems will be slower when

Original: M is large.
Revised: M is large.

Original: Extremely when I t =
Revised: Extremely when I t =

Original: [ M ] , the O ( 1 √ T ) bound is substantially inferior to the smooth casewhich has improved rate of O ( 1 T 2 / 3 ) as discussed in Remark 2.
Revised: [ M ] , the O ( 1 √ T ) bound is substantially inferior to the smooth casewhich has improved rate of O ( 1 T 2 / 3 ) as discussed in Remark 2.

Original:  Convergence of FedProx with Stochastic Minibatching
Revised:  Convergence of FedProx with Stochastic Minibatching

Original: When it comes to the implementation of FedProx , a notable challenge is that the local proximalpoint update oracle (3) is by itself a full-batch ERM problem which would be expensive to solveeven approximately in large-scale settings.
Revised: When it comes to the implementation of FedProx , a notable challenge is that the local proximalpoint update oracle (3) is by itself a full-batch ERM problem which would be expensive to solveeven approximately in large-scale settings.

Original: Moreover, in the settings where the data distribution overdevices is highly imbalanced, the computational overload of local update could vary signiﬁcantlyacross the network, which impairs communication efﬁciency.
Revised: Moreover, in the settings where the data distribution overdevices is highly imbalanced, the computational overload of local update could vary signiﬁcantlyacross the network, which impairs communication efﬁciency.

Original: It is thus desirable to seek stochasticapproximation schemes for hopefully improving the local oracle update efﬁciency and overloadbalance of FedProx .
Revised: It is thus desirable to seek stochasticapproximation schemes for hopefully improving the local oracle update efﬁciency and overloadbalance of FedProx .

Original: To this end, inspired by the recent success of minibatch stochastic proximal pointmethods (MSPP)
Revised: To this end, inspired by the recent success of minibatch stochastic proximal pointmethods (MSPP)

Original: [Asi et al., 2020, Deng and Gao, 2021], we propose to implement FedProx using
Revised: [Asi et al., 2020, Deng and Gao, 2021], we propose to implement FedProx using

Original: MSPP as the local stochastic optimization oracle.
Revised: MSPP as the local stochastic optimization oracle.

Original: More precisely, let B ( m ) t = { z ( m ) i,t } bi =1 i.i.d.
Revised: More precisely, let B ( m ) t = { z ( m ) i,t } bi =1 i.i.d.

Original: ∼ ( D ( m ) ) bbe a minibatch of b i.i.d.
Revised: ∼ ( D ( m ) ) bbe a minibatch of b i.i.d.

Original: samples drawn from the distribution D ( m ) at device m and time instancet ≥ 1 .
Revised: samples drawn from the distribution D ( m ) at device m and time instancet ≥ 1 .

Original: We denote
Revised: We denote

Original: [Equation]
Revised: [Equation]

Original: Input :Minibatch size b ; learning rates { γ t } t ∈ [ T ] .
Revised: Input :Minibatch size b ; learning rates { γ t } t ∈ [ T ] .

Original: Output : w T .
Revised: Output : w T .

Original: Initialization Set w 0 , e.g., typically as a zero vector.
Revised: Initialization Set w 0 , e.g., typically as a zero vector.

Original: for t = 1 , 2 , ..., T do /
Revised: for t = 1 , 2 , ..., T do /

Original: Device selection and model broadcast on the server */
Revised: Device selection and model broadcast on the server */

Original: Server uniformly randomly selects a subset I t ⊆
Revised: Server uniformly randomly selects a subset I t ⊆

Original: [ M ] of devices and sends w t − 1 to all the selected devices; /* Local model updates on the selected devices */ for ξ ∈
Revised: [ M ] of devices and sends w t − 1 to all the selected devices; /* Local model updates on the selected devices */ for ξ ∈

Original: I t in parallel do
Revised: I t in parallel do

Original: Device ξ samples a minibatch B ( ξ ) t
Revised: Device ξ samples a minibatch B ( ξ ) t

Original: [Equation]
Revised: [Equation]

Original: Device ξ inexactly updates the its local model as
Revised: Device ξ inexactly updates the its local model as

Original: [Equation]
Revised: [Equation]

Original: where R ( ξ ) B ( ξ ) t ( w ) is given by (7).
Revised: where R ( ξ ) B ( ξ ) t ( w ) is given by (7).

Original: Device ξ sends w ( ξ ) t back to server.
Revised: Device ξ sends w ( ξ ) t back to server.

Original: end /*
Revised: end /*

Original: Model aggregation on the server */
Revised: Model aggregation on the server */

Original: Sever aggregates the local models received from I t to update the global model as
Revised: Sever aggregates the local models received from I t to update the global model as

Original: [Equation]
Revised: [Equation]

Original: end as the local minibatch empirical risk function over B ( m ) t .
Revised: end as the local minibatch empirical risk function over B ( m ) t .

Original: The only modiﬁcation we propose tomake here is to replace the empirical risk R ( m ) erm ( w ) in the original update form (3) with its minibatchcounterpart R ( m ) B ( m ) t ( w ) .
Revised: The only modiﬁcation we propose tomake here is to replace the empirical risk R ( m ) erm ( w ) in the original update form (3) with its minibatchcounterpart R ( m ) B ( m ) t ( w ) .

Original: The resultant FL framework, which we refer to as FedMSPP (Federated
Revised: The resultant FL framework, which we refer to as FedMSPP (Federated

Original: MSPP), is outlined in Algorithm 1.
Revised: MSPP), is outlined in Algorithm 1.

Original: Clearly, the vanilla FedProx is a special case of FedMSPP whenapplied to the federated ERM form (2) with full data batch B ( m ) t ≡ D ( m ) .
Revised: Clearly, the vanilla FedProx is a special case of FedMSPP whenapplied to the federated ERM form (2) with full data batch B ( m ) t ≡ D ( m ) .

Original: 3.1 Results for Smooth Problems
Revised: 3.1 Results for Smooth Problems

Original: We ﬁrst analyze the convergence rate of FedMSPP for smooth and non-convex problems using thetools borrowed from algorithmic stability theory.
Revised: We ﬁrst analyze the convergence rate of FedMSPP for smooth and non-convex problems using thetools borrowed from algorithmic stability theory.

Original: Analogous to the Deﬁnition 2, we introduce thefollowing deﬁnition of inexact local update oracle for FedMSPP .
Revised: Analogous to the Deﬁnition 2, we introduce thefollowing deﬁnition of inexact local update oracle for FedMSPP .

Original: Deﬁnition 3 (Local inexact oracle of FedMSPP ) .
Revised: Deﬁnition 3 (Local inexact oracle of FedMSPP ) .

Original: Suppose that the local proximal point regularizedobjective Q ( m ) B ( m ) t ( w ; w t − 1 ) (cf.
Revised: Suppose that the local proximal point regularizedobjective Q ( m ) B [Equation] t ( w ; w t − 1 ) (cf.

Original: (8) ) admits a global minimizer.
Revised: (8) ) admits a global minimizer.

Original: For each time instance t , we say thatthe local update oracle of FedMSPP is ε t -inexactly solved with sub-optimality ε t ≥ 

Revised: For each time instance t , we say thatthe local update oracle of FedMSPP is ε t -inexactly solved with sub-optimality ε t ≥

Original: if
Revised: if

Original: [Equation]
Revised: [Equation]

Original: We also assume that the population value gap ¯∆
Revised: We also assume that the population value gap ¯∆

Original: =
Revised: =

Original: ¯ R ( w (0) )
Revised: ¯ R ( w (0) )

Original: − min w ∈ R p ¯ R ( w ) is bounded.
Revised: − min w ∈ R p ¯ R ( w ) is bounded.

Original: Thefollowing theorem is our main result on FedMSPP for smooth and non-convex FL problems.
Revised: Thefollowing theorem is our main result on FedMSPP for smooth and non-convex FL problems.

Original: Theorem 3.
Revised: Theorem 3.

Original: Assume that for each m ∈
Revised: Assume that for each m ∈

Original: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz and L -smoothwith respect to its ﬁrst argument.
Revised: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz and L -smooth withrespect to its ﬁrst argument.

Original: Set | I t | ≡ I and η t ≡ 18 L min (cid:26)
Revised: Set | I t | ≡ I and η t ≡ 18 L min (cid:26)

Original: T 1 / 3 , (cid:113) bIT (cid:27) .
Revised: T 1 / 3 , (cid:113) bIT (cid:27) .

Original: Suppose that thelocal update oracle of FedMSPP is ε t -inexactly solved with ε t ≤ min (cid:110)
Revised: Suppose that the localupdate oracle of FedMSPP is ε t -inexactly solved with ε t ≤ min (cid:110)

Original: G 2 η t 2( Lη t +1) , G 2 η t 8 b 2 , L 2 G 2 η 3 t 2 bI( Lη t +1) (cid:111) . 
Revised: G 2 L , G 2 η t 8 b 2 , Gη t 2 bI (cid:111) . Then we have

Original: Let t ∗ be an index uniformly randomly chosen in { 0 , 1, ..., T − 1 } .
Revised: T T − 1 (cid:88) t = E (cid:2) (cid:107)∇ ¯ R ( w t ) (cid:107) 2 (cid:3) (cid:46) (cid:0) L ¯∆ + G 2 (cid:1) max (cid:26) 1 T 2 / 3 , 1 √

Original: Then it holds that
Revised: TbI

Original: [Equation]
Revised: [Equation]

Original: [Table]

Original: 1 √

Original: TbI (cid:27) .

Original: Proof.
Revised: Proof.

Original: Let us consider d ( m
Revised: Let us consider d ( m

Original: [Equation]
Revised: [Equation]

Original: t
Revised: t

Original: [Equation]
Revised: [Equation]

Original: ) which is roughly the local update direction on devicem , in the sense that w ( m ) t ≈ w t − 1 −
Revised: ) which is roughly the local update direction on devicem , in the sense that w ( m ) t ≈ w t − [Equation] −

Original: [Equation]
Revised: [Equation]

Original: d (
Revised: d (

Original: [Equation]
Revised: [Equation]

Original: t given that the local update oracle is solved to sufﬁcientaccuracy.
Revised: given that the local update oracle is solved to sufﬁcientaccuracy.

Original: As a key ingredient of our proof, we show via some extended uniform stability argumentsin terms of gradients (see Lemma 3) that the averaged directions d t := 1 | I t | (cid:80) ξ ∈
Revised: As a key ingredient of our proof, we show via some extended uniform stability argumentsin terms of gradients (see Lemma 3) that the averaged directions d t := 1 | I t | (cid:80) ξ ∈

Original: I t d ( ξ ) t aligns wellwith the global gradient ∇ ¯ R ( w t − 1 ) in expectation (see Lemma 11).
Revised: I t d ( ξ ) t aligns wellwith the global gradient ∇ ¯ R ( w t − 1 ) in expectation (see Lemma 11).

Original: Therefore, in average it roughlyholds that w t = 1 | I t | (cid:80) ξ ∈
Revised: Therefore, in average it roughlyholds that w t = 1 | I t | (cid:80) ξ ∈

Original: I
Revised: I

Original: t w ( ξ ) t
Revised: t w ( ξ ) t

Original: [Equation]
Revised: [Equation]

Original: R
Revised: R

Original: [Equation]
Revised: [Equation]

Original: 1 ) , which suggests that w t isupdated roughly along the direction of global gradient descent and thus guarantees quick convergence.
Revised: 1 ) , which suggests that w t isupdated roughly along the direction of global gradient descent and thus guarantees quick convergence.

Original: Based on this novel analysis, we are free of imposing any kind of local dissimilarity conditions onlocal objectives.
Revised: Based on this novel analysis, we are free of imposing any kind of local dissimilarity conditions onlocal objectives.

Original: See Appendix C.1 for a full proof of this result.
Revised: See Appendix C.1 for a full proof of this result.

Original: Remark 7.
Revised: Remark 7.

Original: For T ≥ ( bI ) 3 , the bound in Theorem 3 is dominated by O ( 1 √ TbI ) which gives thecommunication complexity 1 bI(cid:15) 2 .
Revised: For T ≥ ( bI ) 3 , the bound in Theorem 3 is dominated by O ( 1 √ TbI ) which gives thecommunication complexity 1 bI(cid:15) 2 .

Original: This shows that FedMSPP enjoys linear speedup both in the size oflocal minibatching and in the size of device sampling.
Revised: This shows that FedMSPP enjoys linear speedup both in the size oflocal minibatching and in the size of device sampling.

Original: Remark 8.
Revised: Remark 8.

Original: While the bound in Theorem 3 is derived for the population form of FL in (1) , identicalbound naturally holds for the empirical form (2) under minibatch sampling according to local dataempirical distribution.
Revised: While the bound in Theorem 3 is derived for the population form of FL in (1) , identicalbound naturally holds for the empirical form (2) under minibatch sampling according to local dataempirical distribution.

Original: 3.2 Results for Non-smooth Problems
Revised: 3.2 Results for Non-smooth Problems

Original: Analogues to FedProx , we can further show that FedMSPP converges reasonably well when appliedto weakly convex and non-smooth problems.
Revised: Analogues to FedProx , we can further show that FedMSPP converges reasonably well when appliedto weakly convex and non-smooth problems.

Original: We assume that the objective value gap ¯∆ ρ :
Revised: We assume that the objective value gap ¯∆ ρ :

Original: =
Revised: =

Original: ¯ R ρ ( w 0 ) − min w ¯ R ρ ( w ) associated with ρ -Moreau-envelope of ¯ R is bounded.
Revised: ¯ R ρ ( w 0 ) − min w ¯ R ρ ( w ) associated with ρ -Moreau-envelope of ¯ R is bounded.

Original: The following is ourmain result in this line.
Revised: The following is ourmain result in this line.

Original: Theorem 4.
Revised: Theorem 4.

Original: Assume that for each m ∈
Revised: Assume that for each m ∈

Original: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz
Revised: [ M ] , the loss function (cid:96) ( m ) is G -Lipschitz

Original: and ν -weaklyconvex with respect to its ﬁrst argument.
Revised: and ν -weaklyconvex with respect to its ﬁrst argument.

Original: Set η t ≡ ρ √ T for arbitrary ρ < 12 ν .
Revised: Set η t ≡ ρ √ T for arbitrary ρ < 12 ν .

Original: Suppose that the localupdate oracle of FedMSPP is exactly solved with ε t ≡ 0 .
Revised: Suppose that the localupdate oracle of FedMSPP is exactly solved with ε t ≡ 0 .

Original: Let t ∗ be an index uniformly randomlychosen in { 0 , 1 , ..., T − 1 } .
Revised: Let t ∗ be an index uniformly randomlychosen in { 0 , 1 , ..., T − 1 } .

Original: Then it holds that
Revised: Then it holds that

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: The proof argument is a slight adaptation of that of Theorem 2 to the population FL setup (1)with FedMSPP .
Revised: The proof argument is a slight adaptation of that of Theorem 2 to the population FL setup (1)with FedMSPP .

Original: For the sake of completeness, a full proof is reproduced in Appendix C.2.
Revised: For the sake of completeness, a full proof is reproduced in Appendix C.2.

Original: We comment in passing that the discussions made in Remarks 4-6 immediately extend to Theorem 4.
Revised: We comment in passing that the discussions made in Remarks 4-6 immediately extend to Theorem 4.

Original:  Conclusions
Revised:  Conclusions

Original: In this paper, we have exposed three shortcomings of the prior analysis for FedProx in unrealisticassumptions about local dissimilarity, inapplicability to non-smooth problems and expensive (andpotentially imbalanced) computational cost of local update.
Revised: In this paper, we have exposed three shortcomings of the prior analysis for FedProx in unrealisticassumptions about local dissimilarity, inapplicability to non-smooth problems and expensive (andpotentially imbalanced) computational cost of local update.

Original: In order to tackle these issues, wedeveloped a novel convergence theory for the vanilla FedProx and its minibatch stochastic variant,
Revised: In order to tackle these issues, wedeveloped a novel convergence theory for the vanilla FedProx and its minibatch stochastic variant,

Original: FedMSPP , through the lens of algorithmic stability theory.
Revised: FedMSPP , through the lens of algorithmic stability theory.

Original: In a nutshell, our results reveal that withminimal modiﬁcations, FedProx is able to kill three birds with one stone: it enjoys favorable ratesof convergence which are simultaneously invariant to local dissimilarity, applicable to smooth ornon-smooth problems, and scaling linearly with respect to local minibatch size and device samplingratio for smooth problems.
Revised: At nutshell, our results reveal that withminimal modiﬁcations, FedProx is able to kill three birds with one stone: it enjoys favorable ratesof convergence which are simultaneously invariant to local dissimilarity, applicable to smooth ornon-smooth problems, and scaling linearly with respect to local minibatch size and device samplingratio for smooth problems.

Original: To the best of our knowledge, the present work is the ﬁrst theoreticalcontribution that achieves all these appealing properties in a single FL framework.
Revised: To the best of our knowledge, the present work is the ﬁrst theoreticalcontribution that achieves all these appealing properties in a single FL framework.

Original: Limitations.
Revised: Limitations.

Original: While our results in Theorem 2 and Theorem 4 for the ﬁrst time guarantee that
Revised: While our results in Theorem 2 and Theorem 4 for the ﬁrst time guarantee that

Original: FedProx and FedMSPP converge non-asymptotically for non-smooth and weakly-convex problems,the corresponding rates of convergence so far cannot demonstrate any linear speedup effort withrespect to device sampling ratio and local minibatch size.
Revised: FedProx and FedMSPP converge asymptotically for non-smooth and weak-convex problems, thecorresponding rates of convergence so far cannot demonstrate any linear speedup effort with respectto device sampling ratio and local minibatch size.

Original: This is as opposed to what have beenshown for smooth problems in Theorem 1 and Theorem 3, and thus we view it as a limitation ofthetechniques used by our analysis.
Revised: This is as opposed to what have been shown forsmooth problems in Theorem 1 and Theorem 3, and thus we view it as a limitation of our analysis.

Original: In the smooth-loss case, the comparison in Table 1 suggests that ourresults in Theorem 1 and Theorem 3 are no stronger in convergence rate than those of theexisting FLmethods based on local SGD update, despite that FedProx requires a more complex local oracle.
Revised: Also for non-smooth FL problems, the current convergence results are only limited to the ideal casethat the local update oracles are exactly solved. We leave a more detailed understanding of how theerrors in the local oracle update would affect the ﬁnal result for future investigation.

Original: Checklist
Revised: Checklist

Original: For all authors
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes] In Section 1.2, we highlighted the core contributions
Revised: [Yes] In Section 1.2, we highlighted the core contributions

Original: made in this paper which are respectively expanded with details in Section 2 and 3.
Revised: made in this paper which are respectively expanded with details in Section 2 and 3.

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] We have provided a remark on
Revised: [Yes] We have provided a remark on

Original: the limitations of our theory at the end of the main paper.
Revised: the limitations of our theory at the end of the main paper.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A]

Original: Our
Revised: Our

Original: contribution is theoretical in nature.
Revised: contribution is theoretical in nature.

Original: As far as we are aware of, there are no foreseeable
Revised: As far as we are aware of, there are no foreseeable

Original: societal or ethical consequences for the present research.
Revised: societal or ethical consequences for the present research.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] The key assumptions on the structure of loss functions (such as Lipschitzness, smoothness,weak-convexity), learning rates and local oracle update sub-optimality have all been
Revised: [Yes] The keyassumptions on the structure of loss functions (such as Lipschitzness, smoothness, weak-convexity), learning rates and local oracle update sub-optimality have all been

Original: explicitly stated in details in Theorems 1, 2, 3, 4.
Revised: explicitly stated in details in Theorems 1, 2, 3, 4.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] Due to space limit, all
Revised: [Yes] Due to space limit, all

Original: [Table]

Original: the full proofs of results are relegated to the appendix sections which can be found in
Revised: the full proofs of results are relegated to the appendix sections which can be found in

Original: [Table]

Original: the supplementary document.
Revised: the supplementary document.

Original: Particularly for Theorem 2 and Theorem 3, we further
Revised: Particularly for Theorem 2 and Theorem 3, we further

Original: [Table]

Original: provide sketched proofs in the main submission to highlight the proof road maps.
Revised: provide sketched proofs in the main submission to highlight the proof road maps.

Original: [Table]

Original: If you ran experiments...
Revised: If you ran experiments...

Original: [N/A]
Revised: [N/A]

Original: Our work is focused on providing deeper theoretical
Revised: Our work is focused on providing deeper theoretical

Original: [Table] understandings of FedProx and its stochastic variants under milder conditions.[Table]
Revised: understandings of FedProx and its stochastic variants under milder conditions.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: [Table] [N/A] We did not use any of such assets in this work.[Table]
Revised: [N/A] We did not use any of such assets in this work.

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: [N/A]
Revised: [N/A]

Original: The present
Revised: The present

Original: [Table] research was carried out with no crowdsourcing or human subjects involved in.[Table]
Revised: research was carried out with no crowdsourcing or human subjects involved in.