Original: [Equation]
Revised: [Equation]

Original: i ,y ( x 1 , x 2 ) and
Revised: i ,y ( x 1 , x 2 ) and

Original: V
Revised: V

Original: [Equation]
Revised: [Equation]

Original: i ,x ( y 1 , y 2 ) , we upper bound
Revised: i ,x ( y 1 , y 2 ) , we upper bound

Original: [Equation]
Revised: [Equation]

Original: in terms of x t , y t , ˜ x t , ˜ y t , V f i ,y ( x 1 , x 2 ) and V − f i ,x ( y 1 , y 2 ) in Lemma G.5.
Revised: in terms of x t , y t , ˜ x t , ˜ y t , V f i ,y ( x 1 , x 2 ) and V − f i ,x ( y 1 , y 2 ) in Lemma G.5.

Original: Note that the upper bound in Lemma G.5 is complicated and different from that of [20] because we have additional terms contributed by dual variable y with different coefﬁcients and terms containing square norms dependent on the reference points.
Revised: Note that the upper bound in Lemma G.5 is complicated and different from that of [20] because we have additional terms contributed by dual variable y with different coefﬁcients and terms containing square norms dependent on the reference points.

Original: This intermediate result generates different bounds and sets of parameters in the subsequent analysis.
Revised: This intermediate result generates different bounds and sets of parameters in the subsequent analysis.

Original: We carefully set the step size and choose algorithm parameters with proven feasibility in Lemma G.7.
Revised: We carefully set the step size and choose algorithm parameters with proven feasibility in Lemma G.7.

Original: We rigorously compute lower and upper bounds on chosen parameters in terms of κ f , κ g and δ in Lemma G.7 and Appendix H. In our work, these derivations are more involved in comparison to [20].
Revised: We rigorously compute lower and upper bounds on chosen parameters in terms of κ f , κ g and δ in Lemma G.7 and Appendix H. In our work, these derivations are more involved in comparison to [20].

Original: Analysis methods of [36] and [23] are based on averaging quantities; for example average of iterates and gradients.
Revised: Analysis methods of [36] and [23] are based on averaging quantities; for example average of iterates and gradients.

Original: The analysis in [36] and [23] requires separate bounds for consensus error and gradient estimation errors and depends in addition on the smoothness of saddle point problem.
Revised: The analysis in [36] and [23] requires separate bounds for consensus error and gradient estimation errors and depends in addition on the smoothness of saddle point problem.

Original: In contrast to [36] and [23], our analysis does not demand any separate bound on consensus error and gradient estimation error and handles non-smooth functions as well.
Revised: In contrast to [36] and [23], our analysis does not demand any separate bound on consensus error and gradient estimation error and handles non-smooth functions as well.

Original: Unlike our compression based communication scheme, the analysis in [5] bounds errors using an accelerated gossip scheme and approximate solution obtained by solving an inner saddle point problem at every node.
Revised: Unlike our compression based communication scheme, the analysis in [5] bounds errors using an accelerated gossip scheme and approximate solution obtained by solving an inner saddle point problem at every node.

Original: I Numerical Experiments
Revised: I Numerical Experiments

Original: We evaluate the effectiveness of proposed algorithms on robust logistic regression problem
Revised: We evaluate the effectiveness of proposed algorithms on robust logistic regression problem

Original: [Equation]
Revised: [Equation]

Original: over a binary classiﬁcation data set D = { ( a i , b i ) } Ni =1 .
Revised: over a binary classiﬁcation data set D = { ( a i , b i ) } Ni =1 .

Original: We consider constraint sets X and Y as (cid:96)ball of radius 100 and 1 respectively.
Revised: We consider constraint sets X and Y as (cid:96)ball of radius 100 and 1 respectively.

Original: We compute smoothness parameters
Revised: We compute smoothness parameters

Original: L xx , L yy , L xy and L yx using Hessian information of the objective function (see Appendix I.5) and set strong convexity and strong concavity parameters to λ and β respectively.
Revised: L xx , L yy , L xy and L yx using Hessian information of the objective function (see Appendix I.5) and set strong convexity and strong concavity parameters to λ and β respectively.

Original: Unless stated otherwise, we set λ
Revised: Unless stated otherwise, we set λ

Original: = β = 10 , number of nodes to m = 20 and number of batches to n = 20 in all our experiments.
Revised: = β = 10 , number of nodes to m = 20 and number of batches to n = 20 in all our experiments.

Original: The initial points x 0 , y 0 are generated randomly and D x , D y are set to 0 .
Revised: The initial points x 0 , y 0 are generated randomly and D x , D y are set to 0 .

Original: We set up the step size of proposed methods and baseline methods using the theoretically values provided in the respective papers.
Revised: We set up the step size of proposed methods and baseline methods using the theoretically values provided in the respective papers.

Original: We implement all the experiments in Python Programming Language.
Revised: We implement all the experiments in Python Programming Language.

Original: I.1 Experimental Setup:
Revised: I.1 Experimental Setup:

Original: Datasets: We rely on four binary classiﬁcation datasets namely, a4a, phishing and ijcnn1 from https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ and sido data from http: //www.causality.inf.ethz.ch/data/SIDO.html .
Revised: Datasets: We rely on four binary classiﬁcation datasets namely, a4a, phishing and ijcnn1 from https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ and sido data from http: //www.causality.inf.ethz.ch/data/SIDO.html .

Original: The characteristics of these datasets are reported in Table 2.
Revised: The characteristics of these datasets are reported in Table 2.

Original: We distribute the samples across 20 nodes and create 20 mini batches of local samples for all datasets.
Revised: We distribute the samples across 20 nodes and create 20 mini batches of local samples for all datasets.

Original: Network Setting: We conduct the experiments for 2D torus topology and ring topology.
Revised: Network Setting: We conduct the experiments for 2D torus topology and ring topology.

Original: We generate weight matrix W with W ij = 1 / 5 and W ij = 1 / 3 for all ( i, j
Revised: We generate weight matrix W with W ij = 1 / 5 and W ij = 1 / 3 for all ( i, j

Original: ) ∈ E ∪ { ( i, i ) } for 2D torus topology and ring topology respectively.
Revised: ) ∈ E ∪ { ( i, i ) } for 2D torus topology and ring topology respectively.

Original: Compression Operator: We consider an unbiased b -bits quantization operator Q ∞ ( x ) [24] throughout our empirical study.
Revised: Compression Operator: We consider an unbiased b -bits quantization operator Q ∞ ( x ) [24] throughout our empirical study.

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: where · represents Hadamard product, | x | denotes elementwise absolute value and u is a random vector uniformly distributed in [0 , 1] d .
Revised: where · represents Hadamard product, | x | denotes elementwise absolute value and u is a random vector uniformly distributed in [0 , 1] d .

Original: Theorem 3 in [24] shows that Q ∞ ( x ) satisﬁes assumption 3.4 with δ = sup x (cid:107) sign ( x )2 − ( b
Revised: Theorem 3 in [24] shows that Q ∞ ( x ) satisﬁes assumption 3.4 with δ = sup x (cid:107) sign ( x )2 − ( b

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: (cid:107) 2 (cid:107)
Revised: (cid:107) 2 (cid:107)

Original: [Equation]
Revised: [Equation]

Original: (cid:107)
Revised: (cid:107)

Original: [Equation]
Revised: [Equation]

Original: (cid:107).
Revised: (cid:107).

Original: We know that (cid:107) x (cid:107) ∞ ≤ (cid:107) x (cid:107) 2 for all x ∈ R d .
Revised: We know that (cid:107) x (cid:107) ∞ ≤ (cid:107) x (cid:107) 2 for all x ∈ R d .

Original: Using this inequality, we can upper bound δ as follows:
Revised: Using this inequality, we can upper bound δ as follows:

Original: [Equation]
Revised: [Equation]

Original: The above bound depends can be made independent of d by choosing b = 1 + log 2 √ d .
Revised: The above bound depends can be made independent of d by choosing b = 1 + log 2 √ d .

Original: We use six different bits value from the set { 1 + log 2 √ d, 2 , 4 , 8 , 16 , 32 } to evaluate the behavior of Algorithm 2 and Algorithm 3 with number of bits.
Revised: We use six different bits value from the set { 1 + log 2 √ d, 2 , 4 , 8 , 16 , 32 } to evaluate the behavior of Algorithm 2 and Algorithm 3 with number of bits.

Original: I.2 Baseline methods
Revised: I.2 Baseline methods

Original: We compare the performance of proposed algorithms C-RDPSG and C-DPSVRG with three noncompression based baseline algorithms: (1) Distributed Min-Max Data similarity algorithm [5] (2)
Revised: We compare the performance of proposed algorithms C-RDPSG and C-DPSVRG with three noncompression based baseline algorithms: (1) Distributed Min-Max Data similarity algorithm [5] (2)

Original: Decentralized Parallel Optimistic Stochastic Gradient (DPOSG) algorithm
Revised: Decentralized Parallel Optimistic Stochastic Gradient (DPOSG) algorithm

Original: [23] and, (3) Decentralized
Revised: [23] and, (3) Decentralized

Original: Minimax Hybrid Stochastic Gradient Descent (DM-HSGD) algorithm
Revised: Minimax Hybrid Stochastic Gradient Descent (DM-HSGD) algorithm

Original: Distributed Min-Max data similarity: This algorithm is based on accelerated gossip scheme employed on model updates and gradient vectors [5].
Revised: Distributed Min-Max data similarity: This algorithm is based on accelerated gossip scheme employed on model updates and gradient vectors [5].

Original: The number of iterates in accelerated gossip scheme and the step size are computed according to the theoretical details provided in [5].
Revised: The number of iterates in accelerated gossip scheme and the step size are computed according to the theoretical details provided in [5].

Original: This algorithm requires approximate solution of an inner saddle point problem at every iterate.
Revised: This algorithm requires approximate solution of an inner saddle point problem at every iterate.

Original: We run extragradient method [16] to solve the inner saddle point problem with a desired precision accuracy provided in [5].
Revised: We run extragradient method [16] to solve the inner saddle point problem with a desired precision accuracy provided in [5].

Original: Throughout this section, we use the shorthand notation for Distributed Min-Max data similarity algorithm as Min-Max similarity.
Revised: Throughout this section, we use the shorthand notation for Distributed Min-Max data similarity algorithm as Min-Max similarity.

Original: Decentralized Parallel Optimistic Stochastic Gradient (DPOSG): DPOSG [23] is a two step algorithm with local model averaging designed for solving unconstrained saddle-point problems in a decentralized fashion.
Revised: Decentralized Parallel Optimistic Stochastic Gradient (DPOSG): DPOSG [23] is a two step algorithm with local model averaging designed for solving unconstrained saddle-point problems in a decentralized fashion.

Original: We include the projection steps to both update sequences of DPOSG as we are solving constrained problem (283).
Revised: We include the projection steps to both update sequences of DPOSG as we are solving constrained problem (283).

Original: The step size and the number of local model averaging steps are tuned according to Theorem 1 in [23].
Revised: The step size and the number of local model averaging steps are tuned according to Theorem 1 in [23].

Original: Decentralized Minimax Hybrid Stochastic Gradient Descent (DM-HSGD): DM-HSGD [36] is a gradient tracking based algorithm designed for solving saddle point problems with a constraint set on dual variable.
Revised: Decentralized Minimax Hybrid Stochastic Gradient Descent (DM-HSGD): DM-HSGD [36] is a gradient tracking based algorithm designed for solving saddle point problems with a constraint set on dual variable.

Original: We incorporate projection step to the model update of primal variable.
Revised: We incorporate projection step to the model update of primal variable.

Original: We use grid search to ﬁnd the best step sizes for primal and dual variable updates.
Revised: We use grid search to ﬁnd the best step sizes for primal and dual variable updates.

Original: Other parameters like initial large batch size and parameters involved in gradient tracking update sequence are chosen according to the experimental setting in [36].
Revised: Other parameters like initial large batch size and parameters involved in gradient tracking update sequence are chosen according to the experimental setting in [36].

Original: I.3 Benchmark Quantities
Revised: I.3 Benchmark Quantities

Original: We run the centralized and uncompressed version of C-DPSVRG for 50 , 000 iterations to ﬁnd saddle point solution z (cid:63) =
Revised: We run the centralized and uncompressed version of C-DPSVRG for 50 , 000 iterations to ﬁnd saddle point solution z (cid:63) =

Original: ( x (cid:63) , y (cid:63) ) of problem (283).
Revised: ( x (cid:63) , y (cid:63) ) of problem (283).

Original: The performance of all the methods is measured using (cid:80) mi =1 (cid:13)(cid:13)
Revised: The performance of all the methods is measured using (cid:80) mi =1 (cid:13)(cid:13)

Original: z it − z
Revised: z it − z

Original: (cid:63) (cid:13)(cid:13) 2 .
Revised: (cid:63) (cid:13)(cid:13) 2 .

Original: Number of gradient computations and communications: We calculate the total number of gradient computations according to the number of samples used in the gradient computation at a given iterate t .
Revised: Number of gradient computations and communications: We calculate the total number of gradient computations according to the number of samples used in the gradient computation at a given iterate t .

Original: The number of communications per iterate are computed as the number of times a node exchanges information with its neighbors.
Revised: The number of communications per iterate are computed as the number of times a node exchanges information with its neighbors.

Original: Number of bits transmitted: We set number of bits b = 4 in compression operator Q ∞ ( x ) for
Revised: Number of bits transmitted: We set number of bits b = 4 in compression operator Q ∞ ( x ) for

Original: C-RDPSG and C-DPSVRG.
Revised: C-RDPSG and C-DPSVRG.

Original: Similar to [15], we assume that on an average 5 bits (1 bit for sign and bits for quantization level) are transmitted at every iterate for C-RDPSG and C-DPSVRG.
Revised: Similar to [15], we assume that on an average 5 bits (1 bit for sign and bits for quantization level) are transmitted at every iterate for C-RDPSG and C-DPSVRG.

Original: We assume that on an average 32 bits are transmitted per communication for DPOSG, DM-HSGD and Min-Max similarity algorithm.
Revised: We assume that on an average 32 bits are transmitted per communication for DPOSG, DM-HSGD and Min-Max similarity algorithm.

Original: I.4 Observations
Revised: I.4 Observations

Original: Comparison to baselines: C-RDPSG converges faster in the beginning and slows down after reaching approximately 10 − 8 accuracy as depicted in Figure 1 and Figure 2.
Revised: Comparison to baselines: C-RDPSG converges faster in the beginning and slows down after reaching approximately 10 − 8 accuracy as depicted in Figure 1 and Figure 2.

Original: C-DPSVRG converges faster than other baseline methods.
Revised: C-DPSVRG converges faster than other baseline methods.

Original: DPOSG and DM-HSGD converge only to a neighbourhood of the saddle point solution and start oscillating after some time.
Revised: DPOSG and DM-HSGD converge only to a neighbourhood of the saddle point solution and start oscillating after some time.

Original: The restart scheme’s inclusion in
Revised: The restart scheme’s inclusion in

Original: C-RDPSG helps mitigate the ﬂat and oscillatory behaviour at the later iterations, unlike DPOSG and
Revised: C-RDPSG helps mitigate the ﬂat and oscillatory behaviour at the later iterations, unlike DPOSG and

Original: DM-HSGD.
Revised: DM-HSGD.

Original: C-RDPSG is faster than C-DPSVRG, DPOSG, DM-HSGD and Min-Max similarity in terms of gradient computations, communications and bits transmitted to achieve saddle points of moderate accuracy.
Revised: C-RDPSG is faster than C-DPSVRG, DPOSG, DM-HSGD and Min-Max similarity in terms of gradient computations, communications and bits transmitted to achieve saddle points of moderate accuracy.

Original: The performance of C-RDPSG is competitive with DPOSG and DM-HSGD in the long run as demonstrated in Figure 1 and Figure 2.
Revised: The performance of C-RDPSG is competitive with DPOSG and DM-HSGD in the long run as demonstrated in Figure 1 and Figure 2.

Original: Compression effect: Plots in Figure 1 depict that C-DPSVRG is 1000 times faster than Min-Max similarity, DPOSG algorithm and 10 times faster than DM-HSGD in terms of transmitted bits.
Revised: Compression effect: Plots in Figure 1 depict that C-DPSVRG is 1000 times faster than Min-Max similarity, DPOSG algorithm and 10 times faster than DM-HSGD in terms of transmitted bits.

Original: We observe that C-RDPSG is also 1000 times faster than Min-max similarity and DPOSG for obtaining saddle point solutions of moderate accuracy, in terms of transmitted bits.
Revised: We observe that C-RDPSG is also 1000 times faster than Min-max similarity and DPOSG for obtaining saddle point solutions of moderate accuracy, in terms of transmitted bits.

Original: Communication efﬁciency: The one-time communication at every iterate in C-DPSVRG speeds up communication and makes C-DPSVRG to be 100 times faster than Min-Max similarity and DPOSG methods as shown in Figure 1.
Revised: Communication efﬁciency: The one-time communication at every iterate in C-DPSVRG speeds up communication and makes C-DPSVRG to be 100 times faster than Min-Max similarity and DPOSG methods as shown in Figure 1.

Original: C-RDPSG is 10 times faster than DM-HSGD and 100 times faster than
Revised: C-RDPSG is 10 times faster than DM-HSGD and 100 times faster than

Original: DPOSG and Min-Max similarity in terms of communications at the initial stages of the algorithm.
Revised: DPOSG and Min-Max similarity in terms of communications at the initial stages of the algorithm.

Original: Different choices of reference probabilities: The full batch gradient computations in C-DPSVRG depends on the reference probability parameter p .
Revised: Different choices of reference probabilities: The full batch gradient computations in C-DPSVRG depends on the reference probability parameter p .

Original: Motivated from [18], we run C-DPSVRG with ﬁve different reference probabilities as 1 /n, 1 / ( κn 3 ) 1 / 4 , 1 / ( κn ) 1 / 2 , 1 / ( κ 3 n ) 1 / 4 and 1 /κ .
Revised: Motivated from [18], we run C-DPSVRG with ﬁve different reference probabilities as 1 /n, 1 / ( κn 3 ) 1 / 4 , 1 / ( κn ) 1 / 2 , 1 / ( κ 3 n ) 1 / 4 and 1 /κ .

Original: From Figure 3, we observe that setting p = 1 /n requires the least number of gradient computations as it corresponds to the less frequent computation of full batch gradients.
Revised: From Figure 3, we observe that setting p = 1 /n requires the least number of gradient computations as it corresponds to the less frequent computation of full batch gradients.

Original: Impact of topology: From Figure 2, we observe that the convergence behaviour of all methods is similar to 2D torus topology.
Revised: Impact of topology: From Figure 2, we observe that the convergence behaviour of all methods is similar to 2D torus topology.

Original: We also note that ring topology requires large number of communications compared to 2D torus due to its sparse connectivity.
Revised: We also note that ring topology requires large number of communications compared to 2D torus due to its sparse connectivity.

Original: Compression error:
Revised: Compression error:

Original: We plot compression error (cid:107) Q ( ν x ) − ν x (cid:107) 

Revised: We plot compression error (cid:107) Q ( ν x ) − ν x (cid:107)

Original: + (cid:107) Q ( ν y )
Revised: + (cid:107) Q ( ν y )

Original: − ν y (cid:107) 2 against number of transmitted bits for C-RDPSG and C-DPSVRG as shown in Figure 4.
Revised: − ν y (cid:107) 2 against number of transmitted bits for C-RDPSG and C-DPSVRG as shown in Figure 4.

Original: We observe that C-DPSVRG with O (log d ) bits achieves compression error 10 − 25 in less than 20,000 transmitted bits.
Revised: We observe that C-DPSVRG with O (log d ) bits achieves compression error 10 − 25 in less than 20,000 transmitted bits.

Original: It shows a clear advantage of using O (log d ) bits in C-DPSVRG while maintaining low compression error.
Revised: It shows a clear advantage of using O (log d ) bits in C-DPSVRG while maintaining low compression error.

Original: In
Revised: In

Original: C-RDPSG, larger the number of bits used in the quantization operator, smaller the compression error.
Revised: C-RDPSG, larger the number of bits used in the quantization operator, smaller the compression error.

Original: There are sharp jumps in the decay of compression error during a restart of C-RDPSG as depicted in
Revised: There are sharp jumps in the decay of compression error during a restart of C-RDPSG as depicted in

Original: Figure 4.
Revised: Figure 4.

Original: Number of bits transmitted: As demonstrated in Figure 5, C-DPSVRG transmits less number of bits when b = 1 + log 2 √ d to achieve highly accurate solution.
Revised: Number of bits transmitted: As demonstrated in Figure 5, C-DPSVRG transmits less number of bits when b = 1 + log 2 √ d to achieve highly accurate solution.

Original: We can observe that the convergence behavior of C-DPSVRG is affected by number of bits less than 1 + log 2 √ d .
Revised: We can observe that the convergence behavior of C-DPSVRG is affected by number of bits less than 1 + log 2 √ d .

Original: For example, the convergence of C-DPSVRG becomes slow for sido data with b = 2 , 4 < 1 + log 2 √ d ≈ 7 as shown in Figure 5.
Revised: For example, the convergence of C-DPSVRG becomes slow for sido data with b = 2 , 4 < 1 + log 2 √ d ≈ 7 as shown in Figure 5.

Original: It shows that Q ∞ ( x ) provides better performance for b
Revised: It shows that Q ∞ ( x ) provides better performance for b

Original: = O (log 2 d ) especially for high-dimensional data points.
Revised: = O (log 2 d ) especially for high-dimensional data points.

Original: The behavior of C-RDPSG is less affected by varying the number of bits as shown in Figure 6.
Revised: The behavior of C-RDPSG is less affected by varying the number of bits as shown in Figure 6.

Original: In the long term, C-RDPSG behavior is almost identical for all chosen values of number of bits except b = 2 .
Revised: In the long term, C-RDPSG behavior is almost identical for all chosen values of number of bits except b = 2 .

Original: Impact of number of nodes: As the number of nodes increases, C-RDPSG requires fewer gradient computations in the initial phase.
Revised: Impact of number of nodes: As the number of nodes increases, C-RDPSG requires fewer gradient computations in the initial phase.

Original: However, smaller number of nodes gives faster convergence at the later iterations for 2D torus and ring topology, as depicted in Figure 8 and Figure 10.
Revised: However, smaller number of nodes gives faster convergence at the later iterations for 2D torus and ring topology, as depicted in Figure 8 and Figure 10.

Original: C-DPSVRG also requires small number of gradient computations in 2D torus with larger number of nodes because it assigns smaller batch sizes to every node.
Revised: C-DPSVRG also requires small number of gradient computations in 2D torus with larger number of nodes because it assigns smaller batch sizes to every node.

Original: As depicted in Figure 7, C-DPSVRG performance does not change much in terms of the number of communications and bits transmitted for 2D torus.
Revised: As depicted in Figure 7, C-DPSVRG performance does not change much in terms of the number of communications and bits transmitted for 2D torus.

Original: The sparsity level of ring topology is higher than that of 2D torus and increases with number of nodes.
Revised: The sparsity level of ring topology is higher than that of 2D torus and increases with number of nodes.

Original: It leads C-DPSVRG to achieve fast convergence eventually in terms of gradient computations with a smaller number of nodes, as demonstrated in Figure 9.
Revised: It leads C-DPSVRG to achieve fast convergence eventually in terms of gradient computations with a smaller number of nodes, as demonstrated in Figure 9.

Original: In contrast to the performance of C-DPSVRG in terms of communications in 2D torus (Figure 7), C-DPSVRG requires more communications for large number of nodes in a ring topology, as shown in Figure 9.
Revised: In contrast to the performance of C-DPSVRG in terms of communications in 2D torus (Figure 7), C-DPSVRG requires more communications for large number of nodes in a ring topology, as shown in Figure 9.

Original: DM-HSGD and C-RDPSG are competitive in terms of gradient computations and communications with larger number of nodes as demonstrated in Figures 11 and 12.
Revised: DM-HSGD and C-RDPSG are competitive in terms of gradient computations and communications with larger number of nodes as demonstrated in Figures 11 and 12.

Original: However it is to be noted that DM-HSGD exhibits an oscillatory behavior in the saddle point solutions, after reaching a moderate solution accuracy.
Revised: However it is to be noted that DM-HSGD exhibits an oscillatory behavior in the saddle point solutions, after reaching a moderate solution accuracy.

Original: I.5 Estimating Lipschitz parameters
Revised: I.5 Estimating Lipschitz parameters

Original: In this section, we estimate Lipschitz parameters L xx , L yy , L xy , L yy of robust logistic regression problem (283).
Revised: In this section, we estimate Lipschitz parameters L xx , L yy , L xy , L yy of robust logistic regression problem (283).

Original: Assume that each node i has N i number of local samples such that (cid:80) mi =1 N i = N .
Revised: Assume that each node i has N i number of local samples such that (cid:80) mi =1 N i = N .

Original: Recall objective function Ψ( x, y ) in equation (283):
Revised: Recall objective function Ψ( x, y ) in equation (283):

Original: [Equation]
Revised: [Equation]

Original: where f i
Revised: where f i

Original: [Equation]
Revised: [Equation]

Original: x,
Revised: x,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Gradients of f i ( x, y ) with respect to x and y are given by
Revised: Gradients of f i ( x, y ) with respect to x and y are given by

Original: [Equation]
Revised: [Equation]

Original: We create n batches { N i 1 , . . .
Revised: We create n batches { N i 1 , . . .

Original: , N in } of local samples N i and write f i ( x, y ) in the form of 1 n f ij ( x, y ) .
Revised: , N in } of local samples N i and write f i ( x, y ) in the form of 1 n f ij ( x, y ) .

Original: [Equation]
Revised: [Equation]

Original: j =   n N
Revised: j =   n N

Original: N ij (cid:88) l = log (cid:16) 1 + exp (cid:16) − b jil x (cid:62) ( a jil + y ) (cid:17)(cid:17) + λ 2 m (cid:107) x (cid:107) 22 − β 2 m (cid:107) y
Revised: N ij (cid:88) l = log (cid:16) 1 + exp (cid:16) − b jil x (cid:62) ( a jil + y ) (cid:17)(cid:17) + λ 2 m (cid:107) x (cid:107) 22 − β 2 m (cid:107) y

Original: (cid:107)  =n n (cid:88)
Revised: (cid:107)  =n n (cid:88)

Original: j = f ij ( x, y ) , where f ij ( x, y ) =
Revised: j = f ij ( x, y ) , where f ij ( x, y ) =

Original: nN (cid:80) N ij l =1 log (cid:16) 1 + exp (cid:16) − b jil x (cid:62) ( a jil + y ) (cid:17)(cid:17) + λ 2 m (cid:107) x (cid:107) 22 − β 2 m (cid:107) y (cid:107) 22 .
Revised: nN (cid:80) N ij l =1 log (cid:16) 1 + exp (cid:16) − b jil x (cid:62) ( a jil + y ) (cid:17)(cid:17) + λ 2 m (cid:107) x (cid:107) 22 − β 2 m (cid:107) y (cid:107) 22 .

Original: We are now ready to ﬁnd required Lipschitz parameters.
Revised: We are now ready to ﬁnd required Lipschitz parameters.

Original: Computing L ijxx :
Revised: Computing L ijxx :

Original: [Equation]
Revised: [Equation]

Original: Computing L ijyy :
Revised: Computing L ijyy :

Original: [Equation]
Revised: [Equation]

Original: Computing L ijxy :
Revised: Computing L ijxy :

Original: [Equation]
Revised: [Equation]

Original: Hence we have
Revised: Hence we have

Original: [Equation]
Revised: [Equation]

Original: We set L xx = max i,j { L ijxx } , L yy = max i,j { L ijyy } and L xy =
Revised: We set L xx = max i,j { L ijxx } , L yy = max i,j { L ijyy } and L xy =

Original: L yx = max
Revised: L yx = max

Original: i,j { L ijxy } .
Revised: i,j { L ijxy } .

Original: The strong convexity and strong concavity parameters are respectively set to λ and β .
Revised: The strong convexity and strong concavity parameters are respectively set to λ and β .