Doubly Accelerated Stochastic Variance Reduced Dual Averaging Method for Regularized Empirical Risk Minimization

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1 Doubly Accelerated Stochastic Variance Reduced Dual Averaging Method for Regularized Empirical Risk Minimization Tomoya Murata Taiji Suzuki More recently, several authors have proposed accelerarxiv: v4 [mathoc] 9 Sep 07 Abstract In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient estimate and can easily make good use of parallel computing The core of our proposed method is the incorporation of our new double acceleration technique and variance reduction technique We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size n mini-batches for achieving the optimal iteration complexities for both nonstrongly and strongly convex objectives, where n is the training set size Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for both non-strongly and strongly convex objectives Introduction We consider a composite convex optimization problem associated with regularized empirical risk minimization, which often arises in machine learning In particular, our goal is to minimize the sum of finite smooth convex functions and a relatively simple possibly non-differentiable convex function by using first order methods in mini-batch settings The use of mini-batches is becoming a golden standard in the machine learning community, because it is generally more efficient to execute matrix-vector multiplications over a mini-batch than an equivalent amount of NTT DATA Mathematical Systems Inc Tokyo, Japan Graduate School of Information Science and Technology, The University of Tokyo Correspondence to: Tomoya Murata <murata@msicojp>, Taiji Suzuki <taiji@mistiu-tokyoacjp> vector-vector ones each over a single instance; and more importantly, mini-batch settings can easily make good use of parallel computing Traditional and effective methods for solving the abovementioned problem are the proximal gradient PG method and accelerated proximal gradient APG method Nesterov et al, 007; Beck & Teboulle, 009; Tseng, 008 These methods are well known to achieve linear convergence for strongly convex objectives Particularly, APG achieves optimal iteration complexities for both non-strongly and strongly convex objectives However, these methods need a per iteration cost of Ond, where n denotes the number of components of the finite sum, and d is the dimension of the solution space In typical machine learning tasks, n and d correspond to the number of instances and features respectively, which can be very large Then, the per iteration cost of these methods can be considerably high A popular alternative is the stochastic gradient descent SGD method Singer & Duchi, 009; Hazan et al, 007; Shalev-Shwartz & Singer, 007 As the per iteration cost of SGD is only Od in non-mini-batch settings, SGD is suitable for many machine learning tasks However, SGD only achieves sublinear rates and is ultimately slower than PG and APG Recently, a number of stochastic gradient methods have been proposed; they use a variance reduction technique that utilizes the finite sum structure of the problem stochastic averaged gradient SAG method Roux et al, 0; Schmidt et al, 03, stochastic variance reduced gradient SVRG method Johnson & Zhang, 03; Xiao & Zhang, 04 and SAGA Defazio et al, 04 Even though the per iteration costs of these methods are same as that of SGD, they achieve a linear convergence for strongly convex objectives Consequently, these methods dramatically improve the total computational cost of PG However, in size b mini-batch settings, the rate is essentially b times worse than in non-mini-batch settings This means that there is little benefit in applying mini-batch scheme to these methods

2 ated stochastic methods for the composite finite sum problem accelerated stochastic dual coordinate ascent ASDCA method Shalev-Shwartz & Zhang, 03, Universal Catalyst UC Lin et al, 05, accelerated proximal coordinate gradient APCG method Lin et al, 04a, stochastic primal-dual coordinate SPDC method Zhang & Xiao, 05, and Katyusha Allen-Zhu, 06 ASDCA UC, APCG, SPDC and Katyusha essentially achieve the optimal total computational cost for strongly convex objectives in non-mini-batch settings However, in size b mini-batch settings, the rate is essentially b times worse than that in non-mini-batch settings, and these methods need size On mini-batches for achieving the optimal iteration complexity, which is essentially the same as APG In addition, Nitanda 04; 05 has proposed the accelerated mini-batch proximal stochastic variance reduced gradient AccProxSVRG method and its variant, the accelerated efficient mini-batch stochastic variance reduced gradient AMSVRG method In nonmini-batch settings, AccProxSVRG only achieves the same rate as SVRG However, in mini-batch settings, AccProx- SVRG significantly improves the mini-batch efficiency of non-accelerated variance reduction methods, and surprisingly, AccProxSVRG essentially only needs size O κ mini-batches for achieving the optimal iteration complexity for strongly convex objectives, where κ is the condition number of the problem However, the necessary size of mini-batches depends on the condition number and gradually increases when the condition number increases and ultimately matches with On for a large condition number Main contribution We propose a new accelerated stochastic variance reduction method that achieves better convergence than existing methods do, and it particularly utilizes mini-batch settings well; it is called the doubly accelerated stochastic variance reduced dual averaging DASVRDA method Our method significantly improves the mini-batch efficiencies of state-of-the-art methods, and our method essentially only needs size O n mini-batches 3 for achieving the More precisely, the rate of ASDCA UC is with extra logfactors, and near but worse than the one of APCG, SPDC and Katyusha This means that ASDCA UC cannot be optimal Katyusha also achieves a near optimal total computational cost for non-strongly convex objectives 3 Actually, when L/ nlog n and L/µ n, our method needs size Õn /L and On µ/l mini-batches, respectively, which are larger than O n, but smaller than On Achieving the optimal iteration complexity for solving high accuracy and bad conditioned problems is much more important than doing ones with low accuracy and well-conditioned ones, because the former needs more overall computational cost than the latter optimal iteration complexities 4 for both non-strongly and strongly convex objectives We list the comparisons of our method with several preceding methods in Table Preliminary In this section, we provide several notations and initions used in this paper Then, we make assumptions for our analysis We use to denote the Euclidean L norm : x x i x i For natural number n, [n] denotes set,, n} Definition We say that a function f : R d R is L-smooth L > 0 if f is differentiable and satisfies fx fy L x y x, y R d If f is convex, is equivalent to the following: see Nesterov 03: fx + fx, y x + L fx fy fy x, y R d Definition A convex function f : R d R is called µ-strongly convex µ 0 if f satisfies µ x y + ξ, y x + fx fy, x, y R d, ξ fx where fx denotes the set of the subgradients of f at x Note that if f is µ-strongly convex, then f has the unique minimizer Definition 3 We say that a function f : R d R is µ- optimally strongly convex µ 0 if f has a minimizer and satisfies µ x x fx fx x R d, x argmin x R dfx If a function f is µ-strongly convex, then f is clearly µ- optimally strongly convex Definition 4 We say that a convex function f : R d R is relatively simple if computing the proximal mapping of f at y, prox f y argmin x R d } x y + fx, takes at most Od computational cost, for any y R d 4 We refer to optimal iteration complexity as the iteration complexity of deterministic Nesterov s acceleration method Nesterov 03

3 SVRG SVRG ++ ASDCA UC APCG SPDC Katyusha AccProxSVRG DASVRDA µ-strongly convex Non-strongly convex Total computational cost Necessary size of mini-batches Total computational cost Necessary size of mini-batches in size b mini-batch settings L/µ n L/µ n in size b mini-batch settings L/ nlog / L/ nlog / O d n + bl µ log Unattainable Unattainable O d nlog + bl Unattainable Unattainable nbl Õ d n + µ log n+ Unattainable Unattainable Õ d nbl Unattainable Unattainable nbl O d n + µ log On On No direct analysis Unattainable Unattainable nbl O d n + µ log On On No direct analysis Unattainable Unattainable nbl O d n + µ log On On O d nlog + nbl On On n b L O d n + n µ + b L µ log L O µ O n µ L No direct analysis Unattainable Unattainable nl O d n + µ + b L µ log O n O n µ L O d nlog + nl + b L O n Õ n L Table Comparisons of our method with SVRG SVRG ++ Allen-Zhu & Yuan, 06, ASDCA UC, APCG, SPDC, Katyusha and AccProxSVRG n is the number of components of the finite sum, d is the dimension of the solution space, b is the mini-batch size, L is the smoothness parameter of the finite sum, µ is the strong convexity parameter of objectives see Def and Def for their initions, and is accuracy Necessary size of mini-batches indicates the order of the necessary size of mini-batches for achieving optimal iteration complexities O L/µlog/ and O L/ for strongly and non-strongly convex objectives, respectively We regard one computation of a full gradient as n/b iterations in size b mini-batch settings, for a fair comparison Unattainable implies that the algorithm cannot achieve the optimal iteration complexity even if it uses size n mini-batches Õ hides extra log-factors The results marked in red denote the main contributions of this paper As f is convex, function / x y +fx is -strongly convex, and function prox f is well-ined Now, we formally describe the problem to be considered in this paper and the assumptions for our theory In this paper, we consider the following composite convex minimization problem: min x R d P x F x + Rx}, 3 where F x n n i f ix Here each f i : R d R is a L i -smooth convex function and R : R d R is a relatively simple and possibly non-differentiable convex function Problems of this form often arise in machine learning and fall under regularized empirical risk minimization ERM In ERM problems, we are given n training examples a i, b i } n i, where each a i R d is the feature vector of example i, and each b i R is the label of example i The following regression and classification problems are typical examples of ERM on our setting: Lasso: f i x a i x b i and Rx λ x Ridge logistic regression: f i x log + exp b i a i x and Rx λ x Support vector machines: Rx λ x f i x h ν i a i x and Here, h ν i is a smooth variant of hinge loss for the inition, for example, see Shalev-Shwartz & Zhang 03 We make the following assumptions for our analysis: Assumption There exists a minimizer x of 3 Assumption Each f i is convex and L i -smooth The above examples satisfy this assumption with L i a i for the squared loss, L i a i /4 for the logistic loss, and L i ν a i for the smoothed hinge loss Assumption 3 Regularization function R is convex and is relatively simple For example, Elastic Net regularizer Rx λ x + λ / x λ, λ 0 satisfies this assumption Indeed, we can analytically compute the proximal mapping of R by prox R z /+λ signz j max z j λ, 0} d j, and this costs only Od We always consider Assumption, and 3 in this paper Assumption 4 There exists µ > 0 such that objective function P is µ-optimally strongly convex If each f i is convex, then for Elastic Net regularization function Rx λ x + λ / x λ 0 and λ > 0, Assumption 4 holds with µ λ We further consider Assumption 4 when we deal with strongly convex objectives 3 Our Approach: Double Acceleration In this section, we provide high-level ideas of our main contribution called double acceleration First, we consider deterministic PG Algorithm and non-mini-batch SVRG Algorithm 3 PG is an extension of the steepest descent to proximal settings SVRG is a stochastic gradient method using the variance reduction technique, which utilizes the finite sum structure of

4 Algorithm PG x 0, η, S Input: x 0 R d, η > 0, S N for s to S do x s One Stage PG x s, η end for Output: S S s x s Algorithm One Stage PG x, η Input: x R d, η > 0 x + prox ηr x η F x Output: x + the problem, and it achieves a faster convergence rate than PG does As SVRG Algorithm 3 matches with PG Algorithm when the number of inner iterations is m, SVRG can be seen as a generalization of PG The key element of SVRG is employing a simple but powerful technique called the variance reduction technique for gradient estimate The variance reduction of the gradient is realized by setting g k f ik x k f ik x + F x rather than vanilla stochastic gradient f ik x k Generally, stochastic gradient f ik x k is an unbiased estimator of F x k, but it may have high variance In contrast, g k is also unbiased, and one can show that its variance is reduced ; that is, the variance converges to zero as x k and x to x Next, we explain to the method of accelerating SVRG and obtaining an even faster convergence rate based on our new but quite natural idea outer acceleration First, we would like to remind you that the procedure of deterministic APG is given as described in Algorithm 5 APG uses the famous momentum scheme and achieves the optimal iteration complexity Our natural idea is replacing One Stage PG in Algorithm 5 with One Stage SVRG With slight modifications, we can show that this algorithm improves the rates of PG, SVRG and APG, and is optimal We call this new algorithm outerly accelerated SVRG Note that the algorithm matches with APG when m and thus, can be seen as a direct generalization of APG However, this algorithm has poor mini-batch efficiency, because in size b mini-batch settings, the rate of this algorithm is essentially b times worse than that of non-mini-batch settings Stateof-the-art methods APCG, SPDC, and Katyusha also suffer from the same problem in the mini-batch setting Now, we illustrate that for improving the mini-batch efficiency, using the inner acceleration technique is beneficial Nitanda Nitanda, 04 has proposed AccProxSVRG in mini-batch settings He applied the momentum scheme to One Stage SVRG, and we call this technique inner acceleration He showed that the inner acceleration could significantly improve the mini-batch efficiency of vanilla SVRG This fact indicates that inner acceleration is essen- Algorithm 3 SVRG x 0, η, m, S Input: x 0 R d, η > 0, m, S N for s to S do x s One Stage SVRG x s, η, m end for Output: S S s x s Algorithm 4 One Stage SVRG x, η, m Input: x R d, η > 0, m N x 0 x for k to m do Pick i k,, n} randomly g k f ik x k f ik x + F x x k prox ηr x k ηg k end for Output: n n k x k tial to fully utilize the mini-batch settings However, AccProxSVRG is not a truly accelerated method, because in non-mini-batch settings, the rate of AccProxSVRG is same as that of vanilla SVRG In this way, we arrive at our main high-level idea called double acceleration, which involves applying momentum scheme to both outer and inner algorithms This enables us not only to lead to the optimal total computational cost in non-mini-batch settings, but also to improving the minibatch efficiency of vanilla acceleration methods We have considered SVRG and its accelerations so far; however, we actually adopt stochastic variance reduced dual averaging SVRDA rather than SVRG itself, because we can construct lazy update rules of innerly accelerated SVRDA for sparse data see Section 6 In Section F of supplementary material, we briefly discuss a SVRG version of our proposed method and provide its convergence analysis 4 Algorithm Description In this section, we describe the concrete procedure of the proposed algorithm in detail DASVRDA for non-strongly convex objectives We provide details of the doubly accelerated SVRDA DASVRDA method for non-strongly convex objectives in Algorithm 6 Our momentum step is slightly different from that of vanilla deterministic accelerated methods: we not only add momentum term θ s / θ s x s x s to the current solution x s but also add term θ s / θ s z s x s, where z s is the current more aggressively updated solution rather than x s ; thus, this

5 Algorithm 5 APG x 0, η, S Input: x 0 R d, η > 0, S N x x 0 θ 0 0 for s to S do θ s s+ ỹ s x s + θ s θ s x s x s x s One Stage PGỹ s, η end for Output: x S term also can be interpreted as momentum 5 Then, we feed ỹ s to One Stage Accelerated SVRDA Algorithm 7 as an initial point Note that Algorithm 6 can be seen as a direct generalization of APG, because if we set m, One Stage Accelerated SVRDA is essentially the same as one iteration PG with initial point ỹ s ; then, we can see that z s x s, and Algorithm 6 essentially matches with deterministic APG Next, we move to One Stage Accelerated SVRDA Algorithm 7 Algorithm 7 is essentially a combination of the accelerated regularized dual averaging AccSDA method Xiao, 009 with the variance reduction technique of SVRG It updates z k by using the weighted average of all past variance reduced gradients ḡ k instead of only using a single variance reduced gradient g k Note that for constructing variance reduced gradient g k, we use the full gradient of F at x s rather than the initial point ỹ s Remark In Algorithm 7, we pick b indexes according to iid non-uniform distribution Q q i } L} L i n Instead of that, we can pick b indexes, such that each index i l k is uniformly picked from B l, where B l } b l is the preined disjoint partition of [n] with size B l n/b If we adopt this scheme, when we parallelize the algorithm using b machines, each machine only needs to store the corresponding partition of the data set rather than the whole dataset, and this can reduce communication cost and memory cost In this setting, the convergence analysis in Section 5 can be easily revised by simply replacing L with L max max i [n] L i DASVRDA for strongly convex objectives Algorithm 8 is our proposed method for strongly convex objectives Instead of directly accelerating the algo- 5 This form also arises in Monotone APG Li & Lin, 05 In Algorithm 7, x x m can be rewritten as /mm + m k kz k, which is a weighted average of z k ; thus, we can say that z is updated more aggressively than x For the outerly accelerated SVRG that is a combination of Algorithm 6 with vanilla SVRG, see section 3, z and x correspond to x m and /m m k x k in Xiao & Zhang, 04, respectively Thus, we can also see that z is updated more aggressively than x rithm using a constant momentum rate, we restart Algorithm 6 Restarting scheme has several advantages both theoretically and practically First, the restarting scheme only requires the optimal strong convexity of the objective Def 3 instead of the ordinary strong convexity Def Whereas, non-restarting accelerated algorithms essentially require the ordinary strong convexity of the objective Second, for restarting algorithms, we can utilize adaptive restart schemes O Donoghue & Candes, 05 The adaptive restart schemes have been originally proposed for deterministic cases The schemes are heuristic but quite effective empirically The most fascinating property of these schemes is that we need not prespecify the strong convexity parameter µ, and the algorithms adaptively determine the restart timings O Donoghue & Candes 05 have proposed two heuristic adaptive restart schemes: the function scheme and gradient scheme We can easily apply these ideas to our method, because our method is a direct generalization of the deterministic APG For the function scheme, we restart Algorithm 6 if P x s > P x s For the gradient scheme, we restart the algorithm if ỹ s x s ỹ s+ x s > 0 Here ỹ s x s can be interpreted as a one stage gradient mapping of P at ỹ s As ỹ s+ x s is the momentum, this scheme can be interpreted such that we restart whenever the momentum and negative one Stage gradient mapping form an obtuse angle this means that the momentum direction seems to be bad We numerically demonstrate the effectiveness of these schemes in Section 7 DASVRDA ns with warm start Algorithms 9 is a combination of DASVRDA ns with warm start scheme At the warm start phase, we repeatedly run One Stage AccSVRDA and increment the number of its inner iterations m u exponentially until m u n/b After that, we run vanilla DASVRDA ns We can show that this algorithm gives a faster rate than vanilla DASVRDA ns Remark For DASVRDA sc, the warm start scheme for DASVRDA ns is not needed because the theoretical rate is identical to the one without warm start Parameter tunings For DASVRDA ns, only learning rate η needs to be tuned, because we can theoretically obtain the optimal choice of γ, and we can naturally use m n/b as a ault epoch length see Section 5 For DASVRDA sc, both learning rate η and fixed restart interval S need to be tuned

6 Algorithm 6 DASVRDA ns x 0, z 0, γ, L i } n i, m, b, S Input: x 0, z 0 R d, γ >, L i > 0} n i, m N, S N, b [n] x z 0, θ 0 γ L n n i L i Q q i } L} L i n η + γm+ b L for s to S do θ s γ s+ ỹ s x s + θ s θ s x s x s + θ s z s θ s x s x s, z s One Stage AccSVRDAỹ s, x s, η, m, b, Q end for Output: x S Algorithm 7 One Stage AccSVRDA ỹ, x, η, m, b, Q Input: ỹ, x, η > 0, m N, b [n], Q x 0 z 0 ỹ, ḡ 0 0, θ 0 for k to m do Pick independently i k,, ib k Q, I k i l k }b l k+ y k x k + z k g k b i I k f i y k f i x + F x ḡ k z k argmin z R d ḡ k + g k ḡ k, z + Rz + η z z 0 } prox ηθk R z 0 η ḡ k x k x k + z k end for Output: x m, z m 5 Convergence Analysis of DASVRDA Method In this section, we provide the convergence analysis of our algorithms First, we consider the DASVRDA ns algorithm Theorem 5 Suppose that Assumptions, and 3 hold Let x 0, z 0 R d, γ 3, m N, b [n] and S N Then DASVRDA ns x 0, z 0, γ, L i } n i, m, b, S satisfies 4 E [P x S P x ] S + P x 0 P x 8 + z 0 x γ ηs + m + m The proof of Theorem 5 is found in the supplementary material Section A We can easily see that the optimal Algorithm 8 DASVRDA sc ˇx 0, γ, L i } n i, m, b, S, T Input: ˇx 0 R d, γ, L i > 0} n i, m N, b [n], S, T N for t to T do ˇx t DASVRDA ns ˇx t, ˇx t, γ, L i } n i, m, b, S end for Output: ˇx T Algorithm 9 DASVRDA ns with warm start x 0, γ, L i } n i, m 0, m, b, U, S z 0 x 0, L n n i L i, Q q i } L L} i n for u to U do m u γm u + m u end for m U m U + m U / /γ η + γm U + b L for u to U do x u, z u One Stage AccSVRDA z u, x u, η, m u, b, Q end for Output: DASVRDA ns x U, z U, γ, L i } n i, m U, b, S choice of γ is b/m + / O + b/m see Section A of supplementary material We denote this value as γ Using Theorem 5, we can establish the convergence rate of DASVRDA ns with warm start Algorithm 9 Theorem 5 Suppose that Assumptions, and 3 hold Let x 0 R d, γ γ, m N, m 0 min + γm + /b L x0 x P x 0 P x, m N, b [n], U log γm/m 0 and S N Then DASVRDA ns with warm start x 0, γ, L i } n i, m 0, m, b, U, S satisfies E [P x S P x ] O S m + L x 0 x mb The proof of Theorem 5 is found in the supplementary material Section B From Theorem 5, we obtain the following corollary: Corollary 53 Suppose that Assumptions,, and 3 hold Let x 0 R d, γ γ, m n/b, } m 0 min + γm + /b L x0 x P x 0 P x, m N, b [n] and U log γm/m 0 If we appropriately choose S O + /m + / mb L x0 x /, then a total computational cost of DASVRDA ns with warm start x 0, γ, L i } n i, m 0, m, b, U, S for E [P x S P x ] is P x0 P x O d nlog + b + n L x0 x }

7 For the proof of Corollary 53, see Section C of supplementary material Remark Corollary 53 implies that if the mini-batch size b is O n, DASVRDA ns with warm start x 0, γ, L i } n i, m 0, n/b, b, U, S still achieves the total computational cost of Odnlog/ + n L/, which is better than Odnlog/ + nb L/ of Katyusha Remark Corollary 53 also implies that DASVRDA ns with warm start only needs size O n mini-batches for achieving the optimal iteration complexity of O L/, when L/ nlog / In contrast, Katyusha needs size On mini-batches for achieving the optimal iteration complexity Note that even when L/ nlog /, our method only needs size Õn /L mini-batches 6, that is typically smaller than On of Katyusha Next, we analyze the DASVRDA sc algorithm for optimally strongly convex objectives Combining Theorem 5 with the optimal strong convexity of the objective function immediately yields the following theorem, which implies that the DASVRDA sc algorithm achieves a linear convergence Theorem 54 Suppose that Assumptions,, 3 and 4 hold Let ˇx 0 R d, γ γ, m N, b [n] and T N Define ρ 4 /γ +4/ηm+mµ}/ /γ S+ } If S is sufficiently large such that ρ 0,, then DASVRDA sc ˇx 0, γ, L i } n i, m, b, S, T satisfies E[P ˇx T P x ] ρ T [P ˇx 0 P x ] From Theorem 54, we have the following corollary Corollary 55 Suppose that Assumptions,, 3 and 4 hold Let ˇx 0 R d, γ γ, m n/b, b [n] There exists S O + b/n + / n L/µ, such that /log/ρ O Moreover, if we appropriately choose T OlogP ˇx 0 P x /, then a total computational cost of DASVRDA sc ˇx 0, γ, L i } n i, m, b, S, T for E [P ˇx T P x ] is O d n + b + n L P ˇx0 P x log µ Remark Corollary 55 implies that if the mini-batch size b is O n, DASVRDA sc ˇx 0, γ, L i } n i, n/b, b, S, T still achieves the total computational cost of Odn + n L/µlog/, which is much better than Odn + nb L/µlog/ of APCG, SPDC, and Katyusha Remark Corollary 55 also implies that DASVRDA sc only needs size O n mini-batches for achieving the optimal iteration complexity O L/µlog/, when L/µ n 6 Note that we regard one computation of a full gradient as n/b iterations in size b mini-batch settings In contrast, APCG, SPDC and Katyusha need size On mini-batches and AccProxSVRG does O L/µ ones for achieving the optimal iteration complexity Note that even when L/µ n, our method only needs size On µ/l mini-batches 7 This size is smaller than On of APCG, SPDC, and Katyusha, and the same as that of AccProx- SVRG 6 Efficient Implementation for Sparse Data: Lazy Update In this section, we briefly comment on the sparse implementations of our algorithms Originally, lazy update was proposed in online settings Duchi et al, 0 Generally, it is difficult for accelerated stochastic variance reduction methods to construct lazy update rules because i generally, variance reduced gradients are not sparse even if stochastic gradients are sparse; ii if we adopt the momentum scheme, the updated solution becomes a convex combination of previous solutions; and iii for non-strongly convex objectives, the momentum rate must not be constant Konečnỳ et al 06 have tackled the problem of i on non-accelerated settings and derived lazy update rules of the mini-batch semi-stochastic gradient descent msgd method Allen-Zhu 06 has only mentioned that lazy updates can be applied to Katyusha but did not give explicit lazy update rules of Katyusha Particularly, for non-strongly convex objectives, it seems to be difficult to derive lazy update rules owing to the difficulty of iii The reason we adopt the stochastic dual averaging scheme Xiao, 009 rather than stochastic gradient descent for our method is to be able to overcome the difficulties faced in i, ii, and iii The lazy update rules of our method support both non-strongly and strongly convex objectives The formal lazy update algorithms of our method can be found in the supplementary material Section D 7 Numerical Experiments In this section, we provide numerical experiments to demonstrate the performance of DASVRDA We numerically compare our method with several wellknown stochastic gradient methods in mini-batch settings: SVRG Xiao & Zhang, 04 and SVRG ++ Allen-Zhu & Yuan, 06, AccProxSVRG Nitanda, 04, Universal Catalyst Lin et al, 05, APCG Lin et al, 04a, and Katyusha Allen-Zhu, 06 The details of the implemented algorithms and their parameter tunings are found in the supplementary material In the experiments, we focus on the regularized logistic regression problem for binary 7 Note that the required size is On µ/l On, which is not On L/µ On

8 a a9a, λ, λ 0 4, 0 b a9a, λ, λ 0 4, 0 6 c a9a, λ, λ 0, 0 6 d rcv, λ, λ 0 4, 0 e rcv, λ, λ 0 4, 0 6 f rcv, λ, λ 0, 0 6 g sido0, λ, λ 0 4, 0 h sido0, λ, λ 0 4, 0 6 i sido0, λ, λ 0, 0 6 Figure Comparisons on a9a top, rcv middle and sido0 bottom data sets, for regularization parameters λ, λ 0 4, 0 left, λ, λ 0 4, 0 6 middle and λ, λ 0, 0 6 right classification, with regularizer λ + λ / We used three publicly available data sets in the experiments Their sizes n and dimensions d, and common minbatch sizes b for all implemented algorithms are listed in Table Data sets n d b a9a 3, rcv 0, 4 47, sido0, 678 4, Table Summary of the data sets and mini-batch size used in our numerical experiments For regularization parameters, we used three settings λ, λ 0 4, 0, 0 4, 0 6, and 0, 0 6 For the former case, the objective is non-strongly convex, and for the latter two cases, the objectives are strongly convex Note that for the latter two cases, the strong convexity of the objectives is µ 0 6 and is relatively small; thus, it makes acceleration methods beneficial Figure shows the comparisons of our method with the different methods described above on several settings Objective Gap means P x P x for the output solution x Gradient Evaluations /n is the number of computations of stochastic gradients f i divided by n Restart DASVRDA means DASVRDA with heuristic adaptive restarting We can observe the following from these results: Our proposed DASVRDA and Restart DASVRDA significantly outperformed all the other methods overall DASVRDA with the heuristic adaptive restart scheme efficiently made use of the local strong convexities of non-strongly convex objectives and significantly outperformed vanilla DASVRDA on a9a and rcv data sets For the other settings, the algorithm was still

9 comparable to vanilla DASVRDA UC+SVRG did not work as well as it did in theory, and its performances were almost the same as that of vanilla SVRG UC+AccProxSVRG sometimes outperformed vanilla AccProxSVRG but was always outperformed by our methods APCG sometimes performed unstably and was outperformed by vanilla SVRG On sido0 data set, for Elastic Net Setting, APCG significantly outperformed all other methods Katyusha outperformed vanilla SVRG overall However, sometimes Katyusha was slower than vanilla SVRG for Elastic Net Settings This is probably because SVRG is almost adaptive to local strong convexities of loss functions, whereas Katyusha is not see the remark in supplementary material 8 Conclusion In this paper, we developed a new accelerated stochastic variance reduced gradient method for regularized empirical risk minimization problems in mini-batch settings: DASVRDA We have shown that DASVRDA achieves the total computational costs of Odnlog/ + nl/ + b L/ and Odn + nl/µ + b L/µlog/ in size b mini-batch settings for non-strongly and strongly convex objectives, respectively In addition, DASVRDA essentially achieves the optimal iteration complexities only with size O n mini-batches for both settings In the numerical experiments, our method significantly outperformed state-of-the-art methods, including Katyusha and AccProxSVRG Acknowledgment This work was partially supported by MEXT kakenhi , 500, , 5H0678 and 5H05707, JST-PRESTO and JST-CREST References Agarwal, Alekh, Negahban, Sahand, and Wainwright, Martin J Fast global convergence rates of gradient methods for high-dimensional statistical recovery Advances in Neural Information Processing Systems, pp 37 45, 00 Agarwal, Alekh, Negahban, Sahand, and Wainwright, Martin J Stochastic optimization and sparse statistical recovery: Optimal algorithms for high dimensions Advances in Neural Information Processing Systems, pp , 0 Allen-Zhu, Zeyuan Katyusha: The first direct acceleration of stochastic gradient methods ArXiv e-prints, abs/ , 06 Allen-Zhu, Zeyuan and Yuan, Yang Univr: A universal variance reduction framework for proximal stochastic gradient method arxiv preprint arxiv:506097, 05 Allen-Zhu, Zeyuan and Yuan, Yang Improved SVRG for Non-Strongly-Convex or Sum-of-Non-Convex Objectives In Proceedings of the 33rd International Conference on Machine Learning, ICML 6, 06 Full version available at Beck, Amir and Teboulle, Marc A fast iterative shrinkagethresholding algorithm for linear inverse problems SIAM journal on imaging sciences, :83 0, 009 Defazio, Aaron, Bach, Francis, and Lacoste-Julien, Simon Saga: A fast incremental gradient method with support for non-strongly convex composite objectives Advances in Neural Information Processing Systems, pp , 04 Duchi, John, Hazan, Elad, and Singer, Yoram Adaptive subgradient methods for online learning and stochastic optimization Journal of Machine Learning Research, Jul: 59, 0 Hazan, Elad, Agarwal, Amit, and Kale, Satyen Logarithmic regret algorithms for online convex optimization Machine Learning, 69-3:69 9, 007 Johnson, Rie and Zhang, Tong Accelerating stochastic gradient descent using predictive variance reduction Advances in Neural Information Processing Systems, pp 35 33, 03 Konečnỳ, Jakub, Liu, Jie, Richtárik, Peter, and Takáč, Martin Mini-batch semi-stochastic gradient descent in the proximal setting IEEE Journal of Selected Topics in Signal Processing, 0:4 55, 06 Li, Huan and Lin, Zhouchen Accelerated proximal gradient methods for nonconvex programming Advances in Neural Information Processing Systems, pp , 05 Lin, Hongzhou, Mairal, Julien, and Harchaoui, Zaid A universal catalyst for first-order optimization Advances in Neural Information Processing Systems, pp , 05

10 Lin, Qihang, Lu, Zhaosong, and Xiao, Lin An accelerated proximal coordinate gradient method Advances in Neural Information Processing Systems, pp , 04a Lin, Qihang, Xiao, Lin, et al An adaptive accelerated proximal gradient method and its homotopy continuation for sparse optimization ICML, pp 73 8, 04b Xiao, Lin and Zhang, Tong A proximal stochastic gradient method with progressive variance reduction SIAM Journal on Optimization, 44: , 04 Zhang, Yuchen and Xiao, Lin Stochastic primal-dual coordinate method for regularized empirical risk minimization Proceedings of the 3nd International Conference on Machine Learning, 95:05, 05 Nesterov, Yurii Introductory lectures on convex optimization: A basic course Springer Science & Business Media, 87, 03 Nesterov, Yurii et al Gradient methods for minimizing composite objective function Technical report, UCL, 007 Nitanda, Atsushi Stochastic proximal gradient descent with acceleration techniques Advances in Neural Information Processing Systems, pp , 04 Nitanda, Atsushi Accelerated stochastic gradient descent for minimizing finite sums arxiv preprint arxiv: , 05 O Donoghue, Brendan and Candes, Emmanuel Adaptive restart for accelerated gradient schemes Foundations of computational mathematics, 53:75 73, 05 Roux, Nicolas L, Schmidt, Mark, and Bach, Francis R A stochastic gradient method with an exponential convergence rate for finite training sets Advances in Neural Information Processing Systems, pp , 0 Schmidt, Mark, Roux, Nicolas Le, and Bach, Francis Minimizing finite sums with the stochastic average gradient arxiv preprint arxiv:309388, 03 Shalev-Shwartz, Shai and Singer, Yoram Logarithmic regret algorithms for strongly convex repeated games The Hebrew University, 007 Shalev-Shwartz, Shai and Zhang, Tong Stochastic dual coordinate ascent methods for regularized loss The Journal of Machine Learning Research, 4: , 03 Singer, Yoram and Duchi, John C Efficient learning using forward-backward splitting Advances in Neural Information Processing Systems, pp , 009 Tseng, Paul On accelerated proximal gradient methods for convex-concave optimization submitted to siam j J Optim, 008 Xiao, Lin Dual averaging method for regularized stochastic learning and online optimization Advances in Neural Information Processing Systems, pp 6 4, 009

11 Supplementary material: Doubly Accelerated Stochastic Variance Reduced Dual Averaging Method for Regularized Empirical Risk Minimization In this supplementary material, we give the proofs of Theorem 5 and the optimality of γ Section A, Theorem 5 Section B and Corollary 53 Section C, the lazy update algorithm of our method Section D and the experimental details Section E Finally, we briefly discuss DASVRG method, which is a variant of DASVRDA method Section F A Proof of Theorem 5 In this section, we give the comprehensive proof of Theorem 5 algorithm Lemma A The sequence } k ined in Algorithm 7 satisfies First we analyze One Stage Accelerated SVRDA for k, where θ 0 Proof Since k+ for k 0, we have that k + k Lemma A The sequence } k ined in Algorithm 7 satisfies m θ m θ m k Proof Observe that θ m θ m mm + 4 m k m k k Lemma A3 For every x, y R d, F y + F y, x y + Rx P x L n Proof Since f i is convex and L i -smooth, we have see Nesterov, 03 n i f i x f i y f i y + f i y, x y f i x L i f i x f i y By the inition of q i }, summing this inequality from i to n and dividing it by n results in F y + F y, x y F x L n Adding Rx to the both sides of this inequality gives the desired result n i f i x f i y

12 Lemma A4 DASVRDA for Regularized ERM ḡ k k g k k k Proof For k, ḡ g k g k by the inition of θ 0 Assume that the claim holds for some k Then ḡ k+ + k + ḡ k + 4 k + k + g k+ k k k+ 4 k + k + g k k + k+ k g k g k + k + g k+ The first equality follows from the inition of ḡ k+ Second equality is due to the assumption of the induction This finishes the proof for Lemma A4 Next we prove the following main lemma for One Stage Accelerated SVRDA The proof is inspired by the one of AccSDA given in Xiao, 009 Lemma A5 Let η < / L For One Stage Accelerated SVRDA, we have that E[P x m P x] ηm + m z 0 x ηm + m E z m x m + k + ke g k F y k m + m k 4 k L n for any x R d, where the expectations are taken with respect to I k k m Proof We ine n E f i y k f i x, i l k x F y k + F y k, x y k + Rx, ˆl k x F y k + g k, x y k + Rx Observe that l k, ˆl k is convex and l k P by the convexity of F and R Moreover, for k we have that k ˆl k z k k F y k + k k g k, z y k + k ḡ k, z + Rz + k Rz k k F y k k k g k, y k The second equality follows from Lemma A4 and Lemma A Thus we see that z k k argmin k ˆl k z + η z z 0 } Observe that F is convex and L-smooth Thus we have that z R d F x k F y k + F y k, x k y k + L x k y k 4 k

13 Hence we see that P x k l k x k + L x k y k l k x k + z k + L θk l k x k + l k z k + θk P x k + θk x k + θk z k y k L θ k z k z k ˆlk z k + L z k z k g k F y k, z k y k θk P x k + ˆlk z k + η z k z k z k z k g k F y k, z k z k g k F y k, z k y k The first inequality follows from 4 The first equality is due to the inition of x k The second inequality is due to the convexity of l k and the inition of y k The third inequality holds because l k P and θk Since η > L, we have that z k z k g k F y k, z k z k g k F y k g k F y k The first inequality is due to Young s inequality The second inequality holds because Using this inequality, we get that P x k θk P x k + ˆlk z k + η z k z k + g k F y k g k F y k, z k y k Multiplying the both sides of the above inequality by yields P x k P x k + ˆlk z k + η z k z k + g k F y k g k F y k, z k y k 5 By the fact that k k ˆl k z+ η z z 0 is η -strongly convex and z k is the minimizer of k k ˆl k z+ η z z 0 for k, we have that k k ˆl k z k + η z k z 0 + k η z k z k k ˆl k z k + η z k z 0

14 for k and, for k, we ine 0 k 0 Using this inequality, we obtain DASVRDA for Regularized ERM P x k k k P x k ˆl k z k η z k z 0 k k g k F y k, z k y k ˆl k z k η z k z 0 + g k F y k Here, the inequality follows from Lemma A we ined θ 0 Summing the above inequality from k to m results in θ m θ m P x m m k ˆlk z m η z m z 0 m g k F y k m k g k F y k, z k y k k Using η -strongly convexity of the function m k ˆl k z + η z z 0 and the optimality of z m, we have that m k ˆlk z m + η z m z 0 m k ˆlk x + η z 0 x η z m x From this inequality, we see that θ m θ m P x m m ˆlk x + η z 0 x η z m x k m g k F y k m + k g k F y k, z k y k k m l k x + η z 0 x η z m x k m g k F y k m + k g k F y k, z k x k By Lemma A3 with x x and y y k, we have that l k x P x L n n i f i x f i y k

15 Applying this inequality to the above inequality yields θ m θ m P x m DASVRDA for Regularized ERM m P x k η z 0 x η z m x m + g k F y k k L n m g k F y k, z k x k n f i x f i y k Using Lemma A and dividing the both sides of the above inequality by θ m θ m result in i P x m P x z 0 x z m x ηθ m θ m ηθ m θ m m + g k F y k θ m θ m k L n m g k F y k, z k x θ m θ m k n f i x f i y k Taking the expectations with respect to I k k m on the both sides of this inequality yields E[P x m P x] z 0 x E z m x ηθ m θ m ηθ m θ m m + E g k F y k θ m θ m k L n i n E f i x f i y k Here we used the fact that E[g k F y k ] 0 for k,, m This finishes the proof of Lemma A5 Now we need the following lemma Lemma A6 For every x R d, n n i Proof From the argument of the proof of Lemma A3, we have n n i i f i x f i x LP x P x f i x f i x LF x F x, x x F x By the optimality of x, there exists ξ Rx such that F x + ξ 0 Then we have F x, x x ξ, x x Rx Rx, and hence n n i f i x f i x LP x P x

16 Proposition A7 Let γ > and η / + γm + /b L For One Pass Accelerated SVRDA, it follows that and E[P x m P x] E[P x m P x ] γ P x P x + where the expectations are taken with respect to I k k m ηm + m ỹ x ηm + m E z m x, ηm + m ỹ x ηm + m E z m x, Proof We bound the variance of the averaged stochastic gradient E g k F y k : E g k F y k E [ E Ik g k F y k [k ] ] b E [ E i Q f i y k f i x/ + F x F y k [k ] ] b E [ E i Q f i y k f i x/ [k ] ] [ ] b E n f i y k f i x n b E [ n i n i [ + b E n n i [ b E n n i f i y k f i x ] f i x f i x f i y k f i x ] ] L b P x P x 7 The second equality follows from the independency of the random variables i,, i b } and the unbiasedness of f i y k f i x/ + F x The first inequality is due to the fact that E X E[X] E X The second inequality follows from Young s inequality The final inequality is due to Lemma A6 Since η + γm+ L and γ >, using 6 yields By Lemma A5 with x x we have b [ k + k 4 E g k F y k k L E n E[P x m P x] Similarly, combining Lemma A5 with x x with 7 results in These are the desired results E[P x m P x ] γ P x P x + n i ] f i y k f i x 0 ηm + m ỹ x ηm + m E z m x ηm + m ỹ x ηm + m E z m x

17 Lemma A8 The sequence θ s } s ined in Algorithm 6 satisfies θ s θ s + γ θ s for any s Proof Since θ s s+ γ for s 0, we have This finishes the proof of Lemma A8 θ s θ s + γ s + γ γ θ s Now we are ready to proof Theorem 5 Proof of Theorem 5 By Proposition A7, we have and E[P x s P x s ] E[P x s P x ] γ E[P x s P x ] + ss + 4 s + γ + γ ηm + m E ỹ s x s ηm + m E z s x s, ηm + m E ỹ s x ηm + m E z s x, where the expectations are taken with respect to the history of all random variables Hence we have and E[P x s P x s ] E[P x s P x ] γ E[P x s P x ] + 4 ηm + m E z s ỹ s, x s ỹ s ηm + m E z s ỹ s, 8 4 ηm + m E z s ỹ s, x ỹ s ηm + m E z s ỹ s 9 Since γ 3, we have θ s for s Multiplying 8 by θ s θ s 0 and adding θ s 9 yield θ se[p x s P x ] θ s θ s + E[P x s P x ] γ 4 ηm + m E θ s z s ỹ s, θ s x s θ s ỹ s + x ηm + m E θ s z s ỹ s By Lemma A8, we have θ s θ s + γ θ s for s

18 Thus we get θ se[p x s P x ] θ s E[P x s P x ] 4 ηm + m E θ s z s ỹ s, θ s x s θ s ỹ s + x ηm + m E θ s z s ỹ s E θ s x s ηm + m θ s ỹ s + x E θ s x s θ s z s + x Since ỹ s x s + θ s θ s x s x s + θ s z s x s, we have θ s θ s x s θ s ỹ s + x θ s x s θ s z s + x Therefore summing the above inequality from s to S, we obtain θ se[p x S P x ] θ 0P x 0 P x + γ P x 0 P x + ηm + m θ 0 x θ 0 z 0 + x ηm + m z 0 x Dividing both sides by θ s finishes the proof of Theorem 5 Optimal choice of γ We can choose the optimal value of γ based on the following lemma Lemma A9 Define gγ + γm+ b γ for γ > Then, γ argmin gγ γ> b m + Proof First observe that Hence we have g γ m+ b γ + γm+ b γ γ γ g γ 0 m + b γ m + γ γ b b + + γ 3γ m + 0 γ b m + γm + b γm + 0 b γ γ γ 0 Here the second and last equivalencies hold from γ > Moreover observe that g γ > 0 for γ > γ and g γ < 0 for < γ < γ This means that γ is the minimizer of g on the region γ >

19 B Proof of Theorem 5 In this section, we give a proof of Theorem 5 Proof of Theorem 5 Since η / + γm U + /b L / + γm u + /b L, from Proposition A7, we have E[P x u P x ] + γ P x u P x + P x u P x + γ Since m u γm u + m u, we have Using this inequality, we obtain that γ ηm u + m u E z u x ηm u + m u z u x γ ηm u + m u z u x γ ηm u + m u ηm u + m u E[P x U P x ] + P x U P x + γ U P x 0 P x + γ U O P x 0 P x + γ U P x 0 P x ηm U + m U E z U x ηm U + m U E z U x z 0 x ηm 0 + m 0 x 0 x ηm 0 + m 0 The last equality is due to the initions of m 0 and η, and the fact m U Om U O γ U m 0 Om see the arguments in the proof of Corollary 53 Since we get m U + m U m U + m U, γ E[P x U P x ] + E z U x γ ηm U + m U O γ U P x 0 P x Using the initions of U and m 0 and combining this inequality with Theorem 5, we obtain that desired result C Proof of Corollary 53 In this section, we give a proof of Corollary 53 Proof of Corollary 53 Observe that the total computational cost at the warm start phase becomes U O dnu + db m u u

20 Since m u γm u + γ + γm u + γ γ m u + γ + γ γ u m 0 + u u γ u O γ u m 0, we have U O dnu + db m u O dnu + db γ U m 0 u Suppose that m m 0 P x0 P x / Then, this condition implies U log γm/m 0 log γ P x 0 P x / Hence we only need to run u Olog γ P x 0 P x / U iterations at the warm start phase and running DASVRDA ns is not needed Then the total computational cost becomes O d nlog P x 0 P x P x0 P x + bm 0 O d nlog P x 0 P x, here we used mb On Next, suppose that m m 0 P x0 P x / In this case, the total computational cost at the warm start phase with full U iterations becomes O d nlog m + mb O d nlog P x 0 P x m 0 Finally, using Theorem 5 yields the desired total computational cost D Lazy Update Algorithm of DASVRDA Method In this section, we discuss how to efficiently compute the updates of the DASVRDA algorithm for sparse data Specifically, we derive lazy update rules of One Stage Accelerated SVRDA for the following empirical risk minimization problem: n n i ψ i a i x + λ x + λ x, λ, λ 0 For the sake of simplicity, we ine the one dimensional soft-thresholding operator as follows: softz, λ sign z max z λ, 0}, for z R Moreover, in this section, we denote [z, z ] as z Z z z z } for integers z, z Z The explicit algorithm of the lazy updates for One Stage Accelerated SVRDA is given by Algorithm 0 Let us analyze the iteration cost of the algorithm Suppose that each feature vector a i is sparse and the expected number of the nonzero elements is Od First note that A k Obd expectedly if d d For updating x k, by Proposition D, we need to compute k K ± / + η λ and j k K ± θk j / + η λ for each j A k For this, we first make lists S k } m k k k / + η λ } m k and S k }m k k k θk / + η λ } m k before running the algorithm This needs only Om Note that these lists are not depend on coordinate j Since K ± j are sets of continuous integers in [k j +, k] or unions of two sets of continuous integers in [k j +, k], we can efficiently compute the above sums For example, if K + j [k j +, s ] [s +, k] for some integers s ± [k j +, k], we can compute k K + / + η λ as S s S kj+ + S k S s+ j and this costs only O Thus, for computing x k and y k, we need only Obd computational cost For computing g k, we need to compute the inner product a i y k for each i I k and this costs Obd expectedly The expected cost of the rest of the updates is apparently Obd Hence, the total expected iteration cost of our algorithm in serial settings becomes Obd rather than Obd Furthermore, we can extend our algorithm to parallel computing settings Indeed, if we have b processors, processor b runs on the set A b k j [d] a ib,j 0} Then the total iteration cost per processor becomes ideally Od Generally the overlap among the sets A b k may cause latency, however for sufficiently sparse data, this latency is negligible The following proposition guarantees that Algorithm 0 is equivalent to Algorithm 7 when Rx λ x + λ / x

21 Algorithm 0 Lazy Updates for One Stage AccSVRDA ỹ, x, η, m, b, Q Input: ỹ, x, η > 0, m N, b [n], Q x 0 z 0 ỹ g0,j sum 0 j [d] θ 0 k j 0 j [d] F x for k to m do Sample independently i,, i b Q I k i,, i b } A k j [d] b [b] : a ib,j 0} k+ for j A k do Update x k,j, y k,j as in Proposition D end for for j A k do g k,j b i I t ψ i a i y ka i,j ψ i a i xa i,j + j g sum k,j gk sum + θ j,j k g k,j + j j j +η λ softz 0,j ηgk,j sum, η λ z k,j x k,j k j k end for end for Output: x m, z m x k,j + z k,j Proposition D Suppose that Rx λ x + λ x with λ, λ 0 Let j [d], k j [m] 0} and k k j + Assume that j f i y k j f i x 0 for any i [b] and k [k j +, k ] In Algorithm 7, the following results hold: x 0,j k j j θ x k,j x kj,j + k k K + j +η λ z 0,j M + k,j k, θ + k k K j +η λ z 0,j M k,j x 0,j k y k,j x k,j + +η λ k, soft z 0,j ηgk sum ηθ j,j k j j j, η λ and where z k,j M ± k,j + η λ softz 0,j ηg sum k,j, η λ, η j ± λ + ηg sum k j,j ηj j j, and K ± j [k j +, k] are ined as follows: Let c η j 4, c ηλ 4 and c 3 ηgk sum ηθ j,j k j θ kj j to simplify the notation Note that c 0 Moreover, we ine D ± c ± c + 4c ± c z 0,j c 3, s + ± s ± c + c ± D +, c + c c c ± D c c,

22 where if s ± ± are not well ined, we simply assign 0 or any number to s ± ± If c > c, then K + j K j D + 0 [k j +, k] [ s +, s + + ] D + > 0 [k j +, k] D 0 [k j +, s ] [ s +, k] D > 0, If c c, then K + j K j c 0 z 0,j c 3 [k j +, k] c 0 z 0,j > c 3 c > 0 D + 0 [k j +, k] [ s +, s + + ] c > 0 D + > 0 [k j +, k] z 0,j < c 3 z 0,j c 3, 3 If c < c, then K + j K j D + 0 [k j +, k] [ s +, s + + ] D + > 0 D 0 [k j +, k] [ s, s + ] D > 0, 4 If c c, then K + j K j z 0,j c 3 [k j +, k] z 0,j > c 3 [k j +, k] c 0 z 0,j < c 3 c 0 z 0,j c 3 c > 0 D 0 [k j +, k] [ s, s + ] c > 0 D > 0 5 If c < c, then K + j K j [k j +, k] D + 0 [k j +, s + ] [ s + +, k] D + > 0 D 0 [k j +, k] [ s, s + ] D > 0, Proof First we consider the case k Observe that y,j θ x 0,j + θ z 0,j z 0,j x 0,j,