Practical Schemes for Finding Near-Stationary Points of Convex Finite-Sums

The problem of finding near-stationary points in convex optimization has not been1 adequately studied yet , unlike other optimality measures such as the function2 value . Even in the deterministic case , the optimal method ( OGM-G , due to Kim3 and Fessler [ 33 ] ) has just been discovered recently . In this work , we conduct a4 systematic study of algorithmic techniques for finding near-stationary points of5 convex finite-sums . Our main contributions are several algorithmic discoveries:6 ( 1 ) we discover a memory-saving variant of OGM-G based on the performance7 estimation problem approach [ 19 ] ; ( 2 ) we design a new accelerated SVRG variant8 that can simultaneously achieve fast rates for minimizing both the gradient norm9 and function value ; ( 3 ) we propose an adaptively regularized accelerated SVRG10 variant , which does not require the knowledge of some unknown initial constants11 and achieves near-optimal complexities . We put an emphasis on the simplicity and12 practicality of the new schemes , which could facilitate future developments.13 1 Introduction14 Classic convex optimization usually focuses on providing guarantees for minimizing function value.15 For this task , the optimal ( up to constant factors ) Nesterov ’ s accelerated gradient method ( NAG ) 16 [ 40 , 41 ] has been known for decades , and there are even methods that can exactly match the lower17 complexity bounds [ 30 , 17 , 55 , 18 ] . On the other hand , in general non-convex optimization , near-18 stationarity is the typical optimality measure , and there has been a flurry of recent research devoted to19 this topic [ 25 , 26 , 23 , 28 , 21 , 60 ] . Recently , there has been growing interest on devising fast schemes20 for finding near-stationary points in convex optimization [ 42 , 2 , 22 , 7 , 31 , 32 , 33 , 27 , 15 , 14 ] . This21 line of research is basically driven by the following facts.22 • Nesterov [ 42 ] studied the problem with a linear constraint : f ( x ? ) = minx∈Q { f ( x ) : Ax = b } ,23 whereQ is a convex set and f is strongly convex . Assuming thatQ and f are simple , we can focus24 on the dual problem φ ( y ? ) = maxy { φ ( y ) , minx∈Q { f ( x ) + 〈y , b−Ax〉 } } . Clearly , the dual25 objective−φ ( y ) is smooth convex . Letting xy be the unique solution to the inner problem , we have26 ∇φ ( y ) = b − Axy . Note that f ( xy ) − f ( x ? ) = φ ( y ) − 〈y , ∇φ ( y ) 〉 − φ ( y ? ) ≤ ‖y‖ ‖∇φ ( y ) ‖ .27 Thus , in this problem , the quantity ‖∇φ ( y ) ‖ serves as a measure of both primal optimality28 f ( xy ) −f ( x ? ) and feasibility ‖b−Axy‖ , which is better than just measuring the function value.29 • Matrix scaling [ 50 ] is a convex problem and its goal is to find near-stationary points [ 4 , 9 ] .30 • Gradient norm is readily available , unlike other optimality measures ( f ( x ) −f ( x ? ) and ‖x− x ? ‖ ) ,31 and is thus usable as a stopping criterion . This fact motivates the design of several parameter-free32 algorithms [ 43 , 39 , 27 ] , and their guarantees are established on the gradient norm.33 • Designing schemes for minimizing the gradient norm can inspire new non-convex optimization34 methods . For example , SARAH [ 46 ] was designed for convex finite-sums with gradient-norm mea-35 sure , but was later discovered to be the near-optimal method for non-convex finite-sums [ 21 , 47 ] .36 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . Moreover , finding near-stationary points is a harder task than minimizing function value , because37 NAG has the optimal guarantee for f ( x ) − f ( x ? ) but is only suboptimal for minimizing ‖∇f ( x ) ‖.38 In this work , we consider the problem minx∈Rd f ( x ) = 1n ∑n i=1 fi ( x ) , where each fi is L-smooth39 and convex . We focus on finding an -stationary point of this objective , i.e. , a point with ‖∇f ( x ) ‖ ≤ .40 We use X ? to denote the set of optimal solutions , which is assumed to be nonempty . There are two41 different assumptions on the initial point x0 , namely , the Initial bounded-Function Condition ( IFC ) :42 f ( x0 ) − f ( x ? ) ≤ ∆0 , and the Initial bounded-Distance Condition ( IDC ) : ‖x0 − x ? ‖ ≤ R0 for some43 x ? ∈ X ? . This subtlety results in drastically different best achievable rates as studied in [ 7 , 22 ] .44 Below we categorize existing algorithmic techniques into three classes ( relating to Table 1 ) .45 ( i ) “ IDC + IFC ” . Nesterov [ 42 ] showed that we can combine the guarantees of a method46 minimizing function value under IDC and a method finding near-stationary points under IFC47 to produce a faster one for minimizing gradient norm under IDC . For example , NAG produces48 f ( xK1 ) − f ( x ? ) = O ( LR20 K21 ) [ 40 ] and GD produces ‖∇f ( xK2 ) ‖ 2 = O ( L ( f ( x0 ) −f ( x ? ) ) K2 ) [ 33 ] 49 under IFC . Letting x0 = xK1 and K = K1 +K2 , by balancing the ratio of K1 and K2 , we50 obtain the guarantee ‖∇f ( xK ) ‖2 = O ( L 2R20 K3 ) for “ NAG + GD ” . We point out that we can use51 this technique to combine the guarantees of Katyusha [ 1 ] and SARAH2 [ 46 ] ; see Appendix E.52 ( ii ) Regularization . Nesterov [ 42 ] used NAG ( strongly convex variant ) to solve the regularized53 objective , and showed that it achieves near-optimal complexity ( optimal up to logarithmic54 factors ) . Inspired by this technique , Allen-Zhu [ 2 ] proposed recursive regularization for55 stochastic approximation algorithms , which also achieves near-optimal complexities [ 22 ] .56 1Table 1 shows that Katyusha+L2S has a slightly better dependence on n than Acc-SVRG-G . It is due to the adoption of n-dependent step size in L2S . As studied in [ 37 ] , despite having a better complexity , n-dependent step size boosts numerical performance only when n is extremely large . If the practically fast n-independent step size is used for L2S , Katyusha+L2S and Acc-SVRG-G have the same complexity . See also Appendix A . 2We adopt the loopless variant of SARAH in [ 37 ] , which has a refined analysis for general convex objectives . ( iii ) Direct methods . Due to the lack of insight , existing direct methods are mostly derived or57 analyzed with the help of computer-aided tools [ 31 , 32 , 54 , 33 ] . The computer-aided approach58 was pioneered by Drori and Teboulle [ 19 ] , who introduced the performance estimation59 problem ( PEP ) . The only known optimal method OGM-G [ 33 ] was designed based on the60 PEP approach.61 Observe that since f ( x ) − f ( x ? ) ≤ ‖∇f ( x ) ‖ ‖x− x ? ‖ , the lower bound for finding near-stationary62 points must be of the same order as for minimizing function value [ 44 ] . Thus , under IDC , the lower63 bound is Ω ( n+ √ n ) due to [ 58 ] . Under IFC , we can establish an Ω ( n+ √ n ) lower bound using64 the techniques in [ 7 , 58 ] . The main contributions of this work are three new algorithmic schemes that65 improve the practicalities of existing methods as summarized below ( highlighted in Table 1 ) .66 • ( Section 3 ) We propose a memory-saving variant of OGM-G for the deterministic case ( n = 1 ) ,67 which does not require a pre-computed and stored parameter sequence . The derivation of the new68 variant is inspired by the numerical solution to a PEP problem.69 • ( Section 4 ) We propose a new accelerated SVRG [ 29 , 59 ] variant that can simultaneously70 achieve fast convergence rates for minimizing both the gradient norm and function value , that is,71 O ( n log 1 + n2/3 2/3 ) complexity for gradient norm and O ( n log 1 + √ n ) complexity for function72 value . Note that other stochastic approaches in Table 1 do not have this property.73 • ( Section 5 ) We propose an adaptively regularized accelerated SVRG variant , which does not74 require the knowledge of R0 or ∆0 and achieves a near-optimal complexity under IDC or IFC.75 We put in extra efforts to make the proposed schemes as simple and elegant as possible . We believe76 that the simplicity makes the extensions of the new schemes easier.77 2 Preliminaries78 Throughout this paper , we use 〈· , ·〉 and ‖·‖ to denote the inner product and the Euclidean norm,79 respectively . We let [ n ] denote the set { 1 , 2 , . . . , n } , E denote the total expectation and Eik denote80 the expectation with respect to a random sample ik . We say that a function f : Rd → R is L-smooth81 if it has L-Lipschitz continuous gradients , i.e.,82 ∀x , y ∈ Rd , ‖∇f ( x ) −∇f ( y ) ‖ ≤ L ‖x− y‖ . A continuously differentiable f is called µ-strongly convex if83 ∀x , y ∈ Rd , f ( x ) − f ( y ) − 〈∇f ( y ) , x− y〉 ≥ µ 2 ‖x− y‖2 . Other equivalent definitions of these two assumptions can be found in the textbook [ 44 ] . The84 following is an important consequence of a function f being L-smooth and convex:85 ∀x , y ∈ Rd , f ( x ) − f ( y ) − 〈∇f ( y ) , x− y〉 ≥ 1 2L ‖∇f ( x ) −∇f ( y ) ‖2 . ( 1 ) We call ( 1 ) the interpolation condition at ( x , y ) following [ 56 ] . If f is both L-smooth and µ-strongly86 convex , we can define a “ shifted ” function h ( x ) = f ( x ) − f ( x ? ) − µ2 ‖x− x ? ‖2 following [ 63 ] . It87 can be easily verified that h is ( L− µ ) -smooth and convex , and thus from ( 1 ) ,88 ∀x , y ∈ Rd , h ( x ) − h ( y ) − 〈∇h ( y ) , x− y〉 ≥ 1 2 ( L− µ ) ‖∇h ( x ) −∇h ( y ) ‖2 , ( 2 ) which is equivalent to the strongly convex interpolation condition discovered in [ 56 ] .89 Oracle complexity ( or simply complexity ) refers to the required number of stochastic gradient∇fi90 computations to find an -accurate solution.91 3 OGM-G : “ Momentum ” Reformulation and a Memory-Saving Variant92 In this section , we focus on the IFC case , i.e. , f ( x0 ) − f ( x ? ) ≤ ∆0 . We use N to denote the total93 iteration number to prevent confusion ( in other sections , we use K ) . Proofs in this section are given in94 Algorithm 1 OGM-G : “ Momentum ” reformulation Input : initial guess x0 ∈ Rd , total iteration number N . Initialize : vector v0 = 0 , scalars θN = 1 and θ2k − θk = θ2k+1 , for k = 0 . . . N − 1 . 1 : for k = 0 , . . . , N − 1 do 2 : vk+1 = vk + 1 Lθkθ2k+1 ∇f ( xk ) . 3 : xk+1 = xk − 1L∇f ( xk ) − ( 2θ 3 k+1 − θ2k+1 ) vk+1 . 4 : end for Output : xN . Appendix B . Recall that OGM-G has the following updates [ 33 ] . Let y0 = x0 . For k = 0 , . . . , N−1,95 96 yk+1 = xk − 1 L ∇f ( xk ) , xk+1 = yk+1 + ( θk − 1 ) ( 2θk+1 − 1 ) θk ( 2θk − 1 ) ( yk+1 − yk ) + 2θk+1 − 1 2θk − 1 ( yk+1 − xk ) , ( 3 ) where { θk } is recursively defined : θN = 1 and { θ2k − θk = θ2k+1 k = 1 . . . N − 1 , θ20 − θ0 = 2θ21 otherwise . 97 OGM-G was discovered from the numerical solution to an SDP problem and its analysis is to show98 that the step coefficients in ( 3 ) specify a feasible solution to the SDP problem . While this analysis is99 natural for the PEP approach , it is hard to understand how each coefficient affects the rate , especially100 if one wants to generalize the scheme . Here we provide a simple algebraic analysis for OGM-G.101 We start with a reformulation3 of OGM-G in Algorithm 1 , which aims to simplify the proof . We102 adopt a consistent { θk } : θN = 1 and θ2k − θk = θ2k+1 , k = 0 . . . N − 1 , which only costs a constant103 factor.4 Interestingly , the reformulated scheme resembles the heavy-ball momentum method [ 49 ] .104 However , it can be shown that Algorithm 1 is not covered by the heavy-ball momentum scheme.105 Defining θ2N+1 = θ 2 N − θN = 0 , we provide the one-iteration analysis in the following proposition:106 Proposition 3.1 . In Algorithm 1 , the following holds at any iteration k ∈ { 0 , . . . , N − 1 } :107 Ak +Bk+1 + Ck+1 + Ek+1 ≤ Ak+1 +Bk + Ck + Ek − θk+1 〈∇f ( xk+1 ) , vk+1〉 + N∑ i=k+1 θi Lθkθ2k+1 〈∇f ( xk ) , ∇f ( xi ) 〉 , ( 4 ) withAk , 1θ2k ( f ( xN ) −f ( x ? ) − 12L ‖∇f ( xN ) ‖ 2 ) , Bk , 1θ2k ( f ( xk ) −f ( x ? ) ) , Ck , 12Lθ2k ‖∇f ( xk ) ‖ 2,108 Ek , θ2k+1 θk 〈∇f ( xk ) , vk〉.109 Remark 3.1.1 . A recent work [ 15 ] also conducted an algebraic analysis of OGM-G under a potential110 function framework . Their potential function decrease can be directly obtained from Proposition 3.1111 by summing up ( 4 ) . By contrast , our “ momentum ” vector { vk } naturally merges into the analysis,112 which significantly simplifies the analysis . Moreover , it provides a better interpretation on how113 OGM-G utilizes the past gradients to achieve acceleration.114 From ( 4 ) , we see that only the last two terms do not telescope . Note that the “ momentum ” vector is a115 weighted sum of the past gradients , i.e. , vk+1 = ∑k i=0 1 Lθiθ2i+1 ∇f ( xi ) . If we sum the terms up from116 k = 0 , . . . , N − 1 , it can be verified that they exactly sum up to 0 . The presence of these special117 terms prevents OGM-G to have a usual potential function ( e.g. , those in [ 6 ] ) . Then , by telescoping118 the remaining terms , we obtain the final convergence guarantee.119 Theorem 3.1 . The output of Algorithm 1 satisfies ‖∇f ( xN ) ‖2 ≤ 8L∆0 ( N+2 ) 2 .120 We observe two drawbacks of OGM-G ( same as the algorithm description in [ 15 ] ) : ( 1 ) it requires121 storing a pre-computed parameter sequence , which costs O ( 1 ) floats ; ( 2 ) except for the last iterate,122 3It can be verified that this scheme is equivalent to the original one ( 3 ) through vk = 1 ( 2θk−1 ) θ2k ( yk − xk ) . 4The original guarantee of OGM-G can be recovered if we set θ20 − θ0 = 2θ21 . Algorithm 2 M-OGM-G : Memory-saving OGM-G Input : initial guess x0 ∈ Rd , total iteration number N . Initialize : vector v0 = 0 . 1 : for k = 0 , . . . , N − 1 do 2 : vk+1 = vk + 12 L ( N−k+1 ) ( N−k+2 ) ( N−k+3 ) ∇f ( xk ) . 3 : xk+1 = xk − 1L∇f ( xk ) − ( N−k ) ( N−k+1 ) ( N−k+2 ) 6 vk+1 . 4 : end for Output : xN or arg minx∈ { x0 , ... , xN } ‖∇f ( x ) ‖ . all other iterates are not known to have guarantees . We resolve these issues by proposing another123 parameterization of Algorithm 1 in the next subsection.124 3.1 Memory-Saving OGM-G125 A straightforward idea to resolve the aforementioned issues is to generalize Algorithm 1 . However,126 we find it rather difficult since the parameters in the analysis are rather strict ( despite that the proof is127 already simple ) . We choose to rely on computer-aided techniques [ 19 ] . The derivation of this variant128 ( Algorithm 2 ) is based on the following numerical experiment.129 Numerical experiment . OGM-G was discovered when considering the relaxed PEP problem [ 33 ] :130 max ∇f ( x0 ) , ... , ∇f ( xN ) ∈Rd f ( x0 ) , ... , f ( xN ) , f ( x ? ) ∈R ‖∇f ( xN ) ‖2 subject to interpolation condition ( 1 ) at ( xk , xk+1 ) , k = 0 , . . . , N − 1 , interpolation condition ( 1 ) at ( xN , xk ) , k = 0 , . . . , N − 1 , interpolation condition ( 1 ) at ( xN , x ? ) , f ( x0 ) − f ( x ? ) ≤ ∆0 , ( P ) where the sequence { xk } is defined as xk+1 = xk − 1L ∑k i=0 hk+1 , i∇f ( xi ) , k = 0 , . . . , N − 1 for131 some step coefficients h ∈ RN ( N+1 ) /2 . Given N , the step coefficients of OGM-G correspond to132 a numerical solution to the problem : arg minh { Lagrangian dual of ( P ) } , which is denoted as ( HD ) .133 Conceptually , solving problem ( HD ) would give us the fastest possible step coefficients under the134 constraints.5 We expect there to be some constant-time slower schemes , which are neglected when135 solving ( HD ) . To identify such schemes , we relax a set of interpolation conditions in problem ( P ) :136 f ( xN ) − f ( xk ) − 〈∇f ( xk ) , xN − xk〉 ≥ 1 2L ‖∇f ( xN ) −∇f ( xk ) ‖2 − ρ ‖∇f ( xk ) ‖2 , for k = 0 , . . . , N − 1 and some ρ > 0 . After this relaxation , solving ( HD ) will no longer give us the137 step coefficients of OGM-G. By trying different ρ and checking the dependence on N , we discover138 Algorithm 2 when ρ = 12L . Similar to our analysis of OGM-G , we provide a simple algebraic analysis139 for the new variant in the following theorem.140 Theorem 3.2 . Define δk+1 , 12 ( N−k+1 ) ( N−k+2 ) ( N−k+3 ) , k = 0 , . . . , N . In Algorithm 2 , it holds that141 142 N∑ k=0 δk+1 2 ‖∇f ( xk ) ‖2 ≤ 12L∆0 ( N + 2 ) ( N + 3 ) . ( 5 ) Remark 3.2.1 . Algorithm 2 converges optimally on the last iterate ( note that δN+1 = 2 ) and the143 minimum gradient since144 min k∈ { 0 , ... , N } ‖∇f ( xk ) ‖2 ≤ 1∑N k=0 δk+1 2 N∑ k=0 δk+1 2 ‖∇f ( xk ) ‖2 ≤ 8L∆0 ( N + 2 ) ( N + 3 ) − 2 . Clearly , the parameters of this variant can be computed on the fly and from ( 5 ) , each iterate has a145 guarantee ( although the guarantee degenerates quickly as k → 0 since 1/δk+1 = Ω ( ( N − k ) 3 ) ) .146 Moreover , we can extend the benefits into the IDC case using the ideas in [ 42 ] as summarized below.147 5However , since problem ( HD ) is non-convex , we can only obtain approximate solutions . Algorithm 3 Acc-SVRG-G : Accelerated SVRG for Gradient minimization Input : parameters { τk } , { pk } , initial guess x0 ∈ Rd , total iteration number K. Initialize : vectors z0 = x̃0 = x0 and scalars αk = Lτk1−τk , ∀k and τ̃ = ∑K−1 k=0 τ −2 k . 1 : for k = 0 , . . . , K − 1 do 2 : yk = τkzk + ( 1− τk ) ( x̃k − 1L∇f ( x̃k ) ) . 3 : zk+1 = arg minx { 〈Gk , x〉+ ( αk/2 ) ‖x− zk‖2 } . 4 : //Gk , ∇fik ( yk ) −∇fik ( x̃k ) +∇f ( x̃k ) , where ik is sampled uniformly in [ n ] . 5 : x̃k+1 = { yk with probability pk , x̃k with probability 1− pk . 6 : end for Output ( for gradient ) : xout is sampled from { Prob { xout = x̃k } = τ−2k τ̃ ∣∣∣ k ∈ { 0 , . . . , K − 1 } } . Output ( for function value ) : x̃K . Corollary 3.2.1 ( IDC case ) . If we first run N/2 iterations of NAG and then continue with N/2148 iterations of Algorithm 2 , we obtain an output satisfying ‖∇f ( xN ) ‖ = O ( LR0N2 ) .149 4 Accelerated SVRG : Fast Rates for Both Gradient Norm and Objective150 In this section , we focus on the IDC case , i.e. , ‖x0 − x ? ‖ ≤ R0 for some x ? ∈ X ? . From the151 development in the previous section , it is natural to ask whether we can use the PEP approach to152 motivate new stochastic schemes . However , due to the exponential growth of the number of possible153 states ( i0 , i1 , . . . ) , we can not directly adopt this approach . A feasible alternative is to first fix an154 algorithmic framework and a family of potential functions , and then use the potential-based PEP155 approach in [ 54 ] . However , this approach is much more restrictive . For example , it can not identify156 special constructions like ( 4 ) in OGM-G. Fortunately , as we will see , we can get some inspiration157 from the recent development of deterministic methods . Proofs in this section are given in Appendix C.158 Our proposed scheme is given in Algorithm 3 . We adopt the elegant loopless design of SVRG in159 [ 34 ] . Note that the full gradient ∇f ( x̃k ) is computed and stored only when x̃k+1 = yk at Step 5 . We160 summarize our main technical novelty as follows.161 Main algorithmic novelty . The design of stochastic accelerated methods is largely inspired by162 NAG . To make it clear , by setting n = 1 , we see that Katyusha [ 1 ] , MiG [ 61 ] , SSNM [ 62 ] , Varag [ 36 ] ,163 VRADA [ 52 ] , ANITA [ 38 ] , the acceleration framework in [ 16 ] and AC-SA [ 35 , 24 ] all reduce to one164 of the following variants of NAG . We say that these methods are under the NAG framework.165 xk = τkzk + ( 1− τk ) yk , zk+1 = zk − αk∇f ( xk ) , yk+1 = τkzk+1 + ( 1− τk ) yk . xk = τkzk + ( 1− τk ) yk , zk+1 = zk − αk∇f ( xk ) , yk+1 = xk − ηk∇f ( xk ) . Auslender and Teboulle [ 5 ] Linear Coupling [ 64 ] See [ 57 , 12 ] for other variants of NAG . When n = 1 , Algorithm 3 reduces to the following scheme:166 { yk = τkzk + ( 1− τk ) ( yk−1 − 1L∇f ( yk−1 ) ) , zk+1 = zk − 1αk∇f ( yk ) . Optimized Gradient Method ( OGM ) [ 19 , 30 ] Algorithm 3 reduces to the scheme of OGM when n = 1 ( this point is clearer in the formulation of167 ITEM in [ 55 ] ) . OGM has a constant-time faster worst-case rate than NAG , which exactly matches168 the lower complexity bound established in [ 17 ] . In the following proposition , we show that the OGM169 framework helps us conduct a tight one-iteration analysis , which gives room for achieving our goal.170 Proposition 4.1 . In Algorithm 3 , the following holds at any iteration k ≥ 0 and ∀x ? ∈ X ? :171 ( 1− τk τ2kpk E [ f ( x̃k+1 ) − f ( x ? ) ] + L 2 E [ ‖zk+1 − x ? ‖2 ] ) + ( 1− τk ) 2 2Lτ2k E [ ‖∇f ( x̃k ) ‖2 ] ≤ ( ( 1− τkpk ) ( 1− τk ) τ2kpk E [ f ( x̃k ) − f ( x ? ) ] + L 2 E [ ‖zk − x ? ‖2 ] ) . ( 6 ) The terms inside the parentheses form the commonly used potential function of SVRG variants . The172 additional E [ ‖∇f ( x̃k ) ‖2 ] term is created by adopting the OGM framework . In other words , we use173 the following potential function for Algorithm 3 ( ak , bk , ck ≥ 0 ) :174 Tk = akE [ f ( x̃k ) − f ( x ? ) ] + bkE [ ‖zk − x ? ‖2 ] + k−1∑ i=0 ciE [ ‖∇f ( x̃i ) ‖2 ] . We first provide a simple parameter choice , which leads to a simple and clean analysis.175 Theorem 4.1 ( Single-stage parameter choice ) . In Algorithm 3 , if we choose pk ≡ 1n , τk = 3 k/n+6 ,176 then the following holds at the outputs:177 E [ ‖∇f ( xout ) ‖2 ] = O ( n3L ( f ( x0 ) − f ( x ? ) ) + n2L2R20 K3 ) , E [ f ( x̃K ) ] − f ( x ? ) = O ( n2 ( f ( x0 ) − f ( x ? ) ) + nLR20 K2 ) . ( 7 ) In other words , to guarantee that E [ ‖∇f ( xout ) ‖ ] ≤ g and E [ f ( x̃K ) ] − f ( x ? ) ≤ f , the oracle com-178 plexities are O ( n ( L ( f ( x0 ) −f ( x ? ) ) ) 1/3 2/3 g + ( nLR0 ) 2/3 2/3 g ) and O ( n √ f ( x0 ) −f ( x ? ) f + √ nLR0√ f ) , respectively.179 From ( 7 ) , we see that Algorithm 3 achieves fast O ( 1K1.5 ) and O ( 1 K2 ) rates for minimizing the180 gradient norm and function value at the same time . However , despite being a simple choice , the oracle181 complexities are not better than the deterministic methods in Table 1 . Below we provide a two-stage182 parameter choice , which is inspired by the idea of including a “ warm-up phase ” in [ 3 , 36 , 52 , 38 ] .183 This theorem corresponds to the reported result in Table 1.184 Theorem 4.2 ( Two-stage parameter choice ) . In Algorithm 3 , let pk = max { 6k+8 , 1 n } , τk = 3 pk ( k+8 ) .185 The oracle complexities needed to guarantee E [ ‖∇f ( xout ) ‖ ] ≤ g and E [ f ( x̃K ) ] − f ( x ? ) ≤ f are186 O ( nmin { log LR0 g , log n } + ( nLR0 ) 2/3 2/3 g ) and O ( nmin { log LR20 f , log n } + √ nLR0√ f ) , respectively.187 If is large or n is very large , the recently proposed ANITA [ 38 ] achieves an O ( n ) complexity , which188 matches the lower complexity bound Ω ( n ) in this case [ 58 ] . Since ANITA uses the NAG framework,189 we show that similar results can be derived under the OGM framework in the following theorem:190 Theorem 4.3 ( Low accuracy parameter choice ) . In Algorithm 3 , let iteration N be the first time191 Step 5 updates x̃k+1 = yk . If we choose pk ≡ 1n , τk ≡ 1 − 1√ n+1 and terminate Algorithm 3 at192 iteration N , then the following holds at x̃N+1 :193 E [ ‖∇f ( x̃N+1 ) ‖2 ] ≤ 8L 2R20 5 ( √ n+ 1 + 1 ) and E [ f ( x̃N+1 ) ] − f ( x ? ) ≤ LR20√ n+ 1 + 1 . In particular , if the required accuracies are low ( or n is very large ) , i.e. , 2g ≥ 8L2R20 5 ( √ n+1+1 ) and194 f ≥ LR 2 0√ n+1+1 , then Algorithm 3 only has an O ( n ) oracle complexity.195 In the low accuracy region ( specified above ) , the choice in Theorem 4.3 removes the O ( log 1 ) factor196 in the complexity of Theorem 4.2 . We include some numerical justifications of Algorithm 3 in197 Appendix A . We believe that the potential-based PEP approach in [ 54 ] can help us identify better198 parameter choices of Algorithm 3 , which we leave for future work.199 Algorithm 4 R-Acc-SVRG-G Input : accuracy > 0 , parameters δ0 = L , β > 1 , initial guess x0 ∈ Rd . 1 : for t = 0 , 1 , 2 , . . . do 2 : Define fδt ( x ) = ( 1/n ) ∑n i=1 f δt i ( x ) , where f δt i ( x ) = fi ( x ) + ( δt/2 ) ‖x− x0‖ 2 . 3 : Initialize vectors z0 = x̃0 = x0 and set τx , τz , α , p , CIDC , CIFC according to Proposition 5.1 . 4 : for k = 0 , 1 , 2 , . . . do 5 : yk = τxzk + ( 1− τx ) x̃k + τz ( δt ( x̃k − zk ) −∇fδt ( x̃k ) ) . 6 : zk+1 = arg minx { 〈 Gδtk , x 〉 + ( α/2 ) ‖x− zk‖2 + ( δt/2 ) ‖x− yk‖2 } . 7 : //Gδtk , ∇f δt ik ( yk ) −∇fδtik ( x̃k ) +∇f δt ( x̃k ) , where ik is sampled uniformly in [ n ] . 8 : x̃k+1 = { yk with probability p , x̃k with probability 1− p. 9 : if 6‖∇f ( x̃k ) ‖ ≤ then output x̃k and terminate the algorithm . 10 : if under IDC and ( 1 + δtα ) k ≥ √ CIDC/δt then break the inner loop . 11 : if under IFC and ( 1 + δtα ) k ≥ √ CIFC/2δt then break the inner loop . 12 : end for 13 : δt+1 = δt/β . 14 : end for 5 Near-Optimal Accelerated SVRG with Adaptive Regularization200 Currently , there is no known stochastic method that directly achieves the optimal rate in . To get near-201 optimal rates , the existing strategy is to use a carefully designed regularization technique [ 42 , 2 ] with202 a method that solves strongly convex problems ; see , e.g. , [ 42 , 2 , 22 , 11 ] . However , the regularization203 parameter requires the knowledge of R0 or ∆0 , which significantly limits its practicality.204 Inspired by the recently proposed adaptive regularization technique [ 27 ] , we develop a near-optimal205 accelerated SVRG variant ( Algorithm 4 ) that does not require the knowledge of R0 or ∆0 . Note206 that this technique was originally proposed for NAG under the IDC assumption . Our development207 extends this technique to the stochastic setting , which brings an O ( √ n ) rate improvement . Moreover,208 we consider both IFC and IDC cases . Proofs in this section are provided in Appendix D.209 Detailed design . Algorithm 4 has a “ guess-and-check ” framework . In the outer loop , we first210 define the regularized objective fδt using the current estimate of regularization parameter δt , and211 then we initialize an accelerated SVRG method ( the inner loop ) to solve the δt-strongly convex fδt .212 If the inner loop breaks at Step 10 or 11 , indicating the poor quality of the current estimate δt , δt will213 be divided by a fixed β . Thus , conceptually , we can adopt any method that solves strongly convex214 finite-sums at the optimal rate as the inner loop . However , since the constructions of Step 10 or 11215 require some algorithm-dependent constants , we have to fix one method as the inner loop.216 The inner loop we adopted is a loopless variant of BS-SVRG [ 63 ] . This is because ( i ) BS-SVRG is217 the fastest known accelerated SVRG variant ( for ill-conditioned problems ) and ( ii ) it has a simple218 scheme , especially after using the loopless construction [ 34 ] . However , its original guarantee is built219 upon { zk } . Clearly , we can not implement the stopping criterion ( Step 9 ) on ‖∇f ( zk ) ‖ . Interestingly,220 we discover that its sequence { x̃k } works perfectly in our regularization framework , even if we can221 neither establish convergence on f ( x̃k ) − f ( x ? ) nor on ‖x̃k − x ? ‖2.7 Moreover , we find that the222 loopless construction significantly simplifies the parameter constraints of BS-SVRG , which originally223 involves Θ ( n ) th-order inequality . We provide the detailed parameter choice as follows:224 Proposition 5.1 ( Parameter choice ) . In Algorithm 4 , we set τx = α+δtα+L+δt , τz = τx δt − α ( 1−τx ) δtL and225 p = 1n . We set α as the ( unique ) positive root of the cubic equation ( 1− p ( α+δt ) α+L+δt ) ( 1 + δtα ) 2 = 1226 and specify CIDC = L2 + Lα 2p L+ ( 1−p ) ( α+δt ) , CIFC = 2L+ 2Lα2p ( L+ ( 1−p ) ( α+δt ) ) δt . Under these choices , we227 have αδt = O ( n+ √ n ( L/δt + 1 ) ) , CIDC = O ( ( L+ δt ) 2 ) , and CIFC = O ( L ) .228 6Note that we maintain the full gradient∇fδt ( x̃k ) and∇f ( x̃k ) = ∇fδt ( x̃k ) − δt ( x̃k − x0 ) . 7It is due to the special potential function of BS-SVRG ( see ( 27 ) ) , which does not contain these two terms . Under the choices of τx and τz , the α above is the optimal choice in our analysis . Then , we can229 characterize the progress of the inner loop in the following proposition:230 Proposition 5.2 ( The inner loop of Algorithm 4 ) . Using the parameters specified in Proposition 5.1,231 after running the inner loop ( Step 4-12 ) of Algorithm 4 for k iterations , we can conclude that232 ( i ) under IDC , i.e. , ‖x0 − x ? ‖ ≤ R0 for some x ? ∈ X ? ,233 E [ ‖∇f ( x̃k ) ‖ ] ≤ ( δt + ( 1 + δt α ) −k√ CIDC ) R0 , ( ii ) under IFC , i.e. , f ( x0 ) − f ( x ? ) ≤ ∆0,234 E [ ‖∇f ( x̃k ) ‖ ] ≤ ( √ 2δt + ( 1 + δt α ) −k√ CIFC ) √ ∆0 . The above results motivate the design of Step 10 and 11 . For example , in the IDC case , when the235 inner loop breaks at Step 10 , using ( i ) above , we obtain E [ ‖∇f ( x̃k ) ‖ ] ≤ 2δtR0 . Then , by discussing236 the relative size of δt and a certain constant , we can estimate the complexity of Algorithm 4 . The237 same methodology is used for the IFC case.238 Theorem 5.1 ( IDC case ) . Denote δ ? IDC = q 2R0 for some q ∈ ( 0 , 1 ) and let the outer iteration t = ` 239 be the first time8 δ ` ≤ δ ? IDC . The following assertions hold:240 ( i ) At outer iteration ` , Algorithm 4 terminates with probability at least 1− q.9241 ( ii ) The total expected oracle complexity of the ` + 1 outer loops is242 O ( ( n log LR0 q + √ nLR0 q ) log LR0 q ) . Theorem 5.2 ( IFC case ) . Denote δ ? IFC = 2q2 8∆0 for some q ∈ ( 0 , 1 ) and let the outer iteration t = ` be243 the first time δ ` ≤ δ ? IFC . The following assertions hold:244 ( i ) At outer iteration ` , Algorithm 4 terminates with probability at least 1− q.245 ( ii ) The total expected oracle complexity of the ` + 1 outer loops is246 O ( ( n log √ L∆0 q + √ nL∆0 q ) log √ L∆0 q ) . Compared with regularized Katyusha in Table 1 , the adaptive regularization approach drops the need247 to estimate R0 or ∆0 at the cost of a mere log 1 factor in the non-dominant term ( if is small ) .248 6 Discussion249 In this work , we proposed several simple and practical schemes that complement existing works250 ( Table 1 ) . Admittedly , the new schemes are currently only limited to the unconstrained Euclidean251 setting , because our techniques heavily rely on the interpolation conditions ( 1 ) and ( 2 ) . On the other252 hand , methods such as OGM [ 30 ] , TM [ 51 ] and ITEM [ 55 , 10 ] , which also rely on these conditions,253 are still not known to have their proximal variants . We list a few future directions as follows.254 ( 1 ) It is not clear how to naturally connect the parameters of M-OGM-G ( Algorithm 2 ) to OGM-G255 ( Algorithm 1 ) . The parameters of both algorithms seem to be quite restrictive and hardly generalizable256 due to the special construction in ( 4 ) . Does there exist an optimal method for minimizing the gradient257 norm that has a proper potential function ( at each iteration ) ? 258 ( 2 ) Is this new “ momentum ” in OGM-G beneficial for training neural nets ? Other classic momentum259 schemes such as NAG [ 40 ] or heavy-ball momentum method [ 49 ] are extremely effective for this260 task [ 53 ] , and they were also originally proposed for convex objectives.261 ( 3 ) Can we directly accelerate SARAH ( L2S ) ? By extending OGM-G ? It seems that existing stochastic262 acceleration techniques fail to accelerate SARAH ( or result in poor dependence on n as in [ 16 ] ) .263 8We assume that is small such that max { δ ? IDC , δ ? 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Summary: This paper studies efficient methods for finding near-stationary points ($||\nabla f(x)|| \le \epsilon$) of a convex (finite-sum) function. The authors nicely introduce main reasons for the need of such research, and their three main contributions are as follows.  First, this paper provides variants of OGM-G that is an optimal method for finding a near-stationary point for a smooth convex function under the initial bounded-function condition (IFC). In specific, Proposition 3.1 complements the complexity of OGM-G (with and without a change of coefficient at the last iteration). In addition, a memory-saving variant of OGM-G was studied, which does not require computing coefficients in advance.   Second, based on OGM and the loopless SVRG, the authors propose Acc-SVRG-G. With the two-stage parameter choice under the initial bounded-distance condition (IDC), it has fast rates for minimizing both the gradient norm and the function value for the first time.   Third, with adaptive regularization, the Acc-SVRG-G achieves the best existing rates (by the regularized Katyusha) up to a logarithmic factor, in terms of minimizing the gradient norm, under both IDC and IFC. The proposed R-Acc-SVRG-G has advantage over the regularized Katyusha, since it does not require the knowledge of $R_0$ and $\Delta_0$.