Stochastic Variance-Reduced Cubic Regularized Newton Method

Transcription

1 Sochasic Variance-Reduced Cubic Regularized Newon Mehod Dongruo Zhou and Pan Xu and Quanquan Gu arxiv: v1 [cs.lg] 13 Feb 2018 February 9, 2018 Absrac We propose a sochasic variance-reduced cubic regularized Newon mehod for non-convex opimizaion. A he core of our algorihm is a novel semi-sochasic gradien along wih a semi-sochasic Hessian, which are specifically designed for cubic regularizaion mehod. We show ha our algorihm is guaraneed o converge o an (ɛ, ɛ)-approximaely local minimum wihin Õ(n4/5 /ɛ 3/2 ) second-order oracle calls, which ouperforms he sae-of-he-ar cubic regularizaion algorihms including subsampled cubic regularizaion. Our work also sheds ligh on he applicaion of variance reducion echnique o high-order non-convex opimizaion mehods. Thorough experimens on various non-convex opimizaion problems suppor our heory. 1 Inroducion We sudy he following finie-sum opimizaion problem: min x R d F (x) = 1 n n f i (x), (1.1) where F (x) and each f i (x) can be non-convex. Such problems are common in machine learning, where each f i (x) is a loss funcion on a raining example (LeCun e al., 2015). Since F (x) is non-convex, finding is global minimum is generally NP-Hard (Hillar and Lim, 2013). As a resul, one possible goal is o find an approximae firs-order saionary poin (ɛ saionary poin): i=1 F (x) ɛ, for some given ɛ > 0. A lo of sudies have been devoed o his problem including gradien descen (GD), sochasic gradien descen (SGD) (Robbins and Monro, 1951), and heir exensions Deparmen of Sysems and Informaion Engineering, Universiy of Virginia, Charloesville, VA 22904, USA; dz4bd@virginia.edu Deparmen of Compuer Science, Universiy of Virginia, Charloesville, VA 22904, USA; px3ds@virginia.edu Deparmen of Compuer Science, Universiy of Virginia, Charloesville, VA 22904, USA; qg5w@virginia.edu 1

2 (Ghadimi and Lan, 2013; Reddi e al., 2016a; Allen-Zhu and Hazan, 2016; Ghadimi and Lan, 2016). Neverheless, firs-order saionary poins can be non-degenerae saddle poins or even local maximum in non-convex opimizaion, which are undesirable. Therefore, a more reasonable objecive is o find an approximae second-order saionary poin (Neserov and Polyak, 2006), which is also known as an (ɛ g, ɛ h )-approximae local minimum of F (x): F (x) 2 < ɛ g, λ min ( 2 F (x)) ɛ h, (1.2) for some given consan ɛ g, ɛ h > 0. In fac, in some machine learning problems like marix compleion (Ge e al., 2016), one finds ha every local minimum is a global minimum, suggesing ha finding an approximae local minimum is a beer choice han a saionary poin, and is good enough in many applicaions. One of he mos popular mehod o achieve his goal is perhaps cubic regularized Newon mehod, which was inroduced by Neserov and Polyak (2006), and solves he following kind of subproblems in each ieraion: h(x) = argmin h R d m(h, x) = F (x), h F (x)h, h + θ 6 h 3 2, (1.3) where θ > 0 is a regularizaion parameer. Neserov and Polyak (2006) proved ha fixing a saring poin x 0, and performing he updaing rule x = x 1 +h(x 1 ), he algorihm can oupu a sequence x i ha converges o a local minimum provided ha he funcion is Hessian Lipschiz. However, i can be seen ha o solve he subproblem (1.3), one needs o calculae he full gradien F (x) and Hessian 2 F (x), which is a big overhead in large scale machine learning problem because n is ofen very large. Some recen sudies presened various algorihms o avoid he calculaion of full gradien and Hessian in cubic regularizaion. Kohler and Lucchi (2017) used subsampling echnique o ge approximae gradien and Hessian insead of exac ones, and Xu e al. (2017b) also used subsampled Hessian. Boh of hem can reduce he compuaional complexiy in some circumsance. However, jus like oher sampling-based algorihm such as subsampled Newon mehod Erdogdu and Monanari (2015); Xu e al. (2016); Roosakhorasani and Mahoney (2016a,b); Ye e al. (2017), heir convergence raes are worse han ha of he Newon mehod, especially when one needs a high-accuracy soluion (i.e., he opimizaion error ɛ is small). This is because he subsampling size one needs o achieve cerain accuracy may be even larger han he full sample size n. Therefore, a naural quesion arises as follows: When we need a high-accuracy local minimum, is here an algorihm ha can oupu an approximae local minimum wih beer second-order oracle complexiy han cubic regularized Newon mehod? In his paper, we give an affirmaive answer o he above quesion. We propose a novel cubic regularizaion algorihm named Sochasic Variance-Reduced Cubic regularizaion (), which incorporaes he variance reducion echniques (Johnson and Zhang, 2013; Xiao and Zhang, 2014; Allen-Zhu and Hazan, 2016; Reddi e al., 2016a) ino he cubic-regularized Newon mehod. The key componen in our algorihm is a novel semi-sochasic gradien, ogeher wih a semisochasic Hessian, ha are specifically designed for cubic regularizaion. Furhermore, we prove 2

3 ha, for Hessian Lipschiz funcions, o aain an approximae (ɛ, ρɛ)-local minimum, our proposed algorihm requires O(n+n 4/5 /ɛ 3/2 ) Second-order Oracle (SO) calls and O(1/ɛ 3/2 ) Cubic Subproblem Oracle (CSO) calls. Here an ISO oracle represens an evaluaion of riple (f i (x), f i (x), 2 f i (x)), and a CSO oracle denoes an evaluaion of he exac soluion (or inexac soluion) of he cubic subproblem (1.3). Compared wih he original cubic regularizaion algorihm (Neserov and Polyak, 2006), which requires O(n/ɛ 3/2 ) ISO calls and O(1/ɛ 3/2 ) CSO calls, our proposed algorihm reduces he SO calls by a facor of Ω(n 1/5 ). We also carry ou experimens on real daa o demonsrae he superior performance of our algorihm. Our major conribuions are summarized as follows: We presen a novel cubic regularizaion mehod wih improved oracle complexiy. To he bes of our knowledge, his is he firs algorihm ha ouperforms cubic regularizaion wihou any loss in convergence rae. This is in sharp conras o subsampled cubic regularizaion mehods (Kohler and Lucchi, 2017; Xu e al., 2017a), which suffer from worse convergence raes han cubic regularizaion. We also exend our algorihm o he case wih inexac soluion o he cubic regularizaion subproblem. Similar o previous work (Caris e al., 2011; Xu e al., 2017a), we also layou a se of sufficien condiions, under which he oupu of he inexac algorihm is sill guaraneed o have he same convergence rae and oracle complexiy as he exac algorihm. This furher sheds ligh on he pracical implemenaion of our algorihm. As far as we know, our work is he firs work, which rigorously demonsraes he advanage of variance reducion for second-order opimizaion algorihms. Alhough here exis a few sudies (Lucchi e al., 2015; Moriz e al., 2016; Rodomanov and Kropoov, 2016) using variance reducion o accelerae Newon mehod, none of hem can deliver faser raes of convergence han sandard Newon mehod. Noaion We use [n] o denoe he index se {1, 2,..., n}. We use v 2 o denoe vecor Euclidean norm. For symmeric marix H R d d, we denoe is eigenvalues by λ 1 (H)... λ d (H), is specral norm by H 2 = max{ λ 1 (H), λ d (H) }, and he Schaen r-norm by H Sr = ( d i=1 λ i(h) r ) 1/r for r 1. We denoe A B if λ 1 (A B) 0 for symmeric marices A, B R d d. We also noe ha A B 2 C A 2 B 2 C I, C > 0. We call ξ a Rademacher random variable if P(ξ = 1) = P(ξ = 1) = 1/2. We use f n = O(g n ) o denoe ha f n Cg n for some consan C > 0 and use f n = Õ(g n) o hide he logarihmic erms of g n. 2 Relaed Work In his secion, we briefly review he relevan work in he lieraure. The mos relaed work o ours is he cubic regularized Newon mehod, which was originally proposed in Neserov and Polyak (2006). Caris e al. (2011) presened an adapive framework of cubic regularizaion, which uses an adapive esimaion of he local Lipschiz consan and approximae soluion o he cubic subproblem. To overcome he compuaional burden of gradien and Hessian marix evaluaions, Kohler and Lucchi (2017); Xu e al. (2017b,a) proposed o use subsampled gradien and Hessian in cubic regularizaion. On he oher hand, in order o solve he 3

4 cubic subproblem (1.3) more efficienly, Carmon and Duchi (2016) proposed o use gradien descen, while Agarwal e al. (2017) proposed a sophisicaed algorihm based on approximae marix inverse and approximae PCA. Tripuraneni e al. (2017) proposed a refined sochasic cubic regularizaion algorihm based on above subproblem solver. However, none of he aforemenioned varians of cubic regularizaion ouperforms he original cubic regularizaion mehod in erms of he oracle complexiy. Anoher imporan line of relaed research is he variance reducion mehod, which has been exensively sudied for large-scale finie-sum opimizaion problems. Variance reducion was firs proposed in convex finie-sum opimizaion (Roux e al., 2012; Johnson and Zhang, 2013; Xiao and Zhang, 2014; Defazio e al., 2014), which uses semi-sochasic gradien o reduce he variance of he sochasic gradien and improves he gradien complexiy of boh sochasic gradien descen (SGD) and gradien descen (GD). Represenaive algorihms include Sochasic Average Gradien (SAG) (Roux e al., 2012), Sochasic Variance Reduced Gradien (SVRG) (Johnson and Zhang, 2013) and SAGA (Defazio e al., 2014), o menion a few. For non-convex finie-sum opimizaion, Garber and Hazan (2015); Shalev-Shwarz (2016) proposed algorihms for he seing where each individual funcion may be non-convex, bu heir sum is sill convex. Laer on, Reddi e al. (2016a) and Allen-Zhu and Hazan (2016) exended he SVRG algorihm o he general non-convex finie-sum opimizaion, which ouperforms SGD and GD in erms of gradien complexiy as well. However, o he bes of our knowledge, i is sill an open problem wheher variance reducion can also improve he oracle complexiy of second-order opimizaion algorihms. Las bu no he leas is he line of research which aims o escape from nondegeneraed saddle poins by finding he negaive curvaure direcion. There is a vas lieraure which focuses on algorihms escaping from saddle poin by using informaion of gradien and negaive curvaure insead of considering he subproblem (1.3). Ge e al. (2015), Jin e al. (2017a) showed ha simple (sochasic) gradien descen wih perurbaion can escape from saddle poins. Carmon e al. (2016); Royer and Wrigh (2017); Allen-Zhu (2017) showed ha by calculaing he negaive curvaure using Hessian informaion, one can find (ɛ, ɛ)-local minimum faser han he firs-order mehods. Recen work (Allen-Zhu and Li, 2017; Jin e al., 2017b; Xu e al., 2017c) proposed firs-order algorihms ha can escape from saddle poins wihou using Hessian informaion. For beer comparison of our algorihm wih he mos relaed algorihms in erms of SO and CSO oracle complexiies, we summarize he resuls in Table 1. I can be seen from Table 1 ha our algorihm () achieves he lowes (SO and CSO) oracle complexiy compared wih he original cubic regularizaion mehod (Neserov and Polyak, 2006) which employs full gradien and Hessian evaluaions and he subsampled cubic mehod (Kohler and Lucchi, 2017; Xu e al., 2017b). In paricular, our algorihm reduces he SO oracle complexiy of cubic regularizaion by a facor of n 1/5 for finding an (ɛ, ɛ)-local minimum. We will provide more deailed discussion in he main heory secion. 1 I is he refined rae proved by Xu e al. (2017b) for he subsampled cubic regularizaion algorihm proposed in Kohler and Lucchi (2017) 4

5 Table 1: Comparisons beween differen mehods o find (ɛ, ɛ)-local minimum on he second-order oracle (SO) complexiy and and he cubic subproblem oracle (CSO) complexiy. Algorihm SO calls CSO calls Gradien Lipschiz Cubic regularizaion (Neserov and Polyak, 2006) Subsampled cubic regularizaion (Kohler and Lucchi, 2017; Xu e al., 2017b) (his paper) O(n/ɛ 3/2 ) O(1/ɛ 3/2 ) no yes Õ(n/ɛ 3/2 + 1/ɛ 5/2 ) 1 O(1/ɛ 3/2 ) yes yes Õ(n + n 4/5 /ɛ 3/2 ) O(1/ɛ 3/2 ) no yes Hessian Lipschiz 3 The Proposed Algorihm In his secion, we presen a novel algorihm, which uilizes sochasic variance reducion echniques o improve cubic regularizaion mehod. To reduce he compuaion burden of gradien and Hessian marix evaluaions in he cubic regularizaion updaes in (1.3), subsampled gradien and Hessian marix have been used in subsampled cubic regularizaion (Kohler and Lucchi, 2017; Xu e al., 2017b) and sochasic cubic regularizaion (Tripuraneni e al., 2017). Neverheless he sochasic gradien and Hessian marix have large variances, which undermine he convergence performance. Inspired by SVRG (Johnson and Zhang, 2013), we propose o use a semi-sochasic version of gradien and Hessian marix, which can conrol he variances auomaically. Specifically, our algorihm has wo loops. A he beginning of he s-h ieraion of he ouer loop, we denoe x s = x s+1 0. We firs calculae he full gradien g s = F ( x s ) and Hessian marix H s = 2 F ( x s ), which are sored for furher references in he inner loop. A he -h ieraion of he inner loop, we calculae he following semi-sochasic gradien and Hessian marix: v s+1 = 1 b g i I g U s+1 = 1 b h j I h ( fi (x s+1 ) f i ( x s ) + g s) 1 ( 2 f i ( x s ) H s) (x s+1 x s), (3.1) b g i I g ( 2 f j (x s+1 ) 2 f j ( x s ) ) + H s, (3.2) where I g and I h are bach index ses, and he bach sizes will be decided laer. In each inner ieraion, we solve he following cubic regularizaion subproblem: h s+1 = argmin m s+1 (h) = v s+1 h Us+1 h, h + M s+1, h (3.3) Then we perform he updae x s+1 +1 = xs+1 + h s+1 in he -h ieraion of he inner loop. The proposed algorihm is displayed in Algorihm 1. 5

6 Algorihm 1 Sochasic Variance Reducion Cubic Regularizaion () 1: Inpu: bach size b g, b h, penaly parameer M s,, s = 1... S, = 0... T, saring poin x 1. 2: Iniializaion 3: for s = 1,..., S do 4: x s+1 0 = x s 5: g s = F ( x s ) = 1 n n i=1 f i( x s ), H s = 1 n n i=1 2 f i ( x s ) 6: for = 0,..., T 1 do 7: Sample index se I g, I h, I g = b g, I h = b h ; 8: v s+1 = 1 b g i I g f i (x s+1 ) f i ( x s ) + g s ( 1 b g i I g 2 f i ( x s ) H s) (x s+1 x s ) 9: U s+1 = 1 b h ( j I h 2 f j (x s+1 ) 2 f j ( x s )) + H s 10: h s+1 = argmin h v s+1, h Us+1 h, h + M s+1, 6 h 3 2, 11: x s+1 +1 = xs+1 + h s+1 12: end for 13: x s+1 = x s+1 T 14: end for 15: Oupu: random choose one x s, for = 0,..., T and s = 1,..., S. There are wo noable feaures of our esimaor of he full gradien and Hessian in each inner loop, compared wih ha used in SVRG (Johnson and Zhang, 2013). The firs is ha our gradien and Hessian esimaors consis of mini-baches of sochasic gradien and Hessian. The second one is ha we use second order informaion when we consruc he gradien esimaor v s+1, while classical SVRG only uses firs order informaion o build i. Inuiively speaking, boh feaures are used o make a more accurae esimaion of he rue gradien and Hessian wih affordable oracle calls. Noe ha similar approximaions of he gradien and Hessian marix have been saged in recen work by Gower e al. (2017) and Wai e al. (2017), where hey used his new kind of esimaor for radiional SVRG in he convex seing, which radically differs from our seing. 4 Main Theory We firs lay down he following Hessian Lipschiz assumpion, which are necessary for our analysis and are widely used in he lieraure (Neserov and Polyak, 2006; Xu e al., 2016; Kohler and Lucchi, 2017). Assumpion 4.1 (Hessian Lipschiz). There exiss a consan ρ > 0, such ha for all x, y and i [n] 2 f i (x) 2 f i (y) 2 ρ x y 2. In fac, his is he only assumpion we need o prove our heoreical resuls. The Hessian Lipschiz assumpion plays a cenral role in conrolling he changing speed of second order informaion. I is obvious ha Assumpion 4.1 implies he Hessian Lipschiz assumpion of F, which, according o Neserov and Polyak (2006), is also equivalen o he following lemma. 6

7 Lemma 4.2. Le funcion F : x R d saisfy ρ-hessian Lipschiz assumpion, hen for any h R d, i holds ha 2 F (x) 2 F (y) 2 ρ x y 2, F (x + h) F (x) + F (x), h F (x)h, h + ρ 6 h 3 2, F (x + h) F (x) 2 F (x)h 2 ρ 2 h 2 2. We hen define he following opimal funcion gap beween iniial poin x 0 and he global minimum of F. Definiion 4.3 (Opimal Gap). For funcion F ( ) and he iniial poin x 0, le F be where F = inf x R d F (x). F = inf{ R : F (x 0 ) F }, Wihou loss of generaliy, we assume F < + hroughou his paper. Before we presen nonasympoic convergence resuls of Algorihm 1, we define { µ(x s+1 ) = max F (x s+1 ) 3/2 2, λ3 min ( 2 F (x s+1 } )) [M s+1, ] 3/2. (4.1) By definiion in (4.1), µ(x s+1 ) < ɛ 3/2 holds if and only if F (x s+1 ) 2 ɛ, λ min ( 2 F (x s+1 ) ) > M s+1, ɛ. (4.2) Therefore, in order o find an (ɛ, ρɛ)-local minimum of he non-convex funcion F, i suffices o find a poin x s+1 which saisfies µ(x s+1 ) < ɛ 3/2, and M s+1, = O(ρ) for all s,. Nex we define our oracles formally: Definiion 4.4 (Second-order Oracle). Given an index i and a poin x, one second-order oracle (SO) call reurns such a riple: [f i (x), f i (x), 2 f i (x)]. (4.3) Definiion 4.5 (Cubic Subproblem Oracle). Given a vecor g R d, a Hessian marix H and a posiive consan θ, one Cubic Subproblem Oracle (CSO) call reurns h sol, where h sol can be solved exacly as follows h sol = argmin h R d g, h h, Hh + θ 6 h 3 2. Remark 4.6. The second-order oracle is a special form of Informaion Oracle which is inroduced by Neserov, which reurns gradien, Hessian and all high order derivaives of objecive funcion F (x). Here, our second-order oracle will only reurns firs and second order informaion a some poin of single objecive f i insead of F. We argue ha i is a reasonable adapion because in his paper we focus on finie-sum objecive funcion. The Cubic Subproblem Oracle will reurn an exac or inexac soluion of (3.3), which plays an imporan role in boh heory and pracice. 7

8 Now we are ready o give a general convergence resul of Algorihm 1: Theorem 4.7. Le A, B, α and β be arbirary posiive consans, choose M s, = M for each s. Define parameer sequences {Θ } T =0 and {c } T =0 as follows ( Θ c = M 3/2 + 1 ) ( ρ 3/2 Θ A 1/2 b 3/4 + M ) g B 2 Cρ3 (log d) 3/2 ( b 3/2 + c α ) h β 1/2, Θ = 3M 2ρ 4A 4B (4.4) c +1 (1 + 2α + β ), 12 c T = 0, where ρ is he Hessian Lipschiz consan, M is he regularizaion parameer of Algorihm 1, and C is an absolue consan. If seing bach size b h > 25 log d, M = O(ρ), and Θ > 0 for all, hen he oupu of Algorihm 1 saisfies where γ n = min Θ /(15M 3/2 ). E[µ(x ou )] E[ F ( x 0 ) F ], (4.5) γ n ST Remark 4.8. To ensure ha x ou is an (ɛ, ρɛ)-local minimum, we can se he righ hand side of (4.5) o be less hen ɛ 3/2. This immediaely implies ha he oal ieraion complexiy of Algorihm 1 is ST = O(E [ F ( x 0 ) F ] /ɛ 3/2 ), which maches he ieraion complexiy of cubic regularizaion Neserov and Polyak (2006). Remark 4.9. Noe ha here is a log d erm in he expression of parameer c, and i is only relaed o Hessian bach size b h. The log d erm comes from marix concenraion inequaliies, which is believed o be unavoidable (Tropp e al., 2015). In oher words, he bach size of Hessian marix b h has a ineviable relaion o dimension d, unlike he bach size of gradien b g. The ieraion complexiy resul in Theorem 4.7 depends on a series of parameer defined as in (4.4). In he following corollary, we will show how o choose hese parameers in pracice o achieve a beer oracle complexiy. Corollary Le bach sizes b g and b h saisfy b g = b h / log d = 1400n 2/5. Se he parameers in Theorem 4.7 as follows A = B = 125ρ, α = 2n 1/10, β = 4n 2/5. Θ and c are defined as in (4.4). Le he cubic regularizaion parameer be M = 2000ρ, and he epoch lengh be T = n 1/5. Then Algorihm 1 converges o a (ɛ, ρɛ)-local minimum wih ( O n + F ρn 4/5 ) ( ) F ρ SO calls and O CSO calls. (4.6) ɛ 3/2 Remark Corollary 4.10 saes ha we can reduce he SO calls by seing he bach size b g, b h relaed o n. In conras, in order o achieve a (ɛ, ρɛ) local minimum, original cubic regularizaion mehod in Neserov and Polyak (2006) needs O(n/ɛ 3/2 ) second-order oracle calls, which is by a facor of n 1/5 worse han ours. And subsampled cubic regularizaion Kohler and Lucchi (2017); Xu e al. (2017b) requires Õ(n/ɛ3/2 + 1/ɛ 5/2 ) SO calls, which is even worse. ɛ 3/2 8

9 5 Pracical Algorihm wih Inexac Oracle In pracice, he exac soluion o he cubic subproblem (3.3) canno be obained. Insead, one can only ge an approximae soluion by some inexac solver. Thus we replace he CSO oracle in (4.5) wih he following inexac CSO oracle h sol argmin h R d g, h h, Hh + θ 6 h 3 2. To analyze he performance of Algorihm 1 wih inexac cubic subproblem solver, we replace he exac solver in Line 10 of Algorihm 1 wih h s+1 argmin m s+1 (h). (5.1) In order o characerize he above inexac soluion, we presen he following sufficien condiion, under which inexac soluion can ensure he same oracle complexiy as he exac soluion: Condiion 5.1 (Inexac condiion). For each s, and given δ > 0, if saisfies h s+1 m s+1 ( h s+1 m s+1 ( h s+1 δ 3/2, h s+1 ) M s+1, h s δ, 12 h s h s δ. saisfies δ- inexac condiion Remark 5.2. Similar inexac condiions have been sudied in he lieraure of cubic regularizaion. For insance, Neserov and Polyak (2006) presened a pracical way o solve he cubic subproblem wihou erminaion condiion. Caris e al. (2011); Kohler and Lucchi (2017) presened erminaion crieria for approximae soluion o cubic subproblem, which is slighly differen from our condiion. In general, he erminaion crieria in Caris e al. (2011); Kohler and Lucchi (2017) conains a non-linear equaion, which is hard o verify and less pracical. In conras, our inexac condiion only conains inequaliy, which is easy o be verified in pracice. Nex we give he convergence resul wih inexac CSO oracle: h s+1 Theorem 5.3. Le o be he oupu in each inner loop of Algorihm 1 which saisfies Condiion 5.1. Le A, B, α, β > 0 be arbirary consans. Le M s, = M for each s, and Θ and c are defined in (4.4), where 1 T. If choosing bach size b h > 25 log d and M = O(ρ), and Θ > 0 for all, hen he oupu of Algorihm 1 wih inexac subproblem solver saisfies: E[µ(x ou )] E[F ( x0 ) F ] γ n ST + δ, where ( δ = δ Θ + Θ ) + 1, γ 2M 3/2 n = min Θ /(15M 3/2 ). 9

10 Remark 5.4. By he definiion of µ(x) in (4.1) and (4.2), in order o aain an (ɛ, ɛ) local minimum, we require E[µ(x ou )] ɛ 3/2 and hus δ < ɛ 3/2, which implies ha δ in Condiion 5.1 should saisfy δ < ɛ 3/2 /(Θ + Θ /(2M 3/2 ) + 1). Thus he oal ieraion complexiy of Algorihm 1 is O( F /(γ n ɛ 3/2 )). By he same choice of parameers, Algorihm 1 wih inexac oracle can achieve a reducion in SO calls. Corollary 5.5. Under Condiion 5.1, and under he same condiions as in Corollary 4.10, he oupu of Algorihm 1 wih he inexac subproblem solver saisfies Eµ(x ou ) ɛ 3/2 + δ f wihin where δ f = O(ρδ). ( O n + F ρn 4/5 ) ( ) F ρ SO calls and O CSO calls, ɛ 3/2 ɛ 3/ (a) a9a (b) covype (c) ijcnn ime in seconds (d) a9a ime in seconds (e) covype ime in seconds (f) ijcnn1 Figure 1: Logarihmic funcion value gap for nonconvex regularized logisic regression on differen daases. (a), (b) and (c) presen he oracle complexiy comparison; (d), (e) and (f) presen he runime comparison. 6 Experimens In his secion, we presen numerical experimens on differen non-convex Empirical Risk Minimizaion (ERM) problems and on differen daases o validae he advanage of our proposed algorihm in finding approximae local minima. 10

11 (a) a9a (b) covype (c) ijcnn ime in seconds (d) a9a ime in seconds (e) covype ime in seconds (f) ijcnn1 Figure 2: Logarihmic funcion value gap for nonlinear leas square on differen daases. (a), (b) and (c) presen he oracle complexiy comparison; (d), (e) and (f) presen he runime comparison. Baselines: We compare our algorihm wih Regularizaion () (Caris e al., 2011), Regularizaion () (Kohler and Lucchi, 2017), Regularizaion () (Tripuraneni e al., 2017) and Gradien Cubic Regularizaion () (Carmon and Duchi, 2016). All of above algorihms are carefully uned for a fair comparison. The Calculaion for SO calls: Here we lis he SO call each algorihm needs for one loop. For, each loop coss (B g +B h ) SO calls, where B g and B h o denoe he subsampling size of gradien and Hessian. For, each loop coss (n g + n h ) SO calls, where we use n g and n h o denoe he subsampling size of gradien and Hessian-vecor operaor. Gradien Cubic and cos n SO calls in each loop. Finally, we define he amoun of is he amoun of SO call divided by n. Parameer uning and subproblem solver: For each algorihm and each daase, we choose differen b g, b h, T for he bes performance. Meanwhile, we also use wo differen sraegies for choosing M s, : he firs one is o fix M s, = M in each ieraion, which is proved o enjoy good convergence performance; he oher one is o choose M s, = α/(1 + β) (s+/t ), α, β > 0 for each ieraion. This choice of parameer is similar o he choice of penaly parameer in Subsampled Cubic and, which someimes makes some algorihms behave beer in our experimen. As o he solver for subproblem (3.3) in each loop, we choose o use he Lanczos-ype mehod inroduced in Caris e al. (2011). Daases: The daases we use are a9a, covype, ijcnn1, which are common daases used in ERM problems. The deailed informaion abou hese daases are in Table 2. Non-convex regularized logisic regression: The firs nonconvex problem we choose is 11

12 (a) a9a (b) covype (c) ijcnn ime in seconds (d) a9a ime in seconds (e) covype ime in seconds (f) ijcnn1 Figure 3: Logarihmic funcion value gap for robus linear regression on differen daases. (a), (b) and (c) presen he oracle complexiy comparison; (d), (e) and (f) presen he runime comparison. Table 2: Overview of he daases used in our experimens Daase sample size n dimension d a9a 32, covype 581, ijcnn1 35, a binary logisic regression problem wih a non-convex regularizer d i=1 λw2 (i) /(1 + w2 (i) ) (Reddi e al., 2016b). More specifically, suppose we are given raining daa {x i, y i } n i=1, where x i R d and y i {0, 1} are feaure vecors and labels corresponding o he i-h daa poins. The minimizaion problem is as follows 1 min w R d n n y i log φ(x T i w) + (1 y i ) log[1 φ(x T i w)] + i=1 d λw(i) 2 /(1 + w2 (i) ), where φ(x) = 1/(1 + exp( x)) is he sigmoid funcion. We fix λ = 10 in our experimens. The experimen resuls are shown in Figure 1. Nonlinear linear squares: The second problem is a non-linear leas squares problem which focuses on he ask of binary linear classificaion (Xu e al., 2017a). Given raining daa {x i, y i } n i=1, where x i R d and y i {0, 1} are feaure vecors and labels corresponding o he i-h daa poins. i=1 12

13 The minimizaion problem is 1 min w R d n n [y i φ(x T i w)] 2 i=1 where φ(x) = 1/(1 + exp( x)) is he sigmoid funcion. The experimen resuls are shown in Figure 2. Robus linear regression: The hird problem is a robus linear regression problem where we use a non-convex robus loss funcion log(x 2 /2 + 1) (Barron, 2017) insead of square loss in leas square regression. Given a raining sample {x i, y i } n i=1, where x i R d and y i {0, 1} are feaure vecors and labels corresponding o he i-h daa poin. The minimizaion problem is 1 min w R d n n η(y i x T i w), i=1 where η(x) = log(x 2 /2 + 1). The experimenal resuls are shown in Figure 3. From Figures 1, 2 and 3, we can see ha our algorihm ouperforms all he oher baseline algorihms on all he daases. The only excepion happens in he non-linear leas square problem and he robus linear regression problem on he covype daase, where our algorihm behaves a lile worse han a he high accuracy regime in erms of epoch couns. However, under his seing, our algorihm sill ouperforms he oher baselines in erms of he cpu ime. 7 Conclusions In his paper, we propose a novel second-order algorihm for non-convex opimizaion called. Our algorihm is he firs algorihm which improves he oracle complexiy of cubic regularizaion and is subsampled varians under cerain regime using variance reducion echniques. We also show ha similar oracle complexiy also holds wih inexac oracle. Under boh seings our algorihm ouperforms he sae-of-he-ar. References Agarwal, N., Allenzhu, Z., Bullins, B., Hazan, E. and Ma, T. (2017). Finding approximae local minima for nonconvex opimizaion in linear ime. Allen-Zhu, Z. (2017). Naasha 2: Faser non-convex opimizaion han sgd. arxiv preprin arxiv: Allen-Zhu, Z. and Hazan, E. (2016). Variance reducion for faser non-convex opimizaion. In Inernaional Conference on Machine Learning. Allen-Zhu, Z. and Li, Y. (2017). Neon2: Finding Local Minima via Firs-Order Oracles. ArXiv e-prins abs/ Full version available a hp://arxiv.org/abs/ Barron, J. T. (2017). A more general robus loss funcion. arxiv preprin arxiv:

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