Improved Optimization of Finite Sums with Minibatch Stochastic Variance Reduced Proximal Iterations

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1 Improved Optimization of Finite Sums with Miniatch Stochastic Variance Reduced Proximal Iterations Jialei Wang University of Chicago Tong Zhang Tencent AI La Astract We present a novel miniatch stochastic optimization method for empirical risk minimization prolems. The method efficiently leverages variance reduced first-order and su-sampled higher-order information to accelerate the convergence. We prove improved iteration complexity over state-of-the-art methods under suitale conditions. Introduction We consider the following optimization prolem of finite-sums: min f(w) := w n n f i (w), () which arises in many machine learning prolems. In the context of regularized loss minimization of linear predictors, we have f i (w) = l(w x i ; i ) + 2 w 2. Let x, x 2,..., x n R d e feature vectors of n data samples, and,..., n R or, } e the corresponding target variales of interest. We note that () covers many popular models used in machine learning: for example when l(w x i ; i ) = log( + exp( i w, x i )) we otain l 2 regularized logistic regression. The uiquitousness of such finite-sum optimization prolems and the massive scale of modern datasets motivate significant research effort on efficient optimization algorithms to solve (). Throughout the paper, we assume each component function is L-smooth and -strongly convex. In recent years there have een significant advances in developing fast optimization algorithms for (), and we refer the readers to ottou206optimization for a comprehensive survey of these developments. For large-scale prolems in the form of (), randomized methods are particularly efficient ecause of their low per iteration computation. Below we riefly review two lines of research: i) randomized variance reduced first-order methods; ii) randomized methods leveraging second-order information. Variance Reduced First-order Methods The key technique for developing fast stochastic first-order methods is variance reduction, which makes the variance of the randomized update direction approching zero when the iterate gets closer to the optimum. Representative methods of this category include SAG roux202stochastic, SVRG johnson203accelerating, SDCA shalev203stochastic and SAGA defazio204saga, etc. In order to find a solution that reaches ɛ-suoptimality, these methods requires O (( ) ( n + L log )) ɛ calls to the first-order oracle to compute the gradient of an individual function. In L the large condition numer regime (e.g. > n), y using an acceleration technique (such as Catalyst shalev206accelerated,frostig205regularizing,lin205universal, APCG lin205accelerated, SPDC zhang205stochastic, (( ) Katyusha allen206katyusha, etc), one can further improve the iteration complexity to O n + n L log ( ) ) ɛ. i= Leveraging Second-order Information Second-order information is often useful in improving the convergence of optimization algorithms. However for large-scale prolems, otaining and inverting the exact Hessian matrix is often computational expensive, making vanilla Newton methods not well suited in solving (). Therefore, there have een emerging studies in designing randomized algorithms that effectively utilize the approximated Hessian information. One line of such research is the su-sampled Newton method which approximates the Hessian matrix ased on a su-sampled miniatch of data. Several work, such as yrd20use,am205su,roosta206su,roosta206su2,ollapragada206exact, estalished local linear OPTML 207: 0th NIPS Workshop on Optimization for Machine Learning (NIPS 207).

2 Algorithm MB-SVRP: Miniatch Stochastic Variance Reduced Proximal Iterations. Parameters η,,, ν, ε. Initialize w 0 = 0. Sampling Sampling items from [n] to form a miniatch B. for s =, 2,... do Calculate ṽ = n n i= f i( w s ). Initialize y 0 = w 0 = w s. for t =, 2,..., m do Sampling items from [n] to form a miniatch B t. Find w t that approximately solve (2) such that f t (w t ) min w ft (w) ε. f t (w) := 2 (w w t ) 2 f i (w t ) (w w t ) + η i B f i (y t ) + ηṽ η f i ( w s ), w i B t i B t + 2 w y t 2. (2) Update: end for Update w s = w m. end for Return w s y t = w t + ν(w t w t ). convergence rates for different variants of su-sampled Newton methods, when the size of the su-sampled data is large enough. All of the aforementioned methods employ the full first-order gradient, and require calling the second-order oracle to compute the Hessian and solving a resulting linear system at every iteration. Therefore the computational complexity (as compared in Tale 2 of xu206su) are often worse than stochastic algorithms such as SVRG. Another line of research is to consider randomness in oth first and second-order information to design lower per-iteration cost algorithms. In particular, yrd206stochastic considered comining miniatch SGD with Limited-memory BFGS (L-BFGS) liu989limited type update. Inspired y this, moritz206linearly,gower206stochastic,wang207stochastic proposed to comine variance reduced stochastic gradient with L-BFGS, and proved linear convergence for strongly convex and smooth ojectives. However, theoretically it is hard to guarantee the quality of approximated Hessian using L-BFGS update, and thus the iteration complexity otained y moritz206linearly,gower206stochastic,wang207stochastic can e pessimistic, and can e much worse than vanilla SVRG. In this paper, we make effort in this direction and make the following contriutions: i) we propose a novel approach that comines the advantages of variance-reduced first-order methods and su-sampled Newton methods, in a efficient way that does not require expensive Hessian matrix computation and inversion. The method can e naturally extended to solve composite optimization prolems with non-smooth regularization; ii) we theoretically show under certain conditions the proposed approach can improve state-of-the-art iteration complexity, and empirically demonstrate it can sustantially improve the convergence of existing methods. 2 Miniatch Stochastic Variance Reduced Proximal Iterations In this section we present the proposed approach for minimizing (). The high-level description is given in Algorithm, which consists of two main innovations: i) simultaneous incorporation of first-order and higher-order information via su-sampling; ii) allowing larger miniatch sizes through acceleration with inexact minimization. We explain these features elow. 2

3 Sampling oth First-order and Higher-order Information We first discuss the main uilding lock of the algorithm. Suppose at iteration t, given the previous iterate w t, we consider the following update rule: w t arg min w 2 (w w t ) 2 f i (w t ) (w w t ) + 2 w w t 2 i B η + f i (w t ) + η f( w) η f i ( w), w (3) i B t i B t ( ) =w t η 2 f i (w t ) + I f i (w t ) + f( w) f i ( w), (4) i B t i B t i B where B, B t are some randomly sampled miniatch from,..., n, oth with miniatch size ; η and are stepsize parameters, and w is a reference predictor used for reducing variance. The main feature of updating rule (3) is it considered oth noisy first-order and higher-order information via miniatch sampling. The term i B f i(w) i B f i(w t ), w in (3) incorporates noisy higher-order information of the ojective function. It which can e treated as a variant of su-sampled Newton method, comined with the miniatch stochastic gradient with variance reduction. And the update rule (3) can e treated as a preconditioned I) miniatch SVRG update rule, with i B 2 f i (w t ) + as a precondition matrix. Large Miniatch Size via Acceleration with Inexact Minimization ( Using (3) as the uilding lock, we propose the MB-SVRP (miniatch stochastic variance reduced proximal iterations) method, which is detailed in Algorithm. At the eginning of the algorithm, we form a miniatch B y sampling from,..., n and fix it for the whole optimization process. Then following the SVRG method johnson203accelerating, the algorithm is divided to multiple stages, indexed y s. At each stage, we iteratively solve a minimization prolem of the form (2) ased on the randomly sampled miniatch B t. Compared with the simple update rule in (3), the major difference in Algorithm is that we consider a momentum scheme y maintaining two sequences w t, y t }, which is inspired y Nesterov s acceleration technique nesterov2003introductory and its recent SVRG variant nitanda204stochastic. The main theoretical advantage over miniatch SVRG without momentum konevcny206mini is that such an acceleration allows us to use a much larger miniatch size (up to a size of O( n)) without slowing down the convergence. Since it is often expensive to find the exact minimizer of (2), we consider an approximate minimizer with ojective suoptimality ε. When we choose the appropriate for f t (w) to otain enough strong convexity, the suprolems can e efficiently solved to high accuracy efficiently. For example, using SVRG-type algorithms johnson203accelerating to solve (2) allows us to find an approximate solution with a small suoptimality ε with a constant numer of passes over the data in the miniatch B B t, if is set appropriately. 3 Theoretical Results Besides the strong convexity and smoothness, we also need the following Hessian Lipschitz condition, and the notion of effective dimension and statistical leverage. Condition. For each component function f i (w), i [n], its Hessian matrix is M-Lipschitz, i.e. 2 f i (w) 2 f i (w ) M w w, w, w R d, i [n]. Definition. (Effective dimension) Let the,..., d e the top-d eigenvalues of H 0 (w ) = (/n) n i= l (w x i ; i )x i x i, define the effective dimension d (for some 0) of H 0 (w ) eing d = d j= j j+. 3

4 Tale : Comparison of iteration complexity of various finite-sum quadratic optimization algorithms when effective dimension and statistical leverage are ounded, where we compare the different relative scale of condition numer κ = L/ and sample size n, ignoring logarithmic factors. κ n κ = n 4/3 κ = n 3/2 κ = n 2 κ = n 3 SVRG n n 4/3 n 3/2 n 2 n 3 MB-SVRP n ρ 2 d n ρ 2 d n ρ 2 d n 3/2 ρ 2 d n 5/2 Acc-SVRG n n 7/6 n 5/4 n 3/2 n 2 ( ) /2 ( ) /2 ( 2 d ) /2 ( 2 d ) /2 Acc-MB-SVRP n ρ 2 d n ρ 2 d n ρ n 5/4 ρ n 7/4 Definition 2. (Statistical leverage at ) Let H (w n ) = n i= l (w x i ; i )x i x i + I, we say the statistical leverage of data matrix X is ounded y ρ at if H /2 (w )l (w x i; i) /2 x i max i [n] (/n) ρ. n j= H /2 (w )l (w x j; j) /2 2 x j Theorem 3. Consider Algorithm on prolems with L-smooth and -strongly convex functions that satisfied Condition with Hessian Lipschitz parameter M. Suppose we sample miniatch B uniformly from,..., n, and set the tuning parameters as = max, L }, min n, ρ 2 d ( ) } /3 L, η min 3 (ρ 2 d ) 2 L, ρ 2 d and( suppose ) we start from the initialization point that is sufficiently close to the optimum: w 0 w R = O 4 L 2 M. Given ɛ > 0, and let ε 0 ( 7 L) ɛ. Then for the MP-SVRP algorithm to find the approximate 5 solution w s of () that reaches the expected ɛ-ojective suoptimality Ef( w s ) min w f(w) ɛ, the total numer of oracle calls to solve (2) is no more than ( (L ) /3 O max, ρ2 d L } ( ) ) n 2 log. ɛ }, Note that in Algorithm, we can use SVRG to solve each suprolem in (2). This leads to the following result. Corollary 4. Assume that the conditions of Theorem 3 hold. Moreover, if we use SVRG to solve each suprolem in (2) up to the suoptimality ε, then the total numer of gradient evaluations used in the whole MP-SVRP algorithm can e upper ounded y O ( max ρ 2 d ( ) 2/3 L, ρ2 d L n } log ( ) ( ) ( ) ) L log 2 + n log. ɛ ɛ Acceleration Corollary 4 stated that if the condition numer is not too large: L n5/4, then MB-SVRP only requires logarithmic passes over data to find a solution with high accuracy. When the condition numer is large, MP-SVRP can still e slow. By using the accelerated proximal point framework proposed in shalev206accelerated, frostig205regularizing,lin205universal, it is possile to otain an accelerated convergence rate which has milder dependence on condition numer, Tale summarized the main results. 4

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