Beyond stochastic gradient descent for large-scale machine learning
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1 Beyond stochastic gradient descent for large-scale machine learning Francis Bach INRIA - Ecole Normale Supérieure, Paris, France Joint work with Eric Moulines, Nicolas Le Roux and Mark Schmidt - ECML-PKDD, September 2014
2 Big data revolution? A new scientific context Data everywhere: size does not (always) matter Science and industry Size and variety Learning from examples n observations in dimension p
3 Search engines - Advertising
4 Marketing - Personalized recommendation
5 Visual object recognition
6 Personal photos
7 Bioinformatics Protein: Crucial elements of cell life Massive data: 2 millions for humans Complex data
8 Context Machine learning for big data Large-scale machine learning: large p, large n p : dimension of each observation (input) n : number of observations Examples: computer vision, bioinformatics, advertising Ideal running-time complexity: O(pn) Going back to simple methods Stochastic gradient methods (Robbins and Monro, 1951) Mixing statistics and optimization Using smoothness to go beyond stochastic gradient descent
9 Context Machine learning for big data Large-scale machine learning: large p, large n p : dimension of each observation (input) n : number of observations Examples: computer vision, bioinformatics, advertising Ideal running-time complexity: O(pn) Going back to simple methods Stochastic gradient methods (Robbins and Monro, 1951) Mixing statistics and optimization Using smoothness to go beyond stochastic gradient descent
10 Context Machine learning for big data Large-scale machine learning: large p, large n p : dimension of each observation (input) n : number of observations Examples: computer vision, bioinformatics, advertising Ideal running-time complexity: O(pn) Going back to simple methods Stochastic gradient methods (Robbins and Monro, 1951) Mixing statistics and optimization Using smoothness to go beyond stochastic gradient descent
11 Outline Introduction: stochastic approximation algorithms Supervised machine learning and convex optimization Stochastic gradient and averaging Strongly convex vs. non-strongly convex Fast convergence through smoothness and constant step-sizes Online Newton steps (Bach and Moulines, 2013) O(1/n) convergence rate for all convex functions More than a single pass through the data Stochastic average gradient (Le Roux, Schmidt, and Bach, 2012) Linear (exponential) convergence rate for strongly convex functions
12 Supervised machine learning Data: n observations (x i,y i ) X Y, i = 1,...,n, i.i.d. Prediction as a linear function θ,φ(x) of features Φ(x) R p (regularized) empirical risk minimization: find ˆθ solution of 1 n min l ( y i, θ,φ(x i ) ) + µω(θ) θ R p n i=1 convex data fitting term + regularizer
13 Usual losses Regression: y R, prediction ŷ = θ,φ(x) quadratic loss 1 2 (y ŷ)2 = 1 2 (y θ,φ(x) )2
14 Usual losses Regression: y R, prediction ŷ = θ,φ(x) quadratic loss 1 2 (y ŷ)2 = 1 2 (y θ,φ(x) )2 Classification : y { 1,1}, prediction ŷ = sign( θ,φ(x) ) loss of the form l(y θ,φ(x) ) True 0-1 loss: l(y θ,φ(x) ) = 1 y θ,φ(x) <0 Usual convex losses: hinge square logistic
15 Supervised machine learning Data: n observations (x i,y i ) X Y, i = 1,...,n, i.i.d. Prediction as a linear function θ,φ(x) of features Φ(x) R p (regularized) empirical risk minimization: find ˆθ solution of 1 n min l ( y i, θ,φ(x i ) ) + µω(θ) θ R p n i=1 convex data fitting term + regularizer
16 Supervised machine learning Data: n observations (x i,y i ) X Y, i = 1,...,n, i.i.d. Prediction as a linear function θ,φ(x) of features Φ(x) R p (regularized) empirical risk minimization: find ˆθ solution of 1 n min l ( y i, θ,φ(x i ) ) + µω(θ) θ R p n i=1 convex data fitting term + regularizer Empirical risk: ˆf(θ) = 1 n n i=1 l(y i, θ,φ(x i ) ) training cost Expected risk: f(θ) = E (x,y) l(y, θ,φ(x) ) testing cost Two fundamental questions: (1) computing ˆθ and (2) analyzing ˆθ May be tackled simultaneously
17 Supervised machine learning Data: n observations (x i,y i ) X Y, i = 1,...,n, i.i.d. Prediction as a linear function θ,φ(x) of features Φ(x) R p (regularized) empirical risk minimization: find ˆθ solution of 1 n min l ( y i, θ,φ(x i ) ) + µω(θ) θ R p n i=1 convex data fitting term + regularizer Empirical risk: ˆf(θ) = 1 n n i=1 l(y i, θ,φ(x i ) ) training cost Expected risk: f(θ) = E (x,y) l(y, θ,φ(x) ) testing cost Two fundamental questions: (1) computing ˆθ and (2) analyzing ˆθ May be tackled simultaneously
18 Smoothness and strong convexity A function g : R p R is L-smooth if and only if it is twice differentiable and θ R p, eigenvalues [ g (θ) ] L smooth non smooth
19 Smoothness and strong convexity A function g : R p R is L-smooth if and only if it is twice differentiable and θ R p, eigenvalues [ g (θ) ] L Machine learning with g(θ) = 1 n n i=1 l(y i, θ,φ(x i ) ) Hessian covariance matrix 1 n n i=1 Φ(x i) Φ(x i ) Bounded data
20 Smoothness and strong convexity A twice differentiable function g : R p R is µ-strongly convex if and only if θ R p, eigenvalues [ g (θ) ] µ convex strongly convex
21 Smoothness and strong convexity A twice differentiable function g : R p R is µ-strongly convex if and only if θ R p, eigenvalues [ g (θ) ] µ Machine learning with g(θ) = 1 n n i=1 l(y i, θ,φ(x i ) ) Hessian covariance matrix 1 n n i=1 Φ(x i) Φ(x i ) Data with invertible covariance matrix (low correlation/dimension)
22 Smoothness and strong convexity A twice differentiable function g : R p R is µ-strongly convex if and only if θ R p, eigenvalues [ g (θ) ] µ Machine learning with g(θ) = 1 n n i=1 l(y i, θ,φ(x i ) ) Hessian covariance matrix 1 n n i=1 Φ(x i) Φ(x i ) Data with invertible covariance matrix (low correlation/dimension) Adding regularization by µ 2 θ 2 creates additional bias unless µ is small
23 Iterative methods for minimizing smooth functions Assumption: g convex and smooth on R p Gradient descent: θ t = θ t 1 γ t g (θ t 1 ) O(1/t) convergence rate for convex functions O(e ρt ) convergence rate for strongly convex functions
24 Iterative methods for minimizing smooth functions Assumption: g convex and smooth on R p Gradient descent: θ t = θ t 1 γ t g (θ t 1 ) O(1/t) convergence rate for convex functions O(e ρt ) convergence rate for strongly convex functions Newton method: θ t = θ t 1 g (θ t 1 ) 1 g (θ t 1 ) O ( e ρ2t ) convergence rate
25 Iterative methods for minimizing smooth functions Assumption: g convex and smooth on R p Gradient descent: θ t = θ t 1 γ t g (θ t 1 ) O(1/t) convergence rate for convex functions O(e ρt ) convergence rate for strongly convex functions Newton method: θ t = θ t 1 g (θ t 1 ) 1 g (θ t 1 ) O ( e ρ2t ) convergence rate Key insights from Bottou and Bousquet (2008) 1. In machine learning, no need to optimize below statistical error 2. In machine learning, cost functions are averages Stochastic approximation
26 Stochastic approximation Goal: Minimizing a function f defined on R p given only unbiased estimates f n(θ n ) of its gradients f (θ n ) at certain points θ n R p
27 Stochastic approximation Goal: Minimizing a function f defined on R p given only unbiased estimates f n(θ n ) of its gradients f (θ n ) at certain points θ n R p Machine learning - statistics f(θ) = Ef n (θ) = El(y n, θ,φ(x n ) ) = generalization error Loss for a single pair of observations: f n (θ) = l(y n, θ,φ(x n ) ) Expected gradient: f (θ) = Ef n(θ) = E { l (y n, θ,φ(x n ) )Φ(x n ) } Beyond convex optimization: see, e.g., Benveniste et al. (2012)
28 Convex stochastic approximation Key assumption: smoothness and/or strong convexity Key algorithm: stochastic gradient descent (a.k.a. Robbins-Monro) θ n = θ n 1 γ n f n(θ n 1 ) Polyak-Ruppert averaging: θ n = 1 n+1 n k=0 θ k Which learning rate sequence γ n? Classical setting: γ n = Cn α
29 Convex stochastic approximation Key assumption: smoothness and/or strong convexity Key algorithm: stochastic gradient descent (a.k.a. Robbins-Monro) θ n = θ n 1 γ n f n(θ n 1 ) Polyak-Ruppert averaging: θ n = 1 n+1 n k=0 θ k Which learning rate sequence γ n? Classical setting: γ n = Cn α Running-time = O(np) Single pass through the data One line of code among many
30 Convex stochastic approximation Existing work Known global minimax rates of convergence for non-smooth problems (Nemirovsky and Yudin, 1983; Agarwal et al., 2012) Strongly convex: O((µn) 1 ) Attainedbyaveragedstochasticgradientdescentwithγ n (µn) 1 Non-strongly convex: O(n 1/2 ) Attained by averaged stochastic gradient descent with γ n n 1/2
31 Convex stochastic approximation Existing work Known global minimax rates of convergence for non-smooth problems (Nemirovsky and Yudin, 1983; Agarwal et al., 2012) Strongly convex: O((µn) 1 ) Attainedbyaveragedstochasticgradientdescentwithγ n (µn) 1 Non-strongly convex: O(n 1/2 ) Attained by averaged stochastic gradient descent with γ n n 1/2 Asymptotic analysis of averaging (Polyak and Juditsky, 1992; Ruppert, 1988) All step sizes γ n = Cn α with α (1/2,1) lead to O(n 1 ) for smooth strongly convex problems A single algorithm with global adaptive convergence rate for smooth problems?
32 Convex stochastic approximation Existing work Known global minimax rates of convergence for non-smooth problems (Nemirovsky and Yudin, 1983; Agarwal et al., 2012) Strongly convex: O((µn) 1 ) Attainedbyaveragedstochasticgradientdescentwithγ n (µn) 1 Non-strongly convex: O(n 1/2 ) Attained by averaged stochastic gradient descent with γ n n 1/2 Asymptotic analysis of averaging (Polyak and Juditsky, 1992; Ruppert, 1988) All step sizes γ n = Cn α with α (1/2,1) lead to O(n 1 ) for smooth strongly convex problems A single algorithm for smooth problems with convergence rate O(1/n) in all situations?
33 Least-mean-square algorithm Least-squares: f(θ) = 1 2 E[ (y n Φ(x n ),θ ) 2] with θ R p SGD = least-mean-square algorithm (see, e.g., Macchi, 1995) usually studied without averaging and decreasing step-sizes with strong convexity assumption E [ Φ(x n ) Φ(x n ) ] = H µ Id
34 Least-mean-square algorithm Least-squares: f(θ) = 1 2 E[ (y n Φ(x n ),θ ) 2] with θ R p SGD = least-mean-square algorithm (see, e.g., Macchi, 1995) usually studied without averaging and decreasing step-sizes with strong convexity assumption E [ Φ(x n ) Φ(x n ) ] = H µ Id New analysis for averaging and constant step-size γ = 1/(4R 2 ) Assume Φ(x n ) R and y n Φ(x n ),θ σ almost surely No assumption regarding lowest eigenvalues of H Main result: Ef( θ n 1 ) f(θ ) 4σ2 p n + 4R2 θ 0 θ 2 n Matches statistical lower bound (Tsybakov, 2003) Non-asymptotic robust version of Györfi and Walk (1996)
35 Markov chain interpretation of constant step sizes LMS recursion for f n (θ) = 1 2 ( yn Φ(x n ),θ ) 2 θ n = θ n 1 γ ( Φ(x n ),θ n 1 y n ) Φ(xn ) The sequence (θ n ) n is a homogeneous Markov chain convergence to a stationary distribution π γ with expectation θ def γ = θπ γ (dθ) - For least-squares, θ γ = θ θ n θ γ θ 0
36 Markov chain interpretation of constant step sizes LMS recursion for f n (θ) = 1 2 ( yn Φ(x n ),θ ) 2 θ n = θ n 1 γ ( Φ(x n ),θ n 1 y n ) Φ(xn ) The sequence (θ n ) n is a homogeneous Markov chain convergence to a stationary distribution π γ with expectation θ def γ = θπ γ (dθ) For least-squares, θ γ = θ θ n θ θ 0
37 Markov chain interpretation of constant step sizes LMS recursion for f n (θ) = 1 2 ( yn Φ(x n ),θ ) 2 θ n = θ n 1 γ ( Φ(x n ),θ n 1 y n ) Φ(xn ) The sequence (θ n ) n is a homogeneous Markov chain convergence to a stationary distribution π γ with expectation θ def γ = θπ γ (dθ) For least-squares, θ γ = θ θ n θ n θ θ 0
38 Markov chain interpretation of constant step sizes LMS recursion for f n (θ) = 1 2 ( yn Φ(x n ),θ ) 2 θ n = θ n 1 γ ( Φ(x n ),θ n 1 y n ) Φ(xn ) The sequence (θ n ) n is a homogeneous Markov chain convergence to a stationary distribution π γ with expectation θ def γ = θπ γ (dθ) For least-squares, θ γ = θ θ n does not converge to θ but oscillates around it oscillations of order γ Ergodic theorem: Averaged iterates converge to θ γ = θ at rate O(1/n)
39 Simulations - synthetic examples Gaussian distributions - p = 20 0 synthetic square log 10 [f(θ) f(θ * )] /2R 2 1/8R 2 4 1/32R 2 1/2R 2 n 1/ log 10 (n)
40 Simulations - benchmarks alpha (p = 500, n = ), news (p = , n = ) log 10 [f(θ) f(θ * )] alpha square C=1 test 1/R 2 1/R 2 n 1/2 SAG log (n) alpha square C=opt test C/R 2 C/R 2 n 1/2 SAG log (n) news square C=1 test 0.2 news square C=opt test log 10 [f(θ) f(θ * )] /R 2 1/R 2 n 1/2 SAG log (n) C/R 2 C/R 2 n 1/2 SAG log 10 (n)
41 Beyond least-squares - Markov chain interpretation Recursion θ n = θ n 1 γf n(θ n 1 ) also defines a Markov chain Stationary distribution π γ such that f (θ)π γ (dθ) = 0 When f is not linear, f ( θπ γ (dθ)) f (θ)π γ (dθ) = 0
42 Beyond least-squares - Markov chain interpretation Recursion θ n = θ n 1 γf n(θ n 1 ) also defines a Markov chain Stationary distribution π γ such that f (θ)π γ (dθ) = 0 When f is not linear, f ( θπ γ (dθ)) f (θ)π γ (dθ) = 0 θ n oscillates around the wrong value θ γ θ θ n θ n θ γ θ 0 θ
43 Beyond least-squares - Markov chain interpretation Recursion θ n = θ n 1 γf n(θ n 1 ) also defines a Markov chain Stationary distribution π γ such that f (θ)π γ (dθ) = 0 When f is not linear, f ( θπ γ (dθ)) f (θ)π γ (dθ) = 0 θ n oscillates around the wrong value θ γ θ moreover, θ θ n = O p ( γ) Ergodic theorem averaged iterates converge to θ γ θ at rate O(1/n) moreover, θ θ γ = O(γ) (Bach, 2013)
44 Simulations - synthetic examples Gaussian distributions - p = 20 0 synthetic logistic 1 log 10 [f(θ) f(θ * )] 1 2 1/2R 2 3 1/8R 2 4 1/32R 2 5 1/2R 2 n 1/ log 10 (n)
45 Restoring convergence through online Newton steps Known facts 1. Averaged SGD with γ n n 1/2 leads to robust rate O(n 1/2 ) for all convex functions 2. Averaged SGD with γ n constant leads to robust rate O(n 1 ) for all convex quadratic functions 3. Newton s method squares the error at each iteration for smooth functions 4. A single step of Newton s method is equivalent to minimizing the quadratic Taylor expansion Online Newton step Rate: O((n 1/2 ) 2 +n 1 ) = O(n 1 ) Complexity: O(p) per iteration
46 Restoring convergence through online Newton steps Known facts 1. Averaged SGD with γ n n 1/2 leads to robust rate O(n 1/2 ) for all convex functions 2. Averaged SGD with γ n constant leads to robust rate O(n 1 ) for all convex quadratic functions O(n 1 ) 3. Newton s method squares the error at each iteration for smooth functions O((n 1/2 ) 2 ) 4. A single step of Newton s method is equivalent to minimizing the quadratic Taylor expansion Online Newton step Rate: O((n 1/2 ) 2 +n 1 ) = O(n 1 ) Complexity: O(p) per iteration
47 Restoring convergence through online Newton steps The Newton step for f = Ef n (θ) def = E [ l(y n, θ,φ(x n ) ) ] at θ is equivalent to minimizing the quadratic approximation g(θ) = f( θ)+ f ( θ),θ θ θ θ,f ( θ)(θ θ) = f( θ)+ Ef n( θ),θ θ θ θ,ef n( θ)(θ θ) [ = E f( θ)+ f n( θ),θ θ θ θ,f n( θ)(θ θ) ]
48 Restoring convergence through online Newton steps The Newton step for f = Ef n (θ) def = E [ l(y n, θ,φ(x n ) ) ] at θ is equivalent to minimizing the quadratic approximation g(θ) = f( θ)+ f ( θ),θ θ θ θ,f ( θ)(θ θ) = f( θ)+ Ef n( θ),θ θ θ θ,ef n( θ)(θ θ) [ = E f( θ)+ f n( θ),θ θ θ θ,f n( θ)(θ θ) ] Complexity of least-mean-square recursion for g is O(p) θ n = θ n 1 γ [ f n( θ)+f n( θ)(θ n 1 θ) ] f n( θ) = l (y n, θ,φ(x n ) )Φ(x n ) Φ(x n ) has rank one New online Newton step without computing/inverting Hessians
49 Choice of support point for online Newton step Two-stage procedure (1) Run n/2 iterations of averaged SGD to obtain θ (2) Run n/2 iterations of averaged constant step-size LMS Reminiscent of one-step estimators (see, e.g., Van der Vaart, 2000) Provable convergence rate of O(p/n) for logistic regression Additional assumptions but no strong convexity
50 Choice of support point for online Newton step Two-stage procedure (1) Run n/2 iterations of averaged SGD to obtain θ (2) Run n/2 iterations of averaged constant step-size LMS Reminiscent of one-step estimators (see, e.g., Van der Vaart, 2000) Provable convergence rate of O(p/n) for logistic regression Additional assumptions but no strong convexity Update at each iteration using the current averaged iterate Recursion: θ n = θ n 1 γ [ f n( θ n 1 )+f n( θ n 1 )(θ n 1 θ n 1 ) ] No provable convergence rate (yet) but best practical behavior Note (dis)similarity with regular SGD: θ n = θ n 1 γf n(θ n 1 )
51 Simulations - synthetic examples Gaussian distributions - p = 20 0 synthetic logistic 1 0 synthetic logistic 2 log 10 [f(θ) f(θ * )] /2R 2 3 1/8R 2 3 every 2 p 4 1/32R 2 every iter. 4 2 step 5 1/2R 2 n 1/2 5 2 step dbl log 10 (n) log 10 (n) log 10 [f(θ) f(θ * )] 1
52 Simulations - benchmarks alpha (p = 500, n = ), news (p = , n = ) alpha logistic C=1 test alpha logistic C=opt test log 10 [f(θ) f(θ * )] /R /R 2 n 1/2 SAG 2 Adagrad Newton log (n) C/R C/R 2 n 1/2 SAG 2 Adagrad Newton log (n) news logistic C=1 test 0.2 news logistic C=opt test log 10 [f(θ) f(θ * )] /R /R 2 n 1/2 SAG 0.8 Adagrad 1 Newton log (n) C/R C/R 2 n 1/2 SAG 0.8 Adagrad 1 Newton log 10 (n)
53 Going beyond a single pass over the data Stochastic approximation Assumes infinite data stream Observations are used only once Directly minimizes testing cost E (x,y) l(y, θ,φ(x) )
54 Going beyond a single pass over the data Stochastic approximation Assumes infinite data stream Observations are used only once Directly minimizes testing cost E (x,y) l(y, θ,φ(x) ) Machine learning practice Finite data set (x 1,y 1,...,x n,y n ) Multiple passes Minimizes training cost 1 n n i=1 l(y i, θ,φ(x i ) ) Need to regularize (e.g., by the l 2 -norm) to avoid overfitting Goal: minimize g(θ) = 1 n n f i (θ) i=1
55 Stochastic vs. deterministic methods Minimizing g(θ) = 1 n n f i (θ) with f i (θ) = l ( y i, θ,φ(x i ) ) +µω(θ) i=1 Batchgradientdescent:θ t = θ t 1 γ t g (θ t 1 ) = θ t 1 γ t n Linear (e.g., exponential) convergence rate in O(e αt ) Iteration complexity is linear in n (with line search) n f i(θ t 1 ) i=1
56 Stochastic vs. deterministic methods Minimizing g(θ) = 1 n n f i (θ) with f i (θ) = l ( y i, θ,φ(x i ) ) +µω(θ) i=1 Batchgradientdescent:θ t = θ t 1 γ t g (θ t 1 ) = θ t 1 γ t n Linear (e.g., exponential) convergence rate in O(e αt ) Iteration complexity is linear in n (with line search) n f i(θ t 1 ) i=1 Stochastic gradient descent: θ t = θ t 1 γ t f i(t) (θ t 1) Sampling with replacement: i(t) random element of {1,...,n} Convergence rate in O(1/t) Iteration complexity is independent of n (step size selection?)
57 Stochastic average gradient (Le Roux, Schmidt, and Bach, 2012) Stochastic average gradient (SAG) iteration Keep in memory the gradients of all functions f i, i = 1,...,n Random selection i(t) {1,...,n} with replacement { Iteration: θ t = θ t 1 γ n t f yi t with yi t n = i (θ t 1 ) if i = i(t) otherwise i=1 y t 1 i
58 Stochastic average gradient (Le Roux, Schmidt, and Bach, 2012) Stochastic average gradient (SAG) iteration Keep in memory the gradients of all functions f i, i = 1,...,n Random selection i(t) {1,...,n} with replacement { Iteration: θ t = θ t 1 γ n t f yi t n with yt i = i (θ t 1 ) if i = i(t) otherwise i=1 y t 1 i Stochastic version of incremental average gradient(blatt et al., 2008) Extra memory requirement Supervised machine learning If f i (θ) = l i (y i, Φ(x i ),θ ), then f i (θ) = l i (y i, Φ(x i ),θ )Φ(x i ) Only need to store n real numbers
59 Stochastic average gradient - Convergence analysis Assumptions Each f i is L-smooth, i = 1,...,n g= 1 n n i=1 f i is µ-strongly convex (with potentially µ = 0) constant step size γ t = 1/(16L) initialization with one pass of averaged SGD
60 Stochastic average gradient - Convergence analysis Assumptions Each f i is L-smooth, i = 1,...,n g= 1 n n i=1 f i is µ-strongly convex (with potentially µ = 0) constant step size γ t = 1/(16L) initialization with one pass of averaged SGD Strongly convex case (Le Roux et al., 2012, 2013) E [ g(θ t ) g(θ ) ] ( 8σ 2 nµ + 4L θ 0 θ 2 n ) exp ( { 1 tmin 8n, µ }) 16L Linear (exponential) convergence rate with ( O(1) iteration { cost 1 After one pass, reduction of cost by exp min 8, nµ }) 16L
61 Stochastic average gradient - Convergence analysis Assumptions Each f i is L-smooth, i = 1,...,n g= 1 n n i=1 f i is µ-strongly convex (with potentially µ = 0) constant step size γ t = 1/(16L) initialization with one pass of averaged SGD Non-strongly convex case (Le Roux et al., 2013) E [ g(θ t ) g(θ ) ] 48 σ2 +L θ 0 θ 2 n n t Improvement over regular batch and stochastic gradient Adaptivity to potentially hidden strong convexity
62 spam dataset (n = , p = )
63 Conclusions Constant-step-size averaged stochastic gradient descent Reaches convergence rate O(1/n) in all regimes Improves on the O(1/ n) lower-bound of non-smooth problems Efficient online Newton step for non-quadratic problems Going beyond a single pass through the data Keep memory of all gradients for finite training sets Randomization leads to easier analysis and faster rates Relationship with Shalev-Shwartz and Zhang (2012); Mairal (2013) Extensions Non-differentiable terms, kernels, line-search, parallelization, etc. Beyond supervised learning, beyond convex problems
64 References A. Agarwal, P. L. Bartlett, P. Ravikumar, and M. J. Wainwright. Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization. Information Theory, IEEE Transactions on, 58(5): , F. Bach. Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression. Technical Report , HAL, F. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence rate o(1/n). Technical Report , HAL, Albert Benveniste, Michel Métivier, and Pierre Priouret. Adaptive algorithms and stochastic approximations. Springer Publishing Company, Incorporated, D. Blatt, A. O. Hero, and H. Gauchman. A convergent incremental gradient method with a constant step size. SIAM Journal on Optimization, 18(1):29 51, L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In Adv. NIPS, L. Györfi and H. Walk. On the averaged stochastic approximation for linear regression. SIAM Journal on Control and Optimization, 34(1):31 61, N. Le Roux, M. Schmidt, and F. Bach. A stochastic gradient method with an exponential convergence rate for strongly-convex optimization with finite training sets. In Adv. NIPS, N. Le Roux, M. Schmidt, and F. Bach. A stochastic gradient method with an exponential convergence rate for strongly-convex optimization with finite training sets. Technical Report , HAL, 2013.
65 O. Macchi. Adaptive processing: The least mean squares approach with applications in transmission. Wiley West Sussex, Julien Mairal. Optimization with first-order surrogate functions. arxiv preprint arxiv: , A. S. Nemirovsky and D. B. Yudin. Problem complexity and method efficiency in optimization. Wiley & Sons, B. T. Polyak and A. B. Juditsky. Acceleration of stochastic approximation by averaging. SIAM Journal on Control and Optimization, 30(4): , H. Robbins and S. Monro. A stochastic approximation method. Ann. Math. Statistics, 22: , ISSN D. Ruppert. Efficient estimations from a slowly convergent Robbins-Monro process. Technical Report 781, Cornell University Operations Research and Industrial Engineering, S. Shalev-Shwartz and T. Zhang. Stochastic dual coordinate ascent methods for regularized loss minimization. Technical Report , Arxiv, A. B. Tsybakov. Optimal rates of aggregation A. W. Van der Vaart. Asymptotic statistics, volume 3. Cambridge Univ. press, 2000.
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