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 ÉCOLE NORMALE SUPÉRIEURE Joint work with Aymeric Dieuleveut, Nicolas Flammarion, Eric Moulines - ERNSI Workshop, 2017
2 Context and motivations Supervised machine learning Goal: estimating a function f : X Y From random observations (x i,y i ) X Y, i = 1,...,n
3 Context and motivations Supervised machine learning Goal: estimating a function f : X Y From random observations (x i,y i ) X Y, i = 1,...,n Specificities Data may come from anywhere From strong to weak prior knowledge Computational constraints Between theory, algorithms and applications
4 Context and motivations 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
5 Search engines - Advertising
6 Visual object recognition
7 Bioinformatics Protein: Crucial elements of cell life Massive data: 2 millions for humans Complex data
8 Context and motivations 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 and motivations 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 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 Explicit features adapted to inputs (can be learned as well) Using Hilbert spaces for non-linear / non-parametric estimation
11 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
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 yi θ,φ(x i ) ) 2 + µω(θ) θ R p 2n i=1 (least-squares regression)
13 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 log ( 1+exp( y i θ,φ(x i ) ) ) + µω(θ) θ R p n i=1 (logistic regression)
14 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
15 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
16 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: Φ(x) R L = O(R 2 )
17 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
18 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 (large µ/l) (small µ/l)
19 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)
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 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
21 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 (µ/l)t ) convergence rate for strongly convex functions
22 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 (µ/l)t ) 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
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 (µ/l)t ) 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
24 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
25 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)
26 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
27 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
28 Convex stochastic approximation Existing analysis Known global minimax rates of convergence for non-smooth problems (Nemirovski 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
29 Convex stochastic approximation Existing analysis Known global minimax rates of convergence for non-smooth problems (Nemirovski 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?
30 Convex stochastic approximation Existing analysis Known global minimax rates of convergence for non-smooth problems (Nemirovski 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 global convergence rate O(1/n) in all situations?
31 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
32 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 ) f(θ ) 4σ2 p n + 4R2 θ 0 θ 2 n Matches statistical lower bound (Tsybakov, 2003) Fewer assumptions than existing bounds for empirical risk min. (
33 Least-squares - Proof technique LMS recursion: θ n θ = [ I γφ(x n ) Φ(x n ) ] (θ n 1 θ )+γε n Φ(x n ) Simplified LMS recursion: with H = E [ Φ(x n ) Φ(x n ) ] θ n θ = [ I γh ] (θ n 1 θ )+γε n Φ(x n ) Direct proof technique of Polyak and Juditsky (1992), e.g., θ n θ = [ I γh ] n (θ0 θ )+γ n k=1 [ I γh ] n kεk Φ(x k ) Infinite expansion of Aguech, Moulines, and Priouret(2000) in powers of γ
34 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
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 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 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 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)
38 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)
39 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)
40 Isn t least-squares regression a regression?
41 Isn t least-squares regression a regression? Least-squares regression Simpler to analyze and understand Explicit relationship to bias/variance trade-offs Many important loss functions are not quadratic Beyond least-squares with online Newton steps Complexity of O(p) per iteration with rate O(p/n) See Bach and Moulines (2013) for details
42 Optimal bounds for least-squares? Least-squares: cannot beat σ 2 p/n (Tsybakov, 2003). Really? Adaptivity to simpler problems
43 Finer assumptions (Dieuleveut and Bach, 2016) Covariance eigenvalues Pessimistic assumption: all eigenvalues λ m less than a constant Actual decay as λ m = o(m α ) with trh 1/α = λ 1/α m small m New result: replace σ2 p n by σ2 (γn) 1/α trh 1/α n 8 6 log( λ m ) log(m)
44 Finer assumptions (Dieuleveut and Bach, 2016) Covariance eigenvalues Pessimistic assumption: all eigenvalues λ m less than a constant Actual decay as λ m = o(m α ) with trh 1/α = λ 1/α m small m New result: replace σ2 p n by σ2 (γn) 1/α trh 1/α n 8 40 log( λ m ) (γ n) 1/α Tr H 1/α / n log(m) alpha
45 Finer assumptions (Dieuleveut and Bach, 2016) Covariance eigenvalues Pessimistic assumption: all eigenvalues λ m less than a constant Actual decay as λ m = o(m α ) with trh 1/α = λ 1/α m small m New result: replace σ2 p n by σ2 (γn) 1/α trh 1/α n Optimal predictor Pessimistic assumption: θ 0 θ 2 finite Finer assumption: H 1/2 r (θ 0 θ ) 2 small Replace θ 0 θ 2 by 4 H1/2 r (θ 0 θ ) 2 γn γ 2r n 2min{r,1} Leads to optimal rates for non-parametric regression
46 Finer assumptions (Dieuleveut and Bach, 2016) Covariance eigenvalues Pessimistic assumption: all eigenvalues λ m less than a constant Actual decay as λ m = o(m α ) with trh 1/α = λ 1/α m small m New result: replace σ2 p n by σ2 (γn) 1/α trh 1/α n Optimal predictor Pessimistic assumption: θ 0 θ 2 finite Finer assumption: H 1/2 r (θ 0 θ ) 2 small Replace θ 0 θ 2 by 4 H1/2 r (θ 0 θ ) 2 γn γ 2r n 2min{r,1} Leads to optimal rates for non-parametric regression
47 Achieving optimal bias and variance terms Averaged gradient descent (Bach and Moulines, 2013) Bias R 2 θ 0 θ 2 n Variance σ 2 p n
48 Achieving optimal bias and variance terms Averaged gradient descent (Bach and Moulines, 2013) Accelerated gradient descent (Nesterov, 1983) Bias R 2 θ 0 θ 2 n R 2 θ 0 θ 2 n 2 Variance σ 2 p n σ 2 p Acceleration is notoriously non-robust to noise (d Aspremont, 2008; Schmidt et al., 2011) For non-structured noise, see Lan (2012)
49 Achieving optimal bias and variance terms Averaged gradient descent (Bach and Moulines, 2013) Accelerated gradient descent (Nesterov, 1983) Between averaging and acceleration (Flammarion and Bach, 2015) Bias R 2 θ 0 θ 2 n R 2 θ 0 θ 2 n 2 R 2 θ 0 θ 2 n 1+α Variance σ 2 p n σ 2 p σ 2 p n 1 α
50 Achieving optimal bias and variance terms Averaged gradient descent (Bach and Moulines, 2013) Accelerated gradient descent (Nesterov, 1983) Between averaging and acceleration (Flammarion and Bach, 2015) Averaging and acceleration (Dieuleveut, Flammarion, and Bach, 2016) Bias R 2 θ 0 θ 2 n R 2 θ 0 θ 2 n 2 R 2 θ 0 θ 2 n 1+α R 2 θ 0 θ 2 n 2 Variance σ 2 p n σ 2 p σ 2 p n 1 α σ 2 p n
51 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
52 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
53 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, 2014) See precise analysis by Dieuleveut, Durmus, and Bach (2017)
54 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)
55 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
56 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
57 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) ]
58 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
59 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
60 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 )
61 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
62 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)
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 Robustness to step-size selection and adaptivity
64 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 Robustness to step-size selection and adaptivity Extensions and future work Going beyond a single pass (Le Roux, Schmidt, and Bach, 2012; Defazio, Bach, and Lacoste-Julien, 2014) Non-differentiable regularization (Flammarion and Bach, 2017) Kernels and nonparametric estimation (Dieuleveut and Bach, 2016) Parallelization Non-convex problems
65 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): , R. Aguech, E. Moulines, and P. Priouret. On a perturbation approach for the analysis of stochastic tracking algorithms. SIAM J. Control and Optimization, 39(3): , F. Bach. Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression. Journal of Machine Learning Research, 15(1): , F. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n). In Adv. NIPS, Albert Benveniste, Michel Métivier, and Pierre Priouret. Adaptive algorithms and stochastic approximations. Springer Publishing Company, Incorporated, L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In Adv. NIPS, A. d Aspremont. Smooth optimization with approximate gradient. SIAM J. Optim., 19(3): , A. Defazio, F. Bach, and S. Lacoste-Julien. Saga: A fast incremental gradient method with support for non-strongly convex composite objectives. In Advances in Neural Information Processing Systems (NIPS), A. Dieuleveut and F. Bach. Non-parametric Stochastic Approximation with Large Step sizes. Annals of Statistics, 44(4): , 2016.
66 A. Dieuleveut, N. Flammarion, and F. Bach. Harder, better, faster, stronger convergence rates for least-squares regression. Technical Report , arxiv, Aymeric Dieuleveut, Alain Durmus, and Francis Bach. Bridging the gap between constant step size stochastic gradient descent and markov chains. Technical report, HAL - to appear, N. Flammarion and F. Bach. From averaging to acceleration, there is only a step-size. arxiv preprint arxiv: , N. Flammarion and F. Bach. Stochastic composite least-squares regression with convergence rate O(1/n). In Proc. COLT, G. Lan. An optimal method for stochastic composite optimization. Math. Program., 133(1-2, Ser. A): , 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 Advances in Neural Information Processing Systems (NIPS), O. Macchi. Adaptive processing: The least mean squares approach with applications in transmission. Wiley West Sussex, A. S. Nemirovski and D. B. Yudin. Problem complexity and method efficiency in optimization. Wiley & Sons, Y. Nesterov. A method for solving a convex programming problem with rate of convergence O(1/k 2 ). Soviet Math. Doklady, 269(3): , B. T. Polyak and A. B. Juditsky. Acceleration of stochastic approximation by averaging. SIAM Journal on Control and Optimization, 30(4): , 1992.
67 H. Robbins and S. Monro. A stochastic approximation method. Ann. Math. Statistics, 22: , D. Ruppert. Efficient estimations from a slowly convergent Robbins-Monro process. Technical Report 781, Cornell University Operations Research and Industrial Engineering, M. Schmidt, N. Le Roux, and F. Bach. Convergence rates for inexact proximal-gradient method. In Adv. NIPS, A. B. Tsybakov. Optimal rates of aggregation. In Proc. COLT, A. W. Van der Vaart. Asymptotic statistics, volume 3. Cambridge Univ. press, 2000.
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