Beyond stochastic gradient descent for large-scale machine learning

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Beyond stochastic gradient descent for large-scale machine learning

Francis Bach

INRIA - Ecole Normale Sup´erieure, Paris, France

Joint work with Eric Moulines, Nicolas Le Roux

and Mark Schmidt - ECML-PKDD, September 2014

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“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

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Search engines - Advertising

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Marketing - Personalized recommendation

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Visual object recognition

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Personal photos

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Bioinformatics

• Protein: Crucial elements of cell life

• Massive data: 2 millions for humans

• Complex data

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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

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Context

Machine learning for “big data”

• Examples: computer vision, bioinformatics, advertising

• Ideal running-time complexity: O(pn) – Going back to simple methods

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Context

Machine learning for “big data”

• Examples: computer vision, bioinformatics, advertising

• Ideal running-time complexity: O(pn)

• Going back to simple methods

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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

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Supervised machine learning

• Data: n observations (xⁱ, y_i) ∈ X × Y, i = 1, . . . , n, i.i.d.

• Prediction as a linear function hθ, Φ(x)i of features Φ(x) ∈ R^p

• (regularized) empirical risk minimization: find ˆθ solution of

θ∈Rmin^p

1 n

n

X

i=1

ℓ y_i, hθ, Φ(xⁱ)i

+ µΩ(θ) convex data fitting term + regularizer

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Usual losses

• Regression: y ∈ R, prediction ˆy = hθ, Φ(x)i – quadratic loss ¹₂(y − ˆy)² = ¹₂(y − hθ, Φ(x)i)²

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Usual losses

• Regression: y ∈ R, prediction ˆy = hθ, Φ(x)i – quadratic loss ¹₂(y − ˆy)² = ¹₂(y − hθ, Φ(x)i)²

• Classification : y ∈ {−1, 1}, prediction ˆy = sign(hθ, Φ(x)i) – loss of the form ℓ(y hθ, Φ(x)i)

– “True” 0-1 loss: ℓ(y hθ, Φ(x)i) = 1y hθ,Φ(x)i<0

– Usual convex losses:

−30 −2 −1 0 1 2 3 4

1 2 3 4 5

0−1 hinge square logistic

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Supervised machine learning

θ∈Rmin^p

1 n

n

X

i=1

+ µΩ(θ) convex data fitting term + regularizer

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Supervised machine learning

θ∈Rmin^p

1 n

n

X

i=1

+ µΩ(θ)

convex data fitting term + regularizer

• Empirical risk: ˆf (θ) = _n¹ Pn

i=1 ℓ(y_i, hθ, Φ(xⁱ)i) training cost

• Expected risk: f(θ) = E^(x,y)ℓ(y, hθ, Φ(x)i) testing cost

• Two fundamental questions: (1) computing ˆθ and (2) analyzing ˆθ – May be tackled simultaneously

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Supervised machine learning

θ∈Rmin^p

1 n

n

X

i=1

+ µΩ(θ)

convex data fitting term + regularizer

• Empirical risk: ˆf (θ) = _n¹ Pn

i=1 ℓ(y_i, hθ, Φ(xⁱ)i) training cost

• Expected risk: f(θ) = E^(x,y)ℓ(y, hθ, Φ(x)i) testing cost

• Two fundamental questions: (1) computing ˆθ and (2) analyzing ˆθ – May be tackled simultaneously

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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, eigenvaluesg^′′(θ) 6 L

smooth non−smooth

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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, eigenvaluesg^′′(θ) 6 L

• Machine learning – with g(θ) = _n¹ Pn

i=1 ℓ(y_i, hθ, Φ(xⁱ)i) – Hessian ≈ covariance matrix _n¹ Pn

i=1 Φ(x_i) ⊗ Φ(xⁱ) – Bounded data

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Smoothness and strong convexity

• A twice differentiable function g : R^p → R is µ-strongly convex if and only if

∀θ ∈ R^p, eigenvaluesg^′′(θ) > µ

convex strongly

convex

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Smoothness and strong convexity

i=1 Φ(x_i) ⊗ Φ(xⁱ)

– Data with invertible covariance matrix (low correlation/dimension)

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Smoothness and strong convexity

i=1 Φ(x_i) ⊗ Φ(xⁱ)

– Data with invertible covariance matrix (low correlation/dimension)

• Adding regularization by ^µ₂kθk²

– creates additional bias unless µ is small

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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

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Iterative methods for minimizing smooth functions

• Newton method: θ^t = θ_t−1 − g^′′(θ_t−1)⁻¹g^′(θ_t−1) – O e^−ρ2^t convergence rate

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Iterative methods for minimizing smooth functions

• Newton method: θ^t = θ_t−1 − g^′′(θ_t−1)⁻¹g^′(θ_t−1) – O e^−ρ2^t 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

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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

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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) = Efn(θ) = E ℓ(yn, hθ, Φ(xⁿ)i) = generalization error

– Loss for a single pair of observations: f_n(θ) = ℓ(y_n, hθ, Φ(xⁿ)i) – Expected gradient:

f^′(θ) = Ef_n^′ (θ) = E ℓ^′(y_n, hθ, Φ(xⁿ)i) Φ(xⁿ)

• Beyond convex optimization: see, e.g., Benveniste et al. (2012)

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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 − γⁿ f_n^′ (θ_n−1)

– Polyak-Ruppert averaging: ¯θ_n = _n+1¹ P_n

k=0 θ_k

– Which learning rate sequence γ_n? Classical setting: γ_n = Cn^−α

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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 − γⁿ f_n^′ (θ_n−1)

– Polyak-Ruppert averaging: ¯θ_n = _n+1¹ P_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

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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)

Attained by averaged stochastic gradient descent with γ_n ∝ (µn)⁻¹ – Non-strongly convex: O(n^−1/2)

Attained by averaged stochastic gradient descent with γ_n ∝ n^−1/2

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Convex stochastic approximation Existing work

• 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?

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Convex stochastic approximation Existing work

• 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?

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Least-mean-square algorithm

• Least-squares: f(θ) = ¹₂E(y_n − hΦ(xⁿ), θi)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 EΦ(x_n) ⊗ Φ(xⁿ) = H < µ · Id

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Least-mean-square algorithm

• Least-squares: f(θ) = ¹₂E(y_n − hΦ(xⁿ), θi)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 EΦ(x_n) ⊗ Φ(xⁿ) = H < µ · Id

• New analysis for averaging and constant step-size γ = 1/(4R²) – Assume kΦ(xⁿ)k 6 R and |yⁿ − hΦ(xⁿ), θ_∗i| 6 σ almost surely – No assumption regarding lowest eigenvalues of H

– Main result: Ef(¯θ_n−1) − f(θ^∗) 6 4σ²p

n + 4R²kθ⁰ − θ^∗k² n

• Matches statistical lower bound (Tsybakov, 2003)

– Non-asymptotic robust version of Gy¨orfi and Walk (1996)

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Markov chain interpretation of constant step sizes

• LMS recursion for fⁿ(θ) = ¹₂ y_n − hΦ(xⁿ), θi2

θ_n = θ_n−1 − γ hΦ(xⁿ), θ_n−1i − yⁿΦ(x_n)

• The sequence (θⁿ)_n is a homogeneous Markov chain – convergence to a stationary distribution π_γ

– with expectation ¯θ_γ ^def= R θπ_γ(dθ) - For least-squares, ¯θ_γ = θ_∗

¯ θγ

θ₀

θn

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Markov chain interpretation of constant step sizes

– with expectation ¯θ_γ ^def= R θπ_γ(dθ)

• For least-squares, ¯θ_γ = θ_∗

θ∗

θ₀

θ_n

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Markov chain interpretation of constant step sizes

θ∗

θ₀

θ_n

¯ θ_n

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Markov chain interpretation of constant step sizes

– 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)

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Simulations - synthetic examples

• Gaussian distributions - p = 20

0 2 4 6

−5

−4

−3

−2

−1 0

log10(n) log 10[f(θ)−f(θ *)]

synthetic square

1/2R² 1/8R² 1/32R² 1/2R²n^1/2

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Simulations - benchmarks

• alpha (p = 500, n = 500 000), news (p = 1 300 000, n = 20 000)

0 2 4 6

−2

−1.5

−1

−0.5 0 0.5 1

log10(n) log 10[f(θ)−f(θ *)]

alpha square C=1 test

1/R² 1/R²n^1/2 SAG

0 2 4 6

−2

−1.5

−1

−0.5 0 0.5 1

log10(n)

alpha square C=opt test

C/R² C/R²n^1/2 SAG

0 2 4

−0.8

−0.6

−0.4

−0.2 0 0.2

log10(n) log 10[f(θ)−f(θ *)]

news square C=1 test

1/R² 1/R²n^1/2 SAG

0 2 4

−0.8

−0.6

−0.4

−0.2 0 0.2

log10(n)

news square C=opt test

C/R² C/R²n^1/2 SAG

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Beyond least-squares - Markov chain interpretation

• Recursion θⁿ = θ_n−1 − γfn^′ (θ_n−1) also defines a Markov chain – Stationary distribution π_γ such that R f^′(θ)π_γ(dθ) = 0

– When f^′ is not linear, f^′(R θπ_γ(dθ)) 6= R f^′(θ)π_γ(dθ) = 0

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Beyond least-squares - Markov chain interpretation

• θ_n oscillates around the wrong value ¯θ_γ 6= θ^∗

¯ θγ

θ₀

θn

¯ θn

θ∗

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Beyond least-squares - Markov chain interpretation

• θ_n oscillates around the wrong value ¯θ_γ 6= θ^∗ – moreover, kθ^∗ − θⁿk = O^p(√γ)

• Ergodic theorem

– averaged iterates converge to ¯θ_γ 6= θ^∗ at rate O(1/n) – moreover, kθ^∗ − ¯θ_γk = O(γ) (Bach, 2013)

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Simulations - synthetic examples

0 2 4 6

−5

−4

−3

−2

−1 0

log10(n) log 10[f(θ)−f(θ *)]

synthetic logistic − 1

1/2R² 1/8R² 1/32R² 1/2R²n^1/2

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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

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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

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Restoring convergence through online Newton steps

• The Newton step for f = Efⁿ(θ) ^def= Eℓ(y_n, hθ, Φ(xⁿ)i)

at ˜θ is equivalent to minimizing the quadratic approximation

g(θ) = f (˜θ) + hf^′(˜θ), θ − ˜θi + ¹₂hθ − ˜θ, f^′′(˜θ)(θ − ˜θ)i

= f (˜θ) + hEf_n^′ (˜θ), θ − ˜θi + ¹₂hθ − ˜θ,Ef_n^′′(˜θ)(θ − ˜θ)i

= Eh

f (˜θ) + hf_n^′ (˜θ), θ − ˜θi + ¹₂hθ − ˜θ, f_n^′′(˜θ)(θ − ˜θ)ii

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Restoring convergence through online Newton steps

• The Newton step for f = Efⁿ(θ) ^def= Eℓ(y_n, hθ, Φ(xⁿ)i)

at ˜θ is equivalent to minimizing the quadratic approximation

g(θ) = f (˜θ) + hf^′(˜θ), θ − ˜θi + ¹₂hθ − ˜θ, f^′′(˜θ)(θ − ˜θ)i

= f (˜θ) + hEf_n^′ (˜θ), θ − ˜θi + ¹₂hθ − ˜θ,Ef_n^′′(˜θ)(θ − ˜θ)i

= Eh

f (˜θ) + hf_n^′ (˜θ), θ − ˜θi + ¹₂hθ − ˜θ, f_n^′′(˜θ)(θ − ˜θ)ii

• Complexity of least-mean-square recursion for g is O(p) θ_n = θ_n−1 − γf_n^′ (˜θ) + f_n^′′(˜θ)(θ_n−1 − ˜θ)

– f_n^′′(˜θ) = ℓ^′′(y_n, h˜θ, Φ(x_n)i)Φ(xⁿ) ⊗ Φ(xⁿ) has rank one

– New online Newton step without computing/inverting Hessians

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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

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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 − γf_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 − γfn^′ (θ_n−1)

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Simulations - synthetic examples

0 2 4 6

−5

−4

−3

−2

−1 0

log10(n) log 10[f(θ)−f(θ *)]

1/2R² 1/8R² 1/32R² 1/2R²n^1/2

0 2 4 6

−5

−4

−3

−2

−1 0

log10(n) log 10[f(θ)−f(θ *)]

every 2^p every iter.

2−step

2−step−dbl.

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Simulations - benchmarks

• alpha (p = 500, n = 500 000), news (p = 1 300 000, n = 20 000)

0 2 4 6

−2.5

−2

−1.5

−1

−0.5 0 0.5

log10(n) log 10[f(θ)−f(θ *)]

alpha logistic C=1 test

1/R² 1/R²n^1/2 SAG Adagrad Newton

0 2 4 6

−2.5

−2

−1.5

−1

−0.5 0 0.5

log10(n)

alpha logistic C=opt test

C/R² C/R²n^1/2 SAG Adagrad Newton

0 2 4

−1

−0.8

−0.6

−0.4

−0.2 0 0.2

log10(n) log 10[f(θ)−f(θ *)]

news logistic C=1 test

1/R² 1/R²n^1/2 SAG Adagrad Newton

0 2 4

−1

−0.8

−0.6

−0.4

−0.2 0 0.2

log10(n)

news logistic C=opt test

C/R² C/R²n^1/2 SAG Adagrad Newton

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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, hθ, Φ(x)i)

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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, hθ, Φ(x)i)

• Machine learning practice

– Finite data set (x₁, y₁, . . . , x_n, y_n) – Multiple passes

– Minimizes training cost _n¹ Pn

i=1 ℓ(y_i, hθ, Φ(xⁱ)i)

– Need to regularize (e.g., by the ℓ₂-norm) to avoid overfitting

• Goal: minimize g(θ) = 1 n

n

X

i=1

f_i(θ)

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Stochastic vs. deterministic methods

• Minimizing g(θ) = 1 n

n

X

i=1

f_i(θ) with f_i(θ) = ℓ y_i, hθ, Φ(xⁱ)i + µΩ(θ)

• Batch gradient descent: θ_t = θ_t−1−γ^tg^′(θ_t−1) = θ_t−1−γ_t n

n

X

i=1

f_i^′(θ_t−1) – Linear (e.g., exponential) convergence rate in O(e^−αt)

– Iteration complexity is linear in n (with line search)

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Stochastic vs. deterministic methods

• Minimizing g(θ) = 1 n

n

X

i=1

f_i(θ) with f_i(θ) = ℓ y_i, hθ, Φ(xⁱ)i + µΩ(θ)

• Batch gradient descent: θ_t = θ_t−1−γ^tg^′(θ_t−1) = θ_t−1−γ_t n

n

X

i=1

f_i^′(θ_t−1) – Linear (e.g., exponential) convergence rate in O(e^−αt)

– Iteration complexity is linear in n (with line search)

• Stochastic gradient descent: θ_t = θ_t−1 − γ^tf_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?)

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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_t n

n

X

i=1

y_i^t with y_i^t =

(f_i^′(θ_t−1) if i = i(t) y_i^t−1 otherwise

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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_t n

n

X

i=1

y_i^t with y_i^t =

(f_i^′(θ_t−1) if i = i(t) y_i^t−1 otherwise

• Stochastic version of incremental average gradient (Blatt et al., 2008)

• Extra memory requirement

– Supervised machine learning

– If f_i(θ) = ℓ_i(y_i, hΦ(xⁱ), θi), then fi^′(θ) = ℓ^′_i(y_i, hΦ(xⁱ), θi) Φ(xⁱ) – Only need to store n real numbers

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Stochastic average gradient - Convergence analysis

• Assumptions

– Each f_i is L-smooth, i = 1, . . . , n – g =_n¹ Pn

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

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Stochastic average gradient - Convergence analysis

• Assumptions

• Strongly convex case (Le Roux et al., 2012, 2013) Eg(θ_t) − g(θ^∗) 6 8σ²

nµ + 4Lkθ⁰−θ^∗k² n

exp

− t min n 1

8n, µ 16L

o

– Linear (exponential) convergence rate with O(1) iteration cost – After one pass, reduction of cost by exp

− min n1

8, nµ 16L

o

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Stochastic average gradient - Convergence analysis

• Assumptions

• Non-strongly convex case (Le Roux et al., 2013) Eg(θ_t) − g(θ^∗) 6 48σ² + Lkθ⁰−θ^∗k²

√n

n t

– Improvement over regular batch and stochastic gradient – Adaptivity to potentially hidden strong convexity

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spam dataset (n = 92 189, p = 823 470)

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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

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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):3235–3249, 2012.

F. Bach. Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression. Technical Report 00804431, HAL, 2013.

F. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence rate o(1/n). Technical Report 00831977, HAL, 2013.

Albert Benveniste, Michel M´etivier, and Pierre Priouret. Adaptive algorithms and stochastic approximations. Springer Publishing Company, Incorporated, 2012.

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, 2008.

L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In Adv. NIPS, 2008.

L. Gy¨orfi and H. Walk. On the averaged stochastic approximation for linear regression. SIAM Journal on Control and Optimization, 34(1):31–61, 1996.

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, 2012.

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 00674995, HAL, 2013.

(65)

O. Macchi. Adaptive processing: The least mean squares approach with applications in transmission.

Wiley West Sussex, 1995.

Julien Mairal. Optimization with first-order surrogate functions. arXiv preprint arXiv:1305.3120, 2013.

A. S. Nemirovsky and D. B. Yudin. Problem complexity and method efficiency in optimization. Wiley

& Sons, 1983.

B. T. Polyak and A. B. Juditsky. Acceleration of stochastic approximation by averaging. SIAM Journal on Control and Optimization, 30(4):838–855, 1992.

H. Robbins and S. Monro. A stochastic approximation method. Ann. Math. Statistics, 22:400–407, 1951. ISSN 0003-4851.

D. Ruppert. Efficient estimations from a slowly convergent Robbins-Monro process. Technical Report 781, Cornell University Operations Research and Industrial Engineering, 1988.

S. Shalev-Shwartz and T. Zhang. Stochastic dual coordinate ascent methods for regularized loss minimization. Technical Report 1209.1873, Arxiv, 2012.

A. B. Tsybakov. Optimal rates of aggregation. 2003.

A. W. Van der Vaart. Asymptotic statistics, volume 3. Cambridge Univ. press, 2000.