Large-scale machine learning and convex …fbach/gradsto_allerton.pdfLarge-scale machine learning and convex optimization Francis Bach INRIA - Ecole Normale Sup´erieure, Paris, France

Large-scale machine learningand convex optimization

Francis Bach

INRIA - Ecole Normale Superieure, Paris, France

Allerton Conference - September 2015

Slides available at www.di.ens.fr/~fbach/gradsto_allerton.pdf

Context

Machine learning for “big data”

• Large-scale machine learning: large d, large n, large k

– d : dimension of each observation (input)

– n : number of observations

– k : number of tasks (dimension of outputs)

• Examples: computer vision, bioinformatics, text processing

– Ideal running-time complexity: O(dn+ kn)

– Going back to simple methods

– Stochastic gradient methods (Robbins and Monro, 1951)

– Mixing statistics and optimization

– Using smoothness to go beyond stochastic gradient descent

Visual object recognition

Personal photos

Learning for bioinformatics - Proteins

• Crucial components of cell life

• Predicting multiple functions and

interactions

• Massive data: up to 1 millions for

humans!

• Complex data

– Amino-acid sequence

– Link with DNA

– Tri-dimensional molecule

Search engines - advertising

Context







• Ideal running-time complexity: O(dn+ kn)

– Going back to simple methods




Context







• Ideal running-time complexity: O(dn+ kn)

• Going back to simple methods




Outline

1. Large-scale machine learning and optimization

• Traditional statistical analysis

• Classical methods for convex optimization

2. Non-smooth stochastic approximation

• Stochastic (sub)gradient and averaging

• Non-asymptotic results and lower bounds

• Strongly convex vs. non-strongly convex

3. Smooth stochastic approximation algorithms

• Asymptotic and non-asymptotic results

• Beyond decaying step-sizes

4. Finite data sets

Supervised machine learning

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

• Prediction as a linear function θ⊤Φ(x) of features Φ(x) ∈ Rd

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

minθ∈Rd

1

n

n∑

i=1

ℓ(

yi, θ⊤Φ(xi)

)

+ µΩ(θ)

convex data fitting term + regularizer

Usual losses

• Regression: y ∈ R, prediction y = θ⊤Φ(x)

– quadratic loss 12(y − y)2 = 1

2(y − θ⊤Φ(x))2

Usual losses

• Regression: y ∈ R, prediction y = θ⊤Φ(x)

– quadratic loss 12(y − y)2 = 1

2(y − θ⊤Φ(x))2

• Classification : y ∈ −1, 1, prediction y = sign(θ⊤Φ(x))

– loss of the form ℓ(y θ⊤Φ(x))– “True” 0-1 loss: ℓ(y θ⊤Φ(x)) = 1y θ⊤Φ(x)<0

– Usual convex losses:

−3 −2 −1 0 1 2 3 40

1

2

3

4

5

0−1hingesquarelogistic

Main motivating examples

• Support vector machine (hinge loss)

ℓ(Y, θ⊤Φ(X)) = max1− Y θ⊤Φ(X), 0

• Logistic regression

ℓ(Y, θ⊤Φ(X)) = log(1 + exp(−Y θ⊤Φ(X)))

• Least-squares regression

ℓ(Y, θ⊤Φ(X)) =1

2(Y − θ⊤Φ(X))2

Usual regularizers

• Main goal: avoid overfitting

• (squared) Euclidean norm: ‖θ‖22 =∑d

j=1 |θj|2

– Numerically well-behaved

– Representer theorem and kernel methods : θ =∑n

i=1αiΦ(xi)

– See, e.g., Scholkopf and Smola (2001); Shawe-Taylor and

Cristianini (2004)

• Sparsity-inducing norms

– Main example: ℓ1-norm ‖θ‖1 =∑d

j=1 |θj|– Perform model selection as well as regularization

– Non-smooth optimization and structured sparsity

– See, e.g., Bach, Jenatton, Mairal, and Obozinski (2012b,a)





minθ∈Rd

1

n

n∑

i=1

ℓ(

yi, θ⊤Φ(xi)

)

+ µΩ(θ)






minθ∈Rd

1

n

n∑

i=1

ℓ(

yi, θ⊤Φ(xi)

)

+ µΩ(θ)


• Empirical risk: f(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi)) training cost

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

• Two fundamental questions: (1) computing θ and (2) analyzing θ

– May be tackled simultaneously





minθ∈Rd

1

n

n∑

i=1

ℓ(

yi, θ⊤Φ(xi)

)

+ µΩ(θ)



∑ni=1 ℓ(yi, θ









minθ∈Rd

1

n

n∑

i=1

ℓ(

yi, θ⊤Φ(xi)

)

such that Ω(θ) 6 D

convex data fitting term + constraint


∑ni=1 ℓ(yi, θ





General assumptions


• Bounded features Φ(x) ∈ Rd: ‖Φ(x)‖2 6 R


∑ni=1 ℓ(yi, θ



• Loss for a single observation: fi(θ) = ℓ(yi, θ⊤Φ(xi))

⇒ ∀i, f(θ) = Efi(θ)

• Properties of fi, f, f

– Convex on Rd

– Additional regularity assumptions: Lipschitz-continuity,

smoothness and strong convexity

Lipschitz continuity

• Bounded gradients of f (Lipschitz-continuity): the function f if

convex, differentiable and has (sub)gradients uniformly bounded by

B on the ball of center 0 and radius D:

∀θ ∈ Rd, ‖θ‖2 6 D ⇒ ‖f ′(θ)‖2 6 B

• Machine learning

– with f(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))

– G-Lipschitz loss and R-bounded data: B = GR

Smoothness and strong convexity

• A function f : Rd → R is L-smooth if and only if it is differentiable

and its gradient is L-Lipschitz-continuous

∀θ1, θ2 ∈ Rd, ‖f ′(θ1)− f ′(θ2)‖2 6 L‖θ1 − θ2‖2

• If f is twice differentiable: ∀θ ∈ Rd, f ′′(θ) 4 L · Id

smooth non−smooth


• A function f : Rd → R is L-smooth if and only if it is differentiable

and its gradient is L-Lipschitz-continuous

∀θ1, θ2 ∈ Rd, ‖f ′(θ1)− f ′(θ2)‖2 6 L‖θ1 − θ2‖2

• If f is twice differentiable: ∀θ ∈ Rd, f ′′(θ) 4 L · Id


– with f(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))

– Hessian ≈ covariance matrix 1n

∑ni=1Φ(xi)Φ(xi)

⊤

– ℓ-smooth loss and R-bounded data: L = ℓR2


• A function f : Rd → R is µ-strongly convex if and only if

∀θ1, θ2 ∈ Rd, f(θ1) > f(θ2) + f ′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22

• If f is twice differentiable: ∀θ ∈ Rd, f ′′(θ) < µ · Id

convexstronglyconvex



∀θ1, θ2 ∈ Rd, f(θ1) > f(θ2) + f ′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22



– with f(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))


∑ni=1Φ(xi)Φ(xi)

⊤

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



∀θ1, θ2 ∈ Rd, f(θ1) > f(θ2) + f ′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖23



– with f(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))


∑ni=1Φ(xi)Φ(xi)

⊤

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

• Adding regularization by µ2‖θ‖2

– creates additional bias unless µ is small

Summary of smoothness/convexity assumptions




∀θ ∈ Rd, ‖θ‖2 6 D ⇒ ‖f ′(θ)‖2 6 B

• Smoothness of f : the function f is convex, differentiable with

L-Lipschitz-continuous gradient f ′:

∀θ1, θ2 ∈ Rd, ‖f ′(θ1)− f ′(θ2)‖2 6 L‖θ1 − θ2‖2

• Strong convexity of f : The function f is strongly convex with

respect to the norm ‖ · ‖, with convexity constant µ > 0:

∀θ1, θ2 ∈ Rd, f(θ1) > f(θ2) + f ′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22

Analysis of empirical risk minimization

• Approximation and estimation errors: Θ = θ ∈ Rd,Ω(θ) 6 D

f(θ)− minθ∈Rd

f(θ) =

[

f(θ)−minθ∈Θ

f(θ)

]

+

[

minθ∈Θ

f(θ)− minθ∈Rd

f(θ)

]

Estimation error Approximation error

– NB: may replace minθ∈Rd

f(θ) by best (non-linear) predictions



f(θ)− minθ∈Rd

f(θ) =

[

f(θ)−minθ∈Θ

f(θ)

]

+

[

minθ∈Θ

f(θ)− minθ∈Rd

f(θ)

]


1. Uniform deviation bounds, with θ ∈ argminθ∈Θ

f(θ)

f(θ)−minθ∈Θ

f(θ) =[

f(θ)− f(θ)]

+[

f(θ)− f(θ∗Θ)]

+[

f(θ∗Θ)− f(θ∗Θ)]

6 supθ∈Θ

|f(θ)− f(θ)|+ 0 + supθ∈Θ

|f(θ)− f(θ)|



f(θ)− minθ∈Rd

f(θ) =

[

f(θ)−minθ∈Θ

f(θ)

]

+

[

minθ∈Θ

f(θ)− minθ∈Rd

f(θ)

]


1. Uniform deviation bounds, with θ ∈ argminθ∈Θ

f(θ)

f(θ)−minθ∈Θ

f(θ) 6 2 · supθ∈Θ

|f(θ)− f(θ)|

– Typically slow rate O(

1/√n)

2. More refined concentration results with faster rates O(1/n)

Motivation from least-squares

• For least-squares, we have ℓ(y, θ⊤Φ(x)) = 12(y − θ⊤Φ(x))2, and

f(θ)− f(θ) =1

2θ⊤

(

1

n

n∑

i=1

Φ(xi)Φ(xi)⊤ − EΦ(X)Φ(X)⊤

)

θ

−θ⊤(

1

n

n∑

i=1

yiΦ(xi)− EY Φ(X)

)

+1

2

(

1

n

n∑

i=1

y2i − EY 2

)

,

sup‖θ‖26D

|f(θ)− f(θ)| 6D2

2

∥

∥

∥

∥

1

n

n∑

i=1

Φ(xi)Φ(xi)⊤ − EΦ(X)Φ(X)⊤

∥

∥

∥

∥

op

+D

∥

∥

∥

∥

1

n

n∑

i=1

yiΦ(xi)− EY Φ(X)

∥

∥

∥

∥

2

+1

2

∣

∣

∣

∣

1

n

n∑

i=1

y2i − EY 2

∣

∣

∣

∣

,

sup‖θ‖26D

|f(θ)− f(θ)| 6 O(1/√n) with high probability

Symmetrization with Rademacher variables

• Let D′ = x′1, y

′1, . . . , x

′n, y

′n an independent copy of the data

D = x1, y1, . . . , xn, yn, with corresponding loss functions f ′i(θ)

E[

supθ∈Θ

∣

∣f(θ)− f(θ)∣

∣

]

= E[

supθ∈Θ

(

f(θ)− 1

n

n∑

i=1

fi(θ)

)

]

= E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

E(

f ′i(θ)− fi(θ)|D

)

∣

∣

∣

∣

]

6 E

[

E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

(

f ′i(θ)− fi(θ)

∣

∣

∣

∣

∣

∣

∣

∣

D]]

= E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

(


)

∣

∣

∣

∣

]

= E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

εi(


)

∣

∣

∣

∣

]

with εi uniform in −1, 1

6 2E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

εifi(θ)

∣

∣

∣

∣

]

= Rademacher complexity

Rademacher complexity

• Define the Rademacher complexity of the class of functions (X,Y ) 7→ℓ(Y, θ⊤Φ(X)) as

Rn = E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

εifi(θ)

∣

∣

∣

∣

]

.

• Note two expectations, with respect to D and with respect to ε

• Main property:

E[

supθ∈Θ

∣

∣f(θ)− f(θ)∣

∣

]

6 2Rn

From Rademacher complexity

to bound in high probability

• Let Z = supθ∈Θ

∣

∣f(θ)− f(θ)∣

∣

• By changing the pair (xi, yi), Z may only change by

2

nsup |ℓ(Y, θ⊤Φ(X))| 6 2

n

(

sup |ℓ(Y, 0)|+GRD)

62

n

(

ℓ0+GRD)

= c

with sup |ℓ(Y, 0)| = ℓ0

• MacDiarmid inequality: with probability greater than 1− δ,

Z 6 EZ +

√

n

2c ·

√

log1

δ6 2Rn +

√2√n

(

ℓ0 +GRD)

√

log1

δ

Bounding the Rademacher average - I• We have, with ϕi(u) = ℓ(yi, u)−ℓ(yi, 0) is almost surelyB-Lipschitz:

Rn = E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

εifi(θ)

∣

∣

∣

∣

]

6 E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

εifi(0)

∣

∣

∣

∣

]

+ E

[

supθ∈Θ

∣

∣

∣

∣

1

n

n∑

i=1

εi[

fi(θ)− fi(0)]

∣

∣

∣

∣

]

6ℓ0√n+ E

[

supθ∈Θ

1

n

n∑

i=1

εi[

fi(θ)− fi(0)]

]

=ℓ0√n+ E

[

supθ∈Θ

1

n

n∑

i=1

εiϕi(θ⊤Φ(xi))

]

• Using Ledoux-Talagrand concentration results for Rademacheraverages (since ϕi is G-Lipschitz), we get:

Rn 6ℓ0√n+ 2G · E

[

sup‖θ‖26D

∣

∣

∣

∣

1

n

n∑

i=1

εiθ⊤Φ(xi)

]∣

∣

∣

∣

Bounding the Rademacher average - II• We have:

Rn 6ℓ0√n+ 2GE

[

sup‖θ‖26D

∣

∣

∣

∣

1

n

n∑

i=1

εiθ⊤Φ(xi)

]∣

∣

∣

∣

=ℓ0√n+ 2GE

∥

∥

∥

∥

D1

n

n∑

i=1

εiΦ(xi)

∥

∥

∥

∥

2

6ℓ0√n+ 2GD

√

√

√

√E

∥

∥

∥

∥

1

n

n∑

i=1

εiΦ(xi)

∥

∥

∥

∥

2

2

by Jensen’s inequality

62(ℓ0 +GRD)√

nby using‖Φ(x)‖2 6 R

• Overall, we get, with probability 1− δ:

supθ∈Θ

∣

∣f(θ)− f(θ)∣

∣ 61√n

(

ℓ0 +GRD)(4 +

√

2 log1

δ

)

Putting it all together

• We have, with probability 1− δ

– For exact minimizer θ ∈ argminθ∈Θ f(θ), we have

f(θ)−minθ∈Θ


|f(θ)− f(θ)|

62√n

(

ℓ0 +GRD)(4 +

√

2 log1

δ

)

– For all θ ∈ Θ

f(θ)−minθ∈Θ


|f(θ)− f(θ)|+[

f(θ)− f(θ)]

• Only need to optimize with precision 2√n(ℓ0 +GRD)

Slow rate for supervised learning (summary)

• Assumptions (f is the expected risk, f the empirical risk)

– Ω(θ) = ‖θ‖2 (Euclidean norm)

– “Linear” predictors: θ(x) = θ⊤Φ(x), with ‖Φ(x)‖2 6 R a.s.

– G-Lipschitz loss: f and f are GR-Lipschitz on Θ = ‖θ‖2 6 D– No assumptions regarding convexity

• With probability greater than 1− δ

supθ∈Θ

|f(θ)− f(θ)| 6 (ℓ0 +GRD)√n

[

2 +

√

2 log2

δ

]

• Expectated estimation error: E[

supθ∈Θ

|f(θ)− f(θ)|]

64(ℓ0 +GRD)√

n

• Using Rademacher averages (see, e.g., Boucheron et al., 2005)

• Lipschitz functions ⇒ slow rate

Motivation from mean estimation

• Estimator θ = 1n

∑ni=1 zi = argminθ∈R

12n

∑ni=1(θ − zi)

2 = f(θ)

• From before:

– f(θ) = 12E(θ − z)2 = 1

2(θ − Ez)2 + 12 var(z) = f(θ) +O(1/

√n)

– f(θ) = 12(θ − Ez)2 + 1

2 var(z) = f(Ez) +O(1/√n)

Motivation from mean estimation

• Estimator θ = 1n

∑ni=1 zi = argminθ∈R

12n

∑ni=1(θ − zi)

2 = f(θ)

• From before:

– f(θ) = 12E(θ − z)2 = 1

2(θ − Ez)2 + 12 var(z) = f(θ) +O(1/

√n)

– f(θ) = 12(θ − Ez)2 + 1

2 var(z) = f(Ez) +O(1/√n)

• More refined/direct bound:

f(θ)− f(Ez) =1

2(θ − Ez)2

E[

f(θ)− f(Ez)]

=1

2E

(

1

n

n∑

i=1

zi − Ez

)2

=1

2nvar(z)

• Bound only at θ + strong convexity

Fast rate for supervised learning


– Same as before (bounded features, Lipschitz loss)

– Regularized risks: fµ(θ) = f(θ)+µ2‖θ‖22 and fµ(θ) = f(θ)+µ

2‖θ‖22– Convexity

• For any a > 0, with probability greater than 1− δ, for all θ ∈ Rd,

fµ(θ)−minη∈Rd

fµ(η) 6 (1+a)(fµ(θ)−minη∈Rd

fµ(η))+8(1 + 1

a)G2R2(32 + log 1

δ)

µn

• Results from Sridharan, Srebro, and Shalev-Shwartz (2008)

– see also Boucheron and Massart (2011) and references therein

• Strongly convex functions ⇒ fast rate

– Warning: µ should decrease with n to reduce approximation error

Complexity results in convex optimization for ML

• Assumption: f convex on Rd

• Classical generic algorithms

– (sub)gradient method/descent

– Accelerated gradient descent

– Newton method

• Key additional properties of f

– Lipschitz continuity, smoothness or strong convexity

• Key insight from Bottou and Bousquet (2008)

– In machine learning, no need to optimize below estimation error

• Key reference: Nesterov (2004)

Summary of smoothness/convexity assumptions




∀θ ∈ Rd, ‖θ‖2 6 D ⇒ ‖f ′(θ)‖2 6 B

• Smoothness of f : the function f is convex, differentiable with

L-Lipschitz-continuous gradient f ′:

∀θ1, θ2 ∈ Rd, ‖f ′(θ1)− f ′(θ2)‖2 6 L‖θ1 − θ2‖2



∀θ1, θ2 ∈ Rd, f(θ1) > f(θ2) + f ′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22

Subgradient method/descent (Shor et al., 1985)

• Assumptions

– f convex and B-Lipschitz-continuous on ‖θ‖2 6 D

• Algorithm: θt = ΠD

(

θt−1 −2D

B√tf ′(θt−1)

)

– ΠD : orthogonal projection onto ‖θ‖2 6 D

• Bound:

f

(

1

t

t−1∑

k=0

θk

)

− f(θ∗) 62DB√

t

• Three-line proof (for “constant” step-size)

• Best possible convergence rate after O(d) iterations

Subgradient method/descent - proof - I

• Iteration: θt = ΠD(θt−1 − γtf′(θt−1)) with γt =

2DB√t

• Assumption: ‖f ′(θ)‖2 6 B and ‖θ‖2 6 D

‖θt − θ∗‖22 6 ‖θt−1 − θ∗ − γtf′(θt−1)‖22 by contractivity of projections

6 ‖θt−1 − θ∗‖22 +B2γ2t − 2γt(θt−1 − θ∗)

⊤f ′(θt−1) because ‖f ′(θt−1)‖2 6 B

6 ‖θt−1 − θ∗‖22 +B2γ2t − 2γt

[

f(θt−1)− f(θ∗)]

(property of subgradients)

• leading to

f(θt−1)− f(θ∗) 6B2γt2

+1

2γt

[

‖θt−1 − θ∗‖22 − ‖θt − θ∗‖22]

Subgradient method/descent - proof - II

• Starting from f(θt−1)− f(θ∗) 6B2γt2

+1

2γt

[

‖θt−1 − θ∗‖22 − ‖θt − θ∗‖22]

t∑

u=1

[

f(θu−1)− f(θ∗)]

6

t∑

u=1

B2γu2

+t

∑

u=1

1

2γu

[

‖θu−1 − θ∗‖22 − ‖θu − θ∗‖22]

=t

∑

u=1

B2γu2

+t−1∑

u=1

‖θu − θ∗‖22( 1

2γu+1− 1

2γu

)

+‖θ0 − θ∗‖22

2γ1− ‖θt − θ∗‖22

2γt

6

t∑

u=1

B2γu2

+t−1∑

u=1

4D2( 1

2γu+1− 1

2γu

)

+4D2

2γ1

=t

∑

u=1

B2γu2

+4D2

2γt6 2DB

√t with γt =

2D

B√t

• Using convexity: f

(

1

t

t−1∑

k=0

θk

)

− f(θ∗) 62DB√

t

Subgradient descent for machine learning


– “Linear” predictors: θ(x) = θ⊤Φ(x), with ‖Φ(x)‖2 6 R a.s.

– f(θ) = 1n

∑ni=1 ℓ(yi,Φ(xi)

⊤θ)

– G-Lipschitz loss: f and f are GR-Lipschitz on Θ = ‖θ‖2 6 D

• Statistics: with probability greater than 1− δ

supθ∈Θ

|f(θ)− f(θ)| 6 GRD√n

[

2 +

√

2 log2

δ

]

• Optimization: after t iterations of subgradient method

f(θ)−minη∈Θ

f(η) 6GRD√

t

• t = n iterations, with total running-time complexity of O(n2d)

Subgradient descent - strong convexity

• Assumptions

– f convex and B-Lipschitz-continuous on ‖θ‖2 6 D– f µ-strongly convex

• Algorithm: θt = ΠD

(

θt−1 −2

µ(t+ 1)f ′(θt−1)

)

• Bound:

f

(

2

t(t+ 1)

t∑

k=1

kθk−1

)

− f(θ∗) 62B2

µ(t+ 1)

• Three-line proof

• Best possible convergence rate after O(d) iterations

Subgradient method - strong convexity - proof - I

• Iteration: θt = ΠD(θt−1 − γtf′(θt−1)) with γt =

2µ(t+1)

• Assumption: ‖f ′(θ)‖2 6 B and ‖θ‖2 6 D and µ-strong convexity of f

‖θt − θ∗‖22 6 ‖θt−1 − θ∗ − γtf′(θt−1)‖22 by contractivity of projections

6 ‖θt−1 − θ∗‖22 +B2γ2t − 2γt(θt−1 − θ∗)

⊤f ′(θt−1) because ‖f ′(θt−1)‖2 6 B

6 ‖θt−1 − θ∗‖22 +B2γ2t − 2γt

[

f(θt−1)− f(θ∗) +µ

2‖θt−1 − θ∗‖22

]

(property of subgradients and strong convexity)

• leading to

f(θt−1)− f(θ∗) 6B2γt2

+1

2

[ 1

γt− µ

]

‖θt−1 − θ∗‖22 −1

2γt‖θt − θ∗‖22

6B2

µ(t+ 1)+

µ

2

[t− 1

2

]

‖θt−1 − θ∗‖22 −µ(t+ 1)

4‖θt − θ∗‖22

Subgradient method - strong convexity - proof - II

• From f(θt−1)− f(θ∗) 6B2

µ(t+ 1)+

µ

2

[t− 1

2

]

‖θt−1 − θ∗‖22 −µ(t+ 1)

4‖θt − θ∗‖22

t∑

u=1

u[

f(θu−1)− f(θ∗)]

6

u∑

t=1

B2u

µ(u+ 1)+

1

4

t∑

u=1

[

u(u− 1)‖θu−1 − θ∗‖22 − u(u+ 1)‖θu − θ∗‖22]

6B2t

µ+

1

4

[

0− t(t+ 1)‖θt − θ∗‖22]

6B2t

µ

• Using convexity: f

(

2

t(t+ 1)

t∑

u=1

uθu−1

)

− f(θ∗) 62B2

µ(t+ 1)

• NB: with step-size γn = 1/(nµ), extra logarithmic factor

(smooth) gradient descent

• Assumptions

– f convex with L-Lipschitz-continuous gradient

– Minimum attained at θ∗

• Algorithm:

θt = θt−1 −1

Lf ′(θt−1)

• Bound:

f(θt)− f(θ∗) 62L‖θ0 − θ∗‖2

t+ 4• Three-line proof

• Not best possible convergence rate after O(d) iterations

(smooth) gradient descent - strong convexity

• Assumptions

– f convex with L-Lipschitz-continuous gradient

– f µ-strongly convex

• Algorithm:

θt = θt−1 −1

Lf ′(θt−1)

• Bound:

f(θt)− f(θ∗) 6 (1− µ/L)t[

f(θ0)− f(θ∗)]

• Three-line proof

• Adaptivity of gradient descent to problem difficulty

• Line search

Accelerated gradient methods (Nesterov, 1983)

• Assumptions

– f convex with L-Lipschitz-cont. gradient , min. attained at θ∗

• Algorithm:θt = ηt−1 −

1

Lf ′(ηt−1)

ηt = θt +t− 1

t+ 2(θt − θt−1)

• Bound:f(θt)− f(θ∗) 6

2L‖θ0 − θ∗‖2(t+ 1)2

• Ten-line proof

• Not improvable

• Extension to strongly convex functions

Optimization for sparsity-inducing norms

(see Bach, Jenatton, Mairal, and Obozinski, 2012b)

• Gradient descent as a proximal method (differentiable functions)

– θt+1 = arg minθ∈Rd

f(θt) + (θ − θt)⊤∇f(θt)+

L

2‖θ − θt‖22

– θt+1 = θt − 1L∇f(θt)

Optimization for sparsity-inducing norms

(see Bach, Jenatton, Mairal, and Obozinski, 2012b)

• Gradient descent as a proximal method (differentiable functions)


f(θt) + (θ − θt)⊤∇f(θt)+

L

2‖θ − θt‖22

– θt+1 = θt − 1L∇f(θt)

• Problems of the form: minθ∈Rd

f(θ) + µΩ(θ)


f(θt) + (θ − θt)⊤∇f(θt)+µΩ(θ)+

L

2‖θ − θt‖22

– Ω(θ) = ‖θ‖1 ⇒ Thresholded gradient descent

• Similar convergence rates than smooth optimization

– Acceleration methods (Nesterov, 2007; Beck and Teboulle, 2009)

Summary: minimizing convex functions

• Assumption: f convex

• Gradient descent: θt = θt−1 − γt f′(θt−1)

– O(1/√t) convergence rate for non-smooth convex functions

– O(1/t) convergence rate for smooth convex functions

– O(e−ρt) convergence rate for strongly smooth convex functions

• Newton method: θt = θt−1 − f ′′(θt−1)−1f ′(θt−1)

– O(

e−ρ2t)

convergence rate

Summary: minimizing convex functions

• Assumption: f convex

• Gradient descent: θt = θt−1 − γt f′(θt−1)

– O(1/√t) convergence rate for non-smooth convex functions

– O(1/t) convergence rate for smooth convex functions

– O(e−ρt) convergence rate for strongly smooth convex functions

• Newton method: θt = θt−1 − f ′′(θt−1)−1f ′(θt−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

Outline











4. Finite data sets

Stochastic approximation

• Goal: Minimizing a function f defined on Rd

– given only unbiased estimates f ′n(θn) of its gradients f ′(θn) at

certain points θn ∈ Rd

• Stochastic approximation

– (much) broader applicability beyond convex optimization

θn = θn−1 − γnhn(θn−1) with E[

hn(θn−1)|θn−1

]

= h(θn−1)

– Beyond convex problems, i.i.d assumption, finite dimension, etc.

– Typically asymptotic results

– See, e.g., Kushner and Yin (2003); Benveniste et al. (2012)

Stochastic approximation

• Goal: Minimizing a function f defined on Rd

– given only unbiased estimates f ′n(θn) of its gradients f ′(θn) at

certain points θn ∈ Rd

• Machine learning - statistics

– loss for a single pair of observations: fn(θ) = ℓ(yn, θ⊤Φ(xn))

– f(θ) = Efn(θ) = E ℓ(yn, θ⊤Φ(xn)) = generalization error

– Expected gradient: f ′(θ) = Ef ′n(θ) = E

ℓ′(yn, θ⊤Φ(xn))Φ(xn)

– Non-asymptotic results

• Number of iterations = number of observations

– “Single” line of code!

Relationship to online learning


– Minimize f(θ) = Ezℓ(θ, z) = generalization error of θ

– Using the gradients of single i.i.d. observations





• Batch learning

– Finite set of observations: z1, . . . , zn– Empirical risk: f(θ) = 1

n

∑nk=1 ℓ(θ, zi)

– Estimator θ = Minimizer of f(θ) over a certain class Θ

– Generalization bound using uniform concentration results





• Batch learning

– Finite set of observations: z1, . . . , zn– Empirical risk: f(θ) = 1

n

∑nk=1 ℓ(θ, zi)

– Estimator θ = Minimizer of f(θ) over a certain class Θ

– Generalization bound using uniform concentration results

• Online learning

– Update θn after each new (potentially adversarial) observation zn– Cumulative loss: 1

n

∑nk=1 ℓ(θk−1, zk)

– Online to batch through averaging (Cesa-Bianchi et al., 2004)

Convex stochastic approximation

• Key properties of f and/or fn

– Smoothness: f B-Lipschitz continuous, f ′ L-Lipschitz continuous

– Strong convexity: f µ-strongly convex





• Key algorithm: Stochastic gradient descent (a.k.a. Robbins-Monro)

θn = θn−1 − γnf′n(θn−1)

– Polyak-Ruppert averaging: θn = 1n

∑n−1k=0 θk

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





• Key algorithm: Stochastic gradient descent (a.k.a. Robbins-Monro)



∑n−1k=0 θk

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

• Desirable practical behavior

– Applicable (at least) to classical supervised learning problems

– Robustness to (potentially unknown) constants (L,B,µ)

– Adaptivity to difficulty of the problem (e.g., strong convexity)

Stochastic subgradient descent/method

• Assumptions

– fn convex and B-Lipschitz-continuous on ‖θ‖2 6 D– (fn) i.i.d. functions such that Efn = f

– θ∗ global optimum of f on ‖θ‖2 6 D

• Algorithm: θn = ΠD

(

θn−1 −2D

B√nf ′n(θn−1)

)

• Bound:

Ef

(

1

n

n−1∑

k=0

θk

)

− f(θ∗) 62DB√

n

• “Same” three-line proof as in the deterministic case

• Minimax rate (Nemirovsky and Yudin, 1983; Agarwal et al., 2012)

• Running-time complexity: O(dn) after n iterations

Stochastic subgradient method - proof - I

• Iteration: θn = ΠD(θn−1 − γnf′n(θn−1)) with γn = 2D

B√n

• Fn : information up to time n

• ‖f ′n(θ)‖2 6 B and ‖θ‖2 6 D, unbiased gradients/functions E(fn|Fn−1) = f

‖θn − θ∗‖22 6 ‖θn−1 − θ∗ − γnf′n(θn−1)‖22 by contractivity of projections

6 ‖θn−1 − θ∗‖22 +B2γ2n − 2γn(θn−1 − θ∗)

⊤f ′n(θn−1) because ‖f ′

n(θn−1)‖2 6 B

E[

‖θn − θ∗‖22|Fn−1

]

6 ‖θn−1 − θ∗‖22 +B2γ2n − 2γn(θn−1 − θ∗)

⊤f ′(θn−1)

6 ‖θn−1 − θ∗‖22 +B2γ2n − 2γn

[

f(θn−1)− f(θ∗)]

(subgradient property)

E‖θn − θ∗‖22 6 E‖θn−1 − θ∗‖22 + B2γ2n − 2γn

[

Ef(θn−1)− f(θ∗)]

• leading to Ef(θn−1)− f(θ∗) 6B2γn2

+1

2γn

[

E‖θn−1 − θ∗‖22 − E‖θn − θ∗‖22]

Stochastic subgradient method - proof - II

• Starting from Ef(θn−1)− f(θ∗) 6B2γn2

+1

2γn

[

E‖θn−1 − θ∗‖22 − E‖θn − θ∗‖22]

n∑

u=1

[

Ef(θu−1)− f(θ∗)]

6

n∑

u=1

B2γu2

+n∑

u=1

1

2γu

[

E‖θu−1 − θ∗‖22 − E‖θu − θ∗‖22]

6

n∑

u=1

B2γu2

+4D2

2γn6

2DB√n

with γn =2D

B√n

• Using convexity: Ef

(

1

n

n−1∑

k=0

θk

)

− f(θ∗) 62DB√

n

Stochastic subgradient descent - strong convexity - I

• Assumptions

– fn convex and B-Lipschitz-continuous

– (fn) i.i.d. functions such that Efn = f

– f µ-strongly convex on ‖θ‖2 6 D– θ∗ global optimum of f over ‖θ‖2 6 D

• Algorithm: θn = ΠD

(

θn−1 −2

µ(n+ 1)f ′n(θn−1)

)

• Bound:

Ef

(

2

n(n+ 1)

n∑

k=1

kθk−1

)

− f(θ∗) 62B2

µ(n+ 1)

• “Same” proof than deterministic case (Lacoste-Julien et al., 2012)

• Minimax rate (Nemirovsky and Yudin, 1983; Agarwal et al., 2012)

Stochastic subgradient descent - strong convexity - II

• Assumptions

– fn convex and B-Lipschitz-continuous

– (fn) i.i.d. functions such that Efn = f

– θ∗ global optimum of g = f + µ2‖ · ‖22

– No compactness assumption - no projections

• Algorithm:

θn = θn−1−2

µ(n+ 1)g′n(θn−1) = θn−1−

2

µ(n+ 1)

[

f ′n(θn−1)+µθn−1

]

• Bound: Eg

(

2

n(n+ 1)

n∑

k=1

kθk−1

)

− g(θ∗) 62B2

µ(n+ 1)

Outline











4. Finite data sets


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 γn ∝ (µn)−1

– Non-strongly convex: O(n−1/2)

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

– Bottou and Le Cun (2005); Bottou and Bousquet (2008); Hazan

et al. (2007); Shalev-Shwartz and Srebro (2008); Shalev-Shwartz

et al. (2007, 2009); Xiao (2010); Duchi and Singer (2009); Nesterov

and Vial (2008); Nemirovski et al. (2009)


Existing work







• Many contributions in optimization and online learning: Bottou

and Le Cun (2005); Bottou and Bousquet (2008); Hazan et al.

(2007); Shalev-Shwartz and Srebro (2008); Shalev-Shwartz et al.

(2007, 2009); Xiao (2010); Duchi and Singer (2009); Nesterov and

Vial (2008); Nemirovski et al. (2009)


Existing work







• Asymptotic analysis of averaging (Polyak and Juditsky, 1992;

Ruppert, 1988)

– All step sizes γn = 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?


Existing work








Ruppert, 1988)



• Non-asymptotic analysis for smooth problems? problems with

convergence rate O(min1/µn, 1/√n)

Smoothness/convexity assumptions

• Iteration: θn = θn−1 − γnf′n(θn−1)


∑n−1k=0 θk

• Smoothness of fn: For each n > 1, the function fn is a.s. convex,

differentiable with L-Lipschitz-continuous gradient f ′n:

– Smooth loss and bounded data



– Invertible population covariance matrix

– or regularization by µ2‖θ‖2

Summary of new results (Bach and Moulines, 2011)

• Stochastic gradient descent with learning rate γn = Cn−α

• Strongly convex smooth objective functions

– Old: O(n−1) rate achieved without averaging for α = 1

– New: O(n−1) rate achieved with averaging for α ∈ [1/2, 1]

– Non-asymptotic analysis with explicit constants

– Forgetting of initial conditions

– Robustness to the choice of C







– Forgetting of initial conditions

– Robustness to the choice of C

• Convergence rates for E‖θn − θ∗‖2 and E‖θn − θ∗‖2

– no averaging: O(σ2γn

µ

)

+O(e−µnγn)‖θ0 − θ∗‖2

– averaging:trH(θ∗)−1

n+ µ−1O(n−2α+n−2+α) +O

(‖θ0 − θ∗‖2µ2n2

)

Classical proof sketch (no averaging)

‖θn − θ∗‖22 = ‖θn−1 − γnf′n(θn−1)− θ∗‖22

= ‖θn−1 − θ∗‖22 − 2γn(θn−1 − θ∗)⊤f ′

n(θn−1) + γ2n‖f ′

n(θn−1)‖226 ‖θn−1 − θ∗‖22 − 2γn(θn−1 − θ∗)

⊤f ′n(θn−1)

+2γ2n‖f ′

n(θ∗)‖22 + 2γ2n‖f ′

n(θn−1)− f ′n(θ∗)‖22

6 ‖θn−1 − θ∗‖22 − 2γn(θn−1 − θ∗)⊤f ′

n(θn−1)

+2γ2n‖f ′

n(θ∗)‖22+2γ2nL[f

′n(θn−1)− f ′

n(θ∗)]⊤(θn−1 − θ∗)

E[

‖θn − θ∗‖22|Fn−1

]

6 ‖θn−1 − θ∗‖22 − 2γn(θn−1 − θ∗)⊤f ′(θn−1)

+2γ2nE‖f ′

n(θ∗)‖22 + 2γ2nL[f

′(θn−1)− 0]⊤(θn−1 − θ∗)

6 ‖θn−1 − θ∗‖22 − 2γn(1− γnL)(θn−1 − θ∗)⊤f ′(θn−1) + 2γ2

nσ2

6 ‖θn−1 − θ∗‖22 − 2γn(1− γnL)1

2µ‖θn−1 − θ∗‖22 + 2γ2

nσ2

=[

1− µγn(1− γnL)]

‖θn−1 − θ∗‖22 + 2γ2nσ

2

E[

‖θn−1 − θ∗‖22]

6[

1− µγn(1− γnL)]

E[

‖θn−1 − θ∗‖22]

+ 2γ2nσ

2

Proof sketch (averaging)

• From Polyak and Juditsky (1992):


⇔ f ′n(θn−1) =

1

γn(θn−1 − θn)

⇔ f ′n(θ∗) + f ′′

n(θ∗)(θn−1 − θ∗) =1

γn(θn−1 − θn) +O(‖θn−1 − θ∗‖2)

⇔ f ′n(θ∗) + f ′′(θ∗)(θn−1 − θ∗) =

1

γn(θn−1 − θn) +O(‖θn−1 − θ∗‖2)

+O(‖θn−1 − θ∗‖)εn

⇔ θn−1 − θ∗ = −f ′′(θ∗)−1f ′

n(θ∗) +1

γnf ′′(θ∗)

−1(θn−1 − θn)

+O(‖θn−1 − θ∗‖2) +O(‖θn−1 − θ∗‖)εn

• Averaging to cancel the term 1γnf ′′(θ∗)−1(θn−1 − θn)

Robustness to wrong constants for γn = Cn−α

• f(θ) = 12|θ|2 with i.i.d. Gaussian noise (d = 1)

• Left: α = 1/2

• Right: α = 1

0 2 4−5

0

5

log(n)

log[

f(θ n)−

f∗ ]

α = 1/2

sgd − C=1/5ave − C=1/5sgd − C=1ave − C=1sgd − C=5ave − C=5

0 2 4−5

0

5

log(n)

log[

f(θ n)−

f∗ ]

α = 1

sgd − C=1/5ave − C=1/5sgd − C=1ave − C=1sgd − C=5ave − C=5

• See also http://leon.bottou.org/projects/sgd













• Non-strongly convex smooth objective functions

– Old: O(n−1/2) rate achieved with averaging for α = 1/2

– New: O(maxn1/2−3α/2, n−α/2, nα−1) rate achieved without

averaging for α ∈ [1/3, 1] (worse than with averaging)

• Take-home message

– Use α = 1/2 with averaging to be adaptive to strong convexity

Beyond stochastic gradient method

• Adding a proximal step

– Goal: minθ∈Rd

f(θ) + Ω(θ) = Efn(θ) + Ω(θ)

– Replace recursion θn = θn−1 − γnf′n(θn) by

θn = minθ∈Rd

∥

∥θ − θn−1 + γnf′n(θn)

∥

∥

2

2+ CΩ(θ)

– Xiao (2010); Hu et al. (2009)

– May be accelerated (Ghadimi and Lan, 2013)

• Related frameworks

– Regularized dual averaging (Nesterov, 2009; Xiao, 2010)

– Mirror descent (Nemirovski et al., 2009; Lan et al., 2012)

Outline











4. Finite data sets


Existing work








Ruppert, 1988)



• A single adaptive algorithm for smooth problems with

convergence rate O(min1/µn, 1/√n) in all situations?

Adaptive algorithm for logistic regression

• Logistic regression: (Φ(xn), yn) ∈ Rd × −1, 1

– Single data point: fn(θ) = log(1 + exp(−ynθ⊤Φ(xn)))

– Generalization error: f(θ) = Efn(θ)





• Cannot be strongly convex ⇒ local strong convexity

– unless restricted to |θ⊤Φ(xn)| 6 M (and with constants eM)

– µ = lowest eigenvalue of the Hessian at the optimum f ′′(θ∗)

logistic loss








• n steps of averaged SGD with constant step-size 1/(

2R2√n)

– with R = radius of data (Bach, 2013):

Ef(θn)− f(θ∗) 6 min

1√n,R2

nµ

(

15 + 5R‖θ0 − θ∗‖)4

– Proof based on self-concordance (Nesterov and Nemirovski, 1994)

Self-concordance

• Usual definition for convex ϕ : R → R: |ϕ′′′(t)| 6 2ϕ′′(t)3/2

– Affine invariant

– Extendable to all convex functions on Rd by looking at rays

– Used for the sharp proof of quadratic convergence of Newton

method (Nesterov and Nemirovski, 1994)

• Generalized notion: |ϕ′′′(t)| 6 ϕ′′(t)

– Applicable to logistic regression (with extensions)

Self-concordance

• Usual definition for convex ϕ : R → R: |ϕ′′′(t)| 6 2ϕ′′(t)3/2

– Affine invariant

– Extendable to all convex functions on Rd by looking at rays

– Used for the sharp proof of quadratic convergence of Newton

method (Nesterov and Nemirovski, 1994)

• Generalized notion: |ϕ′′′(t)| 6 ϕ′′(t)

– Applicable to logistic regression (with extensions)

• Important properties

– Allows global Taylor expansions

– Relates expansions of derivatives of different orders


Proof sketch

• Step 1: use existing result f(θn)−f(θ∗)+R2√n‖θ0−θ∗‖22 = O(1/

√n)

• Step 2: f ′n(θn−1) =

1γ(θn−1−θn) ⇒ 1

n

∑nk=1 f

′k(θk−1) =

1nγ(θ0−θn)

• Step 3:∥

∥

∥f ′(1

n

∑nk=1 θk−1

)

− 1n

∑nk=1 f

′(θk−1)∥

∥

∥

2

= O(

f(θn)− f(θ∗))

= O(1/√n) using self-concordance

• Step 4a: if f µ-strongly convex, f(θn)− f(θ∗) 612µ

∥

∥f ′(θn)∥

∥

2

2

• Step 4b: if f self-concordant, “locally true” with µ = λmin(f′′(θ∗))









2R2√n)



1√n,R2

nµ

(

15 + 5R‖θ0 − θ∗‖)4

– Proof based on self-concordance (Nesterov and Nemirovski, 1994)









2R2√n)



1√n,R2

nµ

(

15 + 5R‖θ0 − θ∗‖)4

– A single adaptive algorithm for smooth problems with

convergence rate O(1/n) in all situations?

Least-mean-square algorithm

• Least-squares: f(θ) = 12E

[

(yn − 〈Φ(xn), θ〉)2]

with θ ∈ Rd

– SGD = least-mean-square algorithm (see, e.g., Macchi, 1995)

– usually studied without averaging and decreasing step-sizes

– with strong convexity assumption E[

Φ(xn)⊗Φ(xn)]

= H < µ · Id

Least-mean-square algorithm

• Least-squares: f(θ) = 12E

[

(yn − 〈Φ(xn), θ〉)2]

with θ ∈ Rd

– SGD = least-mean-square algorithm (see, e.g., Macchi, 1995)

– usually studied without averaging and decreasing step-sizes

– with strong convexity assumption E[

Φ(xn)⊗Φ(xn)]

= H < µ · Id

• New analysis for averaging and constant step-size γ = 1/(4R2)

– Assume ‖Φ(xn)‖ 6 R and |yn − 〈Φ(xn), θ∗〉| 6 σ almost surely

– No assumption regarding lowest eigenvalues of H

– Main result: Ef(θn−1)− f(θ∗) 64σ2d

n+

4R2‖θ0 − θ∗‖2n

• Matches statistical lower bound (Tsybakov, 2003)

– Non-asymptotic robust version of Gyorfi and Walk (1996)

Least-squares - Proof technique

• LMS recursion:

θn − θ∗ =[

I − γΦ(xn)⊗ Φ(xn)]

(θn−1 − θ∗) + γ εnΦ(xn)

• Simplified LMS recursion: with H = E[

Φ(xn)⊗ Φ(xn)]

θn − θ∗ =[

I − γH]

(θn−1 − θ∗) + γ εnΦ(xn)

– Direct proof technique of Polyak and Juditsky (1992), e.g.,

θn − θ∗ =[

I − γH]n(θ0 − θ∗) + γ

n∑

k=1

[

I − γH]n−k

εkΦ(xk)

• Infinite expansion of Aguech, Moulines, and Priouret (2000) in powers

of γ

Markov chain interpretation of constant step sizes

• LMS recursion for fn(θ) =12

(

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)

• The sequence (θn)n is a homogeneous Markov chain

– convergence to a stationary distribution πγ

– with expectation θγdef=

∫

θπγ(dθ)

- For least-squares, θγ = θ∗

θγ

θ0

θn



(

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)




∫

θπγ(dθ)

• For least-squares, θγ = θ∗

θ∗

θ0

θn



(

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)




∫

θπγ(dθ)


θ∗

θ0

θn

θn



(

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)




∫

θπγ(dθ)


– θn does not converge to θ∗ but oscillates around it

– oscillations of order√γ

• Ergodic theorem:

– Averaged iterates converge to θγ = θ∗ at rate O(1/n)

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

1/2R2n1/2

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/R2n1/2

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/R2n1/2

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/R2n1/2

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/R2n1/2

SAG

Beyond least-squares - Markov chain interpretation

• Recursion θn = θn−1 − γf ′n(θn−1) also defines a Markov chain

– Stationary distribution πγ such that∫

f ′(θ)πγ(dθ) = 0

– When f ′ is not linear, f ′(∫

θπγ(dθ)) 6=∫

f ′(θ)πγ(dθ) = 0




f ′(θ)πγ(dθ) = 0


θπγ(dθ)) 6=∫

f ′(θ)πγ(dθ) = 0

• θn oscillates around the wrong value θγ 6= θ∗

θγ

θ0

θn

θn

θ∗




f ′(θ)πγ(dθ) = 0


θπγ(dθ)) 6=∫

f ′(θ)πγ(dθ) = 0

• θn oscillates around the wrong value θγ 6= θ∗

– moreover, ‖θ∗ − θn‖ = Op(√γ)

– Linear convergence up to the noise level for strongly-convex

problems (Nedic and Bertsekas, 2000)

• Ergodic theorem

– averaged iterates converge to θγ 6= θ∗ at rate O(1/n)

– moreover, ‖θ∗ − θγ‖ = O(γ) (Bach, 2013)



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

1/2R2n1/2

Restoring convergence through online Newton steps

• Known facts

1. Averaged SGD with γn ∝ n−1/2 leads to robust rate O(n−1/2)

for all convex functions

2. Averaged SGD with γn 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


• Known facts

1. Averaged SGD with γn ∝ n−1/2 leads to robust rate O(n−1/2)

for all convex functions

2. Averaged SGD with γn 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


• The Newton step for f = Efn(θ)def= E

[

ℓ(yn, 〈θ,Φ(xn)〉)]

at θ is

equivalent to minimizing the quadratic approximation

g(θ) = f(θ) + 〈f ′(θ), θ − θ〉+ 12〈θ − θ, f ′′(θ)(θ − θ)〉

= f(θ) + 〈Ef ′n(θ), θ − θ〉+ 1

2〈θ − θ,Ef ′′n(θ)(θ − θ)〉

= E

[

f(θ) + 〈f ′n(θ), θ − θ〉+ 1

2〈θ − θ, f ′′n(θ)(θ − θ)〉

]


• The Newton step for f = Efn(θ)def= E

[

ℓ(yn, 〈θ,Φ(xn)〉)]

at θ is

equivalent to minimizing the quadratic approximation

g(θ) = f(θ) + 〈f ′(θ), θ − θ〉+ 12〈θ − θ, f ′′(θ)(θ − θ)〉

= f(θ) + 〈Ef ′n(θ), θ − θ〉+ 1

2〈θ − θ,Ef ′′n(θ)(θ − θ)〉

= E

[

f(θ) + 〈f ′n(θ), θ − θ〉+ 1

2〈θ − θ, f ′′n(θ)(θ − θ)〉

]

• Complexity of least-mean-square recursion for g is O(p)

θn = θn−1 − γ[

f ′n(θ) + f ′′

n(θ)(θn−1 − θ)]

– f ′′n(θ) = ℓ′′(yn, 〈θ,Φ(xn)〉)Φ(xn)⊗ Φ(xn) has rank one

– New online Newton step without computing/inverting Hessians

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

Logistic regression - Proof technique

• Using generalized self-concordance of ϕ : u 7→ log(1 + e−u):

|ϕ′′′(u)| 6 ϕ′′(u)

– NB: difference with regular self-concordance: |ϕ′′′(u)| 6 2ϕ′′(u)3/2

• Using novel high-probability convergence results for regular averaged

stochastic gradient descent

• Requires assumption on the kurtosis in every direction, i.e.,

E〈Φ(xn), η〉4 6 κ[

E〈Φ(xn), η〉2]2

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: θn = θn−1 − γ[

f ′n(θn−1) + f ′′

n(θn−1)(θn−1 − θn−1)]

– No provable convergence rate (yet) but best practical behavior

– Note (dis)similarity with regular SGD: θn = θn−1 − γf ′n(θn−1)



0 2 4 6−5

−4

−3

−2

−1

0

log10

(n)

log 10

[f(θ)

−f(

θ *)]


1/2R2

1/8R2

1/32R2

1/2R2n1/2

0 2 4 6−5

−4

−3

−2

−1

0

log10

(n)lo

g 10[f(

θ)−

f(θ *)]


every 2p

every iter.2−step2−step−dbl.

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/R2n1/2

SAGAdagradNewton

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/R2n1/2

SAGAdagradNewton

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/R2n1/2

SAGAdagradNewton

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/R2n1/2

SAGAdagradNewton

Optimal bounds for least-squares?

• Least-squares: cannot beat σ2d/n (Tsybakov, 2003). Really?

• Refined assumptions with adaptivity (Dieuleveut and Bach, 2014)

f(θn)− f(θ∗) 616σ2 tr Σ1/α

n(γn)1/α +

4‖H1/2−r(θ0 − θ∗)‖2γ2rn2minr,1

– Extension to non-parametric estimation

– Previous results: α = 0 and r = 1/2

• Asymptotic analysis (Defossez and Bach, 2015)

– Bias vs. variance: which one is dominating?

• Acceleration (Flammarion and Bach, 2015)

Bias-variance decomposition

(Defossez and Bach, 2015)

• Simplification: dominating term when n → ∞ and γ → 0

• Variance (e.g., starting from the solution)

f(θn)− f(θ∗) ∼1

nE

[

ε2Φ(x)⊤H−1Φ(x)]

– NB: f noise ε is independent, then we obtain dσ2

n

– Exponentially decaying remainder terms (strongly convex problems)

• Bias (e.g., no noise)

f(θn)− f(θ∗) ∼1

n2γ2(θ0 − θ∗)

⊤H−1(θ0 − θ∗)

Bias-variance decomposition

100 101 102 103 104 105 106 107

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

(slope=-2)

Iteration n

f(θ

n)−

f(θ

∗ )

(bias) γ = γ0/10(var.) γ = γ0/10(bias) γ = γ0

(var.) γ = γ0

Acceleration (Flammarion and Bach, 2015)

• Existing results (Bach and Moulines, 2013)

– Variance =σ2d

n

– Bias 6 minR2‖θ0 − θ∗‖2

n,R4〈θ0 − θ∗,H−1(θ0 − θ∗)〉

n2

• Is it possible to get a bias term inR2‖θ0 − θ∗‖2

n2?

– Corresponds to acceleration (Nesterov, 1983)

– Best (current) result:

σ2d

n1−α+

R2‖θ0 − θ∗‖2n1+α

Outline











4. Finite data sets

Going beyond a single pass over the data


– Assumes infinite data stream

– Observations are used only once

– Directly minimizes testing cost E(x,y) ℓ(y, θ⊤Φ(x))

Going beyond a single pass over the data


– 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 (x1, y1, . . . , xn, yn)

– Multiple passes

– Minimizes training cost 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))

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

• Goal: minimize g(θ) =1

n

n∑

i=1

fi(θ)

Stochastic vs. deterministic methods

• Minimizing g(θ) =1

n

n∑

i=1

fi(θ) with fi(θ) = ℓ(

yi, θ⊤Φ(xi)

)

+ µΩ(θ)

• Batch gradient descent: θt = θt−1−γtg′(θt−1) = θt−1−

γtn

n∑

i=1

f ′i(θt−1)

– Linear (e.g., exponential) convergence rate in O(e−αt)

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



n

n∑

i=1


yi, θ⊤Φ(xi)

)

+ µΩ(θ)


γtn

n∑

i=1

f ′i(θt−1)



n

n∑

i=1


yi, θ⊤Φ(xi)

)

+ µΩ(θ)


γtn

n∑

i=1

f ′i(θt−1)

– Linear (e.g., exponential) convergence rate in O(e−αt)

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

• Stochastic gradient descent: θt = θt−1 − γtf′i(t)(θt−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?)



n

n∑

i=1


yi, θ⊤Φ(xi)

)

+ µΩ(θ)


γtn

n∑

i=1

f ′i(θt−1)

• Stochastic gradient descent: θt = θt−1 − γtf′i(t)(θt−1)


• Goal = best of both worlds: Linear rate with O(1) iteration cost

Robustness to step size

time

log

(exc

ess

co

st)

stochastic

deterministic


• Goal = best of both worlds: Linear rate with O(1) iteration cost

Robustness to step size

hybridlog

(exc

ess

co

st)

stochastic

deterministic

time

Accelerating gradient methods - Related work

• Nesterov acceleration

– Nesterov (1983, 2004)

– Better linear rate but still O(n) iteration cost

• Hybrid methods, incremental average gradient, increasing

batch size

– Bertsekas (1997); Blatt et al. (2008); Friedlander and Schmidt

(2011)

– Linear rate, but iterations make full passes through the data.

Accelerating gradient methods - Related work

• Momentum, gradient/iterate averaging, stochastic version of

accelerated batch gradient methods

– Polyak and Juditsky (1992); Tseng (1998); Sunehag et al. (2009);

Ghadimi and Lan (2010); Xiao (2010)

– Can improve constants, but still have sublinear O(1/t) rate

• Constant step-size stochastic gradient (SG), accelerated SG

– Kesten (1958); Delyon and Juditsky (1993); Solodov (1998); Nedic

and Bertsekas (2000)

– Linear convergence, but only up to a fixed tolerance.

• Stochastic methods in the dual

– Shalev-Shwartz and Zhang (2012)

– Similar linear rate but limited choice for the fi’s

Stochastic average gradient

(Le Roux, Schmidt, and Bach, 2012)

• Stochastic average gradient (SAG) iteration

– Keep in memory the gradients of all functions fi, i = 1, . . . , n

– Random selection i(t) ∈ 1, . . . , n with replacement

– Iteration: θt = θt−1 −γtn

n∑

i=1

yti with yti =

f ′i(θt−1) if i = i(t)

yt−1i otherwise


(Le Roux, Schmidt, and Bach, 2012)

• Stochastic average gradient (SAG) iteration

– Keep in memory the gradients of all functions fi, i = 1, . . . , n

– Random selection i(t) ∈ 1, . . . , n with replacement

– Iteration: θt = θt−1 −γtn

n∑

i=1

yti with yti =

f ′i(θt−1) if i = i(t)

yt−1i otherwise

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

• Extra memory requirement

– Supervised machine learning

– If fi(θ) = ℓi(yi,Φ(xi)⊤θ), then f ′

i(θ) = ℓ′i(yi,Φ(xi)⊤θ)Φ(xi)

– Only need to store n real numbers

Stochastic average gradient - Convergence analysis

• Assumptions

– Each fi is L-smooth, i = 1, . . . , n

– g= 1n

∑ni=1 fi is µ-strongly convex (with potentially µ = 0)

– constant step size γt = 1/(16L)

– initialization with one pass of averaged SGD


• Assumptions


– g= 1n




• Strongly convex case (Le Roux et al., 2012, 2013)

E[

g(θt)− g(θ∗)]

6

(8σ2

nµ+

4L‖θ0−θ∗‖2n

)

exp(

− tmin 1

8n,

µ

16L

)

– Linear (exponential) convergence rate with O(1) iteration cost

– After one pass, reduction of cost by exp(

−min1

8,nµ

16L

)


• Assumptions


– g= 1n




• Non-strongly convex case (Le Roux et al., 2013)

E[

g(θt)− g(θ∗)]

6 48σ2 + L‖θ0−θ∗‖2√

n

n

t

– Improvement over regular batch and stochastic gradient

– Adaptivity to potentially hidden strong convexity

Convergence analysis - Proof sketch

• Main step: find “good” Lyapunov function J(θt, yt1, . . . , y

tn)

– such that E[

J(θt, yt1, . . . , y

tn)|Ft−1

]

< J(θt−1, yt−11 , . . . , yt−1

n )

– no natural candidates

• Computer-aided proof

– Parameterize function J(θt, yt1, . . . , y

tn) = g(θt)−g(θ∗)+quadratic

– Solve semidefinite program to obtain candidates (that depend on

n, µ, L)

– Check validity with symbolic computations

Rate of convergence comparison

• Assume that L = 100, µ = .01, and n = 80000

– Full gradient method has rate(

1− µL

)

= 0.9999

– Accelerated gradient method has rate(

1−√

µL

)

= 0.9900

– Running n iterations of SAG for the same cost has rate(

1− 18n

)n= 0.8825

– Fastest possible first-order method has rate(√

L−√µ√

L+√µ

)2

= 0.9608

• Beating two lower bounds (with additional assumptions)

– (1) stochastic gradient and (2) full gradient


Implementation details and extensions

• The algorithm can use sparsity in the features to reduce the storage

and iteration cost

• Grouping functions together can further reduce the memory

requirement

• We have obtained good performance when L is not known with a

heuristic line-search

• Algorithm allows non-uniform sampling

• Possibility of making proximal, coordinate-wise, and Newton-like

variants

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

Extensions and related work

• Exponential convergence rate for strongly convex problems

• Need to store gradients

– SVRG (Johnson and Zhang, 2013)

• Adaptivity to non-strong convexity

– SAGA (Defazio, Bach, and Lacoste-Julien, 2014)

• Simple proof

– SVRG, SAGA

• Lower bounds

– Agarwal and Bottou (2014)

Summary and future work

• 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

• Going beyond a single pass through the data

Summary and future work

• 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

• Going beyond a single pass through the data

• Extensions and future work

– Pre-conditioning

– Proximal extensions fo non-differentiable terms

– kernels and non-parametric estimation

– line-search

– parallelization

Outline











4. Finite data sets

Conclusions

Machine learning and convex optimization

• Statistics with or without optimization?

– Significance of mixing algorithms with analysis

– Benefits of mixing algorithms with analysis

• Open problems

– Non-parametric stochastic approximation

– Going beyond a single pass over the data (testing performance)

– Characterization of implicit regularization of online methods

– Distributed processing

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.

Alekh Agarwal and Leon Bottou. A lower bound for the optimization of finite sums. arXiv preprint

arXiv:1410.0723, 2014.

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):872–899, 2000.

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-asymptotic analysis of stochastic approximation algorithms for machine

learning. In Adv. NIPS, 2011.

F. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence

rate o(1/n). Technical Report 00831977, HAL, 2013.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski. Structured sparsity through convex optimization,

2012a.

Francis Bach, Rodolphe Jenatton, Julien Mairal, and Guillaume Obozinski. Optimization with sparsity-

inducing penalties. Foundations and Trends R© in Machine Learning, 4(1):1–106, 2012b.

A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems.

SIAM Journal on Imaging Sciences, 2(1):183–202, 2009.

Albert Benveniste, Michel Metivier, and Pierre Priouret. Adaptive algorithms and stochastic

approximations. Springer Publishing Company, Incorporated, 2012.

D. P. Bertsekas. A new class of incremental gradient methods for least squares problems. SIAM

Journal on Optimization, 7(4):913–926, 1997.

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. Bottou and Y. Le Cun. On-line learning for very large data sets. Applied Stochastic Models in

Business and Industry, 21(2):137–151, 2005.

S. Boucheron and P. Massart. A high-dimensional wilks phenomenon. Probability theory and related

fields, 150(3-4):405–433, 2011.

S. Boucheron, O. Bousquet, G. Lugosi, et al. Theory of classification: A survey of some recent

advances. ESAIM Probability and statistics, 9:323–375, 2005.

N. Cesa-Bianchi, A. Conconi, and C. Gentile. On the generalization ability of on-line learning algorithms.

Information Theory, IEEE Transactions on, 50(9):2050–2057, 2004.

Aaron Defazio, Francis Bach, and Simon Lacoste-Julien. Saga: A fast incremental gradient method

with support for non-strongly convex composite objectives. In Advances in Neural Information

Processing Systems, pages 1646–1654, 2014.

A. Defossez and F. Bach. Constant step size least-mean-square: Bias-variance trade-offs and optimal

sampling distributions. 2015.

B. Delyon and A. Juditsky. Accelerated stochastic approximation. SIAM Journal on Optimization, 3:

868–881, 1993.

A. Dieuleveut and F. Bach. Non-parametric Stochastic Approximation with Large Step sizes. Technical

report, ArXiv, 2014.

J. Duchi and Y. Singer. Efficient online and batch learning using forward backward splitting. Journal

of Machine Learning Research, 10:2899–2934, 2009. ISSN 1532-4435.

N. Flammarion and F. Bach. From averaging to acceleration, there is only a step-size. arXiv preprint

arXiv:1504.01577, 2015.

M. P. Friedlander and M. Schmidt. Hybrid deterministic-stochastic methods for data fitting.

arXiv:1104.2373, 2011.

S. Ghadimi and G. Lan. Optimal stochastic approximation algorithms for strongly convex stochastic

composite optimization. Optimization Online, July, 2010.

Saeed Ghadimi and Guanghui Lan. Optimal stochastic approximation algorithms for strongly convex

stochastic composite optimization, ii: shrinking procedures and optimal algorithms. SIAM Journal

on Optimization, 23(4):2061–2089, 2013.

L. Gyorfi and H. Walk. On the averaged stochastic approximation for linear regression. SIAM Journal

on Control and Optimization, 34(1):31–61, 1996.

E. Hazan, A. Agarwal, and S. Kale. Logarithmic regret algorithms for online convex optimization.

Machine Learning, 69(2):169–192, 2007.

Chonghai Hu, James T Kwok, and Weike Pan. Accelerated gradient methods for stochastic optimization

and online learning. In NIPS, volume 22, pages 781–789, 2009.

Rie Johnson and Tong Zhang. Accelerating stochastic gradient descent using predictive variance

reduction. In Advances in Neural Information Processing Systems, pages 315–323, 2013.

H. Kesten. Accelerated stochastic approximation. Ann. Math. Stat., 29(1):41–59, 1958.

H. J. Kushner and G. G. Yin. Stochastic approximation and recursive algorithms and applications.

Springer-Verlag, second edition, 2003.

S. Lacoste-Julien, M. Schmidt, and F. Bach. A simpler approach to obtaining an o (1/t) convergence

rate for projected stochastic subgradient descent. Technical Report 1212.2002, ArXiv, 2012.

Guanghui Lan, Arkadi Nemirovski, and Alexander Shapiro. Validation analysis of mirror descent

stochastic approximation method. Mathematical programming, 134(2):425–458, 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. 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.

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

Wiley West Sussex, 1995.

A. Nedic and D. Bertsekas. Convergence rate of incremental subgradient algorithms. Stochastic

Optimization: Algorithms and Applications, pages 263–304, 2000.

A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to

stochastic programming. SIAM Journal on Optimization, 19(4):1574–1609, 2009.

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

& Sons, 1983.

Y. Nesterov. A method for solving a convex programming problem with rate of convergence O(1/k2).

Soviet Math. Doklady, 269(3):543–547, 1983.

Y. Nesterov. Introductory lectures on convex optimization: a basic course. Kluwer Academic Publishers,

2004.

Y. Nesterov. Gradient methods for minimizing composite objective function. Center for Operations

Research and Econometrics (CORE), Catholic University of Louvain, Tech. Rep, 76, 2007.

Y. Nesterov. Primal-dual subgradient methods for convex problems. Mathematical programming, 120

(1):221–259, 2009.

Y. Nesterov and A. Nemirovski. Interior-point polynomial algorithms in convex programming. SIAM

studies in Applied Mathematics, 1994.

Y. Nesterov and J. P. Vial. Confidence level solutions for stochastic programming. Automatica, 44(6):

1559–1568, 2008. ISSN 0005-1098.

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.

B. Scholkopf and A. J. Smola. Learning with Kernels. MIT Press, 2001.

S. Shalev-Shwartz and N. Srebro. SVM optimization: inverse dependence on training set size. In Proc.

ICML, 2008.

S. Shalev-Shwartz and T. Zhang. Stochastic dual coordinate ascent methods for regularized loss

minimization. Technical Report 1209.1873, Arxiv, 2012.

S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal estimated sub-gradient solver for svm.

In Proc. ICML, 2007.

S. Shalev-Shwartz, O. Shamir, N. Srebro, and K. Sridharan. Stochastic convex optimization. In proc.

COLT, 2009.

J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press,

2004.

Naum Zuselevich Shor, Krzysztof C. Kiwiel, and Andrzej Ruszcay?ski. Minimization methods for

non-differentiable functions. Springer-Verlag New York, Inc., 1985.

M.V. Solodov. Incremental gradient algorithms with stepsizes bounded away from zero. Computational

Optimization and Applications, 11(1):23–35, 1998.

K. Sridharan, N. Srebro, and S. Shalev-Shwartz. Fast rates for regularized objectives. 2008.

P. Sunehag, J. Trumpf, SVN Vishwanathan, and N. Schraudolph. Variable metric stochastic

approximation theory. International Conference on Artificial Intelligence and Statistics, 2009.

P. Tseng. An incremental gradient(-projection) method with momentum term and adaptive stepsize

rule. SIAM Journal on Optimization, 8(2):506–531, 1998.

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

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

L. Xiao. Dual averaging methods for regularized stochastic learning and online optimization. Journal

of Machine Learning Research, 9:2543–2596, 2010. ISSN 1532-4435.

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Large-scale machine learning and convex …fbach/gradsto_allerton.pdfLarge-scale machine learning and convex optimization Francis Bach INRIA - Ecole Normale Sup´erieure, Paris, France