Francis Bach INRIA - Ecole Normale Sup´erieure, Paris, Francefbach/fbach_mlss_2018.pdf ·...

Statistical machine learningand convex optimization

Francis Bach

INRIA - Ecole Normale Superieure, Paris, France

ÉCOLE NORMALE

S U P É R I E U R E

Machine Learning Summer School - Madrid, 2018

Slides available: www.di.ens.fr/~fbach/fbach_mlss_2018.pdf

1

“Big data/AI” 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 d

2

“Big data/AI” 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 d

3

Advertising

4

Marketing - Personalized recommendation

5

Visual object recognition

6

Personal photos

• Recognizing people and places

- Emile and Francis

- Chocolateria San Gines

7

Personal photos

• Recognizing people and places

– Emile and Francis

– Chocolateria San Gines

8

Bioinformatics

• Protein: Crucial elements of cell life

• Massive data: 2 millions for humans

• Complex data

9

Context

Machine learning for “big data”

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

– d : dimension of each observation (or number of features)

– n : number of observations

• Examples: computer vision, bioinformatics, advertising

– Ideal running-time complexity: O(dn)

– Going back to simple methods

– Stochastic gradient methods (Robbins and Monro, 1951b)

– Mixing statistics and optimization

– Using smoothness to go beyond stochastic gradient descent

10

Context






• Ideal running-time complexity: O(dn)





11

Context






• Ideal running-time complexity: O(dn)

• Going back to simple methods




12

Scaling to large problems

“Retour aux sources”

• 1950’s: Computers not powerful enough

IBM “1620”, 1959

CPU frequency: 50 KHz

Price > 100 000 dollars

• 2010’s: Data too massive

- Stochastic gradient methods (Robbins and Monro, 1951a)

- Going back to simple methods

13

Scaling to large problems

“Retour aux sources”

• 1950’s: Computers not powerful enough

IBM “1620”, 1959

CPU frequency: 50 KHz

Price > 100 000 dollars

• 2010’s: Data too massive

• Stochastic gradient methods (Robbins and Monro, 1951a)


14

Outline - II

1. Introduction

• Large-scale machine learning and optimization

• Classes of functions (convex, smooth, etc.)

• Traditional statistical analysis (regardless of optimization)

2. Classical methods for convex optimization

• Smooth optimization (gradient descent, Newton method)

• Non-smooth optimization (subgradient descent)

• Proximal methods

3. Non-smooth stochastic approximation

• Stochastic (sub)gradient and averaging

• Non-asymptotic results and lower bounds

• Strongly convex vs. non-strongly convex

15

Outline - II

4. Classical stochastic approximation (not covered)

• Asymptotic analysis

• Robbins-Monro algorithm and Polyak-Rupert averaging

5. Smooth stochastic approximation algorithms

• Non-asymptotic analysis for smooth functions

• Least-squares regression without decaying step-sizes

6. Finite data sets (partially covered)

• Gradient methods with exponential convergence rates

• (Dual) stochastic coordinate descent

• Frank-Wolfe

7. Non-convex problems (“open” / not covered)

16

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

– NB: non-linear problems (on the board)

17




• (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

18

Usual losses

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

– quadratic loss 12(y − y)2 = 1

2(y − θ⊤Φ(x))2

19

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

20

Main motivating examples

• Support vector machine (hinge loss): non-smooth

ℓ(Y, θ⊤Φ(X)) = max1− Y θ⊤Φ(X), 0

• Logistic regression: smooth

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

• Structured output regression

– See Tsochantaridis et al. (2005); Lacoste-Julien et al. (2013)

21

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) and references therein

22

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) and references therein

• 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 et al. (2012b,a) and references therein

23





minθ∈Rd

1

n

n∑

i=1

ℓ(

yi, θ⊤Φ(xi)

)

+ µΩ(θ)


24





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 simultaneously25





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

28

Convexity

• Global definitions

g(θ)

θ

29

Convexity

• Global definitions (full domain)

g(θ)

θ

g(θ1)g(θ2)

θ2θ1

– Not assuming differentiability:

∀θ1, θ2, α ∈ [0, 1], g(αθ1 + (1− α)θ2) 6 αg(θ1) + (1− α)g(θ2)

30

Convexity


g(θ)

θ

g(θ1)g(θ2)

θ2θ1

– Assuming differentiability:

∀θ1, θ2, g(θ1) > g(θ2) + g′(θ2)⊤(θ1 − θ2)

• Extensions to all functions with subgradients / subdifferential

31

Convexity


g(θ)

θ

• Local definitions

– Twice differentiable functions

– ∀θ, g′′(θ) < 0 (positive semi-definite Hessians)

32

Convexity


g(θ)

θ

• Local definitions

– Twice differentiable functions

– ∀θ, g′′(θ) < 0 (positive semi-definite Hessians)

• Why convexity?

33

Why convexity?

• Local minimum = global minimum

– Optimality condition (smooth): g′(θ) = 0

– Most algorithms do not need convexity for their definitions

– Local convexity

• Convex duality

– See Boyd and Vandenberghe (2003)

• Recognizing convex problems


34

Why convexity?

• Local minimum = global minimum

– Optimality condition (smooth): g′(θ) = 0

– Most algorithms do not need convexity for their definitions

– Local convexity around a local optimum

• Convex duality


• Recognizing convex problems


35

Lipschitz continuity

• Bounded gradients of g (⇔ Lipschitz-continuity): the function

g 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 ⇒ ‖g′(θ)‖2 6 B

⇔

∀θ, θ′ ∈ Rd, ‖θ‖2, ‖θ′‖2 6 D ⇒ |g(θ)− g(θ′)| 6 B‖θ − θ′‖2

• Machine learning

– with g(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))

– G-Lipschitz loss and R-bounded data: B = GR (see board)

36

Smoothness and strong convexity

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

and its gradient is L-Lipschitz-continuous

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

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

smooth non−smooth

37


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

and its gradient is L-Lipschitz-continuous

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

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

• Machine learning (see board)

– with g(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))

– Hessian ≈ covariance matrix 1n

∑ni=1Φ(xi)Φ(xi)

⊤

– Lloss-smooth loss and R-bounded data: L = LlossR2

38


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

∀θ1, θ2 ∈ Rd, g(θ1) > g(θ2) + g′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22

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

convexstronglyconvex

39



∀θ1, θ2 ∈ Rd, g(θ1) > g(θ2) + g′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22


(large µ/L) (small µ/L)

40



∀θ1, θ2 ∈ Rd, g(θ1) > g(θ2) + g′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22



– with g(θ) = 1n

∑ni=1 ℓ(yi, θ

⊤Φ(xi))


∑ni=1Φ(xi)Φ(xi)

⊤

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

41



∀θ1, θ2 ∈ Rd, g(θ1) > g(θ2) + g′(θ2)

⊤(θ1 − θ2) +µ2‖θ1 − θ2‖22



– with g(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

42

Summary of smoothness/convexity assumptions

• Bounded gradients of g (Lipschitz-continuity): the function g 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 ⇒ ‖g′(θ)‖2 6 B

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

L-Lipschitz-continuous gradient g′ (e.g., bounded Hessians):

∀θ ∈ Rd, g′′(θ) 4 L · Id

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

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

∀θ ∈ Rd, g′′(θ) < µ · Id

43

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

44



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)

45



f(θ)− minθ∈Rd

f(θ) =

[

f(θ)−minθ∈Θ

f(θ)

]

+

[

minθ∈Θ

f(θ)− minθ∈Rd

f(θ)

]



f(θ)

f(θ)−minθ∈Θ

f(θ) 6 supθ∈Θ

f(θ)− f(θ) + supθ∈Θ

f(θ)− f(θ)

– Typically slow rate O(

1/√n)

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

46



f(θ)− minθ∈Rd

f(θ) =

[

f(θ)−minθ∈Θ

f(θ)

]

+

[

minθ∈Θ

f(θ)− minθ∈Rd

f(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)

47

Slow rate for supervised learning

• 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

48

Slow rate for supervised learning


– Ω(θ) = ‖θ‖2 (Euclidean norm)


– 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)|]

64ℓ0 + 4GRD√

n

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

• Lipschitz functions ⇒ slow rate

49

Motivation from mean estimation

• Estimator θ = 1n

∑ni=1 zi = argminθ∈R

12n

∑ni=1(θ − zi)

2 = f(θ)

– θ∗ = Ez = argminθ∈R12E(θ − z)2 = f(θ)

– From before (estimation error): f(θ)− f(θ∗) = O(1/√n)

50

Motivation from mean estimation

• Estimator θ = 1n

∑ni=1 zi = argminθ∈R

12n

∑ni=1(θ − zi)

2 = f(θ)

– θ∗ = Ez = argminθ∈R12E(θ − z)2 = f(θ)

– From before (estimation error): f(θ)− f(θ∗) = O(1/√n)

• Direct computation:

– f(θ) = 12E(θ − z)2 = 1

2(θ − Ez)2 + 12 var(z)

• 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 (instead of uniform bound)

51

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

52

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) 68G2R2(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

53

Outline - II

1. Introduction












54

Outline - II










• Frank-Wolfe


55

Complexity results in convex optimization

• Assumption: g convex on Rd

• Classical generic algorithms

– Gradient descent and accelerated gradient descent

– Newton method

– Subgradient method (and ellipsoid algorithm)

56

Complexity results in convex optimization

• Assumption: g convex on Rd

• Classical generic algorithms

– Gradient descent and accelerated gradient descent

– Newton method

– Subgradient method (and ellipsoid algorithm)

• Key additional properties of g

– Lipschitz continuity, smoothness or strong convexity

• Key insight from Bottou and Bousquet (2008)

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

• Key references: Nesterov (2004), Bubeck (2015)

57

Several criteria for characterizing convergence

• Objective function values

g(θ)− infη∈Rd

g(η)

– Usually weaker condition

• Iterates

infη∈argmin g

∥

∥θ − η∥

∥

2

– Typically used for strongly-convex problems

58

Several criteria for characterizing convergence

• Objective function values

g(θ)− infη∈Rd

g(η)

– Usually weaker condition

• Iterates

infη∈argmin g

∥

∥θ − η∥

∥

2

– Typically used for strongly-convex problems

• NB 1: relationships between the two types in several situations

• NB 2: similarity with prediction vs. estimation in statistics

59

(smooth) gradient descent

• Assumptions

– g convex with L-Lipschitz-continuous gradient (e.g., L-smooth)

– Minimum attained at θ∗

• Algorithm:

θt = θt−1 −1

Lg′(θt−1)

60

(smooth) gradient descent - strong convexity

• Assumptions


– g µ-strongly convex

• Algorithm:

θt = θt−1 −1

Lg′(θt−1)

• Bound:

g(θt)− g(θ∗) 6 (1− µ/L)t[

g(θ0)− g(θ∗)]

• Three-line proof

• Line search, steepest descent or constant step-size

61

(smooth) gradient descent - slow rate

• Assumptions


– Minimum attained at θ∗

• Algorithm:

θt = θt−1 −1

Lg′(θt−1)

• Bound:

g(θt)− g(θ∗) 62L‖θ0 − θ∗‖2

t+ 4• Four-line proof

• Adaptivity of gradient descent to problem difficulty

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

62

Gradient descent - Proof for quadratic functions

• Quadratic convex function: g(θ) = 12θ

⊤Hθ − c⊤θ

– µ and L are smallest largest eigenvalues of H

– Global optimum θ∗ = H−1c (or H†c) such that Hθ∗ = c

63



⊤Hθ − c⊤θ



• Gradient descent with γ = 1/L:

θt = θt−1 −1

L(Hθt−1 − c) = θt−1 −

1

L(Hθt−1 −Hθ∗)

θt − θ∗ = (I − 1

LH)(θt−1 − θ∗) = (I − 1

LH)t(θ0 − θ∗)

64



⊤Hθ − c⊤θ




θt = θt−1 −1

L(Hθt−1 − c) = θt−1 −

1


θt − θ∗ = (I − 1

LH)(θt−1 − θ∗) = (I − 1

LH)t(θ0 − θ∗)

• Strong convexity µ > 0: eigenvalues of (I − 1LH)t in [0, (1− µ

L)t]

– Convergence of iterates: ‖θt − θ∗‖2 6 (1− µ/L)2t‖θ0 − θ∗‖2– Function values: g(θt)− g(θ∗) 6 (1− µ/L)2t

[

g(θ0)− g(θ∗)]

– Function values: g(θt)− g(θ∗) 6Lt ‖θ0 − θ∗‖2

65



⊤Hθ − c⊤θ




θt = θt−1 −1

L(Hθt−1 − c) = θt−1 −

1


θt − θ∗ = (I − 1

LH)(θt−1 − θ∗) = (I − 1

LH)t(θ0 − θ∗)

• Convexity µ = 0: eigenvalues of (I − 1LH)t in [0, 1]

– No convergence of iterates: ‖θt − θ∗‖2 6 ‖θ0 − θ∗‖2– Function values: g(θt)−g(θ∗) 6 maxv∈[0,L] v(1−v/L)2t‖θ0−θ∗‖2– Function values: g(θt)− g(θ∗) 6

Lt ‖θ0 − θ∗‖2

66

Accelerated gradient methods (Nesterov, 1983)

• Assumptions

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

• Algorithm:θt = ηt−1 −

1

Lg′(ηt−1)

ηt = θt +t− 1

t+ 2(θt − θt−1)

• Bound:g(θt)− g(θ∗) 6

2L‖θ0 − θ∗‖2(t+ 1)2

• Ten-line proof (see, e.g., Schmidt, Le Roux, and Bach, 2011)

• Not improvable

• Extension to strongly-convex functions

67

Accelerated gradient methods - strong convexity

• Assumptions

– g convex with L-Lipschitz-cont. gradient , min. attained at θ∗– g µ-strongly convex

• Algorithm:

θt = ηt−1 −1

Lg′(ηt−1)

ηt = θt +1−

√

µ/L

1 +√

µ/L(θt − θt−1)

• Bound: g(θt)− f(θ∗) 6 L‖θ0 − θ∗‖2(1−√

µ/L)t

– Ten-line proof (see, e.g., Schmidt, Le Roux, and Bach, 2011)

– Not improvable

– Relationship with conjugate gradient for quadratic functions

68

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)

69

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)

70

Soft-thresholding for the ℓ1-norm

• Example: quadratic problem in 1D, i.e. minx∈R

1

2x2 − xy + λ|x|

71



1

2x2 − xy + λ|x|

• Piecewise quadratic function with a kink at zero

– Derivative at 0+: g+ = λ− y and 0−: g− = −λ− y

– x = 0 is the solution iff g+ > 0 and g− 6 0 (i.e., |y| 6 λ)

– x > 0 is the solution iff g+ 6 0 (i.e., y > λ) ⇒ x∗ = y − λ

– x 6 0 is the solution iff g− 6 0 (i.e., y 6 −λ) ⇒ x∗ = y + λ

• Solution x∗ = sign(y)(|y| − λ)+ = soft thresholding

72



1

2x2 − xy + λ|x|

• Piecewise quadratic function with a kink at zero

• Solution x∗ = sign(y)(|y| − λ)+ = soft thresholding

x

−λ

x*(y)

λ y

73

Projected gradient descent

• Problems of the form: minθ∈K

f(θ)

– θt+1 = argminθ∈K

f(θt) + (θ − θt)⊤∇f(θt)+

L

2‖θ − θt‖22

– θt+1 = argminθ∈K

1

2

∥

∥

∥θ −

(

θt −1

L∇f(θt)

)

∥

∥

∥

2

2– Projected gradient descent

• Similar convergence rates than smooth optimization

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

74

Newton method

• Given θt−1, minimize second-order Taylor expansion

g(θ) = g(θt−1)+g′(θt−1)⊤(θ−θt−1)+

1

2(θ−θt−1)

⊤g′′(θt−1)⊤(θ−θt−1)

• Expensive Iteration: θt = θt−1 − g′′(θt−1)−1g′(θt−1)

– Running-time complexity: O(d3) in general

75

Newton method

• Given θt−1, minimize second-order Taylor expansion

g(θ) = g(θt−1)+g′(θt−1)⊤(θ−θt−1)+

1

2(θ−θt−1)

⊤g′′(θt−1)⊤(θ−θt−1)

• Expensive Iteration: θt = θt−1 − g′′(θt−1)−1g′(θt−1)

– Running-time complexity: O(d3) in general

• Quadratic convergence: If ‖θt−1 − θ∗‖ small enough, for some

constant C, we have

(C‖θt − θ∗‖) = (C‖θt−1 − θ∗‖)2


76

Summary: minimizing smooth convex functions

• Assumption: g convex

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

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

– O(e−tµ/L) convergence rate for strongly smooth convex functions

– Optimal rates O(1/t2) and O(e−t√

µ/L)

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

– O(

e−ρ2t)

convergence rate

77

Summary: minimizing smooth convex functions

• Assumption: g convex



– O(e−tµ/L) convergence rate for strongly smooth convex functions

– Optimal rates O(1/t2) and O(e−t√

µ/L)

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

– O(

e−ρ2t)

convergence rate

• From smooth to non-smooth

– Subgradient method (and ellipsoid)

78

Counter-example (Bertsekas, 1999)

Steepest descent for nonsmooth objectives

• g(θ1, θ2) =

−5(9θ21 + 16θ22)1/2 if θ1 > |θ2|

−(9θ1 + 16|θ2|)1/2 if θ1 6 |θ2|

• Steepest descent starting from any θ such that θ1 > |θ2| >

(9/16)2|θ1|

−5 0 5−5

0

5

79

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

• Assumptions

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

• Algorithm: θt = ΠD

(

θt−1 −2D

B√tg′(θt−1)

)

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

Constraints

80

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

• Assumptions

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


(

θt−1 −2D

B√tg′(θt−1)

)

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

• Bound:

g

(

1

t

t−1∑

k=0

θk

)

− g(θ∗) 62DB√

t


• Best possible convergence rate after O(d) iterations (Bubeck, 2015)

81

Subgradient method/“descent” - proof - I

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

2DB√t

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

82

Subgradient method/“descent” - proof - I


2DB√t

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

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

= ‖θt−1 − θ∗‖22 + γ2t ‖g′(θt−1)‖22 − 2γt(θt−1 − θ∗)

⊤g′(θt−1)

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

⊤g′(θt−1) because ‖g′(θt−1)‖2 6 B

6 ‖θt−1 − θ∗‖22 +B2γ2t − 2γt

[

g(θt−1)− g(θ∗)]

(property of subgradients)

• leading to

g(θt−1)− g(θ∗) 6B2γt2

+1

2γt

[

‖θt−1 − θ∗‖22 − ‖θt − θ∗‖22]

83

Subgradient method/“descent” - proof - II

• Starting from g(θt−1)− g(θ∗) 6B2γt2

+1

2γt

[

‖θt−1 − θ∗‖22 − ‖θt − θ∗‖22]

• Constant step-size γt = γ

t∑

u=1

[

g(θu−1)− g(θ∗)]

6

t∑

u=1

B2γ

2+

t∑

u=1

1

2γ

[

‖θu−1 − θ∗‖22 − ‖θu − θ∗‖22]

6 tB2γ

2+

1

2γ‖θ0 − θ∗‖22 6 t

B2γ

2+

2

γD2

• Optimized step-size γt =2DB√tdepends on “horizon” t

– Leads to bound of 2DB√t

• Using convexity: g

(

1

t

t−1∑

k=0

θk

)

−g(θ∗) 61

t

t−1∑

k=0

g(θk)−g(θ∗) 62DB√

t

84

Subgradient method/“descent” - proof - III

• Starting from g(θt−1)− g(θ∗) 6B2γt2

+1

2γt

[

‖θt−1 − θ∗‖22 − ‖θt − θ∗‖22]

• Decreasing step-size

t∑

u=1

[

g(θu−1)− g(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 3DB

√t with γt =

2D

B√t

• Using convexity: g(

1t

∑t−1k=0 θk

)

− g(θ∗) 63DB√

t

85

Subgradient descent for machine learning



– 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)

86

Subgradient descent - strong convexity

• Assumptions

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


(

θt−1 −2

µ(t+ 1)g′(θt−1)

)

• Bound:

g

(

2

t(t+ 1)

t∑

k=1

kθk−1

)

− g(θ∗) 62B2

µ(t+ 1)


• Best possible convergence rate after O(d) iterations (Bubeck, 2015)

87

Subgradient method - strong convexity - proof - I


2µ(t+1)

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

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

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

⊤g′(θt−1) because ‖g′(θt−1)‖2 6 B

6 ‖θt−1 − θ∗‖22 +B2γ2t − 2γt

[

g(θt−1)− g(θ∗)+µ

2‖θt−1 − θ∗‖22

]

(property of subgradients and strong convexity)

• leading to

g(θt−1)− g(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

88

Subgradient method - strong convexity - proof - II

• From g(θt−1)− g(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[

g(θu−1)− g(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: g

(

2

t(t+ 1)

t∑

u=1

uθu−1

)

− g(θ∗) 62B2

t+ 1

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

89

Summary: minimizing convex functions


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


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

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

– O(

e−ρ2t)

convergence rate

- Key insights from Bottou and Bousquet (2008)

- In machine learning, no need to optimize below statistical error

- In machine learning, cost functions are averages

- Testing errors are more important than training errors

⇒ Stochastic approximation

90

Summary of rates of convergence

• Problem parameters

– D diameter of the domain

– B Lipschitz-constant

– L smoothness constant

– µ strong convexity constant

convex strongly convex

nonsmooth deterministic: BD/√t deterministic: B2/(tµ)

stochastic: BD/√n stochastic: B2/(nµ)

smooth deterministic: LD2/t2 deterministic: exp(−t√

µ/L)

stochastic: LD2/√n stochastic: L/(nµ)

finite sum: n/t finite sum: exp(−min1/n, µ/Lquadratic deterministic: LD2/t2 deterministic: exp(−t

√

µ/L)

stochastic: d/n+ LD2/n stochastic: d/n+ LD2/n

91

Summary: minimizing convex functions


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


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


– 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

3. Testing errors are more important than training errors

⇒ Stochastic approximation

92

Outline - II

1. Introduction












93

Outline - II










• Frank-Wolfe


94

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

95





• 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

96





• 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 next lecture)

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

97

Relationship to online learning


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

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

98





• 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

99





• 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)

100

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

101





• 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

102





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

θn = θn−1 − γnf′n(θn−1)


∑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)

103

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 C = ‖θ‖2 6 D

• Algorithm: θn = ΠD

(

θn−1 −2D

B√nf ′n(θn−1)

)

104


• Assumptions


– θ∗ global optimum of f on C = ‖θ‖2 6 D


(

θn−1 −2D


)

• 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

105

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

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 propert

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]

106

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

107

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


(

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)


108

Stochastic subgradient - strong convexity - proof - I

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

µ(n+1)

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

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

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

⊤f ′n(θt−1) because ‖f ′

n(θt−1)‖2E(·|Fn−1) 6 ‖θn−1 − θ∗‖22 +B2γ2

n − 2γn[

f(θn−1)− f(θ∗)+µ

2‖θn−1 − θ∗‖22

]

(property of subgradients and strong convexity)

• leading to

Ef(θn−1)− f(θ∗) 6B2γn2

+1

2

[ 1

γn− µ

]

‖θn−1 − θ∗‖22 −1

2γn‖θn − θ∗‖22

6B2

µ(n+ 1)+

µ

2

[n− 1

2

]

‖θn−1 − θ∗‖22 −µ(n+ 1)

4‖θn − θ∗‖22

109

Stochastic subgradient - strong convexity - proof - II

• FromEf(θn−1)−f(θ∗)6B2

µ(n+ 1)+µ

2

[n− 1

2

]

E‖θn−1−θ∗‖22−µ(n+ 1)

4E‖θn−θ∗‖22

n∑

u=1

u[

Ef(θu−1)− f(θ∗)]

6

n∑

u=1

B2u

µ(u+ 1)+

1

4

n∑

u=1

[

u(u− 1)E‖θu−1 − θ∗‖22 − u(u+ 1)E‖θu

6B2n

µ+

1

4

[

0− n(n+ 1)E‖θn − θ∗‖22]

6B2n

µ

• Using convexity: Ef

(

2

n(n+ 1)

n∑

u=1

uθu−1

)

− g(θ∗) 62B2

n+ 1

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

110

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)

• Minimax convergence rate

111

Beyond convergence in expectation

• Typical result: Ef

(

1

n

n−1∑

k=0

θk

)

− f(θ∗) 62DB√

n

– Obtained with simple conditioning arguments

• High-probability bounds

– Markov inequality: P(

f(

1n

∑n−1k=0 θk

)

− f(θ∗) > ε)

62DB√nε

112

Beyond convergence in expectation

• Typical result: Ef

(

1

n

n−1∑

k=0

θk

)

− f(θ∗) 62DB√

n

– Obtained with simple conditioning arguments

• High-probability bounds

– Markov inequality: P(

f(

1n

∑n−1k=0 θk

)

− f(θ∗) > ε)

62DB√nε

– Deviation inequality (Nemirovski et al., 2009; Nesterov and Vial,

2008)

P

(

f(1

n

n−1∑

k=0

θk

)

− f(θ∗) >2DB√

n(2 + 4t)

)

6 2 exp(−t2)

• See also Bach (2013) for logistic regression

113

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)

114

Minimax rates (Agarwal et al., 2012)

• Model of computation (i.e., algorithms): first-order oracle

– Queries a function f by obtaining f(θk) and f ′(θk) with zero-mean

bounded variance noise, for k = 0, . . . , n− 1 and outputs θn

• Class of functions

– convex B-Lipschitz-continuous (w.r.t. ℓ2-norm) on a compact

convex set C containing an ℓ∞-ball

• Performance measure

– for a given algorithm and function εn(algo, f) = f(θn)−infθ∈C f(θ)– for a given algorithm: sup

functions f

εn(algo, f)

• Minimax performance: infalgo

supfunctions f

εn(algo, f)

115

Minimax rates (Agarwal et al., 2012)

• Convex functions: domain C that contains an ℓ∞-ball of radius D

infalgo

supfunctions f

ε(algo, f) > cst ×min

BD

√

d

n,BD

– Consequences for ℓ2-ball of radius D: BD/√n

– Upper-bound through stochastic subgradient

• µ-strongly-convex functions:

infalgo

supfunctions f

εn(algo, f) > cst ×minB2

µn,B2

µd,BD

√

d

n,BD

116











µ/L)



√

µ/L)


117

Outline - II

1. Introduction












118

Outline - II










• Frank-Wolfe


119


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)

120


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)

121


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?

122


Existing work








Ruppert, 1988)



• Non-asymptotic analysis for smooth problems? problems with

convergence rate O(min1/µn, 1/√n)123

Algorithm

• Iteration: θn = θn−1 − γnf′n(θn−1)


∑n−1k=0 θk

124

Summary of results (Bach and Moulines, 2011)

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

• Strongly convex smooth objective functions

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

– New: O(n−1µ−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

125

Summary of results (Bach and Moulines, 2011)



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

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




• 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

)

126

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

127

Summary of new results (Bach and Moulines, 2011)



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

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


128







• 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]

• Take-home message

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

129

Robustness to lack of strong convexity

• Left: f(θ) = |θ|2 between −1 and 1

• Right: f(θ) = |θ|4 between −1 and 1

• affine outside of [−1, 1], continuously differentiable.

0 1 2 3 4 5

−4

−3

−2

−1

0

1

log(n)

log[

f(θ n)−

f∗ ]

power 2

sgd − 1/3ave − 1/3sgd − 1/2ave − 1/2sgd − 2/3ave − 2/3sgd − 1ave − 1

0 1 2 3 4 5

−6

−4

−2

0

2

log(n)

log[

f(θ n)−

f∗ ]

power 4

sgd − 1/3ave − 1/3sgd − 1/2ave − 1/2sgd − 2/3ave − 2/3sgd − 1ave − 1

130


Existing work








Ruppert, 1988)



• A single adaptive algorithm for smooth problems with

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

131

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

132



[

(yn − 〈Φ(xn), θ〉)2]

with θ ∈ Rd




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)

133

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

134



(

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

135



(

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)




∫

θπγ(dθ)


θ∗

θ0

θn

θn

136



(

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)

137

Simulations - synthetic examples

• Gaussian distributions - d = 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

138

Simulations - benchmarks

• alpha (d = 500, n = 500 000), news (d = 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

139

Isn’t least-squares regression a “regression”?

140

Isn’t least-squares regression a “regression”?

• Least-squares regression

– Simpler to analyze and understand

– Explicit relationship to bias/variance trade-offs

– See Defossez and Bach (2015); Dieuleveut et al. (2016)

• Many important loss functions are not quadratic

– Beyond least-squares with online Newton steps

– Complexity of O(d) per iteration with rate O(d/n)

– See Bach and Moulines (2013) for details

141

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

142




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∗

143




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)

144



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

145

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(d) per iteration

146


• 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(d) per iteration

147


• 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)〉

]

148


• 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(d)

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

149

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(d/n) for logistic regression

– Additional assumptions but no strong convexity

150








• 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)

151



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.

152

Simulations - benchmarks

• alpha (d = 500, n = 500 000), news (d = 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

153











µ/L)



√

µ/L)


154











µ/L)


finite sum: n/t finite sum: exp(−t/(n+L/µ))

quadratic deterministic: LD2/t2 deterministic: exp(−t√

µ/L)


155

Outline - II

1. Introduction












156

Outline - II










• Frank-Wolfe


157

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))

158

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)

159

Iterative methods for minimizing smooth functions

• Assumption: g convex and L-smooth on Rd


g(θt)− g(θ∗) 6 O(1/t)

g(θt)− g(θ∗) 6 O(e−t(µ/L)) = O(e−t/κ)

(small κ = L/µ) (large κ = L/µ)

160




g(θt)− g(θ∗) 6 O(1/t)

g(θt)− g(θ∗) 6 O((1−µ/L)t) = O(e−t(µ/L)) if µ-strongly convex

(small κ = L/µ) (large κ = L/µ)

161




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

– O(e−t/κ) linear if strongly-convex


– O(

e−ρ2t)

quadratic rate

162





– O(e−t/κ) linear if strongly-convex ⇔ O(κ log 1ε) iterations


– O(

e−ρ2t)

quadratic rate ⇔ O(log log 1ε) iterations

163





– O(e−t/κ) linear if strongly-convex ⇔ complexity = O(nd · κ log 1ε)


– O(

e−ρ2t)

quadratic rate ⇔ complexity = O((nd2 + d3) · log log 1ε)

164







– O(

e−ρ2t)


• Key insights for machine learning (Bottou and Bousquet, 2008)

1. No need to optimize below statistical error

2. Cost functions are averages

3. Testing error is more important than training error

165







– O(

e−ρ2t)


• Key insights for machine learning (Bottou and Bousquet, 2008)

1. No need to optimize below statistical error

2. Cost functions are averages

3. Testing error is more important than training error

166

Stochastic gradient descent (SGD) for finite sums

minθ∈Rd

g(θ) =1

n

n∑

i=1

fi(θ)

• Iteration: θt = θt−1 − γtf′i(t)(θt−1)

– Sampling with replacement: i(t) random element of 1, . . . , n– Polyak-Ruppert averaging: θt =

1t+1

∑tu=0 θu

167

Stochastic gradient descent (SGD) for finite sums

minθ∈Rd

g(θ) =1

n

n∑

i=1

fi(θ)

• Iteration: θt = θt−1 − γtf′i(t)(θt−1)

– Sampling with replacement: i(t) random element of 1, . . . , n– Polyak-Ruppert averaging: θt =

1t+1

∑tu=0 θu

• Convergence rate if each fi is convex L-smooth and g µ-strongly-

convex:

Eg(θt)− g(θ∗) 6

O(1/√t) if γt = 1/(L

√t)

O(L/(µt)) = O(κ/t) if γt = 1/(µt)

– No adaptivity to strong-convexity in general

– Adaptivity with self-concordance assumption (Bach, 2013)

– Running-time complexity: O(d · κ/ε)168

Stochastic vs. deterministic methods

• Minimizing g(θ) =1

n

n∑

i=1

fi(θ) with fi(θ) = ℓ(

yi, h(xi, θ))

+ λΩ(θ)

169



n

n∑

i=1


yi, h(xi, θ))

+ λΩ(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−t/κ)

– Iteration complexity is linear in n

170



n

n∑

i=1


yi, h(xi, θ))

+ λΩ(θ)


γtn

n∑

i=1

f ′i(θt−1)

171



n

n∑

i=1


yi, h(xi, θ))

+ λΩ(θ)


γtn

n∑

i=1

f ′i(θt−1)

– Linear (e.g., exponential) convergence rate in O(e−t/κ)

– Iteration complexity is linear in n

• 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(κ/t)

– Iteration complexity is independent of n

172



n

n∑

i=1


yi, h(xi, θ))

+ λΩ(θ)


γtn

n∑

i=1

f ′i(θt−1)

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

173


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

Simple choice of step size

time

log(excess

cost)

deterministic

stochastic

174


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

Simple choice of step size

time

log(excess

cost)

deterministic

stochastic

new

175

Accelerating gradient methods - Related work

• Generic acceleration (Nesterov, 1983, 2004)

θt = ηt−1 − γtg′(ηt−1) and ηt = θt + δt(θt − θt−1)

176




– Good choice of momentum term δt ∈ [0, 1)

g(θt)− g(θ∗) 6 O(1/t2)

g(θt)− g(θ∗) 6 O(e−t√

µ/L) = O(e−t/√κ) if µ-strongly convex

– Optimal rates after t = O(d) iterations (Nesterov, 2004)

177




– Good choice of momentum term δt ∈ [0, 1)

g(θt)− g(θ∗) 6 O(1/t2)

g(θt)− g(θ∗) 6 O(e−t√

µ/L) = O(e−t/√κ) if µ-strongly convex

– Optimal rates after t = O(d) iterations (Nesterov, 2004)

– Still O(nd) iteration cost: complexity = O(nd · √κ log 1ε)

178


• Constant step-size stochastic gradient

– Solodov (1998); Nedic and Bertsekas (2000)

– Linear convergence, but only up to a fixed tolerance

179





• Stochastic methods in the dual (SDCA)

– Shalev-Shwartz and Zhang (2012)

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

– Extensions without duality: see Shalev-Shwartz (2016)

180





• Stochastic methods in the dual (SDCA)

– Shalev-Shwartz and Zhang (2012)

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

– Extensions without duality: see Shalev-Shwartz (2016)

• Stochastic version of accelerated batch gradient methods

– Tseng (1998); Ghadimi and Lan (2010); Xiao (2010)

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

181

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

182







n∑

i=1

yti with yti =


yt−1i otherwise

yt1

yt2

yt3

yt4

ytn−1

ytn

f1 f2 f3 f4 fnfn−1

1

n

∑n

i=1yti

g =1

n

∑n

i=1fifunctions

gradients ∈ Rd

183







n∑

i=1

yti with yti =


yt−1i otherwise

yt1

yt2

yt3

yt4

ytn−1

ytn


1

n

∑n

i=1yti

g =1

n

∑n

i=1fifunctions

gradients ∈ Rd

184







n∑

i=1

yti with yti =


yt−1i otherwise

yt1

yt2

yt3

yt4

ytn−1

ytn


1

n

∑n

i=1yti

g =1

n

∑n

i=1fifunctions

gradients ∈ Rd

185







n∑

i=1

yti with yti =


yt−1i otherwise

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

186







n∑

i=1

yti with yti =


yt−1i otherwise


• Extra memory requirement: n gradients in Rd in general

• Linear supervised machine learning: only n real numbers

– If fi(θ) = ℓ(yi,Φ(xi)⊤θ), then f ′

i(θ) = ℓ′(yi,Φ(xi)⊤θ)Φ(xi)

187

Stochastic average gradient - Convergence analysis

• Assumptions

– Each fi is L-smooth, i = 1, . . . , n - link with R2

– g= 1n

∑ni=1 fi is µ-strongly convex

– constant step size γt = 1/(16L) - no need to know µ

188

Stochastic average gradient - Convergence analysis

• Assumptions

– Each fi is L-smooth, i = 1, . . . , n - link with R2

– g= 1n

∑ni=1 fi is µ-strongly convex

– constant step size γt = 1/(16L) - no need to know µ

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

E[

g(θt)− g(θ∗)]

6 cst ×(

1−min 1

8n,

µ

16L

)t

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

– After one pass, reduction of cost by exp(

−min

18,

nµ16L

)

– NB: in machine learning, may often restrict to µ > L/n

⇒ constant error reduction after each effective pass

189

Running-time comparisons (strongly-convex)

• Assumptions: g(θ) = 1n

∑ni=1 fi(θ)

– Each fi convex L-smooth and g µ-strongly convex

Stochastic gradient descent d×∣

∣

∣

Lµ × 1

ε

Gradient descent d×∣

∣

∣nLµ × log 1

ε

Accelerated gradient descent d×∣

∣

∣n√

Lµ × log 1

ε

SAG d×∣

∣

∣(n+ L

µ) × log 1ε

– NB-1: for (accelerated) gradient descent, L = smoothness constant of g

– NB-2: with non-uniform sampling, L = average smoothness constants of all fi’s

190

Running-time comparisons (strongly-convex)


∑ni=1 fi(θ)

– Each fi convex L-smooth and g µ-strongly convex


∣

∣

Lµ × 1

ε


∣

∣nLµ × log 1

ε


∣

∣n√

Lµ × log 1

ε

SAG d×∣

∣

∣(n+ L

µ) × log 1ε

• Beating two lower bounds (Nemirovsky and Yudin, 1983; Nesterov,

2004): with additional assumptions

(1) stochastic gradient: exponential rate for finite sums

(2) full gradient: better exponential rate using the sum structure

191

Running-time comparisons (non-strongly-convex)


∑ni=1 fi(θ)

– Each fi convex L-smooth

– Ill conditioned problems: g may not be strongly-convex (µ = 0)


∣

∣

∣

1/ε2


∣

∣

∣

n/ε


∣

∣

∣

n/√ε

SAG d×∣

∣

∣

∣

√n/ε

• Adaptivity to potentially hidden strong convexity

• No need to know the local/global strong-convexity constant

192


Implementation details and extensions

• Sparsity in the features

– Just-in-time updates ⇒ replace O(d) by number of non zeros

– See also Leblond, Pedregosa, and Lacoste-Julien (2016)

• Mini-batches

– Reduces the memory requirement + block access to data

• Line-search

– Avoids knowing L in advance

• Non-uniform sampling

– Favors functions with large variations

• See www.cs.ubc.ca/~schmidtm/Software/SAG.html

193

Experimental results (logistic regression)

quantum dataset rcv1 dataset

(n = 50 000, d = 78) (n = 697 641, d = 47 236)

AFG

L−BFGS

SGASG

IAG

AFG

L−BFGS

SG

ASG

IAG

194

Experimental results (logistic regression)

quantum dataset rcv1 dataset

(n = 50 000, d = 78) (n = 697 641, d = 47 236)

AFG

L−BFGS

SG

ASG

IAG

SAG−LS

AFGL−BFGS

SG

ASG

IAG

SAG−LS

195

Before non-uniform sampling

protein dataset sido dataset

(n = 145 751, d = 74) (n = 12 678, d = 4 932)

IAG

AGD

L−BFG

S

SG−C

ASG−C

PCD−L

DCA

SAG

IAG

AGD

L−BFGS

SG−CASG−C

PCD−L

DCA

SAG

196

After non-uniform sampling

protein dataset sido dataset

(n = 145 751, d = 74) (n = 12 678, d = 4 932)

IAG

AGDL−BFGS SG−CASG−C

PCD−L

DC

A

SAG

SAG−LS (Lipschitz)

IAG AGD L−BFGS

SG−C ASG−CPCD−L

DCA

SAG

SAG−LS (Lipschitz)

197

Linearly convergent stochastic gradient algorithms

• Many related algorithms

– SAG (Le Roux, Schmidt, and Bach, 2012)

– SDCA (Shalev-Shwartz and Zhang, 2012)

– SVRG (Johnson and Zhang, 2013; Zhang et al., 2013)

– MISO (Mairal, 2015)

– Finito (Defazio et al., 2014a)

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

– · · ·

• Similar rates of convergence and iterations

– Different interpretations and proofs / proof lengths

– Lazy gradient evaluations

– Variance reduction

198

Linearly convergent stochastic gradient algorithms

• Many related algorithms

– SAG (Le Roux, Schmidt, and Bach, 2012)

– SDCA (Shalev-Shwartz and Zhang, 2012)

– SVRG (Johnson and Zhang, 2013; Zhang et al., 2013)

– MISO (Mairal, 2015)

– Finito (Defazio et al., 2014a)

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

– · · ·

• Similar rates of convergence and iterations

• Different interpretations and proofs / proof lengths

– Lazy gradient evaluations

– Variance reduction

199

Variance reduction

• Principle: reducing variance of sample of X by using a sample from

another random variable Y with known expectation

Zα = α(X − Y ) + EY

– EZα = αEX + (1− α)EY

– var(Zα) = α2[

var(X) + var(Y )− 2cov(X,Y )]

– α = 1: no bias, α < 1: potential bias (but reduced variance)

– Useful if Y positively correlated with X

200

Variance reduction

• Principle: reducing variance of sample of X by using a sample from

another random variable Y with known expectation

Zα = α(X − Y ) + EY

– EZα = αEX + (1− α)EY

– var(Zα) = α2[

var(X) + var(Y )− 2cov(X,Y )]

– α = 1: no bias, α < 1: potential bias (but reduced variance)

– Useful if Y positively correlated with X

• Application to gradient estimation (Johnson and Zhang, 2013;

Zhang, Mahdavi, and Jin, 2013)

– SVRG: X = f ′i(t)(θt−1), Y = f ′

i(t)(θ), α = 1, with θ stored

– EY = 1n

∑ni=1 f

′i(θ) full gradient at θ,X − Y = f ′

i(t)(θt−1)− f ′i(t)(θ)

201

Stochastic variance reduced gradient (SVRG)

(Johnson and Zhang, 2013; Zhang et al., 2013)

• Initialize θ ∈ Rd

• For iepoch = 1 to # of epochs

– Compute all gradients f ′i(θ) ; store g

′(θ) = 1n

∑ni=1 f

′i(θ)

– Initialize θ0 = θ

– For t = 1 to length of epochs

θt = θt−1 − γ[

g′(θ) +(

f ′i(t)(θt−1)− f ′

i(t)(θ))

]

– Update θ = θt

• Output: θ

– No need to store gradients - two gradient evaluations per inner step

– Two parameters: length of epochs + step-size γ

– Same linear convergence rate as SAG, simpler proof

202

Stochastic variance reduced gradient (SVRG)

(Johnson and Zhang, 2013; Zhang et al., 2013)

• Initialize θ ∈ Rd

• For iepoch = 1 to # of epochs

– Compute all gradients f ′i(θ) ; store g

′(θ) = 1n

∑ni=1 f

′i(θ)

– Initialize θ0 = θ

– For t = 1 to length of epochs

θt = θt−1 − γ[

g′(θ) +(

f ′i(t)(θt−1)− f ′

i(t)(θ))

]

– Update θ = θt

• Output: θ

– No need to store gradients - two gradient evaluations per inner step

– Two parameters: length of epochs + step-size γ

– Same linear convergence rate as SAG, simpler proof

203

Interpretation of SAG as variance reduction

• SAG update: θt = θt−1−γ

n

n∑

i=1

yti with yti =


yt−1i otherwise

– Interpretation as lazy gradient evaluations

204



n

n∑

i=1

yti with yti =


yt−1i otherwise


• SAG update: θt = θt−1 − γ[

1n

∑ni=1 y

t−1i + 1

n

(

f ′i(t)(θt−1)− yt−1

i(t)

)

]

– Biased update (expectation w.r.t. to i(t) not equal to full gradient)

205



n

n∑

i=1

yti with yti =


yt−1i otherwise



1n

∑ni=1 y

t−1i + 1

n

(

f ′i(t)(θt−1)− yt−1

i(t)

)

]


• SVRG update: θt = θt−1−γ[

1n

∑ni=1 f

′i(θ)+

(

f ′i(t)(θt−1)−f ′

i(t)(θ))

]

– Unbiased update

206



n

n∑

i=1

yti with yti =


yt−1i otherwise



1n

∑ni=1 y

t−1i + 1

n

(

f ′i(t)(θt−1)− yt−1

i(t)

)

]



1n

∑ni=1 f

′i(θ)+

(

f ′i(t)(θt−1)−f ′

i(t)(θ))

]

– Unbiased update

• SAGA update: θt = θt−1 − γ[

1n

∑ni=1 y

t−1i +

(

f ′i(t)(θt−1)− yt−1

i(t)

)

]

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

– Unbiased update without epochs

207

SVRG vs. SAGA

• SAGA update: θt = θt−1 − γ[

1n

∑ni=1 y

t−1i +

(

f ′i(t)(θt−1)− yt−1

i(t)

)

]


1n

∑ni=1 f

′i(θ)+

(

f ′i(t)(θt−1)−f ′

i(t)(θ))

]

SAGA SVRG

Storage of gradients yes no

Epoch-based no yes

Parameters step-size step-size & epoch lengths

Gradient evaluations per step 1 at least 2

Adaptivity to strong-convexity yes no

Robustness to ill-conditioning yes no

– See Babanezhad et al. (2015)

208

Proximal extensions

• Composite optimization problems: minθ∈Rd

1

n

n∑

i=1

fi(θ)+h(θ)

– fi smooth and convex

– h convex, potentially non-smooth

209

Proximal extensions


1

n

n∑

i=1

fi(θ)+h(θ)



– Constrained optimization: h(θ) = 0 if θ ∈ K, and +∞ otherwise

– Sparsity-inducing norms, e.g., h(θ) = ‖θ‖1

210

Proximal extensions


1

n

n∑

i=1

fi(θ)+h(θ)





• Proximal methods (a.k.a. splitting methods)

– Extra projection / soft thresholding step after gradient update

– See, e.g., Combettes and Pesquet (2011); Bach, Jenatton, Mairal,

and Obozinski (2012b); Parikh and Boyd (2014)

211

Proximal extensions


1

n

n∑

i=1

fi(θ)+h(θ)





• Proximal methods (a.k.a. splitting methods)

– Extra projection / soft thresholding step after gradient update

– See, e.g., Combettes and Pesquet (2011); Bach, Jenatton, Mairal,

and Obozinski (2012b); Parikh and Boyd (2014)

• Directly extends to variance-reduced gradient techniques

– Same rates of convergence

212

Acceleration

• Similar guarantees for finite sums: SAG, SDCA, SVRG (Xiao and

Zhang, 2014), SAGA, MISO (Mairal, 2015)


∣

∣nLµ × log 1

ε


∣

∣n√

Lµ × log 1

ε

SAG(A), SVRG, SDCA, MISO d×∣

∣

∣(n+ L

µ) × log 1ε

213

Acceleration

• Similar guarantees for finite sums: SAG, SDCA, SVRG (Xiao and

Zhang, 2014), SAGA, MISO (Mairal, 2015)


∣

∣nLµ × log 1

ε


∣

∣n√

Lµ × log 1

ε

SAG(A), SVRG, SDCA, MISO d×∣

∣

∣(n+ L

µ) × log 1ε

Accelerated versions d×∣

∣

∣

∣

(n+√

nLµ) × log 1

ε

• Acceleration for special algorithms (e.g., Shalev-Shwartz and

Zhang, 2014; Nitanda, 2014; Lan, 2015)

• Catalyst (Lin, Mairal, and Harchaoui, 2015)

– Widely applicable generic acceleration scheme

214

From training to testing errors

• rcv1 dataset (n = 697 641, d = 47 236)

– NB: IAG, SG-C, ASG with optimal step-sizes in hindsight

Training cost Testing cost

Effective Passes

0 10 20 30 40 50

Objective minus Optimum

10-20

10-15

10-10

10-5

100

AGDL-BFGS

SG-CASG

IAG

SAG

215

From training to testing errors

• rcv1 dataset (n = 697 641, d = 47 236)

– NB: IAG, SG-C, ASG with optimal step-sizes in hindsight

Training cost Testing cost

Effective Passes

0 10 20 30 40 50

Objective minus Optimum

10-20

10-15

10-10

10-5

100

AGDL-BFGS

SG-CASG

IAG

SAG

Effective Passes

0 10 20 30 40 50

Test Logistic Loss

105

0

0.5

1

1.5

2

2.5

AGD

L-BFGS

SG-C

ASG

IAG

SAG

216

SGD minimizes the testing cost!

• Goal: minimize f(θ) = Ep(x,y)ℓ(y, θ⊤Φ(x))

– Given n independent samples (xi, yi), i = 1, . . . , n from p(x, y)

– Given a single pass of stochastic gradient descent

– Bounds on the excess testing cost Ef(θn)− infθ∈Rd f(θ)

217






• Optimal convergence rates: O(1/√n) and O(1/(nµ))

– Optimal for non-smooth losses (Nemirovsky and Yudin, 1983)

– Attained by averaged SGD with decaying step-sizes

218






• Optimal convergence rates: O(1/√n) and O(1/(nµ))

– Optimal for non-smooth losses (Nemirovsky and Yudin, 1983)

– Attained by averaged SGD with decaying step-sizes

• Constant-step-size SGD

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

problems (Solodov, 1998; Nedic and Bertsekas, 2000)

– Full convergence and robustness to ill-conditioning?

219

Robust averaged stochastic gradient

(Bach and Moulines, 2013)

• Constant-step-size SGD is convergent for least-squares

– Convergence rate in O(1/n) without any dependence on µ

– Simple choice of step-size (equal to 1/L)

! " #

!$%

!$&

!$#

!$"

!

!$"

'()*!+,-

'()*!./+

/+-0

*1

*1,*1"

345

- Convergence in O(1/n) for smooth losses with O(d) online Newton

step220

Robust averaged stochastic gradient

(Bach and Moulines, 2013)

• Constant-step-size SGD is convergent for least-squares

– Convergence rate in O(1/n) without any dependence on µ

– Simple choice of step-size (equal to 1/L)

! " #

!$%

!$&

!$#

!$"

!

!$"

'()*!+,-

'()*!./+

/+-0

*1

*1,*1"

345

• Convergence in O(1/n) for smooth losses with O(d) online Newton

step221

Conclusions - variance reduction

• Linearly-convergent stochastic gradient methods

– Provable and precise rates

– Improves on two known lower-bounds (by using structure)

– Several extensions / interpretations / accelerations

222






• Extensions and future work

– Extension to saddle-point problems (Balamurugan and Bach, 2016)

– Lower bounds for finite sums (Agarwal and Bottou, 2014; Lan,

2015; Arjevani and Shamir, 2016)

– Sampling without replacement (Gurbuzbalaban et al., 2015;

Shamir, 2016)

223






• Extensions and future work

– Extension to saddle-point problems (Balamurugan and Bach, 2016)

– Lower bounds for finite sums (Agarwal and Bottou, 2014; Lan,

2015; Arjevani and Shamir, 2016)

– Sampling without replacement (Gurbuzbalaban et al., 2015;

Shamir, 2016)

– Bounds on testing errors for incremental methods (Frostig et al.,

2015; Babanezhad et al., 2015)

224

Frank-Wolfe - conditional gradient - I

• Goal: minimize smooth convex function f(θ) on compact set C

• Standard result: accelerated projected gradient descent with optimal

rate O(1/t2)

– Requires projection oracle: argminθ∈C12‖θ − η‖2

• Only availability of the linear oracle: argminθ∈C θ⊤η

– Many examples (sparsity, low-rank, large polytopes, etc.)

– Iterative Frank-Wolfe algorithm (see, e.g., Jaggi, 2013, and

references therein) with geometric interpretation (see board)

θt ∈ argminθ∈C

θ⊤f ′(θt−1)

θt = (1− ρt)θt−1 + ρtθt

225

Frank-Wolfe - conditional gradient - II

• Convergence rates: f(θt)− f(θ∗) 62Ldiam(C)2

t+ 1

Iteration:

θt ∈ argminθ∈C

θ⊤f ′(θt−1)

θt = (1− ρt)θt−1 + ρtθt

From smoothness: f(θt) 6 f(θt−1)+f ′(θt−1)⊤[ρt(θt−θt−1)

]

+L

2

∥

∥ρt(θt−θt−1)∥

∥

2

From compactness: f(θt) 6 f(θt−1) + f ′(θt−1)⊤[ρt(θt − θt−1)

]

+L

2ρ2tdiam(C)2

From convexity, f(θt)− f(θ∗) 6 f ′(θt−1)⊤(θt−1− θ∗) 6 max

θ∈Cf ′(θt−1)

⊤(θt−1− θ),

which is equal to f ′(θt−1)⊤(θt−1 − θt)

Thus, f(θt) 6 f(θt−1)− ρt[

f(θt−1)− f(θ∗)]

+L

2ρ2tdiam(C)2

With ρt = 2/(t+ 1): f(θt) 62Ldiam(C)2

t+1 by direct expansion

226

Frank-Wolfe - conditional gradient - II

• Convergence rates: f(θt)− f(θ∗) 62Ldiam(C)2

t

• Remarks and extensions

– Affine-invariant algorithms

– Certified duality gaps and dual interpretations (Bach, 2015)

– Adapted to very large polytopes

– Line-search extensions: minimize quadratic upper-bound

– Stochastic extensions (Lacoste-Julien et al., 2013)

– Away and pairwise steps to avoid oscillations (Lacoste-Julien and

Jaggi, 2015)

227

Outline - II

1. Introduction












228

Outline - II










• Frank-Wolfe


229

Subgradient descent for machine learning



– 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)

230


• Assumptions


– θ∗ global optimum of f on ‖θ‖2 6 D


(

θn−1 −2D


)

• Bound:

Ef

(

1

n

n−1∑

k=0

θk

)

− f(θ∗) 62DB√

n

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


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

231









• 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

)

232



[

(yn − 〈Φ(xn), θ〉)2]

with θ ∈ Rd




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)

233








• 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)

234







n∑

i=1

yti with yti =


yt−1i otherwise


• 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

235











µ/L)


finite sum: n/t finite sum: exp(−t/(n+L/µ))

quadratic deterministic: LD2/t2 deterministic: exp(−t√

µ/L)


236

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 (see, e.g. Dieuleveut and

Bach, 2014)

– Characterization of implicit regularization of online methods

– Structured prediction

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

– Parallel and distributed optimization

– Non-convex optimization (see, e.g. Reddi et al., 2016)

237

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