Stochastic Optimization Introduction + Sparse...

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Stochastic OptimizationIntroduction + Sparse regularization + Convex analysis

† ‡ Taiji Suzuki

†Tokyo Institute of TechnologyGraduate School of Information Science and EngineeringDepartment of Mathematical and Computing Sciences

‡JST, PRESTO

Intensive course @ Nagoya University

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Outline

1 Introduction

2 Short course to convex analysisConvexity and related conceptsDualitySmoothness and strong convexity

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Lecture plan

Day 1:

Convex analysisFirst order method“Online” stochastic optimization method: SGD, SRDA

Day 2:

AdaGrad, acceleration of SGD“Batch” stochastic optimization method: SDCA, SVRG, SAGDistributed optimization (if possible)

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Outline

1 Introduction

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Machine learning as optimization

Machine learning is a methodology to deal with a lot of uncertain data.

Generalization error minimization minθ∈Θ

EZ [ℓθ(Z )]

Empirical approximation minθ∈Θ

n∑i=1

ℓθ(zi )

Stochastic optimization is an intersection of learning and optimization.5 / 55

New data inputMassive data

x1x2x3x4....

Recently stochastic optimization is used to treat huge data.

n∑i=1

ℓθ(zi )︸︷︷︸Huge

+ψ(θ)

How to optimize this in efficient way?Do we need to go through the whole data at every iteration? 6 / 55

History of stochastic optimization for ML

1951 Robbins and Monro Stochastic approximationfor root finding problem

1957 Rosenblatt Perceptron

1978 Nemirovskii and Yudin Robustification for non-smooth obj.1983 and optimality

1988 Ruppert Robust step size policy and averaging1992 Polyak and Juditsky for smooth obj.

1998 Bottou Online stochastic optimization2004 Bottou and LeCun for large scale ML task

2009-2012

Singer and Duchi; Duchi

et al.; Xiao

FOBOS, AdaGrad, RDA

2012- Le Roux et al. Linear convergence on batch data2013 Shalev-Shwartz and Zhang (SAG,SDCA,SVRG)

Johnson and Zhang7 / 55

Overview of stochastic optimization

Stochastic approximation (SA)Optimization for systems with uncertainty,e.g., machine control, traffic management, social science, and so on.gt = ∇f (x (t)) + ξt is observed where ξt is noise (typically i.i.d.).

Stochastic approximation for machine learning and statisticsTypically generalization error minimization:

f (x) = minx

EZ [ℓ(Z , x)].

ℓ(z , x) is a loss function:e.g., logistic loss ℓ((w , y), x) = log(1 + exp(−yw⊤x)) forz = (w , y) ∈ Rp × {±1}.gt = ∇ℓ(zt , x (t)) is observed where zt ∼ P(Z ) is i.i.d. data.Used for huge dataset.We don’t need exact optimization. Optimization with certainprecision (typically O(1/n)) is sufficient.

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Two types of stochastic optimization

Online type stochastic optimization:

We observe data sequentially.Each observation is used just once (basically).

EZ [ℓ(Z , x)]

Batch type stochastic optimization

The whole sample has been already observed.We may use training data multiple times.

n∑i=1

ℓ(zi , x)

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Summary of convergence rates

Online methods (expected risk minimization):GR√T

(non-smooth, non-strongly convex)

µT(non-smooth, strongly convex)

σR√T

T 2(smooth, non-strongly convex)

µT+ exp

(−√µ

)(smooth, strongly convex)

Batch methods (empirical risk minimization)

exp(− 1

n+µLT)(smooth loss, strongly convex reg)

(− 1

)(smooth loss, strongly convex reg with acceleration)

G : upper bound of norm of gradient, R: diameter of the domain,L: smoothness, µ: strong convexity, σ: variance of the gradient

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Example of empirical risk minimization:High dimensional data analysis

Redundant information deteriorates the estimation accuracy.

Bio-informatics Text data Image data11 / 55

Example of empirical risk minimization:High dimensional data analysis

Redundant information deteriorates the estimation accuracy.

Bio-informatics Text data Image data11 / 55

Sparse estimation

Cut off redundant information → sparsity

R. Tsibshirani (1996). Regression shrinkage and selection via the lasso. J. Royal.

Statist. Soc B., Vol. 58, No. 1, pages 267–288.

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Variable selection (linear regression)

Design matrix X = (Xij) ∈ Rn×p.p (dimension) ≫ n (number of samples).The true vector β∗ ∈ Rp: At most d non-zero elements (sparse).

Linear model : Y = Xβ∗ + ξ.

Estimate β∗ from (Y ,X )．The number of parameters that we need to estimate is d → variableselection.

AIC:βAIC = argmin

β∈Rp∥Y − Xβ∥2 + 2σ2∥β∥0

where ∥β∥0 = |{j | βj = 0}|.→ 2p candidates. NP-hard → Convex approximation．

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Variable selection (linear regression)

Design matrix X = (Xij) ∈ Rn×p.p (dimension) ≫ n (number of samples).The true vector β∗ ∈ Rp: At most d non-zero elements (sparse).

Linear model : Y = Xβ∗ + ξ.

Estimate β∗ from (Y ,X )．The number of parameters that we need to estimate is d → variableselection.

AIC:βAIC = argmin

β∈Rp∥Y − Xβ∥2 + 2σ2∥β∥0

where ∥β∥0 = |{j | βj = 0}|.→ 2p candidates. NP-hard → Convex approximation．

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Lasso estimator

Lasso [L1 regularization]

βLasso = argminβ∈Rp

∥Y − Xβ∥2 + λ∥β∥1

where ∥β∥1 =∑p

j=1 |βj |.

→ Convex optimization！L1-norm is the convex hull of L0-normon [−1, 1]p (the largest convex functionwhich supports from below).

L1-norm is the Lovasz extension of thecardinality function.

More generally for a loss function ℓ (logistic loss, hinge loss, ...)

ℓ(zi , x) + λ∥x∥1

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Lasso estimator

Lasso [L1 regularization]

∥Y − Xβ∥2 + λ∥β∥1

where ∥β∥1 =∑p

j=1 |βj |.

→ Convex optimization！L1-norm is the convex hull of L0-normon [−1, 1]p (the largest convex functionwhich supports from below).

L1-norm is the Lovasz extension of thecardinality function.

More generally for a loss function ℓ (logistic loss, hinge loss, ...)

ℓ(zi , x) + λ∥x∥1

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Sparsity of Lasso estimator

Suppose p = n and X = I .

2∥Y − β∥2 + C∥β∥1

⇒ βLasso,i = argminb∈R

2(yi − b)2 + C |b|

{sign(yi )(yi − C ) (|yi | > C )

0 (|yi | ≤ C ).

Small signal is shrunk to 0→ sparse !

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Sparsity of Lasso estimator (fig)

β = argminβ∈Rp

n∥Xβ − Y ∥22 + λn

p∑j=1

|βj |.

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Example

Y = Xβ + ϵ.

n = 1, 000, p = 10, 000, d = 500.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

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Example

Y = Xβ + ϵ.

n = 1, 000, p = 10, 000, d = 500.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-2

TrueLasso

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Example

Y = Xβ + ϵ.

n = 1, 000, p = 10, 000, d = 500.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-2

TrueLassoLeastSquares

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Benefit of sparsity

β = argminβ∈Rp

n∥Xβ − Y ∥22 + λn

p∑j=1

|βj |.

Theorem (Lasso’s convergence rate)

Under some conditions，there exists a constant C such that

∥β − β∗∥22 ≤ Cd log(p)

※ The overall dimension p affects just in O(log(p)) !The actual dimension d is dominant．

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Extensions of sparse regularization

β = argminβ∈Rp

n∥Xβ − Y ∥22 + λn

p∑j=1

|βj |

β = argminβ∈Rp

nℓ(yi , x

⊤i β) + ψ(β)

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Examples

Overlapped group lasso

ψ(β) = C∑g∈G

∥βg∥

The groups may overlap.More aggressive sparsity.

Genome Wide Association Study (GWAS)

(Balding ‘06, McCarthy et al. ‘08)20 / 55

Application of group reg. (1)

Multi-task learning (Lounici et al., 2009)

Estimate simultaneously across T tasks:

y(t)i = x

(t)⊤i β(t) + ϵ

(t)i (i = 1, . . . , n(t), t = 1, . . . ,T ).

minβ(t)

T∑t=1

n(t)∑i=1

(yi − x(t)⊤i β(t))2 + C

p∑k=1

∥(β(1)k , . . . , β(T )k )∥︸︷︷︸

Group regularization

β(1)β(2) β(T)

Select non-zero elements across tasks21 / 55

Multi-task learning (Lounici et al., 2009)

Estimate simultaneously across T tasks:

y(t)i = x

(t)⊤i β(t) + ϵ

(t)i (i = 1, . . . , n(t), t = 1, . . . ,T ).

minβ(t)

T∑t=1

n(t)∑i=1

(yi − x(t)⊤i β(t))2 + C

p∑k=1

∥(β(1)k , . . . , β(T )k )∥︸︷︷︸

Group regularization

β(1)β(2) β(T)

Select non-zero elements across tasks21 / 55

Sentence regularization for text classification (Yogatama and Smith,

The words occurred in the same sentence is grouped:

ψ(β) =D∑

Sd∑s=1

λd ,s∥β(d ,s)∥2,

(d expresses a document, s expresses a sentence).

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Trace norm regularization

W : M × N matrix.

∥W ∥Tr = Tr[(WW⊤)12 ] =

min{M,N}∑j=1

σj(W )

σj(W ) is the j-th singular value of W (non-negative)．

Sum of singular values = L1-regularization on singular values→ Singular values are sparse

Sparse singular values = Low rank

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Application of trace norm reg.:Recommendation system

Assuming the rank is 1.

Movie A Movie B Movie C · · · Movie X

User 1 4 8 * · · · 2

User 2 2 * 2 · · · *

User 3 2 4 * · · · *

(e.g., Srebro et al. (2005), NetFlix Bennett and Lanning (2007))

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Assuming the rank is 1.

Movie A Movie B Movie C · · · Movie X

User 1 4 8 4 · · · 2

User 2 2 4 2 · · · 1

User 3 2 4 2 · · · 1

(e.g., Srebro et al. (2005), NetFlix Bennett and Lanning (2007))

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→ Low rank matrix completion:

Rademacher complexity of low rank matrices: Srebro et al. (2005).Compressed sensing: Candes and Tao (2009), Candes and Recht(2009).

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Example: Reduced rank regression

Reduced rank regression (Anderson, 1951, Burket, 1964, Izenman, 1975)

Multi-task learning (Argyriou et al., 2008)

Reduced rank regression

W ∗ is low rank.

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(Generalized) Fused Lasso

ψ(β) = C∑

(i ,j)∈E

|βi − βj |.

(Tibshirani et al. (2005), Jacob et al. (2009))

Genome data analysis by Fused lasso (Tibshi-

rani and Taylor ‘11)

TV-denoising (Chambolle ‘04)

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Sparse covariance selection

xk ∼ N(0,Σ) (i.i.d.,Σ ∈ Rp×p), Σ = 1n

∑nk=1 xkx

⊤k .

S = argminS⪰O

{− log(det(S)) + Tr[SΣ] + λ

p∑i ,j=1

|Si ,j |}.

(Meinshausen and B uhlmann, 2006, Yuan and Lin, 2007, Banerjee et al., 2008)

Estimating the inverse S of Σ.Si ,j = 0 ⇔ X(i),X(j) is conditionally independent.Gaussian graphical model can be estimated by convex optimization.

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Covariance selection on the stock data of 50 randomly selected companiesin NASDAQ list from 4 January 2011 to 31 December 2014.

(Lie Michael, Bachelor thesis)

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Other examples

Robust PCA (Candes et al. 2009).

Low rank tensor estimation (Signoretto et al., 2010; Tomioka et al.,2011).

Dictionary learning (Kasiviswanathan et al., 2012; Rakotomamonjy,2013).

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Outline

1 Introduction

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Regularized empirical risk minimization

Basically, we want to solve

Empirical risk minimization:

minx∈Rp

n∑i=1

ℓ(zi , x).

Regularized empirical risk minimization:

minx∈Rp

n∑i=1

ℓ(zi , x) + ψ(x).

In this lecture, we assume ℓ and ψ are convex.→ convex analysis to exploit the properties of convex functions.

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Outline

1 Introduction

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Convex set

Definition (Convex set)

A convex set is a set that contains the segment connecting two points inthe set:

x1, x2 ∈ C =⇒ θx1 + (1− θ2)x2 ∈ C (θ ∈ [0, 1]).

Convex set Non-convex set Non-convex set

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Epigraph and domainLet R := R ∪ {∞}.

Definition (Epigraph and domain)

The epigraph of a function f : Rp → R is given by

epi(f ) := {(x , µ) ∈ Rp+1 : f (x) ≤ µ}.

The domain of a function f : Rp → R is given by

dom(f ) := {x ∈ Rp : f (x) <∞}.

epigraph

domain( ]

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Convex function

Let R := R ∪ {∞}.

Definition (Convex function)

A function f : Rp → R is a convex function if f satisfies

θf (x) + (1− θ)f (y) ≥ f (θx + (1− θ)y) (∀x , y ∈ Rp, θ ∈ [0, 1]),

where ∞+∞ = ∞, ∞ ≤ ∞.

Convex Non-convex

f is convex ⇔ epi(f ) is a convex set.36 / 55

Proper and closed convex function

If the domain of a function f is not empty (dom(f ) = ∅), f is calledproper.

If the epigraph of a convex function f is a closed set, then f is calledclosed.(We are interested in only a proper closed function in this lecture.)

Even if f is closed, it’s domain is not necessarily closed (even for 1D).

“f is closed” does not imply“f is continuous.”

Closed convex function is continuous on a segment in its domain.

Closed function is “lower semicontinuity.”

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Convex loss functions (regression)

All well used loss functions are (closed) convex. The followings are convexw.r.t. u with a fixed label y ∈ R.

Squared loss: ℓ(y , u) = 12(y − u)2.

τ -quantile loss: ℓ(y , u) = (1− τ)max{u − y , 0}+ τ max{y − u, 0}.for some τ ∈ (0, 1). Used for quantile regression.ϵ-sensitive loss: ℓ(y , u) = max{|y − u| − ϵ, 0} for some ϵ > 0. Usedfor support vector regression.

f-y-3 -2 -1 0 1 2 30

3 τ-quantile

-sensitive

Squared

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Convex surrogate loss (classification)

y ∈ {±1}Logistic loss: ℓ(y , u) = log((1 + exp(−yu))/2).Hinge loss: ℓ(y , u) = max{1− yu, 0}.Exponential loss: ℓ(y , u) = exp(−yu).Smoothed hinge loss:

ℓ(y , u) =

0, (yu ≥ 1),12 − yu, (yu < 0),12(1− yu)2, (otherwise).

yf-3 -2 -1 0 1 2 30

40-1Logistic

expHinge

Smoothed-hinge

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Convex regularization functions

Ridge regularization: R(x) = ∥x∥22 :=∑p

j=1 x2j .

L1 regularization: R(x) = ∥x∥1 :=∑p

j=1 |xj |.

Trace norm regularization: R(X ) = ∥X∥tr =∑min{q,r}

k=1 σj(X )where σj(X ) ≥ 0 is the j-th singular value.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

Bridge (=0.5)L1RidgeElasticnet

∑ni=1(yi − z⊤i x)2 + λ∥x∥1: Lasso

∑ni=1 log(1 + exp(−yiz

⊤i x)) + λ∥X∥tr: Low rank matrix recovery

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Other definitions of sets

Convex hull: conv(C ) is the smallest convex set that contains a setC ⊆ Rp.

Affine set: A set A is an affine set if and only if ∀x , y ∈ A, the line thatintersects x and y lies in A: λx + (1− λ)y ∀λ ∈ R.

Affine hull: The smallest affine set that contains a set C ⊆ Rp.Relative interior: ri(C ). Let A be the affine hull of a convex set C ⊆ Rp.

ri(C ) is a set of internal points with respect to the relativetopology induced by the affine hull A.

Convex hullAffine hull

Relative interior

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Continuity of a closed convex function

Theorem

For a (possibly non-convex) function f : Rp → R, the following threeconditions are equivalent to each other.

1 f is lower semi-continuous.

2 For any converging sequence {xn}∞n=1 ⊆ Rp s.t. x∞ = limn xn,lim infn f (xn) ≥ f (x∞).

3 f is closed.

Remark: Any convex function f is continuous in ri(dom(f )). The continuity

could be broken on the boundary of the domain.

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Outline

1 Introduction

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Subgradient

We want to deal with non-differentiable function such as L1 regularization.To do so, we need to define something like gradient.

Definition (Subdifferential, subgradient)

For a proper convex function f : Rp → R, the subdifferential of f atx ∈ dom(f ) is defined by

∂f (x) := {g ∈ Rp | ⟨x ′ − x , g⟩+ f (x) ≤ f (x ′) (∀x ′ ∈ Rp)}.

An element of the subdifferential is called subgradient.

Subgradient

Figure: Subgraient

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Properties of subgradient

Subgradient does not necessarily exist (∂f (x) could be empty).f (x) = x log(x) (x ≥ 0) is proper convex but not subdifferentiable at x = 0.

Subgradient always exists on ri(dom(f )).

If f is differentiable at x , its gradient is the unique element of subdiff.

∂f (x) = {∇f (x)}.

If ri(dom(f )) ∩ ri(dom(h)) = ∅, then∂(f + h)(x) = ∂f (x) + ∂h(x)

= {g + g ′ | g ∈ ∂f (x), g ′ ∈ ∂h(x)}(∀x ∈ dom(f ) ∩ dom(h)).

For all g ∈ ∂f (x) and all g ′ ∈ ∂f (x ′) (x , x ′ ∈ dom(f )),

⟨g − g ′, x − x ′⟩ ≥ 0.

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Subgradient always exists on ri(dom(f )).If f is differentiable at x , its gradient is the unique element of subdiff.

∂f (x) = {∇f (x)}.

⟨g − g ′, x − x ′⟩ ≥ 0.

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∂f (x) = {∇f (x)}.

⟨g − g ′, x − x ′⟩ ≥ 0.

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∂f (x) = {∇f (x)}.

⟨g − g ′, x − x ′⟩ ≥ 0.

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Legendre transform

Defines the other representation on the dual space (the space of gradients).

Definition (Legendre transform)

Let f be a (possibly non-convex) function f : Rp → R s.t. dom(f ) = ∅.Its convex conjugate is given by

f ∗(y) := supx∈Rp

{⟨x , y⟩ − f (x)}.

The map from f to f ∗ is called Legendre transform.

line with gradient y

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Examples

f (x) f ∗(y)

Squared loss 12x2 1

Hinge loss max{1− x , 0}

{y (−1 ≤ y ≤ 0),

∞ (otherwise).

Logistic loss log(1 + exp(−x)){(−y) log(−y) + (1 + y) log(1 + y) (−1 ≤ y ≤ 0),

∞ (otherwise).

L1 regularization ∥x∥1

{0 (maxj |yj | ≤ 1),

∞ (otherwise).

Lp regularization∑d

j=1 |xj |p ∑d

j=1p−1

p−1|yj |

pp−1

(p > 1)

logistic dual of logistic

L1-norm dual

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Properties of Legendre transformf ∗ is a convex function even if f is not.f ∗∗ is the closure of the convex hull of f :

f ∗∗ = cl(conv(f )).

Corollary

Legendre transform is a bijection from the set of proper closed convexfunctions onto that defined on the dual space.

f (proper closed convex) ⇔ f ∗ (proper closed convex)

0 1 2 3 4

6f(x)cl.conv.

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Connection to subgradient

y ∈ ∂f (x) ⇔ f (x) + f ∗(y) = ⟨x , y⟩ ⇔ x ∈ ∂f ∗(y).

∵ y ∈ ∂f (x) ⇒ x = argmaxx ′∈Rp

{⟨x ′, y⟩ − f (x ′)}

(take the “derivative” of ⟨x ′, y⟩ − f (x ′))

⇒ f ∗(y) = ⟨x , y⟩ − f (x).

Remark: By definition, we always have

f (x) + f ∗(y) ≥ ⟨x , y⟩.

→ Young-Fenchel’s inequality.

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⋆ Fenchel’s duality theorem

Theorem (Fenchel’s duality theorem)

Let f : Rp → R, g : Rq → R be proper closed convex, and A ∈ Rq×p.Suppose that either of condition (a) or (b) is satisfied, then it holds that

infx∈Rp

{f (x) + g(Ax)} = supy∈Rq

{−f ∗(A⊤y)− g∗(−y)}.

� �(a) ∃x ∈ Rp s.t. x ∈ ri(dom(f )) and Ax ∈ ri(dom(g)).

(b) ∃y ∈ Rq s.t. A⊤y ∈ ri(dom(f ∗)) and −y ∈ ri(dom(g∗)).� �If (a) is satisfied, there exists y∗ ∈ Rq that attains sup of the RHS.If (b) is satisfied, there exists x∗ ∈ Rp that attains inf of the LHS.Under (a) and (b), x∗, y∗ are the optimal solutions of the each side iff

A⊤y∗ ∈ ∂f (x∗), Ax∗ ∈ ∂g∗(−y∗).

→ Karush-Kuhn-Tucker condition.50 / 55

Equivalence to the separation theorem

Convex

Concave

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Applying Fenchel’s duality theorem to RERM

RERM (Regularized Empirical Risk Minimizatino):Let ℓi (z

⊤i x) = ℓ(yi , z

⊤i x) where (zi , yi ) is the input-output pair of the i-th

observation.� �(Primal) inf

x∈Rp

ℓi (z⊤i x)︸︷︷︸ + ψ(x)

f (Zx)� �[Fenchel’s duality theorem]

infx∈Rp

{f (Zx) + ψ(x)} = − infy∈Rn

{f ∗(y) + ψ∗(−Z⊤y)}

� �(Dual) sup

y∈Rn

ℓ∗i (yi ) + ψ∗(−Z⊤y)

}� �This fact will be used to derive dual coordinate descent alg. 52 / 55

Outline

1 Introduction

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Smoothness and strong convexity

Definition

Smoothness: the gradient is Lipschitz continuous:

∥∇f (x)−∇f (x ′)∥ ≤ L∥x − x ′∥.

Strong convexity: ∀θ ∈ (0, 1), ∀x , y ∈ dom(f ),

2θ(1− θ)∥x − y∥2 + f (θx + (1− θ)y) ≤ θf (x) + (1− θ)f (y).

Smooth butnot strongly convex

Smooth andStrongly convex

Strongly convex butnot smooth

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Duality between smoothness and strong convexity

Smoothness and strong convexity is in a relation of duality.

Theorem

Let f : Rp → R be proper closed convex.

f is L-smooth ⇐⇒ f ∗ is 1/L-strongly convex.

logistic loss its dual function

Smooth butnot strongly convex

Strongly convex butnot smooth

(gradient → ∞)

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