stochastic optimization part i: convex analysis and...

Stochastic OptimizationPart I: Convex analysis and

online stochastic optimization

† Taiji Suzuki

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

MLSS2015@Kyoto

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Outline

1 Introduction

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

3 First order methodProximal gradient descentNesterov’s acceleration and optimal convergence

4 Online stochastic optimizationStochastic gradient descentStochastic regularized dual averaging

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Lecture plan

Part I: introduction to stochastic optimization

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

Part II: faster stochastic gradient methods and “batch” methods

AdaGrad, acceleration of SGD“Batch” stochastic optimization method: SDCA, SVRG, SAG

Part III: stochastic ADMM and distributed optimization

Stochastic ADMM for structured sparsityDistributed optimization (very shortly)

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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

1

n

n∑i=1

ℓθ(zi )

Stochastic optimization is an intersection of learning and optimization.5 / 73

New data inputMassive data

x1x2x3x4....

Recently stochastic optimization is used to treat huge data.

1

n

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 / 73

History of stochastic optimization for ML

1951 Robbins and Monro Stochastic approximationfor root finding problem

1957 Rosenblatt Perceptron

1978 Nemirovskii and Yudin Robustification via averagingfor non-smooth obj.

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 / 73

Overview of stochastic optimization

minx

f (x)

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:

minx

f (x) = minx

EZ [ℓ(Z , x)].

gt = ∇ℓ(zt , x (t)) is observed where zt ∼ P(Z ) is i.i.d. data.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}.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).

minx

EZ [ℓ(Z , x)]

Batch type stochastic optimization

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

minx

1

n

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)

G 2

µT(non-smooth, strongly convex)

σR√T

+R2L

T 2(smooth, non-strongly convex)

σ2

µT+ exp

(−√µ

LT

)(smooth, strongly convex)

Batch methods (empirical risk minimization)

exp(− 1

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

exp

(− 1

n+√

nµL

T

)(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 / 73

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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Lasso estimator

Lasso [L1 regularization]

βLasso = argminβ∈Rp

∥Y − Xβ∥2 + λ∥β∥1

where ∥β∥1 =∑p

j=1 |βj |.

Convex optimization

L1norm is the convex hull of L0normon [−1, 1]p (the largest convex functionwhich supports from below).

L1norm is the Lovasz extension of thecardinality function.

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

minx

{n∑

i=1

ℓ(zi , x) + λ∥x∥1

}

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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

L1norm is the convex hull of L0normon [−1, 1]p (the largest convex functionwhich supports from below).

L1norm is the Lovasz extension of thecardinality function.

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

minx

{n∑

i=1

ℓ(zi , x) + λ∥x∥1

}13 / 73

Outline

1 Introduction




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Regularized empirical risk minimization

Basically, we want to solve

Empirical risk minimization:

minx∈Rp

1

n

n∑i=1

ℓ(zi , x).

Regularized empirical risk minimization:

minx∈Rp

1

n

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.19 / 73

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

1

2

3 τ-quantile

-sensitive

Squared

Huber

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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

1

2

3

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

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Bridge (=0.5)L1RidgeElasticnet

1n

∑ni=1(yi − z⊤i x)2 + λ∥x∥1: Lasso

1n

∑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.

f(x)

x

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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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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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.

f(x)

f*(y)

line with gradient y

xx*

0

f(x*)

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Examples

f (x) f ∗(y)

Squared loss 12x2 1

2y 2

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

pp

p−1|yj |

pp−1

(p > 1)

0

0 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

-6

-4

-2

0

2

4

6f(x)cl.conv.

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Connection to subgradient

Lemma

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.33 / 73

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

{n∑

i=1

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

{n∑

i=1

ℓ∗i (yi ) + ψ∗(−Z⊤y)

}� �This fact will be used to derive dual coordinate descent alg. 35 / 73

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 ),

µ

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

0 0 0

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

0

0 1

Smooth butnot strongly convex

Strongly convex butnot smooth

(gradient → ∞)

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Outline

1 Introduction




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Regularized learning problem

Lasso:

minx∈Rp

1

n

n∑i=1

(yi − z⊤i x)2 + ∥x∥1︸︷︷︸regularization

.

General regularized learning problem:

minx∈Rp

1

n

n∑i=1

ℓ(zi , x) + ψ(x).

Difficulty: Sparsity inducing regularization is usually non-smooth.

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First order optimization

xt-1xtxt+1

Optimization methods that use only the function value f (x) and thefirst order gradient g ∈ ∂f (x).Computation per iteration is light, and suited for high dimensionalproblems.Newton method is a second order method.

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Outline

1 Introduction




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Gradient descent

Let f (x) =∑n

i=1 ℓ(zi , x).minx

f (x).

Subgradient method

Differentiable f (x):xt = xt−1 − ηt∇f (xt−1).

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Gradient descent

Let f (x) =∑n


f (x).

Subgradient method

Subdifferentiable f (x):

gt ∈ ∂f (xt−1),

xt = xt−1 − ηtgt .

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Gradient descent

Let f (x) =∑n


f (x).

Subgradient method (equivalent formula)

Subdifferentiable f (x):

xt = argminx

{⟨x , gt⟩+

1

2ηt∥x − xt−1∥2

},

where gt ∈ ∂f (xt−1).

Proximal point algorithm:

xt = argminx

{f (x) +

1

2ηt∥x − xt−1∥2

}.

f (xt) → optimum for any convex f and ηt = η > 0 (Guler, 1991).

If f (x) is strongly convex: f (xt)− f (x∗) ≤ 12η

(1

1+ση

)t−1

∥x0 − x∗∥2.43 / 73

Proximal gradient descent

Let f (x) =∑n

i=1 ℓ(zi , x).

minx

f (x) + ψ(x).


xt = argminx

{⟨x , gt⟩+ ψ(x) +

1

2ηt∥x − xt−1∥2

}= argmin

x

{ηtψ(x) +

1

2∥x − (xt−1 − ηtgt)∥2

}where gt ∈ ∂f (xt−1).

The update rule is given by proximal mapping:

prox(q|ψ) = argminx

{ψ(x) +

1

2∥x − q∥2

}→ By using the proximal mapping, we can avoid bad properties (e.g.,non-smoothness) of ψ. 44 / 73

Example

L1 regularization: ψ(x) = C∥x∥1.

xt,j = STCηt (xt−1,j − ηtgt,j) (j-th component)

whereSTC (q) = sign(q)max{|q| − C , 0}.

→ Unimportant elements are forced to be 0.

For many practically used regularizations, analytic form is obtained.

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Example of proximal mapping (cont.)

Trace norm: ψ(X ) = C∥X∥tr = C∑

j σj(X ) (sum of singular values).Let

Xt−1 − ηtGt = U diag(σ1, . . . , σd)V ,

then

Xt = U

STCηt (σ1). . .

STCη(σd)

V .

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Convergence of proximal gradient descent

Strong convexity and smoothness of f determines the convergence rate.

xt = prox(xt−1 − ηtgt |ηtψ(x)).

property of f µ-Strongly convex non-strongly conv

γ-Smooth exp

(−t

µ

γ

)γ

t

Non-smooth1

µt

1√t

The step size ηt should be appropriately chosen.Setting of ηt Strongly conv non-strongly conv

Smooth 1γ

1γ

Non-smooth 2µt

1√t

To achieve this convergence rate, we need to take an average of {xt}tappropriately; Polyak-Ruppert averaging, polynomially decaying averaging.

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Convergence of proximal gradient descent

Strong convexity and smoothness of f determines the convergence rate.

xt = prox(xt−1 − ηtgt |ηtψ(x)).

property of f µ-Strongly convex non-strongly conv

γ-Smooth exp

(−t

√µ

γ

)γ

t2

Non-smooth1

µt

1√t

The step size ηt should be appropriately chosen.Setting of ηt Strongly conv non-strongly conv

Smooth 1γ

1γ

Non-smooth 2µt

1√t

To achieve this convergence rate, we need to take an average of {xt}tappropriately; Polyak-Ruppert averaging, polynomially decaying averaging.Convergence for smooth loss can be improved by Nesterov’sacceleration.→ Optimal rate

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Outline

1 Introduction




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Nesterov’s acceleration (non-strongly convex)minx{f (x) + ψ(x)}Assumption: f (x) is γ-smooth.

Nesterov’s acceleration scheme

Let s1 = 1 and η = 1γ , and iterate the following for t = 1, 2, . . .

1 Let gt ∈ ∂f (yt), and update xt = prox(yt − ηgt |ηψ).

2 Set st+1 =1+√

1+4s2t2 .

3 Update yt+1 = xt +(st−1st+1

)(xt − xt−1).

If f is γ-smooth, then

f (xt)− f (x∗) ≤ 2γ∥xt − x∗∥t2

.

This is also called Fast Iterative Shrinkage Thresholding Algorithm (FISTA)(Beck and Teboulle, 2009).

The step size η = 1/γ can be adaptively determined: back-trackingtechnique.

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Nesterov’s acceleration (strongly convex)minx{f (x) + ψ(x)}Assumption: f (x) is γ-smooth and µ-strongly convex. (it must be γ > µ)

Nesterov’s acceleration scheme

Let A1 = 1, α1 = γ/µ and η = 1γ , and iterate the following for

t = 1, 2, . . .

1 Let gt ∈ ∂f (yt), and update xt = prox(yt − ηgt |ηψ).2 Set αt+1 > 1 so that (γ − µ)α2

t+1 − (2γ + At)αt+1 + γ = 0, and letAt+1 = At/αt+1.

3 Update yt+1 = xt +(

µ+At

(γ−µ)(αt+1−1)(αt−1)

)(xt − xt−1).

If f is γ-smooth and µ-strongly convex, then

f (xt)− f (x∗) ≤ γ

(1−

√γ

µ

)t

∥x0 − x∗∥2.

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Iteration0 20 40 60 80 100 120

Rel

ativ

e ob

ject

ive

(f(x t)

- f* )

10-8

10-6

10-4

10-2

100

102

104

NormalNesterov

Nesterov’s acceleration v.s. normal gradient descentLasso: n = 8, 000, p = 500.

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Outline

1 Introduction




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Two types of stochastic optimization

Online type stochastic optimization:

We observe data sequentially.We don’t need to wait until the whole sample is obtained.Each observation is obtained just once (basically).

minx

EZ [ℓ(Z , x)]

Batch type stochastic optimization

The whole sample has been already observed.We can make use of the (finite and fixed) sample size.We may use sample multiple times.

minx

1

n

n∑i=1

ℓ(zi , x)

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Online method

You don’t need to wait until the whole sample arrives.Update the parameter at each data observation.

zt-1 z

txt

xt+1

P(z)

Observe ObserveUpdate Update

storage New data input

minx

E[ℓ(Z , x)] + ψ(x) ≃ minx

1

T

T∑t=1

ℓ(zt , x) + ψ(x)

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The objective of online stochastic optimization

Let ℓ(z , x) be a loss of x for an observation z .

(Expected loss) L(x) = EZ [ℓ(Z , x)]

or

(EL with regularization) Lψ(x) = EZ [ℓ(Z , x)] + ψ(x)

The distribution of Z could be

the true population→ L(x) is the generalization error.

an empirical distribution of stored data in a storage→ L (or Lψ) is the (regularized) empirical risk.

Online stochastic optimization itself is learning!

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Outline

1 Introduction




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Three steps to stochastic gradient descent

E[ℓ(Z , x)] =

∫ℓ(Z , x)dP(Z )

≃(1)ℓ(zt , xt−1) (sampling)

≃(2)

⟨∇xℓ(zt , xt−1), x⟩ (linearlization)

The approximation is correct just around xt−1.

minx

E[ℓ(Z , x)] ≃(3)

minx

{⟨∇xℓ(zt , xt−1), x⟩+

1

2ηt∥x − xt−1∥2

}(proximation)

xt-158 / 73

Stochastic gradient descent (SGD)

SGD (without regularization)

Observe zt ∼ P(Z ), and let ℓt(x) := ℓ(zt , x).

Calculate subgradient:

gt ∈ ∂xℓt(xt−1).

Update x asxt = xt−1 − ηtgt .

We just need to observe one training data zt at each iteration.→ O(1) computation per iteration (O(n) for batch gradient descent).

We do not need to go through the whole sample {zi}ni=1

Reminder: prox(q|ψ) := argminx{ψ(x) + 1

2∥x − q∥2}.

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Stochastic gradient descent (SGD)

SGD (with regularization)


Calculate subgradient:

gt ∈ ∂xℓt(xt−1).

Update x asxt = prox(xt−1 − ηtgt |ηtψ).

We just need to observe one training data zt at each iteration.→ O(1) computation per iteration (O(n) for batch gradient descent).

We do not need to go through the whole sample {zi}ni=1

Reminder: prox(q|ψ) := argminx{ψ(x) + 1

2∥x − q∥2}.

59 / 73

Convergence analysis of SGD

Assumption

(A1) E[∥gt∥2] ≤ G 2.

(A2) E[∥xt − x∗∥2] ≤ D2.

Theorem

Let xT = 1T+1

∑Tt=0 xt . For ηt =

η0√t, it holds

Ez1:T [Lψ(xT )− Lψ(x∗)] ≤ η0G

2 + D2/η0√T

.

For η0 =DG , we have

2GD√T.

This is minimax optimal (up to constant).

G is independent of ψ thanks to the proximal mapping. Note that∥∂ψ(x)∥ ≤ C

√p for L1-reg.

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Convergence analysis of SGD (strongly convex)

Assumption

(A1) E[∥gt∥2] ≤ G 2.

(A3) Lψ is µ-strongly convex.

Theorem

Let xT = 1T+1

∑Tt=0 xt . For ηt =

1µt , it holds

Ez1:T [Lψ(xT )− Lψ(x∗)] ≤ G 2 log(T )

Tµ.

Better than non-strongly convex situation.But, this is not minimax optimal.The bound is tight (Rakhlin et al., 2012).

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Polynomial averaging for strongly convex risk

Assumption

(A1) E[∥gt∥2] ≤ G 2.

(A3) Lψ is µ-strongly convex.

Modify the update rule as

xt = prox

(xt−1 − ηt

t

t + 1gt |ηtψ

),

and take the weighted average xT =2

(T + 1)(T + 2)

T∑t=0

(t + 1)xt .

Theorem

For ηt =2µt , it holds Ez1:T [Lψ(xT )− Lψ(x

∗)] ≤ 2G 2

Tµ.

log(T ) is removed.This is minimax optimal (explained later).

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Remark on polynomial averaging

xT =2

(T + 1)(T + 2)

T∑t=0

(t + 1)xt

O(T ) computation? No.xT can be efficiently updated:

xt =t

t + 2xt−1 +

2

t + 2xt .

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General step size and weighting policy

Let st (t = 1, 2, . . . ,T +1) be a positive sequence such that∑T+1

t=1 st = 1.

xt = prox

(xt−1 − ηt

stst+1

gt |ηtψ)

(t = 1, . . . ,T )

xT =T∑t=0

st+1xt .

Assumption: (A1) E[∥gt∥2] ≤ G 2, (A2) E[∥xt − x∗∥2] ≤ D2, (A3) Lψ is

µ-strongly convex.

Theorem

Ez1:T [Lψ(xT )− Lψ(x∗)]

≤T∑t=1

st+1ηt+1

2G 2 +

T−1∑t=0

max{ st+2

ηt+1− st+1(

1ηt

+ µ), 0}D2

2

As for t = 0, we set 1/η0 = 0 .64 / 73

Special case

Let the weight proportion to the step size (step size could be seen asimportance):

st =ηt∑T+1τ=1 ητ

.

In this setting, the previous theorem gives

Ez1:T [Lψ(xT )− Lψ(x∗)] ≤

∑Tt=1 η

2tG

2 + D2

2∑T

t=1 ηt� �∞∑t=1

ηt = ∞

∞∑t=1

η2t <∞

ensures the convergence.� �65 / 73

Outline

1 Introduction




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Stochastic regularized dual averaging (SRDA)

The second assumption E[∥xt − x∗∥2] ≤ D2 can be removed by using dualaveraging (Nesterov, 2009, Xiao, 2009).

SRDA


Calculate gradient: gt ∈ ∂xℓt(xt−1).

Take the average of the gradients:

gt =1

t

t∑τ=1

gτ .

Update as

xt = argminx∈Rp

{⟨gt , x⟩+ ψ(x) +

1

2ηt∥x∥2

}= prox(−ηt gt |ηtψ).

The information of old observations is maintained by taking average ofgradients.

67 / 73

Convergence analysis of SRDA

Assumption(A1) E[∥gt∥2] ≤ G 2.(A2) E[∥xt − x∗∥2] ≤ D2.

Theorem

Let xT = 1T+1

∑Tt=0 xt . For ηt = η0

√t, it holds

Ez1:T [Lψ(xT )− Lψ(x∗)] ≤ η0G

2 + ∥x∗ − x0∥2/η0√T

.

If ∥x∗ − x0∥ ≤ R, then for η0 =RG , we have

2RG√T.

This is minimax optimal (up to constant).The norm of intermediate solution xt is well controlled. Thus (A2) isnot required.

68 / 73

Convergence analysis of SRDA (strongly convex)

Assumption

(A1) E[∥gt∥2] ≤ G 2.

(A2) E[∥xt − x∗∥2] ≤ D2.

(A3) ψ is µ-strongly convex.

Modify the update rule as

gt =2

(t + 1)(t + 2)

t∑τ=1

τgτ , xt = prox(− ηt gt |ηtψ

),

and take the weighted average xT = 2(T+1)(T+2)

∑Tt=0(t + 1)xt .

Theorem

For ηt = (t + 1)(t + 2)/ξ, it holds

Ez1:T [Lψ(xT )− Lψ(x∗)] ≤ ξ∥x∗ − x0∥2

T 2+

2G 2

Tµ.

ξ → 0 yields 2G2

Tµ : minimax optimal.

69 / 73

General convergence analysis

Let the weight st > 0 (t = 1, . . . ) is any positive sequence. We generalizethe update rule as

gt =

∑tτ=1 sτgτ∑t+1τ=1 sτ

,

xt = prox(−ηt gt |ηtψ) (t = 1, . . . ,T ).

Let the weighted average of (xt)t be xT =∑T

τ=0 sτ+1xτ∑Tτ=0 sτ+1

.

Assumption: (A1) E[∥gt∥2] ≤ G 2, (A3) Lψ is µ-strongly convex (µ can be 0).

Theorem

Suppse that ηt/(∑t+1

τ=1 sτ ) is non-decreasing, then

Ez1:T [Lψ(xT )− Lψ(x∗)]

≤ 1∑T+1t=1 st

(T+1∑t=1

s2t2[(∑t

τ=1 sτ )(µ+ 1/ηt−1)]G 2 +

∑T+2t=1 st

2ηT+1∥x∗ − x0∥2

).

70 / 73

Computational cost and generalization error

The optimal learning rate for a strongly convex expected risk(generalization error) is O(1/n) (n is the sample size).

To achieve O(1/n) generalization error, we need to decrease thetraining error to O(1/n).

Normal gradient des. SGD

Time per iteration n 1Number of iterations until ϵ error log(1/ϵ) 1/ϵ

Time until ϵ error n log(1/ϵ) 1/ϵTime until 1/n error n log(n) n

(Bottou, 2010)

SGD is O(log(n)) faster with respect to the generalization error.

71 / 73

Typical behavior

Elaplsed time (s)0 0.5 1 1.5 2 2.5 3 3.5

Tra

inig

err

or0

0.1

0.2

0.3

0.4

0.5

SGDBatch

Elaplsed time (s)0 0.5 1 1.5 2 2.5 3 3.5

Gen

eral

izat

ion

erro

r

0

0.1

0.2

0.3

0.4

0.5

Normal gradient descent v.s. SGDLogistic regression with L1-regularization: n = 10, 000, p = 2.

SGD decreases the objective rapidly, and after a while, the batch gradientmethod catches up and slightly surpasses.

72 / 73

Summary of part I

Overview of stochastic optimization

Convex analysis

First order method


Online stochastic optimization

Stochastic gradient descent (SGD)Stochastic regularized dual averaging (SRDA)RG√T

convergence in general

G 2

µT convergence for strongly convex risk

Next part: faster algorithm of online stochastic optimization, and batchstochastic optimization method

73 / 73

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