Stochastic Optimization Online and batch stochastic...

Stochastic OptimizationOnline and batch stochastic optimization methods

† ‡ 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 Online stochastic optimizationStochastic gradient descentStochastic regularized dual averaging

2 Getting stochastic gradient methods fasterBregman divergence and AdaGradAcceleration of stochastic gradient methodsMinimax optimality of first order online stochastic methods

3 Batch stochastic methodsDual method: stochastic dual coordinate ascentPrimal method: SVRG, SAG and SAGAMinimax optimality of first order batch stochastic methods

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Outline

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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).

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.

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

Observe ObserveUpdate Update

storage New data input

E[ℓ(Z , x)] + ψ(x) ≃ minx

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)]

(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

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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.

E[ℓ(Z , x)] ≃(3)

⟨∇xℓ(zt , xt−1), x⟩+

2ηt∥x − xt−1∥2

(proximation)

xt-18 / 78

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 zini=1

Reminder: prox(q|ψ) := argminxψ(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 zini=1

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

2∥x − q∥2.

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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 (Polyak-Ruppert averaging). 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 )

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 + 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

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

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Remark on polynomial averaging

(T + 1)(T + 2)

T∑t=0

(t + 1)xt

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

t + 2xt−1 +

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(

+ µ), 0D2

As for t = 0, we set 1/η0 = 0 .14 / 78

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

2 + D2

t=1 ηt ∞∑t=1

ηt =∞

∞∑t=1

η2t <∞

ensures the convergence. 15 / 78

Outline

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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).

Calculate gradient: gt ∈ ∂xℓt(xt−1).

Take the average of the gradients:

t∑τ=1

Update as

xt = argminx∈Rp

⟨gt , x⟩+ ψ(x) +

2ηt∥x∥2

= prox(−ηt gt |ηtψ).

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

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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.

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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

(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

ξ → 0 yields 2G2

Tµ : minimax optimal.

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General convergence analysis

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

∑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

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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.

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Typical behavior

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

SGDBatch

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

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.

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Outline

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Outline

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Changing the metric (divergence)

L(x) + ψ(x)

x (t) = argminx∈Rp

⟨gt , x⟩+ ψ(x) +

2η∥x − x (t−1)∥2

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L(x) + ψ(x)

⟨gt , x⟩+ ψ(x) +

2η∥x − x (t−1)∥2Ht

∥x∥2H := x⊤Hx .

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L(x) + ψ(x)

⟨gt , x⟩+ ψ(x) +

2η∥x − x (t−1)∥2Ht

∥x∥2H := x⊤Hx .

Choice of Ht

Hessian Ht = ∇∇⊤L(x (t−1)): Newton methodFisher information matrix Ht = EZ |x(t−1) [−∇x∇⊤

x px(Z )|x=x(t−1) ]:Natural gradient(x is a parameter of a parametric model pxx)

c.f. Bregman divergence.

Bϕ(x ||x ′) := ϕ(x)− ϕ(x ′)− ⟨∇ϕ(x ′), x − x ′⟩.→ Mirror descent 25 / 78

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AdaGrad (Duchi et al. (2011))

Ht = G12t + δI

for some δ ≥ 0, where Gt is either of the followings:

(Full) Gt =t∑

gτg⊤τ ,

(Diag) Gt = diag

gτg⊤τ

AdaGrad stretches flat directions and shrinks steep directions.

Ada-SGD:

⟨gt , x⟩+ ψ(x) +

2η∥x − x (t−1)∥2Ht

Ada-SRDA: for gt =1t

∑tτ=1 gτ ,

x (t) = argminx

⟨gt , x⟩+ ψ(x) +

2tη∥x∥2Ht

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Analysis of AdaGrad

Theorem

Let q = 2 for FULL, and q =∞ for Diag. Define the regret as

Q(T ) :=1

T∑t=1

(ℓt+1(x

(t)) + ψ(x (t))− ℓt(β∗)− ψ(β∗)).

Ada-SGD: ∀δ ≥ 0,

Q(T ) ≤ δ

Tη∥x∗∥22 +

maxt≤T∥x∗ − x (t)∥2q/η + 2η

2Ttr[G

Ada-SRDA: for δ ≥ maxt ∥gt∥2,

Q(T ) ≤ δ

Tη∥β∗∥22 +

∥x∗∥2q/η + 2η

2Ttr[G

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Analysis of AdaGrad

Suppose

The gradient is unbalanced:

|gt,j |2 ≤ Gj−2 (j = 1, . . . , p, ∀t).

(Ada-SGD) E[L(x (T ))− L(x∗)] ≤ Clog(p)√

(ordinary SGD) E[L(x (T ))− L(x∗)] ≤ CGE[maxt ∥x (t)∥]√

T≤ C

√p√T

√p → log(p)

Much improvement.29 / 78

AdaGrad is used in various applications including sparse learning and deeplearning.

In deep learning, we often encounter a phenomenon called plateau, that is,we are stuck in a flat region.

It is hard to get out from plateau by standard SGD. AdaGrad adaptivelyadjust the search space to get out of plateau.

AdaGrad is one of the standard optimization methods for deep learning.Related methods: AdaDelta (Zeiler, 2012), RMSProp (Tieleman andHinton, 2012), Adam (Kingma and Ba, 2014).

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Outline

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Nesterov’s acceleration of SGD

Assumption:

the expected loss L(x) is γ-smooth.

the variance of gradient is bounded by σ2:

EZ [∥∇βℓ(Z , β)−∇L(β)∥2] ≤ σ2.

→ combining with Nesterov’s acceleration, the convergence can be gotfaster.

Acceleration for SGD: Hu et al. (2009)

Accleration for SRDA: Xiao (2010), Chen et al. (2012)

General method and analysis (including non-convex): Lan (2012),Ghadimi and Lan (2012, 2013)

Ez1:T [Lψ(x(T ))]− Lψ(x

∗) ≤ C(σD√T+ D2γ

)(D is the diameter: E[∥x (t) − x∗∥2] ≤ D2 (∀t))

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Speed up of accelerated SGD

∗) ≤ C

(σD√T

)σ2 is the variance of the gradient estimate:

EZ [∥∇βℓ(Z , β)−∇L(β)∥2] ≤ σ2.

The variance can be reduced by simply taking average:

g = ∇ℓ(z , x (t−1)) ⇒ g =1

K∑k=1

∇ℓ(zk , x (t−1))

(Variance) σ2 σ2/K

Computing independent gradients can be parallelized.

As σ → 0, the bound goes to O(1/T 2): non-stochastic Nesterov’sacceleration.

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(a) Objective for L1 (b) Objective for Elastic-net

Numerical comparison on synthetic data with (a) L1 regularization (Lasso)and (b) Elastic-net regularization (figure is from Chen et al. (2012)).

SAGE: Accelerated SGD (Hu et al., 2009), AC-RDA: Accelerated stochastic RDA (Xiao,

2010), AC-SA: Accelerated stochastic approximation Ghadimi and Lan (2012), ORDA:

Optimal stochastic RDA (Chen et al., 2012)34 / 78

Accelerated SA for strongly convex objective

Assumption: Objective is µ-strongly convex and γ-smooth.

Accelerated stochastic approximation: Hu et al. (2009), Ghadimi andLan (2012)

∗) ≤ C

µT+γR2

Multi-stage accelerated stochastic approximation: Chen et al. (2012),Ghadimi and Lan (2013)

∗) ≤ C

µT+ exp

(−C√µ

))σ = 0 gives the batch optimal rate.

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Summary of convergence rates

Online methods (expected risk minimization):GR√T

(non-smooth, non-strongly convex) Polyak-Ruppert averaging

µT(non-smooth, strongly convex) Polynomial averaging

σR√T

T 2(smooth, non-strongly convex) Acceleration

µT+ exp

(−√µ

)(smooth, strongly convex) 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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Outline

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Minimax optimal rate of stochastic first order methods

minx∈B

L(x) = minx∈B

EZ [ℓ(Z , x)]

Condition

gx ∈ ∂xℓ(Z , x) is bounded as ∥E[gx ]∥ ≤ G (∀x ∈ B).The domain B contains a ball with radius R.L(x) is µ-strongly convex (µ = 0 is allowed).

Theorem (Minimax optimality (Agarwal et al., 2012, Nemirovsky and Yudin,

1983))

For any first order algorithm, there exist loss function ℓ and distributionP(Z ) satisfying the assumption on which the algorithm must suffer

E[L(x (T ))− L(x∗)] ≥ c min

GR√T,G 2

µT,GR√p

SGD and SRDA achieve this optimal rate.First order algorithm: an algorithm that depends on only the loss and itsgradient (ℓ(Z , x), gx) for a query point x . (SGD, SRDA are included.)

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Outline

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From expectation to finite sum

Online:

P(x) = E[ℓ(Z , x)] =

∫ℓ(Z , x)dP(Z )

Batch:

P(x) =1

n∑i=1

ℓ(zi , x)

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From online to batch

In the batch setting, the data are fixed. We just minimize the objective

function defined by

P(x) =1

n∑i=1

ℓi(x) + ψ(x).

We construct a method that

uses few observations per iteration (like online method),

converges linearly (unlike online method):

T > (n + γ/λ)log(1/ϵ)

to achieve ϵ accuracy for γ-smooth loss andλ-strongly convex regularization. 41 / 78

Three methods that must be remembered

Stochastic Average Gradient descent, SAG (Le Roux et al., 2012,

Schmidt et al., 2013, Defazio et al., 2014)

Stochastic Variance Reduced Gradient descent, SVRG (Johnson

and Zhang, 2013, Xiao and Zhang, 2014)

Stochastic Dual Coordinate Ascent, SDCA (Shalev-Shwartz and

Zhang, 2013a)

- SAG and SVRG are methods performed on the primal.

- SDCA is on the dual.

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Assumptions

P(x) = 1n

∑ni=1 ℓi (x)︸︷︷︸

smooth

+ ψ(x)︸︷︷︸strongly convex

Assumption:

ℓi : Loss is γ-smooth.ψ: reg func is λ-strongly convex. Typically λ = O(1/n) or O(1/

√n).

Example:Loss function

smoothed hinge loss

logistic loss

Regularization function

L2 regularization

Elastic net regularization

ψ(x) + λ∥x∥2 (with small h)

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Outline

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Coordinate Descent

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Note on CD

failure success

Left hand side: CD fails. No descent direction.

To make CD success, the objective should have descent direction.Ideally, separable f (x) =

∑pj=1 fj(xj).

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Coordinate descent in primal

minxP(x) = min

xf (x) + ψ(x) = min

xf (x) +

p∑j=1

ψj(xj)

Coordinate descent (sketch)

1 Choose j ∈ 1, . . . , p in some way．(typically, random choice)

2 j-th coordinate xj is updated so that the objective is decreased,

Usually a block of coordinates are updated instead of one coordinate(block coordinate descent).

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Coordinate descent in primal

minxP(x) = min

xf (x) + ψ(x) = min

xf (x) +

p∑j=1

ψj(xj)

Coordinate descent (sketch)

1 Choose j ∈ 1, . . . , p in some way．(typically, random choice)

2 j-th coordinate xj is updated so that the objective is decreased, e.g.,

x(t)j ← argminxj P(x

(t−1)1 , . . . , xj , . . . , x

(t−1)p ),

orfor gj =

∂f (x (t))∂xj

x(t+1)j ← argminxj ⟨gj , xj⟩+ ψj(xj) +

12ηt∥xj − x

(t−1)j ∥2.

Usually a block of coordinates are updated instead of one coordinate(block coordinate descent).

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Convergence of primal CD method

We consider a separable regularization:

minxP(x) = minxf (x) + ψ(x) = minxf (x) +∑p

j=1 ψj(xj).

Assumption: f is γ-smooth (∥∇f (x)−∇f (x ′)∥ ≤ γ∥x − x ′∥)

Cyclic (Saha and Tewari, 2013, Beck and Tetruashvili, 2013)

P(x (t))− R(x∗) ≤ γp∥x (0) − x∗∥2

2t= O(1/t) (with isotonicity).

Random choice (Nesterov, 2012, Richtarik and Takac, 2014)No acceleration: O(1/t).Nesterov’s acceleration: O(1/t2) (Fercoq and Richtarik, 2013).f is α-strongly convex: O(exp(−C (α/γ)t)).f is α-strongly conv + acceleration: O(exp(−C

√α/γt)) (Lin et al.,

2014).

Nice review is given by Wright (2015).48 / 78

Stochastic Dual Coordinate Ascent, SDCA

Suppose that ∃fi : R→ R such that ℓ(zi , x) = fi (a⊤i x).

Let A = [a1, . . . , an]. (Primal) inf

x∈Rp

n∑i=1

fi (a⊤i x) + ψ(x)

[Fenchel’s duality theorem]

infx∈Rpf (A⊤x) + nψ(x) = − inf

y∈Rnf ∗(y) + nψ∗(−Ay/n)

(Dual) inf

y∈Rn

n∑i=1

f ∗i (yi ) + ψ∗(−1

) We used the following facts:

For f (α) =∑n

i=1 fi (αi ), we have f ∗(β) =∑n

i=1 f∗i (βi ).

For ψ(x) = nψ(x), we have ψ∗(y) = nψ∗(y/n).

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Stochastic Dual Coordinate Ascent, SDCA

Suppose that ∃fi : R→ R such that ℓ(zi , x) = fi (a⊤i x).

Let A = [a1, . . . , an]. (Primal) inf

x∈Rp

n∑i=1

fi (a⊤i x) + ψ(x)

[Fenchel’s duality theorem]

infx∈Rpf (A⊤x) + nψ(x) = − inf

y∈Rnf ∗(y) + nψ∗(−Ay/n)

(Dual) inf

y∈Rn

n∑i=1

f ∗i (yi ) + ψ∗(−1

) We used the following facts:

For f (α) =∑n

i=1 fi (αi ), we have f ∗(β) =∑n

i=1 f∗i (βi ).

For ψ(x) = nψ(x), we have ψ∗(y) = nψ∗(y/n).49 / 78

Remarks

supy∈Rn

n∑i=1

f ∗i (yi ) + ψ∗(−1

The dual loss term∑n

i=1 f∗i (yi ) is separable.

Each coordinate yi affects the objective through only the i-th data:

f ∗i (yi ),

ψ∗(−1

n(a1y1 + · · ·+ aiyi + · · ·+ anyn)

→ Coordinate descent behaves like online methods!

The loss fi is smooth ⇔ f ∗i is strongly convex.

The reg func ψ is strongly convex ⇔ ψ∗ is smooth.

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Algorithm of SDCA

SDCA (Shalev-Shwartz and Zhang, 2013a)

Iterate the following for t = 1, 2, . . .

1 Pick up an index i ∈ 1, . . . , n uniformly at random.

2 Update the i-th coordinate yi so that the objective function isdecreased.

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Algorithm of SDCA

SDCA (Shalev-Shwartz and Zhang, 2013a)

2 Update the i-th coordinate yi :(let A\i = [a1, . . . , ai−1, ai+1, . . . , an], and y\i = (yj)j =i )

• y(t)i ∈ argmin

yi∈R

f ∗i (yi ) + nψ∗

(− 1

n(aiyi + A\iy

(t−1)\i )

2η∥yi − y

(t−1)i ∥2

• y(t)j =y

(t−1)j (for j = i).

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Algorithm of SDCA

SDCA (linearized version) (Shalev-Shwartz and Zhang, 2013a)

2 Calculate x (t−1) = ∇ψ∗(−Ay (t−1)/n).

3 Update the i-th coordinate yi :

yi∈R

f ∗i (yi )− ⟨x (t−1), aiyi ⟩+

2η∥yi − y

(t−1)i ∥2

• y(t)j =y

(t−1)j (for j = i).

If the reg func ψ is λ-strongly covnex, ψ∗ is 1/λ-smooth and thusdifferentiable: x (t) = ∇ψ∗(−Ay (t)/n).x (t) is actually the primal variable.Computational complexity per iteration is same as online methods!Important relation: prox(q|g∗) = q − prox(q|g). primal!

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Algorithm of SDCA

SDCA (linearized version) (Shalev-Shwartz and Zhang, 2013a)

3 Update the i-th coordinate yi :

yi∈R

f ∗i (yi )− ⟨x (t−1), aiyi ⟩+

2η∥yi − y

(t−1)i ∥2

= prox(y

(t−1)i + ηa⊤i x

(t−1)|ηf ∗i ),

• y(t)j =y

(t−1)j (for j = i).

If the reg func ψ is λ-strongly covnex, ψ∗ is 1/λ-smooth and thusdifferentiable: x (t) = ∇ψ∗(−Ay (t)/n).x (t) is actually the primal variable.Computational complexity per iteration is same as online methods!Important relation: prox(q|g∗) = q − prox(q|g). primal!

51 / 78

Convergence analysis of SDCA

Assuption:

fi is γ-smooth.ψ is λ-strongly convex.

Theorem

Suppose there exists R such that ∥ai∥ ≤ R. Then, for η = λn/R2, we have

E[P(x (T )) + D(y (T ))] ≤(n +

(− T

n + R2γλ

)(D(y (0))− D(y∗)).

E[·] is taken w.r.t. the choice of coordinates.

Linear convergence!Required number of iterations to achieve ϵ:

T ≥ C

)log ((n + γ/λ)/ϵ) .

52 / 78

Comparison with the non-stochastic method

How much computation is required to achieve E[P(x (T ))− P(x∗)] ≤ ϵ?Let κ = γ/λ (condition number).

SDCA:(n + κ) log ((n + κ)/ϵ)

Ω((n + κ) log (1/ϵ)) iterations × Ω(1) per iteration

Non-stochastic first order method:

nκ log (1/ϵ)

Ω(κ log (1/ϵ)) iterations × Ω(n) per iteration

Sample size n = 100, 000, reg param λ = 1/1000, smoothness γ = 1:

n×κ = 108, n+κ = 105.

53 / 78

Numerical comparison between SDCA, SDCA-perm (randomly shuffledcyclic), SGD (figure is from Shalev-Shwartz and Zhang (2013a)).

54 / 78

Nesterov’s acceleration of SDCA

Accelerated SDCA (Lin et al., 2014)

Set α = 1n

√λγ .

1 y (t) = y (t−1)+αw (t−1)

4 Update the i-th coordinate:

• w(t)i ∈ argminwi∈R

f ∗i (wi )−⟨x(t−1),aiwi ⟩+αn

2γ∥wi−y

(t)i −(1−α)w (t−1)

i ∥2

• w(t)j =(1− α)w (t−1)

j + y(t)i (for j = i).

5 y(t)i = y

(t)i + nα(w

(t)i − (1− α)w (t−1)

i − y(t)i ),

y(t)j = y

(t)j (for j = i).

Shalev-Shwartz and Zhang (2014) also proposed a double-loop acceleration

method.55 / 78

Convergence of accelerated SDCA

∥ai∥ ≤ R (∀i) ∥A∥: spectral norm of A

Theorem

Convergence of acc. SDCA If

√γnR2

(Cγ∥A∥22λnϵ

then(Duality gap) E[P(x (T ))− D(y (T ))] ≤ ϵ.

(normal)(n +

)log((n + κ)/ϵ)

(accelerated)

√γn

)log((n + κ)/ϵ)

56 / 78

Mini-batch SDCA

Instead of choosing one coordinate yi , we may choose a block ofcoordinates yI where I ⊆ 1, . . . , n.Typically, 1, . . . , n is divided into K equally sized groups: I1, . . . , IK s.t.|Ik | = n/K ,

∪k Ik = 1, . . . , n, Ik ∩ Ik′ = ∅.

Mini-batch technique (Takac et al., 2013, Shalev-Shwartz and Zhang, 2013b).

If K = n, we observe only one data at each iteration.57 / 78

Mini-batch SDCA

Mini-batch SDCA (stochastic block coordinate descent)

For t = 1, 2, . . . ,, iterate the following:

1 Randomly pick up a mini-batch I ⊆ 1, . . . , n so thatP(i ∈ I ) = 1/K (∀i).

2 x (t−1) = ∇ψ∗(−Ay (t−1)/n).

3 Update y (t) as

• y (t)I ∈ argminyi (i∈I )

|I |∑i=1

f ∗i (yi )− ⟨x (t−1),AI yI ⟩+1

2η∥yI − y

(t−1)I ∥2

• y (t)i =y(t−1)i (i ∈ I ).

The update of yi can be parallelized:

yi = prox(y(t−1)i + ηa⊤i x

(t−1)|ηf ∗i ) (i ∈ I ).

58 / 78

Mini-batch SDCA

Mini-batch SDCA (stochastic block coordinate descent)

For t = 1, 2, . . . ,, iterate the following:

1 Randomly pick up a mini-batch I ⊆ 1, . . . , n so thatP(i ∈ I ) = 1/K (∀i).

2 x (t−1) = ∇ψ∗(−Ay (t−1)/n).

3 Update y (t) as

• y (t)I ∈ argminyi (i∈I )

|I |∑i=1

[f ∗i (yi )− ⟨x (t−1),Aiyi ⟩+

2η∥yi − y

(t−1)i ∥2

• y (t)i =y(t−1)i (i ∈ I ).

The update of yi can be parallelized:

yi = prox(y(t−1)i + ηa⊤i x

(t−1)|ηf ∗i ) (i ∈ I ).

58 / 78

Convergence of mini-batch SDCA

Assuption:

fi is γ-smooth.ψ is λ-strongly convex.

Theorem

Suppose there exists R such that ∥A⊤I AI∥ ≤ R2 (∀I ). Then, for

η = λn/R2, we have

E[P(x (T ))− D(y (T ))] ≤(K +

(− T

K + R2γλ

)(D(y (0))− D(y∗)).

E[·] is taken w.r.t. the choice of coordinates.

T ≥ C

)log((n + κ)/ϵ)

achieves ϵ accuracy. → iteration complexity is improved (if R2 is not largeand parallelization is used).

59 / 78

Outline

60 / 78

Primal methods

The key idea: reduce the variance of gradient estimate.

n∑i=1

ℓi (x)︸︷︷︸How to approximate this?

+ψ(x)

61 / 78

Primal methods

The key idea: reduce the variance of gradient estimate.

n∑i=1

ℓi (x)︸︷︷︸How to approximate this?

⟨g ,x⟩

+ψ(x)

Online method: pick up i ∈ 1, . . . , n randomly, and use linearapproximation.

g = ∇ℓi (x) ⇒ E[g ] = 1n

∑ni=1∇ℓi (x)

This is an unbiased estimator of the full gradient.

How about variance?→ Variance is the problem!→ In the batch setting, it is easy to reduce the variance.

61 / 78

Stochastic Variance Reduced Gradient descent,SVRG (Johnson and Zhang, 2013, Xiao and Zhang, 2014)

minxL(x) + ψ(x) = minx 1n∑n

i=1 ℓi (x) + ψ(x)

With fixed reference point x which is close to x , a reduced variancegradient estimator is given as

g = ∇ℓi (x)−∇ℓi (x) + 1n

∑nj=1∇ℓj(x)︸︷︷︸∇L(x)

Bias: unbiased,

E[g ] =1

n∑i=1

[∇ℓi (x)−∇ℓi (x) +∇L(x)] =1

n∑i=1

∇ℓi (x) = ∇L(x).

Variance ?

62 / 78

A key observation

g = ∇ℓi (x)−∇ℓi (x) +∇L(x).Variance:

Var[g ] = 1n

∑ni=1 ∥∇ℓi (x)−∇ℓi (x) +∇L(x)−∇L(x)∥

∑ni=1

∥∥∇ℓi (x)−∇ℓi (x)∥2 − ∥∇L(x)−∇L(x)∥∥2(∵ Var[X ] = E[∥X∥2]− ∥E[X ]∥2)

≤ 1n

∑ni=1 ∥∇ℓi (x)−∇ℓi (x)∥2

≤ γ∥x − x∥2.

The variance could be small if x and x are close and ℓi is smooth.

Main strategy:

Calculate the full gradient at x .Update xt several times, say, O(n) times.Set x = xt .

63 / 78

Variance

64 / 78

Variance

64 / 78

Algorithm of SVRG

The algorithm consists of inner loop and outer loop.

For t = 1, 2, . . . , iterate the following:

1 Set x = x (t−1), x[0] = x ,

g = ∇L(x) = 1n

∑ni=1∇ℓi (x). (full gradient)

2 For k = 1, . . . ,m, execute the following:1 Uniformly sample i ∈ 1, . . . , n．2 Set

g = ∇ℓi (x[k−1])−∇ℓi (x) + g . (variance reduction)

3 Update x[k] asx[k] = prox(x[k−1] − ηg |ηψ).

3 Set x (t) = 1m

∑mk=1 x[k].

Computational complexity until t iteration: O(t × (n +m)).65 / 78

Convergence analysis

Assuption: ℓi is γ-smooth, and ψ is λ-strongly convex.

Theorem

If η and m satisfy η > 4γ and

ρ := ηλ(1−4γ/η)m + 4γ(m+1)

η(1−4γ/η)m < 1,

then, after T iteration the objective is bounded by

E[P(x (T ))− P(x∗)] ≤ ρT (P(x (0))− P(x∗))

The assumption of the theorem is satisfied by

m ≥ Ω(γλ

Inner loop computation O(n +m) for each t.Outer loop iteration T = O(log(1/ϵ)) until ϵ accuracy.

⇒ The whole computation :

O ((n +m) log(1/ϵ)) = O((n +

λ) log(1/ϵ)

)66 / 78

(c) rcv1 (d) covertype

(e) sido0

Numerical comparison between several stochastic methods on a batchsetting (figure is from Xiao and Zhang (2014)). 67 / 78

Related method: SAGA

SAGA (Defazio et al., 2014) does not require the double-loop,but requires more memory.Difference: the gradient estimate g

(SAGA) g = ∇ℓi (x (t−1))−∇ℓi (xi ) +1

n∑j=1

∇ℓj(xj)

(SVRG) g = ∇ℓi (x (t−1))−∇ℓi (x) +1

n∑j=1

∇ℓi (x)

x depends on the data index i ∈ 1, . . . , n.xi is updated at every iteration:

xi = x (t−1) (if i is chosen at the t-th round),

xj is not changed (∀j = i).

- Update rule of x (t) is same: x (t) = prox(x (t−1) − ηg |ηψ).- We need to store all gradients ∇ℓi (xi ) (i = 1, . . . , n).

68 / 78

Algorithm of SAGA

SAGA (Defazio et al., 2014)

1 Pick up i ∈ 1, . . . , n uniformly at random.

2 Update g(t)j (j = 1, . . . , n) as

g(t)j =

∇ℓi (x (t−1)) (i = j),

g(t−1)j (otherwise).

3 Update x (t) as

vt = g(t)i − g

(t−1)i +

n∑j=1

g(t−1)j ,

x (t) = prox(x (t−1) − ηvt |ηψ).

69 / 78

Convergence of SAGA

Assumption: ℓi is γ-smooth and λ-strongly convex (λ = 0 is allowed)(∀i = 1, . . . , n)．

Theorem

Set η = 1/3γ. Then

λ = 0: for x (T ) = 1T

∑Tt=1 x

(t), it holds that

E[P(x (T ))− P(x∗)] ≤ 4n

λ > 0:

E[∥x (T ) − x∗∥2] ≤(1−min

SAGA is adaptive to the strong convexity λ.

70 / 78

Stochastic Average Gradient descent, SAG

Like SAGA, SAG is also a single-loop method (Le Roux et al., 2012,Schmidt et al., 2013).Historically SAG was proposed earlier than SVRG and SAGA.Proximal technique can not be involved in SAG→ SAGA was proposed to overcome this drawback.

P(x) =1

n∑i=1

ℓi (x) ≃ ⟨g , x⟩.

(SAG) g =∇ℓi (x (t−1))−∇ℓi (xi )

n∑j=1

∇ℓj(xj)

g is biased.

(SAGA) g = ∇ℓi (x (t−1))−∇ℓi (xi ) + 1n

∑nj=1∇ℓj(xj)

(SVRG) g = ∇ℓi (x (t−1))−∇ℓi (x) + 1n

∑nj=1∇ℓi (x)

71 / 78

Algorithm of SAG

Initialize g(0)i = 0 (i = 1, . . . , n).

For t = 1, 2, . . . , iterate the following:

1 Pick up i ∈ 1, . . . , n uniformly at random.

2 Update g(t)i ′ (i ′ = 1, . . . , n) as

g(t)i ′ =

∇ℓi (x (t−1)) (i = i ′),

g(t−1)i ′ (otherwise).

3 Update x (t) as

x (t) = x (t−1) − η

n∑j=1

g(t)j .

72 / 78

Convergence analysis of SAG

Assumption: ℓi is γ-smooth and P(x) = 1n

∑ni=1 ℓi (x) is λ-strongly

convex (λ = 0 is allowed)．Milder condition than SAGA because the strong convexity is about P(x)rather than the loss function ℓi .

Theorem (Convergence rate of SAG)

Set η = 116γ . Then SAG converges as

λ = 0: x (T ) = 1T

∑Tt=1 x

(t)に対し，

E[P(x (T ))− P(x∗)] ≤ 32n

λ > 0:

E[∥x (T ) − x∗∥2] ≤(1−min

SAG also has adaptivity. 73 / 78

Catalyst: Acceleration of SVRG, SAG, SAGA

Catalyst (Lin et al., 2015)

Iterate the following for t = 1, 2, . . . :

1 Find an approximated solution of a modified problem which hashigher strong convexity:

x (t) ≃ argminxP(x) + α

2 ∥x − y (t−1)∥2

(up to ϵt precision).

2 Accelerate the solution: y (t) = x (t) + βt(x(t) − x (t−1)).

− Catalyst is an acceleration method of an inexact proximal point alg.− For ϵt = C (1−

√λ/2(λ+ α))t ,

P(x (t))− P(x∗) ≤ C ′(1−

2(λ+α)

− Using SVRG, SAG, SAGA with α = maxc γn , λ in the inner loop

achieves (n +√

γnλ ) log(1/ϵ) overall computation.

− This is a universal method but is sensitive to the choice of the innerloop iteration number and α. 74 / 78

Summary and comparison of batch methods

Properties of the batch methodsMethod SDCA SVRG SAG

P/D Dual Primal PrimalMemory efficiency

Acceleration (µ > 0) Catalyst CatalystOther remark ℓi (β) = fi (x

⊤i β) double loop smooth reg.

Convergence rate (up to log term of γ, µ)Method λ > 0 λ = 0 Acceleration (µ > 0)

SDCA (n + γλ) log(1/ϵ) - (n +

√nγλ ) log(1/ϵ)

SVRG (n + γλ) log(1/ϵ) - (n +

SAG (n + γλ) log(1/ϵ) γn/ϵ (n +

: Catalyst.

As for µ = 0, Catalyst gives an acceleration with convergence rate O(n√

γϵ ).

75 / 78

Summary and comparison of batch methods

Properties of the batch methodsMethod SDCA SVRG SAG

P/D Dual Primal PrimalMemory efficiency

Acceleration (µ > 0) Catalyst CatalystOther remark ℓi (β) = fi (x

⊤i β) double loop smooth reg.

Convergence rate (up to log term of γ, µ)Method λ > 0 λ = 0 Acceleration (µ > 0)

SDCA (n + γλ) log(1/ϵ) - (n +

SVRG (n + γλ) log(1/ϵ) - (n +

SAG (n + γλ) log(1/ϵ) γn/ϵ (n +

: Catalyst.

As for µ = 0, Catalyst gives an acceleration with convergence rate O(n√

γϵ ).

75 / 78

Outline

76 / 78

Minimax optimal convergence rate

Let κ = γλ be the condition number.

The iteration numberT ≥ (n + κ) log(1/ϵ)

is almost minimax, but not minimax.

The accelerated version

T ≥(n +√nκ)log(1/ϵ)

is minimax up to log(1/ϵ) (Agarwal and Bottou, 2015).

77 / 78

Minimax optimality in the batch setting

P(x) =1

n∑i=1

ℓi (x) +λ

2∥x∥2

Assumption: ℓi is (γ − λ)-smooth (γ > λ).

First order oracle: for an input (x , i), it returns the pair (ℓi (x),∇ℓi (x)).First order algorithm: an algorithm that depends on only the return of thefirst order oracle for a query point x .(SAG, SAGA, SVRG are included. SDCA is not included.)

Theorem (Minimax optimal rate for FOA (Agarwal and Bottou, 2015))

For any first order algorithm, there exist functions ℓi (i = 1, . . . , n)satisfying the assumption on which the algorithm must perform at least

T ≥ Ω(n +√

n(κ− 1) log(1/ϵ))

calls for the first order oracle to get ∥x (T ) − x∗∥ ≤ ϵ∥x∗∥.78 / 78

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