theory and algorithms for forecasting non-stationary time series

Theory and Algorithms for Forecasting Non-Stationary

Time Series

MEHRYAR MOHRI MOHRI@ COURANT INSTITUTE & GOOGLE RESEARCH..

GOOGLE RESEARCH..VITALY KUZNETSOV KUZNETSOV@

pageTheory and Algorithms for Forecasting Non-Stationary Time Series

MotivationTime series prediction:

stock values. economic variables. weather: e.g., local and global temperature. sensors: Internet-of-Things. earthquakes. energy demand. signal processing. sales forecasting.

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Google Trends

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Google Trends

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Google Trends

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ChallengesStandard Supervised Learning:

IID assumption. Same distribution for training and test data. Distributions fixed over time (stationarity).

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none of these assumptions holds for time series!


OutlineIntroduction to time series analysis.

Learning theory for forecasting non-stationary time series.

Algorithms for forecasting non-stationary time series.

Time series prediction and on-line learning.

7

Introduction to Time Series Analysis


Classical FrameworkPostulate a particular form of a parametric model that is assumed to generate data.

Use given sample to estimate unknown parameters of the model.

Use estimated model to make predictions.

9


Autoregressive (AR) ModelsDefinition: AR( ) model is a linear generative model based on the th order Markov assumption:where

s are zero mean uncorrelated random variables with variance .

are autoregressive coefficients. is observed stochastic process.

10

t

pp

Yt

8t, Yt =pX

i=1

aiYti + t

a1, . . . , ap


Moving Averages (MA)Definition: MA( ) model is a linear generative model for the noise term based on the th order Markov assumption:where

are moving average coefficients.

11

q

q

8t, Yt = t +qX

j=1

bjtj

b1, . . . , bq


ARMA modelDefinition: ARMA( , ) model is a generative linear model that combines AR( ) and MA( ) models:

12

p

p

q

q

.

(Whittle, 1951; Box & Jenkins, 1971)

8t, Yt =pX

i=1

aiYti + t +qX

j=1

bjtj


ARMA

13


StationarityDefinition: a sequence of random variables is stationary if its distribution is invariant to shifting in time.

14

Z = {Zt}+11

(Zt, . . . , Zt+m) (Zt+k, . . . , Zt+m+k)

same distribution


Weak StationarityDefinition: a sequence of random variables is weakly stationary if its first and second moments are invariant to shifting in time, that is,

is independent of . for some function .

15

Z = {Zt}+11

f

tE[Zt]E[ZtZtj ] = f(j)


Lag OperatorLag operator is defined by

ARMA model in terms of the lag operator:

Characteristic polynomialcan be used to study properties of this stochastic process.

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L LYt = Yt1.

P (z) = 1pX

i=1

aizi

1

pX

i=1

aiLi

!Yt =

1 +

qX

j=1

bjLj

!t


Weak Stationarity of ARMATheorem: an ARMA( , ) process is weakly stationary if the roots of the characteristic polynomial are outside the unit circle.

17

P (z)

p q


ProofIf roots of the characteristic polynomial are outside the unit circle then: where for all and are constants.

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| i| > 1 i = 1, . . . , p c, c0

P (z) = 1pX

i=1

aizi = c( 1 z) ( p z)

= c0(1 11 z) (1 1p z)


ProofTherefore, the ARMA( , ) process admits MA( ) representation:where is well-defined since

19

1

1 1i L

!1=

1X

k=0

1i L

k

| 1i | < 1.

p q 1

pX

i=1

aiLi

!Yt =

1 +

qX

j=1

bjLj

!t

Yt =

1 11 L

1

1 1p L

11 +

qX

j=1

bjLj

t


ProofTherefore, it suffices to show thatis weakly stationary.

The mean is constant

Covariance function only depends on the lag :

20

Yt =1X

j=0

jtj

lE[YtYtl]

E[YtYtl] =1X

k=0

1X

j=0

kjE[tjtlk] =1X

j=0

jj+l

E[Yt] =1X

j=0

jE[tj ] = 0.

.


ARIMANon-stationary processes can be modeled using processes whose characteristic polynomial has unit roots.

Characteristic polynomial with unit roots can be factored:where has no unit roots.

Definition: ARIMA( , , ) model is an ARMA( , ) modelfor :

21

P (z) = R(z)(1 z)D

R(z)

(1 L)DYtp pq qD

.

1

pX

i=1

aiLi

! 1 L

!DYt =

1 +

qX

j=1

bjLj

!t


ARIMANon-stationary processes can be modeled using processes whose characteristic polynomial has unit roots.

Characteristic polynomial with unit roots can be factored:where has no unit roots.

Definition: ARIMA( , , ) model is an ARMA( , ) modelfor :

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Other ExtensionsFurther variants:

models with seasonal components (SARIMA). models with side information (ARIMAX). models with long-memory (ARFIMA). multi-variate time series models (VAR). models with time-varying coefficients. other non-linear models.

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Modeling VarianceDefinition: the generalized autoregressive conditional heteroscedasticity GARCH( , ) model is an ARMA( , ) model for the variance of the noise term :where

s are zero mean Gaussian random variables with variance conditioned on .

is the mean parameter.

24

t t

8t, 2t+1 = ! +p1X

i=0

i2ti +

q1X

j=0

j2tj

tt {Yt1, Yt2, . . .}

! > 0

p pq q

(Engle, 1982; Bollerslev, 1986)


GARCH ProcessDefinition: the generalized autoregressive conditional heteroscedasticity (GARCH( , )) model is an ARMA( , ) model for the variance of the noise term :where

s are zero mean Gaussian random variables with variance conditioned on .

is the mean parameter.

25


State-Space ModelsContinuous state space version of Hidden Markov Models:where

is an -dimensional state vector. is an observed stochastic process. and are model parameters. and are noise terms.

26

Xt+1 = BXt +Ut,

Yt = AXt + t

Xt n

Yt

A B

Ut t


State-Space Models

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EstimationDifferent methods for estimating model parameters:

Maximum likelihood estimation: Requires further parametric assumptions on the noise

distribution (e.g. Gaussian).

Method of moments (Yule-Walker estimator). Conditional and unconditional least square estimation.

Restricted to certain models.

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Invertibility of ARMADefinition: an ARMA( , ) process is invertible if the roots of the polynomial are outside the unit circle.

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p q

Q(z) = 1 +qX

j=1

bjzj


Learning guaranteeTheorem: assume ARMA( , ) is weakly stationary and invertible. Let denote the least square estimate of and assume that is known. Then, converges in probability to zero.

Similar results hold for other estimators and other models.

30

baTkbaT ak

p

p

qYt

a = (a1, . . . , ap)


NotesMany other generative models exist.

Learning guarantees are asymptotic.

Model needs to be correctly specified.

Non-stationarity needs to be modeled explicitly.

31

Theory


Time Series ForecastingTraining data: finite sample realization of some stochastic process,

Loss function: , where is a hypothesis set of functions mapping from to .

Problem: find with small path-dependent expected loss,

33

H

X Y

h 2 H

L : H Z ! [0, 1]

(X1, Y1), . . . , (XT , YT ) 2 Z = X Y.

L(h,ZT1 ) = EZT+1

L(h, ZT+1)|ZT1

.


Standard AssumptionsStationarity:

Mixing:

34

(Zt, . . . , Zt+m) (Zt+k, . . . , Zt+m+k)

same distribution

B A

n n+ k

dependence between events decaying with k.


Learning TheoryStationary and -mixing process: generalization bounds.

PAC-learning preserved in that setting (Vidyasagar, 1997). VC-dimension bounds for binary classification (Yu, 1994). covering number bounds for regression (Meir, 2000). Rademacher complexity bounds for general loss

functions (MM and Rostamizadeh, 2000).

PAC-Bayesian bounds (Alquier et al., 2014).

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Learning TheoryStationarity and mixing: algorithm-dependent bounds.

AdaBoost (Lozano et al., 1997). general stability bounds (MM and Rostamizadeh, 2010). regularized ERM (Steinwart and Christmann, 2009). stable on-line algorithms (Agarwal and Duchi, 2013).

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ProblemStationarity and mixing assumptions:

often do not hold (think trend or periodic signals). not testable. estimating mixing parameters can be hard, even if

general functional form known.

hypothesis set and loss function ignored.

37


QuestionsIs learning with general (non-stationary, non-mixing) stochastic processes possible?

Can we design algorithms with theoretical guarantees?

need a new tool for the analysis.

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Key Quantity - Fixed h

39

1 Tt T + 1

key difference

Key average quantity:1

T

TX

t=1

hL(h,ZT1 ) L(h,Zt11 )

i.

L(h,Zt11 ) L(h,ZT1 )


DiscrepancyDefinition:

captures hypothesis set and loss function. can be estimated from data, under mild assumptions. in IID case or for weakly stationary processes with

linear hypotheses and squared loss (K and MM, 2014).

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= 0

= suph2H

L(h,ZT1 )

1

T

TX

t=1

L(h,Zt11 )

.


Weighted DiscrepancyDefinition: extension to weights .

strictly extends discrepancy definition in drifting (MM and Muoz Medina, 2012) or domain adaptation (Mansour, MM, Rostamizadeh 2009; Cortes and MM 2011, 2014); or for binary loss (Devroye et al., 1996; Ben-David et al., 2007).

admits upper bounds in terms of relative entropy, or in terms of -mixing coefficients of asymptotic stationarity for an asymptotically stationary process.

41

(q1, . . . , qT ) = q

(q) = suph2H

L(h,ZT1 )

TX

t=1

qt L(h,Zt11 )

.


EstimationDecomposition: .

42

1 Ts T T + 1

(q) suph2H

1

s

TX

t=Ts+1L(h,Zt11 )

TX

t=1

qt L(h,Zt11 )

+ suph2H

L(h,ZT1 )

1

s

TX

t=Ts+1L(h,Zt11 )

.

(q) 0(q) +s


Learning GuaranteeTheorem: for any , with probability at least , for all and ,

43

> 0 1 h 2 H

where G = {z 7! L(h, z) : h 2 H}.

> 0

L(h,ZT1 ) TX

t=1

qtL(h, Zt) +(q) + 2+ kqk2

r2 log

E[N1(,G, z)]

,


Bound with Emp. DiscrepancyCorollary: for any , with probability at least , for all and ,

44

> 0 1 h 2 H > 0

where

8>>>>>>>:

b(q) = sup

h2H

1

s

TX

t=Ts+1L(h, Zt)

TX

t=1

qtL(h, Zt)

us unif. dist. over [T s, T ]G = {z 7! L(h, z) : h 2 H}.

L(h,ZT1 ) TX

t=1

qtL(h, Zt) + b(q) +s + 4

+

hkqk2 + kq usk2

is

2 log

2Ez

N1(,G, z)

,


Weighted Sequential -CoverDefinition: let be a -valued full binary tree of depth . Then, a set of trees is an -norm -weighted -cover of a function class on if

45

z Z TV

G zl1 q

" v1g(z1)v3g(z3)v6g(z6)v12g(z12)

#1

kqk1.

(Rakhlin et al., 2010; K and MM, 2015)

g(z1), v1

g(z2), v2 g(z3), v3

g(z4), v4 g(z5), v5 g(z6), v6 g(z7), v7

g(z12), v12

+11

+1+11 1

1

8g 2 G, 8 2 {1}T , 9v 2 V :TX

t=1

|vt() g(zt())|

kqk1.


Sequential Covering NumbersDefinitions:

sequential covering number:

expected sequential covering number:

46

N1(,G, z) = min{|V| : V l1-norm q-weighted -cover of G}.

Z1, Z01 D1

Z3, Z03 D2(|Z 01)Z2, Z 02 D2(|Z1)

Z5, Z05

D3(|Z1, Z 02)Z4, Z

04

D3(|Z1, Z2) ZT : distribution based on Zts.

EzZT

[N1(,G, z)].


ProofKey quantities:

Chernoff technique: for any ,

47

(q) = suph2H

L(h,ZT1 )TX

t=1

qt L(h,Zt11 ).

(ZT1 ) = suph2H

L(h, ZT )

TX

t=1

qtL(h, Zt)

P(ZT1 (q) > )

P"sup

h2H

TX

t=1

qtL(h,Zt11 ) L(h, Zt)

>

#(sub-add. of sup)

= P"exp

t suph2H

TX

t=1


> et

#(t > 0)

et E"exp

t suph2H

TX

t=1


#. (Markovs ineq.)

t > 0


SymmetrizationKey tool: decoupled tangent sequence associated to .

and i.i.d. given .

48

Z0T1 Z

T1

Zt Z 0t Zt11

P(ZT1 (q) > )

et E"exp

t suph2H

TX

t=1


#

= et E"exp

t suph2H

TX

t=1

qtE[L(h, Z 0t)|Zt11 ] L(h, Zt)

#

(tangent seq.)

= et E"exp

t suph2H

Eh TX

t=1

qtL(h, Z 0t) L(h, Zt)

ZT1i#

(lin. of expectation)

et E"exp

t suph2H

TX

t=1


#. (Jensens ineq.)


Symmetrization

49

P(ZT1 (q) > )

et E"exp

t suph2H

TX

t=1


#

= et E(z,z0)

E

"exp

t suph2H

TX

t=1

qttL(h, z0t()) L(h, zt())

#

(tangent seq. prop.)

= et E(z,z0)

E

"exp

t suph2H

TX

t=1

qttL(h, z0t()) + t sup

h2H

TX

t=1

qttL(h, zt())#

(sub-add. of sup)

et E(z,z0)

E

"1

2

exp

2t sup

h2H

TX

t=1

qttL(h, z0t())

+

1

2

exp

2t sup

h2H

TX

t=1

qttL(h, zt())#

(convexity. of exp)

= et EzE

"exp

2t sup

h2H

TX

t=1

qttL(h, zt())#

.


Covering Number

50

P(ZT1 (q) > )

et EzE

"exp

2t sup

h2H

TX

t=1

qttL(h, zt())#

et EzE

"exp

2thmaxv2V

TX

t=1

qttvt() + i#

(-covering)

et(2) Ez

"X

v2VE

"exp

2t

TX

t=1

qttvt()##

(monotonicity of exp)

et(2) Ez

"X

v2Vexp

t2kqk2

2

##(Hoedings ineq.)

Ez

hN1(,G, z)

iexp

t( 2) + t

2kqk2

2

.

Algorithms


ReviewTheorem: for any , with probability at least , for all and ,

This bound can be extended to hold uniformly over at the price of the additional term: .

Data-dependent learning guarantee.

52

> 0 1 h 2 H

q

eO(kq uk1p

log2 log2(1 kq uk)1)

> 0

L(h,ZT1 ) TX

t=1

qtL(h, Zt) + b(q) +s + 4

+

hkqk2 + kq usk2

is

2 log

2Ez

N1(,G, z)

.


Discrepancy-Risk MinimizationKey Idea: directly optimize the upper bound on generalization over and .

This problem can be solved efficiently for some and .

53

q h

L H


Kernel-Based RegressionSquared loss function:

Hypothesis set: for PDS kernel ,

Complexity term can be bounded by .

54

H =n

x 7! w K(x) : kwkH o

.

K

O(log

3/2 T ) supx

K(x, x)kqk2

L(y, y0) = (y y0)2


Instantaneous DiscrepancyEmpirical discrepancy can be further upper bounded in terms of instantaneous discrepancies:where and .

55

b(q) TX

t=1

qtdt +Mkq uk1

M = supy,y0

L(y, y0)

dt = suph2H

1

s

TX

t=Ts+1L(h, Zt) L(h, Zt)

!


ProofBy sub-additivity of supremum

56

b(q) = suph2H

(1

s

TX

t=Ts+1L(h, Zt)

TX

t=1

qtL(h, Zt)

)

= suph2H

(TX

t=1

qt

1

s

TX

t=Ts+1L(h, Zt) L(h, Zt)

!

+TX

t=1

1T

qt1s

TX

t=Ts+1L(h, Zt)

)

TX

t=1

qt suph2H

1

s

TX

t=Ts+1L(h, Zt) qtL(h, Zt)

!+Mku qk1.


Computing DiscrepanciesInstantaneous discrepancy for kernel-based hypothesis with squared loss: .

Difference of convex (DC) functions.

Global optimum via DC-programming: (Tao and Ahn, 1998).

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dt = supkw0k

TX

s=1

us(w0 K(xs) ys)2 (w0 K(xt) yt)2

!


Discrepancy-Based ForecastingTheorem: for any , with probability at least , for all kernel-based hypothesis and all .

Corresponding optimization problem: .

58

> 0 1 h 2 H 0 < kq uk1 1

minq2[0,1]T ,w

(TX

t=1

qt(w K(xt) yt)2 + 1TX

t=1

qtdt + 2kwkH + 3kq uk1

)

L(h,ZT1 ) TX

t=1

qt

L(h, Zt

) +

b(q) +

s

+

eOlog

3/2 T supx

K(x, x)+ kq uk1




59

> 0 1 h 2 H 0 < kq uk1 1

minq2[0,1]T ,w

(TX

t=1


t=1


)

L(h,ZT1 ) TX

t=1

qt

L(h, Zt

) +

b(q) +

s

+

eOlog

3/2 T supx

K(x, x)+ kq uk1




60

> 0 1 h 2 H 0 < kq uk1 1

minq2[0,1]T ,w

(TX

t=1


t=1


)

L(h,ZT1 ) TX

t=1

qt

L(h, Zt

) +

b(q) +

s

+

eOlog

3/2 T supx

K(x, x)+ kq uk1




61

> 0 1 h 2 H 0 < kq uk1 1

minq2[0,1]T ,w

(TX

t=1


t=1


)

L(h,ZT1 ) TX

t=1

qt

L(h, Zt

) +

b(q) +

s

+

eOlog

3/2 T supx

K(x, x)+ kq uk1




62

> 0 1 h 2 H 0 < kq uk1 1

minq2[0,1]T ,w

(TX

t=1


t=1


)

L(h,ZT1 ) TX

t=1

qt

L(h, Zt

) +

b(q) +

s

+

eOlog

3/2 T supx

K(x, x)+ kq uk1


Convex ProblemChange of variable: .

Upper bound: .

where . convex optimization problem.

63

rt = 1/qt

D = {r : rt 1}

.

|r1t 1/T | T1|rt T |

minr2D,w

(TX

t=1

(w K(xt) yt)2 + 1dtrt

+ 2kwkH + 3TX

t=1

|rt T |)


Two-Stage AlgorithmMinimize empirical discrepancy over (convex optimization).

Solve (weighted) kernel-ridge regression problem: where is the solution to discrepancy minimization problem.

64

q

minw

(TX

t=1

q

t (w K(xt) yt)2 + kwkH

)

q

b(q)


Preliminary ExperimentsArtificial data sets:

65

,


True vs. Empirical Discrepancies

66

Discrepancies Weights


Running MSE

67


Real-world DataCommodity prices, exchange rates, temperatures & climate.

68

Time Series Prediction & On-line Learning


Two Learning ScenariosStochastic scenario:

distributional assumption. performance measure: expected loss. guarantees: generalization bounds. On-line scenario:

no distributional assumption. performance measure: regret. guarantees: regret bounds. active research area: (Cesa-Bianchi and Lugosi, 2006; Anava et al.

2013, 2015, 2016; Bousquet and Warmuth, 2002; Herbster and Warmuth, 1998, 2001; Koolen et al., 2015).

70


On-Line Learning SetupAdversarial setting with hypothesis/action set .

For to do

player receives . player selects . adversary selects . player incurs loss . Objective: minimize (external) regret

71

ht 2 H

H

Tt = 1

xt 2 X

yt 2 Y

L(ht(xt), yt)

RegT =TX

t=1

L(ht(xt), yt) minh2H

TX

t=1

L(h(xt), yt).


Example: Exp. Weights (EW)Expert set , .

72

EW({E1, . . . ,EN})1 for i 1 to N do2 w1,i 13 for t 1 to T do4 Receive(x

t

)

5 ht

PN

i=1 wt,iEiPNi=1 wt,i

6 Receive(yt

)7 Incur-Loss(L(h

t

(xt

), yt

))8 for i 1 to N do9 w

t+1,i wt,i eL(Ei(xt),yt) . (parameter > 0)10 return h

T

H = {E1, . . . ,EN} H = conv(H)


EW GuaranteeTheorem: assume that is convex in its first argument and takes values in . Then, for any and any sequence , the regret of EW at time satisfies

73

For

L

[0, 1] >0y1, . . . , yT 2 Y T

RegT logN

+

T

8

.

=p

8 logN/T ,

RegT p

(T/2) logN.

RegTT

= O

rlogN

T

.


EW - ProofPotential:

Upper bound:

74

t = logPN

i=1 wt,i.

t

t1 = log

PN

i=1 wt1,i eL(Ei(xt),yt)

PN

i=1 wt1,i

= log

Ewt1

[eL(Ei(xt),yt)]

= log

E

wt1

exp

L(E

i

(xt

), yt

) Ewt1

[L(Ei

(xt

), yt

)] E

wt1[L(E

i

(xt

), yt

)]

Ewt1

[L(Ei

(xt

), yt

)] +

2

8(Hoedings ineq.)

L

Ewt1

[Ei

(xt

)], yt

+

2

8(convexity of first arg. of L)

= L(ht

(xt

), yt

) +

2

8.


EW - ProofUpper bound: summing up the inequalities yields

Lower bound:

Comparison:

75

T 0 TX

t=1

L(ht(xt), yt) +

2T

8.

T

0 = logNX

i=1

e

PT

t=1 L(Ei(xt),yt) logN

log Nmaxi=1

e

PT

t=1 L(Ei(xt),yt) logN

= N

min

i=1

TX

t=1

L(Ei

(x

t

), y

t

) logN.

TX

t=1

L(ht(xt), yt)Nmin

i=1

TX

t=1

L(Ei(xt), yt) logN

+

T

8

.


QuestionsCan we exploit both batch and on-line to

design flexible algorithms for time series prediction with stochastic guarantees?

tackle notoriously difficult time series problems e.g., model selection, learning ensembles?

76


Model SelectionProblem: given time series models, how should we use sample to select a single best model?

in i.i.d. case, cross-validation can be shown to be close to the structural risk minimization solution.

but, how do we select a validation set for general stochastic processes?

use most recent data? use the most distant data? use various splits?

models may have been pre-trained on .

77

NZT1

ZT1


Learning EnsemblesProblem: given a hypothesis set and a sample , find accurate convex combination with and .

in most general case, hypotheses may have been pre-trained on .

78

H ZT1h =

PTt=1 qtht h 2 HA

q 2

ZT1

on-line-to-batch conversion for general non-stationary non-mixing processes.


On-Line-to-Batch (OTB)Input: sequence of hypotheses returned after rounds by an on-line algorithm minimizing general regret

79

h = (h1, . . . , hT )

AT

RegT =TX

t=1

L(ht, Zt) infh2H

TX

t=1

L(h, Zt).


On-Line-to-Batch (OTB)Problem: use to derive a hypothesis with small path-dependent expected loss,

i.i.d. problem is standard: (Littlestone, 1989), (Cesa-Bianchi et al., 2004).

but, how do we design solutions for the general time-series scenario?

80

h = (h1, . . . , hT ) h 2 H

LT+1(h,ZT1 ) = EZT+1

L(h, ZT+1)|ZT1

.


QuestionsIs OTB with general (non-stationary, non-mixing) stochastic processes possible?

Can we design algorithms with theoretical guarantees?

need a new tool for the analysis.

81


Relevant Quantity

82

Average difference:

1 T T + 1

key difference

LT+1(ht,ZT1 )

t+ 1

Lt(ht,Zt11 )

1

T

TX

t=1

hLT+1(ht,ZT1 ) Lt(ht,Zt11 )

i.


On-line DiscrepancyDefinition:

: sequences that can return. : arbitrary weight vector.

natural measure of non-stationarity or dependency. captures hypothesis set and loss function. can be efficiently estimated under mild assumptions. generalization of definition of (Kuznetsov and MM, 2015) .

83

HA Aq = (q1, . . . , qT )

disc(q) = suph2HA

TX

t=1

qthLT+1(ht,ZT1 ) Lt(ht,Zt11 )

i.


Discrepancy EstimationBatch discrepancy estimation method.

Alternative method:

assume that the loss is -Lipschitz. assume that there exists an accurate hypothesis :

84

h

= infh

EhL(ZT+1, h

(XT+1))|ZT1i 1.


Discrepancy EstimationLemma: fix sequence in . Then, for any , with probability at least , the following holds for all :

85

ZT1 Z > 01 > 0

where

ddiscH(q) = suph2H,h2HA

TX

t=1

qthLht(XT+1), h(XT+1)

L

ht, Zt

i.

disc(q) ddiscHT (q) + + 2+Mkqk2

r2 log

E[N1(,G, z)]

,


Proof Sketch

86

disc(q) = suph2HA

TX

t=1


i

suph2HA

TX

t=1

qt

LT+1(ht,ZT1 ) E

hL(ht(XT+1), h

(XT+1))ZT1

i

+ suph2HA

TX

t=1

qthEhL(ht(XT+1), h

(XT+1))ZT1

i Lt(ht,Zt11 )

i

suph2HA

TX

t=1

qt EhL(h(XT+1), YT+1)

ZT1i

+ suph2HA

TX

t=1

qthEhL(ht(XT+1), h

(XT+1))ZT1

i Lt(ht,Zt11 )

i

= suph2HA

EhL(h(XT+1), YT+1)

ZT1i

+ suph2HA

TX

t=1

qthEhL(ht(XT+1), h

(XT+1))ZT1

i Lt(ht,Zt11 )

i.


Learning GuaranteeLemma: let be a convex loss bounded by and a hypothesis sequence adapted to . Fix . Then, for any , the following holds with probability at least for the hypothesis :

87

L M

> 0 1 h =

PTt=1 qtht

ZT1 q 2 hT1

LT+1(h,ZT1 ) TX

t=1

qtL(ht, Zt) + disc(q) +Mkqk2

r2 log

1

.


ProofBy definition of the on-line discrepancy,

is a martingale difference, thus by Azumas inequality, whp,

By convexity of the loss:

88

At = qthLt(ht, Zt11 ) L(ht, Zt)

i

TX

t=1

qtLt(ht, Zt11 ) TX

t=1

qtL(ht, Zt) + kqk2q

2 log

1 .

TX

t=1


i disc(q).

LT+1(h,ZT1 ) TX

t=1

qtLT+1(ht,ZT1 ).


Learning GuaranteeTheorem: let be a convex loss bounded by and a set of hypothesis sequences adapted to . Fix . Then, for any , the following holds with probability at least for the hypothesis :

89

L M H

> 0 1 h =

PTt=1 qtht

ZT1 q 2

LT+1(h,ZT1 )

infh2H

TX

t=1

LT+1(h,ZT1 ) + 2disc(q) +RegT

T

+Mkq uk1 + 2Mkqk2

r2 log

2

.


ConclusionTime series forecasting:

key learning problem in many important tasks. very challenging: theory, algorithms, applications. new and general data-dependent learning guarantees

for non-mixing non-stationary processes.

algorithms with guarantees. Time series prediction and on-line learning:

proof for flexible solutions derived via OTB. application to model selection. application to learning ensembles.

90


Time Series Workshop

91

Join us in Room 117 Friday, December 9th.


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102


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103


-MixingDefinition: a sequence of random variables is -mixing if

104

Z = {Zt}+11

B A

n n+ k

dependence between events decaying with k.

(k) = supn

EB2n1

sup

A21n+k

P[A | B] P[A]! 0.

theory and algorithms for forecasting non-stationary time series

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