Online Learning with Kernel Losses11-11-00)-11... · 2019-06-11 · Online Learning with Kernel Losses Aldo Pacchiano UC Berkeley Joint work with Niladri Chatterji and Peter Bartlett

Online Learning with Kernel Losses

Aldo Pacchiano

UC Berkeley

Joint work with Niladri Chatterji and Peter Bartlett 1

Talk Overview

• Intro to Online Learning

• Linear Bandits

• Kernel Bandits

2

Online Learning

3

Online Learning

3

Learner Adversaryt = 1, · · · , n

Online Learning

Learner chooses an action at 2 A

3


Online Learning


Adversary reveals loss (or reward) `t 2 W

3


Online Learning


Adversary reveals loss (or reward) Can be i.i.d or

adversarial`t 2 W

3


Online Learning



adversarial`t 2 W

3

Learner Adversary

nX

t=1

`t(at)

t = 1, · · · , n

Online Learning



adversarial`t 2 W

3

Learner Adversary

R(n) =nX

t=1

`t(at) � mina⇤2A

nX

t=1

`t(at)

t = 1, · · · , n

Online Learning

Learner chooses an action

The learner’s objective is to minimize Regret

at 2 A


adversarial`t 2 W

3

Learner Adversary

R(n) =nX

t=1

`t(at) � mina⇤2A

nX

t=1

`t(at)

t = 1, · · · , n

Full information vs Bandit feedback

4


Full Information: Learner gets to sees all of

`t(·)

4


Full Information:

Bandit Feedback:

Learner gets to sees all of

Learner only sees the value

`t(·)

`t(at)

4


Full Information:

Bandit Feedback:

Learner gets to sees all of

Learner only sees the value

`t(·)

`t(at)

4

Multi Armed Bandits

5

P1

µ1

P2 P3

µ2 µ3

Multi Armed Bandits

at 2 {1, · · · ,K}

5

Learner choosesP1

µ1

P2 P3

µ2 µ3

Multi Armed Bandits

Xat ⇠ Pat

at 2 {1, · · · ,K}

5

Learner choosesP1

µ1

Gets reward

P2 P3

µ2 µ3

Multi Armed Bandits

Xat ⇠ Pat

at 2 {1, · · · ,K}

5

Learner choosesP1

µ1

Gets reward

P2 P3

µ2 µ3

R(n) = maxa⇤2{1,···K}

nµa⇤ � E"

nX

t=1

Xat

#

Multi Armed Bandits

Xat ⇠ Pat

at 2 {1, · · · ,K}

5

Learner choosesP1

µ1

Gets reward

P2 P3

µ2 µ3

R(n) = maxa⇤2{1,···K}

nµa⇤ � E"

nX

t=1

Xat

#

R(n) = O(pKn log(n))MAB regret

[Auer et al. 2002]

Structured losses

6

Arms = Paths

MAB regret ExponentialPacket routing

Network (V,E)

at 2 A ⇢ {0, 1}E

wt 2 W = [0, 1]ELoss = delay

hat, wtiDelay is linear

R(n) = O(

p|num paths| · n log(n))

Structured losses

6

Arms = Paths

MAB regret ExponentialPacket routing

Network (V,E)

at 2 A ⇢ {0, 1}E

wt 2 W = [0, 1]ELoss = delay

hat, wtiDelay is linear

R(n) = O(

p|num paths| · n log(n))

Linear Bandits

7

Linear Bandits

Learner chooses an action at 2 A ⇢ Rd

7

Linear Bandits


Adversary’s loss `t(a) = hwt, ai for wt 2 W ⇢ Rd

7

Linear Bandits


Adversary’s loss `t(a) = hwt, ai for Can be i.i.d or

adversarial

wt 2 W ⇢ Rd

7

Linear Bandits


Adversary’s loss `t(a) = hwt, ai for

Learner only experienceshwt, ati Can be i.i.d or adversarial

wt 2 W ⇢ Rd

7

Linear Bandits




wt 2 W ⇢ Rd

7

Expected regret:

R(n) = E"

nX

t=1

hwt, ati � infa2A

nX

t=1

hwt, ai#

Linear Bandits




wt 2 W ⇢ Rd

MAB reduces to Linear Bandits

7

A = {e1, · · · , ed} W = [0, 1]d

Expected regret:

,

R(n) = E"

nX

t=1

hwt, ati � infa2A

nX

t=1

hwt, ai#

Exponential weights for adversarial linear bandits

8


8

For t = 1, · · · , n :

Sample mixture at ⇠ pt = (1� �)qt| {z }Exploitation

+ �⌫|{z}Exploration


8

For t = 1, · · · , n :




8

For t = 1, · · · , n :




8

For t = 1, · · · , n :

See hwt, ati




8

For t = 1, · · · , n :

See hwt, ati



Build loss estimator wt


8

For t = 1, · · · , n :

See hwt, ati



Update

Build loss estimator wt

qt(a) / exp(�⌘hwt, ai)qt�1(a)| {z }Exponential weights

Exponential weights

9

qt(a) / exp(�⌘htX

i=1

wi, ai)

Exponential weights

9

A

tX

i=1

wi


i=1

wi, ai)

Exponential weights

9

A A

tX

i=1

wi

qt


i=1

wi, ai)

Unbiased estimator of the loss

⌃t = Ea⇠pt

⇥aa>

⇤wt = (⌃t)

�1 athwt, atiLet and set

10


⌃t = Ea⇠pt

⇥aa>

⇤wt = (⌃t)


10


⌃t = Ea⇠pt

⇥aa>

⇤wt = (⌃t)


is an unbiased estimator of :wt wt

10


⌃t = Ea⇠pt

⇥aa>

⇤wt = (⌃t)


is an unbiased estimator of :

Eat⇠pt [wt|Ft�1] =�Ea⇠pt

⇥aaT

⇤��1 Eat⇠pt [athwt, ati|Ft�1]

=�Ea⇠pt

⇥aaT

⇤��1 Eat⇠pt

⇥ata

>t |Ft�1

⇤wt

= wt

wt wt

10

Linear bandits regret

11

Theorem. (Linear Bandits Regret).

R(n) �n+log(|A|)

⌘+ ⌘

nX

t=1

EEa⇠pt(hwt, ai)2

[See for example Bubeck ‘11]

Uniform over , [Cesa-Bianchi, Lugosi, ’12]

John’s distribution [Bubeck, Cesa-Bianchi, Kakade ’12]

A

O(dpn)

O(p

dn log(|A|)) = O(dpn)

Exploration over Barycentric Spanner, [Dani, Hayes, Kakade ’08]

O(dpn log(|A|)) = O(d3/2

pn)

Linear bandits regret Dimension dependence

Variance bound:E⇥Eat⇠pt

⇥(hwt, ai)2

⇤⇤ d

12



⇥(hwt, ai)2

⇤⇤ d

Dimension dependence

12



⇥(hwt, ai)2

⇤⇤ d

Dimension dependence

12

⌘dn

R(n) �n+log(|A|)

⌘+ ⌘

nX

t=1

EEa⇠pt(hwt, ai)2

| {z }

Recap

13

• Intro to Online Learning

• Linear Bandits

• Kernel Bandits

Covfefe

Online Quadratic losses

Bt

`t(a) = hbt, ai+ a>Bta

Offline problem haspolytime solution

14

min `t(a)

a 2 A

Symmetric and possibly non convex

Strong Duality

z = x2 � .5 ⇤ y2 + x ⇤ y � .5 ⇤ x+ .5y + 1Peter Bartlett Niladri Chatterji

at 2 A = {a s.t. kak2 1}

Linearization of Quadratic losses

Quadratic losses are linear in the space of

`(a) = hbt, ai+ a>Bta `(a) =

⌧✓Bt

bt

◆,

✓aa>

a

◆�

matricesvector( )

We can use the linear bandits machinery

Exponential weights for quadratic bandits

15

Exponential weights for adversarial quadratic bandits

16

For t = 1, · · · , n :

See



Update

Build loss estimator

hbt, ati+ a>t Btat

✓Bt

bt

◆

qt(a) / exp(�⌘(hbt, ai+ a>Bta))qt�1(a)| {z }Exponential weights

Sampling is poly time

Beyond “Finite Dimensional” Losses

17

Evasion games:

Gaussian kernel -`t(a) = exp(�ka� wtk2)

Infinite dimensional

Obstacle avoidance

Space of Quadratics as a Reproducing Kernel Hilbert Space

The Reproducing Kernel Hilbert Space of HK

Quadratics losses lie in an RKHS K(x, y) = hx, yi+ (hx, yi)2

K

Feature mapx ! �(x) 2 RD

Dot productK(x, y) = h�(x),�(y)i

18

Kernel Bandits

19

Kernel Bandits

If `t 2 HK can we leverage linearity?

19

Kernel Bandits


Main challenge:

Dimension of HK might be infinite.

Naive Linear regret

19

O(p

dn log(|A|))

Kernel Bandits


Main challenge:

Dimension of HK might be infinite.

Naive Linear regret

Encouraging facts:

Kernel spaces are “small”

19

O(p

dn log(|A|))

Towards an Algorithm

Algorithm Strategy:

20


Algorithm Strategy:

1) Construct an dimensional proxy kernel that uniformly approximates the original kernel over .

20

Km(x, y) ⇡ K(x, y)

m < 1A⇥A


Algorithm Strategy:


II) Exponential weights using the proxy kernel. Control the bias.

20

Km(x, y) ⇡ K(x, y)

m < 1A⇥A

K(wt, at)


Algorithm Strategy:



20

Km(x, y) ⇡ K(x, y)

Receive

m < 1A⇥A

K(wt, at)


Algorithm Strategy:



20

Km(x, y) ⇡ K(x, y)

Receive

m < 1A⇥A

K(wt, at)


Algorithm Strategy:



20

Km(x, y) ⇡ K(x, y)

Km(wt, at)Receive Pretend it was

m < 1A⇥A

Kernel functions are “small”

Kernel function evaluations can be uniformly approximated by a small number of basis functions.

21

Kernel functions are “small”

Kernel function evaluations can be uniformly approximated by a small number of basis functions.

Mercer’s Theorem

There exist functions {�i}1i=1 and nonnegative values {µi}1i=1

K(x, y) =1X

i=1

µi�i(x)�i(y)

such that:

8x, y 2 A⇥A

21

Eigendecay

22

K(x, y) =1X

i=1

µi�i(x)�i(y)

Eigendecay

22

Gaussian KernelSobolev Kernel

K(x, y) = exp(�kx� yk2) K(x, y) = min(x, y)

K(x, y) =1X

i=1

µi�i(x)�i(y)

Eigendecay

22


µj Ce��jExponential decay

{�j(x)} = {sin(j⇡x), cos(j⇡x)}

µj ⇡ e�cj log(j)


K(x, y) =1X

i=1

µi�i(x)�i(y)

Eigendecay

22


µj Ce��j µj Cj��Exponential decay Polynomial decay




µj ⇡1

j2

�j(x) ⇡ sin

✓2j⇡x

2

◆

K(x, y) =1X

i=1

µi�i(x)�i(y)

Eigendecay

22


µj Ce��j µj Cj��Exponential decay Polynomial decay




µj ⇡1

j2

�j(x) ⇡ sin

✓2j⇡x

2

◆

K(x, y) =1X

i=1

µi�i(x)�i(y)

Construction of a finite dimensional Proxy Kernel

Eigenfunctions {�i}1i=1 with eigenvalues {µi}1i=1

K(x, y) =1X

i=1

µi�i(x)�i(y)

23



K(x, y) =1X

i=1

µi�i(x)�i(y)

Truncate at i = m

23



K(x, y) =1X

i=1

µi�i(x)�i(y)

Truncate at i = m

Ko(x, y) =mX

i=1

µi�i(x)�i(y)

23



K(x, y) =1X

i=1

µi�i(x)�i(y)

Truncate at i = m

Deterministic Proxy KernelKo(x, y) =

mX

i=1

µi�i(x)�i(y)

23



K(x, y) =1X

i=1

µi�i(x)�i(y)

Truncate at i = m

Deterministic Proxy KernelKo(x, y) =

mX

i=1

µi�i(x)�i(y)

23

Building proxy Kernel with samplesKernel PCA

Exponential weights for adversarial kernel bandits

24


24

Km(x, y) = h�m(x),�m(y)iBuild from P


24

For t = 1, · · · , n :





24

For t = 1, · · · , n :

See




K(wt, at)


24

For t = 1, · · · , n :

See





K(wt, at)

wt


24

For t = 1, · · · , n :

See



Update



K(wt, at)

wt

qt(a) / exp(�⌘hwt,�m(a)i)qt�1(a)| {z }Exponential weights


24

For t = 1, · · · , n :

See



Update


Sampling might not be poly time


K(wt, at)

wt

qt(a) / exp(�⌘hwt,�m(a)i)qt�1(a)| {z }Exponential weights

Biased loss estimates

is biased:

Eat⇠pt [wt|Ft�1] = EhK(at, yt)

⇣(⌃(t)

m )�1�m(at)⌘ ��Ft�1

i

= �m(yt) + E

2

664⇣K(at, yt)� Km(at, yt)

⌘⇣(⌃(t)

m )�1�m(at)⌘

| {z }=:⇠t, the bias

��Ft�1

3

775

25

Let and set⌃(t)m = Ea⇠pt

⇥�m(a)�m(a)>

⇤

wt

wt := K(at, yt)⇣(⌃(t)

m )�1�m(at)⌘

Kernel Bandits Regret

26

Expected regret

game vs Km K game

⇠t bias

R(n) = E"

nX

t=1

K(wt, at)� infa2A

nX

t=1

K(wt, a⇤)

#

Theorem.

R(n) 2�n+ ⌘mn+2✏n

⌘| {z }Bias variance

+2✏n+1

⌘log(|A|).


26

Expected regret

game vs Km K game

⇠t bias

R(n) = E"

nX

t=1

K(wt, at)� infa2A

nX

t=1

K(wt, a⇤)

#

Theorem.

R(n) 2�n+ ⌘mn+2✏n


+2✏n+1

⌘log(|A|).


26

Expected regret

game vs Km K game

⇠t bias

R(n) = E"

nX

t=1

K(wt, at)� infa2A

nX

t=1

K(wt, a⇤)

#

Theorem.

R(n) 2�n+ ⌘mn+2✏n


+2✏n+1

⌘log(|A|).


26

Expected regret

game vs Km K game

⇠t bias

R(n) = E"

nX

t=1

K(wt, at)� infa2A

nX

t=1

K(wt, a⇤)

#

Theorem.

R(n) 2�n+ ⌘mn+2✏n


+2✏n+1

⌘log(|A|).


27

Corollary.

1 Polynomial Decay. µj Cj�� and � > 2:

R(n) O

⇣log(|A|)

��22(��1)n

�2(��1)

⌘

2 Exponential Decay. µj Ce��j:

R(n) O

⇣plog(|A|) log(n)n

⌘

Experiments

28

Lower Bound

29

Lower Bound

29

Polynomial decay µj Cj��<latexit sha1_base64="ychUrnvNonFPPlbYXzEDvS1xm1g=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWAR3FgSEXRZ7MZlBfuAJobJ9KaddvJwZiKU2IW/4saFIm79DXf+jdM2C209cOFwzr3ce4+fcCaVZX0bhaXlldW14nppY3Nre8fc3WvKOBUUGjTmsWj7RAJnETQUUxzaiQAS+hxa/rA28VsPICSLo1s1SsANSS9iAaNEackzD5ww9QbY4XCPa4O77NTxQZGxZ5atijUFXiR2TsooR90zv5xuTNMQIkU5kbJjW4lyMyIUoxzGJSeVkBA6JD3oaBqREKSbTe8f42OtdHEQC12RwlP190RGQilHoa87Q6L6ct6biP95nVQFl27GoiRVENHZoiDlWMV4EgbuMgFU8ZEmhAqmb8W0TwShSkdW0iHY8y8vkuZZxbYq9s15uXqVx1FEh+gInSAbXaAqukZ11EAUPaJn9IrejCfjxXg3PmatBSOf2Ud/YHz+ADjylZQ=</latexit><latexit sha1_base64="ychUrnvNonFPPlbYXzEDvS1xm1g=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWAR3FgSEXRZ7MZlBfuAJobJ9KaddvJwZiKU2IW/4saFIm79DXf+jdM2C209cOFwzr3ce4+fcCaVZX0bhaXlldW14nppY3Nre8fc3WvKOBUUGjTmsWj7RAJnETQUUxzaiQAS+hxa/rA28VsPICSLo1s1SsANSS9iAaNEackzD5ww9QbY4XCPa4O77NTxQZGxZ5atijUFXiR2TsooR90zv5xuTNMQIkU5kbJjW4lyMyIUoxzGJSeVkBA6JD3oaBqREKSbTe8f42OtdHEQC12RwlP190RGQilHoa87Q6L6ct6biP95nVQFl27GoiRVENHZoiDlWMV4EgbuMgFU8ZEmhAqmb8W0TwShSkdW0iHY8y8vkuZZxbYq9s15uXqVx1FEh+gInSAbXaAqukZ11EAUPaJn9IrejCfjxXg3PmatBSOf2Ud/YHz+ADjylZQ=</latexit><latexit sha1_base64="ychUrnvNonFPPlbYXzEDvS1xm1g=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWAR3FgSEXRZ7MZlBfuAJobJ9KaddvJwZiKU2IW/4saFIm79DXf+jdM2C209cOFwzr3ce4+fcCaVZX0bhaXlldW14nppY3Nre8fc3WvKOBUUGjTmsWj7RAJnETQUUxzaiQAS+hxa/rA28VsPICSLo1s1SsANSS9iAaNEackzD5ww9QbY4XCPa4O77NTxQZGxZ5atijUFXiR2TsooR90zv5xuTNMQIkU5kbJjW4lyMyIUoxzGJSeVkBA6JD3oaBqREKSbTe8f42OtdHEQC12RwlP190RGQilHoa87Q6L6ct6biP95nVQFl27GoiRVENHZoiDlWMV4EgbuMgFU8ZEmhAqmb8W0TwShSkdW0iHY8y8vkuZZxbYq9s15uXqVx1FEh+gInSAbXaAqukZ11EAUPaJn9IrejCfjxXg3PmatBSOf2Ud/YHz+ADjylZQ=</latexit><latexit sha1_base64="ychUrnvNonFPPlbYXzEDvS1xm1g=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWAR3FgSEXRZ7MZlBfuAJobJ9KaddvJwZiKU2IW/4saFIm79DXf+jdM2C209cOFwzr3ce4+fcCaVZX0bhaXlldW14nppY3Nre8fc3WvKOBUUGjTmsWj7RAJnETQUUxzaiQAS+hxa/rA28VsPICSLo1s1SsANSS9iAaNEackzD5ww9QbY4XCPa4O77NTxQZGxZ5atijUFXiR2TsooR90zv5xuTNMQIkU5kbJjW4lyMyIUoxzGJSeVkBA6JD3oaBqREKSbTe8f42OtdHEQC12RwlP190RGQilHoa87Q6L6ct6biP95nVQFl27GoiRVENHZoiDlWMV4EgbuMgFU8ZEmhAqmb8W0TwShSkdW0iHY8y8vkuZZxbYq9s15uXqVx1FEh+gInSAbXaAqukZ11EAUPaJn9IrejCfjxXg3PmatBSOf2Ud/YHz+ADjylZQ=</latexit>

:

Lower Bound

29


Rn � ⌦⇣n

�+12�

⌘

<latexit sha1_base64="BYJ7uAoCHWKs/Xbmcb2avX6dlxA=">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</latexit><latexit sha1_base64="BYJ7uAoCHWKs/Xbmcb2avX6dlxA=">AAACNHicbVBNSxxBFOxRE80ak4055tK4CAZhmZGAHsVcBA8xwVVhe7O86X0z22xPz9j9RliG+VFe8kNykUAOCcGrvyG94x78Kmgoqt7jdVVcaOUoDH8FC4tLL14ur7xqrb5ee/O2/W791OWlldiTuc7teQwOtTLYI0UazwuLkMUaz+LJ55l/donWqdyc0LTAQQapUYmSQF4ato9EBjSWoKtv9dBwkeIFF18yTEFoTGiLe5jvFReJBVmJGAm2o7raaVjNG9TCqnRMH4ftTtgNG/CnJJqTDpvjeNj+KUa5LDM0JDU414/CggYVWFJSY90SpcMC5ARS7HtqIEM3qJrQNd/0yognufXPEG/U+xsVZM5Ns9hPziK6x95MfM7rl5TsDSplipLQyLtDSak55XzWIB8pi5L01BOQVvm/cjkGXw/5nlu+hOhx5KfkdKcbhd3o66fO/sG8jhX2gW2wLRaxXbbPDtkx6zHJrtg1+8P+Bj+C38G/4OZudCGY77xnDxDc/gdfaqlg</latexit><latexit sha1_base64="BYJ7uAoCHWKs/Xbmcb2avX6dlxA=">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</latexit><latexit sha1_base64="BYJ7uAoCHWKs/Xbmcb2avX6dlxA=">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</latexit>

:

Lower Bound

29


Rn � ⌦⇣n

�+12�

⌘


Exponential decay µj C exp(��j)<latexit sha1_base64="6KrhvBMH2GSczXQefXPt+SwlH88=">AAACBnicbVC7TkJBEJ2LL8QXamlMNqIJFpJ7bbQk0lhiIo+ES8jeZYCFvQ939xoJobLxV2wsNGrLN9j5IfYuj0LRk0xycs5MZuZ4keBK2/anlVhYXFpeSa6m1tY3NrfS2ztlFcaSYYmFIpRVjyoUPMCS5lpgNZJIfU9gxesVxn7lFqXiYXCt+xHWfdoOeIszqo3USO+7ftzoElfgDSkQF++iLDlxPdSUdMlxI52xc/YE5C9xZiSTP/x6GwFAsZH+cJshi30MNBNUqZpjR7o+oFJzJnCYcmOFEWU92saaoQH1UdUHkzeG5MgoTdIKpalAk4n6c2JAfaX6vmc6fao7at4bi/95tVi3zusDHkSxxoBNF7ViQXRIxpmQJpfItOgbQpnk5lbCOlRSpk1yKROCM//yX1I+zTl2zrkyaVzAFEnYgwPIggNnkIdLKEIJGNzDIzzDi/VgPVmv1vu0NWHNZnbhF6zRN/dDmck=</latexit><latexit sha1_base64="sfWIAO7uvw7TRl0fvNwYBT6YgNU=">AAACBnicbVC7SgNBFJ2Nr5j4WLUUYTAKsTDs2mgZTGMZwTwgG8Ls5CaZZPbhzGwwLKls/BUbC0VtBf/Azg/R2smj0MQDFw7n3Mu997ghZ1JZ1qeRWFhcWl5JrqbSa+sbm+bWdlkGkaBQogEPRNUlEjjzoaSY4lANBRDP5VBxe4WRX+mDkCzwr9QghLpH2j5rMUqUlhrmnuNFjS52OFzjAnbgJsziY8cFRXAXHzXMjJWzxsDzxJ6STP7g6+W9n/4uNswPpxnQyANfUU6krNlWqOoxEYpRDsOUE0kICe2RNtQ09YkHsh6P3xjiQ600cSsQunyFx+rviZh4Ug48V3d6RHXkrDcS//NqkWqd1WPmh5ECn04WtSKOVYBHmeAmE0AVH2hCqGD6Vkw7RBCqdHIpHYI9+/I8KZ/kbCtnX+o0ztEESbSL9lEW2egU5dEFKqISougW3aNH9GTcGQ/Gs/E6aU0Y05kd9AfG2w/v45tD</latexit><latexit sha1_base64="sfWIAO7uvw7TRl0fvNwYBT6YgNU=">AAACBnicbVC7SgNBFJ2Nr5j4WLUUYTAKsTDs2mgZTGMZwTwgG8Ls5CaZZPbhzGwwLKls/BUbC0VtBf/Azg/R2smj0MQDFw7n3Mu997ghZ1JZ1qeRWFhcWl5JrqbSa+sbm+bWdlkGkaBQogEPRNUlEjjzoaSY4lANBRDP5VBxe4WRX+mDkCzwr9QghLpH2j5rMUqUlhrmnuNFjS52OFzjAnbgJsziY8cFRXAXHzXMjJWzxsDzxJ6STP7g6+W9n/4uNswPpxnQyANfUU6krNlWqOoxEYpRDsOUE0kICe2RNtQ09YkHsh6P3xjiQ600cSsQunyFx+rviZh4Ug48V3d6RHXkrDcS//NqkWqd1WPmh5ECn04WtSKOVYBHmeAmE0AVH2hCqGD6Vkw7RBCqdHIpHYI9+/I8KZ/kbCtnX+o0ztEESbSL9lEW2egU5dEFKqISougW3aNH9GTcGQ/Gs/E6aU0Y05kd9AfG2w/v45tD</latexit><latexit sha1_base64="rxOBCj0G4MeW/w28ys/mhsY83/8=">AAACBnicbVDLSsNAFJ34rPUVdSnCYBHqwpK40WWxG5cV7AOaECbTm3baycOZiVhCV278FTcuFHHrN7jzb5y2WWjrgQuHc+7l3nv8hDOpLOvbWFpeWV1bL2wUN7e2d3bNvf2mjFNBoUFjHou2TyRwFkFDMcWhnQggoc+h5Q9rE791D0KyOLpVowTckPQiFjBKlJY888gJU2+AHQ53uIYdeEjK+MzxQRE8wKeeWbIq1hR4kdg5KaEcdc/8croxTUOIFOVEyo5tJcrNiFCMchgXnVRCQuiQ9KCjaURCkG42fWOMT7TSxUEsdEUKT9XfExkJpRyFvu4MierLeW8i/ud1UhVcuhmLklRBRGeLgpRjFeNJJrjLBFDFR5oQKpi+FdM+EYQqnVxRh2DPv7xImucV26rYN1apepXHUUCH6BiVkY0uUBVdozpqIIoe0TN6RW/Gk/FivBsfs9YlI585QH9gfP4ALOKW+g==</latexit>

:

:

Lower Bound

29


Rn � ⌦⇣n

�+12�

⌘



Rn � ⌦⇣n1/2

⌘

<latexit sha1_base64="nsGSg0R6M5uCBYoBTJSubG1VG7g=">AAACHHicbVDBbtNAFHwuhZaUggtHLqtGlcIltdNDeqzaCzdaRNpKcYjWm2dn1fXa7D5XiixLfAPiwoVf4cKhCHHhgMTfdJ30UBJGWmk0857ezsSFkpaC4K+39mD94aONzcetrSfbT5/5O8/PbV4agQORq9xcxtyikhoHJEnhZWGQZ7HCi/jqpPEvrtFYmet3NCtwlPFUy0QKTk4a+wdRxmkquKre1mPNohQ/sOhNhimPFCbUYQ76fcXC/R6rIyPTKb0a++2gG8zBVkl4R9pH/c+fPgLA6dj/HU1yUWaoSShu7TAMChpV3JAUCutWVFosuLjiKQ4d1TxDO6rm4Wq255QJS3LjniY2V+9vVDyzdpbFbrKJYpe9RvyfNywpORxVUhcloRaLQ0mpGOWsaYpNpEFBauYIF0a6vzIx5YYLcn22XAnhcuRVct7rhkE3PHNtHMMCm/ASdqEDIfThCF7DKQxAwBf4Bjfww/vqffd+er8Wo2ve3c4L+Afen1vQ6aIV</latexit><latexit sha1_base64="bgg4x9+jU1xGbJqPS40MZx+45cU=">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</latexit><latexit sha1_base64="bgg4x9+jU1xGbJqPS40MZx+45cU=">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</latexit><latexit sha1_base64="P7MkYu2AcvooIzrrzmOwU1Nkj5c=">AAACHHicbVA9T8MwFHT4LOWrwMhiUSHBUpIywFjBwkZBFJCaUjnuS2rVcYL9glRF/SEs/BUWBhBiYUDi3+CWDlA4ydLp7j093wWpFAZd99OZmp6ZnZsvLBQXl5ZXVktr65cmyTSHBk9koq8DZkAKBQ0UKOE61cDiQMJV0Dse+ld3oI1I1AX2U2jFLFIiFJyhldqlfT9m2OVM5ueDtqJ+BLfUP40hYr6EEHeohbrJqbdXpQNfi6iLu+1S2a24I9C/xBuTMhmj3i69+52EZzEo5JIZ0/TcFFs50yi4hEHRzwykjPdYBE1LFYvBtPJRuAHdtkqHhom2TyEdqT83chYb048DOzmMYia9ofif18wwPGzlQqUZguLfh8JMUkzosCnaERo4yr4ljGth/0p5l2nG0fZZtCV4k5H/kstqxXMr3plbrh2N6yiQTbJFdohHDkiNnJA6aRBO7skjeSYvzoPz5Lw6b9+jU854Z4P8gvPxBfZln/g=</latexit>

:

:

Lower Bound

29


Rn � ⌦⇣n

�+12�

⌘



Rn � ⌦⇣n1/2

⌘


:

:

Lower Bound

29


Rn � ⌦⇣n

�+12�

⌘



Rn � ⌦⇣n1/2

⌘


:

:

A = {(Aj)1j=1 s.t. |Aj | = 1 8j}

<latexit 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Lower Bound

29


Rn � ⌦⇣n

�+12�

⌘



Rn � ⌦⇣n1/2

⌘


:

:

A = {(Aj)1j=1 s.t. |Aj | = 1 8j}

<latexit 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W = {(wj)1j=1 s.t. |wj | = µj 8j}

<latexit 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Final thoughts

30

Thanks for your attention

Documents

Online Learning with Kernel Losses11-11-00)-11... · 2019-06-11 · Online Learning with Kernel Losses Aldo Pacchiano UC Berkeley Joint work with Niladri Chatterji and Peter Bartlett