Bandit Problems Part III - Bandits for Optimization · I observe conversion indicator X t ∼B(µ A...

Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019

Bandit ProblemsPart III - Bandits for OptimizationRLSS, Lille, July 2019

K arms ↔ K probability distributions : νa has mean µa

ν1 ν2 ν3 ν4 ν5

At round t, an agent:I chooses an arm AtI receives a reward Rt = XAt ,t ∼ νAt

Sequential sampling strategy (bandit algorithm):

At+1 = Ft(A1,R1, . . . ,At ,Rt).

Goal: Maximize E[∑T

t=1 Rt]

Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 2

.The Stochastic Multi-Armed Bandit Setup

K arms ↔ K probability distributions : νa has mean µa

ν1 ν2 ν3 ν4 ν5

At round t, an agent:I chooses an arm AtI receives a sample Xt = XAt ,t ∼ νAt

Sequential sampling strategy (bandit algorithm):

At+1 = Ft(A1,X1, . . . ,At ,Xt).

Goal: Maximize E[∑T

t=1 Xt]→ not the only possible goal!

.The Stochastic Multi-Armed Bandit Setup

B(µ1) B(µ2) B(µ3) B(µ4) B(µ5)

For the t-th patient in a clinical study,I chooses a treatment AtI observes a response Xt ∈ {0, 1}: P(Xt = 1) = µAt

Maximize rewards ↔ cure as many patients as possible

Alternative goal: identify as quickly as possible the best treatment(without trying to cure patients during the study)

.Should we maximize rewards?

B(µ1) B(µ2) B(µ3) B(µ4) B(µ5)

For the t-th patient in a clinical study,I chooses a treatment AtI observes a response Xt ∈ {0, 1}: P(Xt = 1) = µAt

Maximize rewards ↔ cure as many patients as possible

Alternative goal: identify as quickly as possible the best treatment(without trying to cure patients during the study)

Probability that some version of a website generates a conversion:

µ1 µ2 µK

Best version: a? = argmaxa=1,...,K

Sequential protocol: for the t-th visitor:I display version AtI observe conversion indicator Xt ∼ B(µAt ).

Maximize rewards ↔ maximize the number of conversions

Alternative goal: identify the best version(without trying to maximize conversions during the test)

Probability that some version of a website generates a conversion:

µ1 µ2 µK

Best version: a? = argmaxa=1,...,K

Sequential protocol: for the t-th visitor:I display version AtI observe conversion indicator Xt ∼ B(µAt ).

Maximize rewards ↔ maximize the number of conversions

Alternative goal: identify the best version(without trying to maximize conversions during the test)

Goal: find the best version.

.A/B Testing (K = 2)

A way to do A/B Testing:I allocate nA users to page A and nB users to page B

(decided in advance, often nA = nB)I perform a statistical test of “A better than B”

Alternative: fully adaptive A/B TestingI sequentially choose which version to allocate to each visitorI (adaptively choose when to stop the experiment)

Ü best arm identification in a bandit model

.Improve your A/B tests?

A way to do A/B Testing:I allocate nA users to page A and nB users to page B

(decided in advance, often nA = nB)I perform a statistical test of “A better than B”

Alternative: fully adaptive A/B TestingI sequentially choose which version to allocate to each visitorI (adaptively choose when to stop the experiment)

Ü best arm identification in a bandit model

.Improve your A/B tests?

Finding the best armin a bandit model

Goal: identify an arm with large mean as quickly and accurately aspossible ' identify

a? = argmaxa=1,...,K

Algorithm: made of three components:Ü sampling rule: At (arm to explore)Ü recommendation rule: Bt (current guess for the best arm)Ü stopping rule τ (when do we stop exploring?)

Probability of error [Even-Dar et al., 2006, Audibert et al., 2010]The probability of error after n rounds is

pν(n) = Pν (Bn 6= a?) .

blaEmilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 8

.Pure Exploration in Bandit Models

Simple regret [Bubeck et al., 2009]The simple regret after n rounds is

rν(n) = µ? − µBn .

Simple regret [Bubeck et al., 2009]The simple regret after n rounds is

rν(n) = µ? − µBn .

∆minpν(n) ≤ Eν [rν(n)] ≤ ∆maxpν(n)

I Objectives studied in the literature:

Fixed-confidence setting Fixed-budget setting Anytime explorationinput: risk parameter δ input: budget T no input(tolerance parameter ε)

minimize E[τ ] τ = T for all t,P(Bτ 6= a?) ≤ δ minimize P(BT 6= a?) minimize pν(t)

or P(rν(τ) < ε) ≤ δ or E[rT (ν)] or E[rν(t)][Even-Dar et al., 2006] [Bubeck et al., 2009] [Jun and Nowak, 2016]

[Audibert et al., 2010]

.Several objectives

Finding the Best Arm in a Bandit ModelAlgorithms for Best Arm IdentificationPerformance Lower BoundsAn asymptotically optimal algorithm

Beyond Best Arm IdentificationActive Identification in Bandit ModelsExamples

Bandit for Optimization in a Larger SpaceBlack-Box OptimizationHierarchical BanditsBayesian Optimization

Bandit Tools for Planning in GamesUpper Confidence Bounds for TreesBAI tools for Planning in Games

.Outline

Context: bounded rewards (νa supported in [0, 1])We know good algorithms to maximize rewards, for example UCB(α)

At+1 = argmaxa=1,...,K

µa(t) +√α

ln(t)Na(t)

I How good is it for best arm identification?

Possible recommendation rules:

Empirical Best Arm Bt = argmaxa µa(t)(EBA)

Most Played Arm Bt = argmaxa Na(t)(MPA)

Empirical Distribution of Plays Bt ∼ pt , where(EDP) pt =

(N1(t)

t , . . . , NK (t)t

)[Bubeck et al., 2009]

.Can we use UCB?

Context: bounded rewards (νa supported in [0, 1])We know good algorithms to maximize rewards, for example UCB(α)

At+1 = argmaxa=1,...,K

µa(t) +√α

ln(t)Na(t)

I How good is it for best arm identification?

Possible recommendation rules:

Empirical Best Arm Bt = argmaxa µa(t)(EBA)

Most Played Arm Bt = argmaxa Na(t)(MPA)

Empirical Distribution of Plays Bt ∼ pt , where(EDP) pt =

(N1(t)

t , . . . , NK (t)t

)[Bubeck et al., 2009]

.Can we use UCB?

I UCB + Empirical Distribution of Plays

E [rν(n)] = E [µ? − µBn ] = E[ K∑

b=1(µ? − µb)1(Bn=b)

= E[ K∑

b=1(µ? − µb)P(Bn = b|Fn)

= E[ K∑

b=1(µ? − µb)Nb(n)

K∑b=1

(µ? − µb)E[Nb(n)]

= Rν(n)n .

Ü a conversion from cumulated regret to simple regret!Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 12

.Can we use UCB?

E [rν (UCB(α), n)] ≤ Rν(UCB(α), n)n ≤ C(ν) ln(n)

I UCB + Most Played Arm

Theorem [Bubeck et al., 2009]With the Most Play Armed as a recommendation rule, for n large enough,

E [rν (UCB(α), n)] ≤ C

√Kα ln(n)

(more precise problem-dependent analysis in [Bubeck et al., 2009])

.Can we use UCB?

E [rν (UCB(α), n)] ≤ Rν(UCB(α), n)n ≤ C

√Kα ln(n)

.Can we use UCB?

E [rν (UCB(α), n)] ≤ Rν(UCB(α), n)n ≤ C

√Kα ln(n)

.Can we use UCB?

Eν [rν (UCB(α), n)] '

√Kα ln(n)

I the uniform allocation strategy can beat UCB!

Theorem [Bubeck et al., 2009]For n ≥ 4K ln(K )/∆2

min, the simple regret decays exponentially :

Eν [rν (Unif, n)] ≤ ∆max exp(−1

8nK ∆2

I the smaller the cumulative regret, the larger the simple regret

Theorem [Bubeck et al., 2009]

′′E[rν(A, n)] ≥ ∆min2 exp (−C ×Rν(A, n))′′

.Are those results good?

Answer:I UCB has to be coupled with an appropriate recommendation ruleI it is not guaranteed to perform better than uniform exploration...

Variants of UCB?I UCB-E for the fixed-budget setting [Audibert et al., 2010]I LIL-UCB for the fixed-confidence setting [Jamieson et al., 2014]

Other algorithms?I many specific algorithm for best arm identification

.Can we use UCB?

Answer:I UCB has to be coupled with an appropriate recommendation ruleI it is not guaranteed to perform better than uniform exploration...

Variants of UCB?I UCB-E for the fixed-budget setting [Audibert et al., 2010]I LIL-UCB for the fixed-confidence setting [Jamieson et al., 2014]

Other algorithms?I many specific algorithm for best arm identification

.Can we use UCB?

Input: total number of plays TIdea: split the budget in log2(K ) phases of equal length, eliminate theworst half of the remaining arms after each phase.

Initialisation: S0 = {1, . . . ,K};For r = 0 to dln2(K )e − 1, do

sample each arm a ∈ Sr tr =⌊

T|Sr |dlog2(K)e

⌋times;

let µra be the empirical mean of arm a;

let Sr+1 be the set of d|Sr |/2e arms with largest µra

Output: BT the unique arm in Sdlog2(K)e

Theorem [Karnin et al., 2013]Letting H2(ν) = maxa 6=a? a∆−2

[a] , for any bounded bandit instance,

Pν (BT 6= a?) ≤ 3 log2(K ) exp(− T

8 log2(K )H2(ν)

.Fixed Budget: Sequential Halving

Input: risk parameter δ ∈ (0, 1).Idea: sample all remaining arm uniformly and perform eliminations ofarms which look sub-optimal

Initialization: S = {1, . . . ,K}While |S| > 1

Draw all arms in S. t ← t + |S|.S ← S\{a} if maxi∈S µi (t)− µa(t) ≥ 2

√ln(Kt2/δ)

t .Output: the unique arm Bτ ∈ S.

Theorem [Even-Dar et al., 2006]Successive Elimination satisfies Pν (Bτ = a?) ≥ 1− δ. Moreover,

(τδ = O

( K∑a=2

aln( Kδ∆a

)))≥ 1− δ.

.Fixed Confidence: Successive Elimination

Ia(t) = [LCBa(t),UCBa(t)].

771 459 200 45 48 23

I At round t, drawBt = argmax

bµb(t)

Ct = argmaxc 6=Bt

UCBc(t)

I Stop at round t if

LCBBt (t) > UCBCt (t)− ε

Theorem [Kalyanakrishnan et al., 2012]For well-chosen confidence intervals, Pν(µBτ > µ? − ε) ≥ 1− δ and

E [τδ] = O([

2 ∨ ε2 +K∑

a ∨ ε2

]ln(1δ

.Fixed-confidence: LUCB

.Outline

Context: Exponential family bandit modelν = (ν1, . . . , νK ) ↔ µ = (µ1, . . . , µK )

(Bernoulli, Gaussian with known variance, Poisson...)

Algorithm: made of three components:Ü sampling rule: At (arm to explore)Ü stopping rule τ (when do we stop exploring?)Ü recommendation rule: Bτ (guess for the best arm when stopping)

ObjectiveI a δ-correct strategy: for all µ with a unique optimal arm,

Pµ (Bτ = a?) ≥ 1− δ.I with a small sample complexity Eµ[τδ].

Ü minimal sample complexity?Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 20

.Minimizing the sample complexity

Divergence function: kl(µ, µ′) = KL(νµ, νµ′).

Change of distribution lemma [Kaufmann et al., 2016]µ and λ be such that a?(µ) 6= a?(λ). For any δ-correct algorithm,

K∑a=1

Eµ[Na(τ)]kl(µa, λa) ≥ klBer(δ, 1− δ).

I For any a ∈ {2, . . . ,K}, introducing λ:

µa µ1 µ1+ εµ2µK

{λa = µ1 + ελi = µi , if i 6= a

Eµ[Na(τ)]kl(µa, µ1 + ε) ≥ klBer(δ, 1− δ)

Eµ[Na(τ)] ≥ 1kl(µa, µ1) ln

( 13δ

.A first lower bound

Divergence function: kl(µ, µ′) = (µ−µ′)2

2σ2 (Gaussian distributions).

K∑a=1

I For any a ∈ {2, . . . ,K}, introducing λ:

µa µ1 µ1+ εµ2µK

{λa = µ1 + ελi = µi , if i 6= a

Eµ[Na(τ)]kl(µa, µ1 + ε) ≥ klBer(δ, 1− δ)

Eµ[Na(τ)] ≥ 1kl(µa, µ1) ln

( 13δ

K∑a=1

I One obtains

Theorem [Kaufmann et al., 2016]For any δ-correct algorithm,

Eµ[τ ] ≥(

1kl(µ1, µ2) +

K∑a=2

1kl(µa, µ1)

)ln( 1

K∑a=1

I One obtains

Theorem [Kaufmann et al., 2016]For any δ-correct algorithm,

Eµ[τ ] ≥(

1kl(µ1, µ2) +

K∑a=2

1kl(µa, µ1)

)ln( 1

Ü not tight enough...

K∑a=1

I Let Alt(µ) = {λ : a?(λ) 6= a?(µ)}.

infλ∈Alt(µ)

K∑a=1

Eµ[Na(τ)]kl(µa, λa) ≥ klBer(δ, 1− δ)

Eµ[τ ]× infλ∈Alt(µ)

K∑a=1

Eµ[Na(τ)]Eµ[τ ] kl(µa, λa) ≥ ln

( 13δ

Eµ[τ ]×(

supw∈ΣK

infλ∈Alt(µ)

K∑a=1

wakl(µa, λa))≥ ln

( 13δ

.The best possible lower bound

Theorem [Garivier and Kaufmann, 2016]For any δ-PAC algorithm,

Eµ[τ ] ≥ T?(µ) ln( 1

whereT?(µ)−1 = sup

w∈ΣK

infλ∈Alt(µ)

( K∑a=1

wakl(µa, λa)).

Moreover, the vector of optimal proportions,

w?(µ) = argmaxw∈ΣK

infλ∈Alt(µ)

( K∑a=1

wakl(µa, λa))

is well-defined, and can be computed efficiently.

.The best possible lower bound

.Outline

µ(t) = (µ1(t), . . . , µK (t)): vector of empirical means

I Introducing

a : Na(t) <√

one has

At+1 ∈

argmin

a∈UtNa(t) if Ut 6= ∅ (forced exploration)

argmax1≤a≤K

[(w?(µ(t)))a − Na(t)

](tracking)

LemmaUnder the Tracking sampling rule,

t→∞Na(t)

t = (w?(µ))a

.How to match the lower bound? Sampling rule.

I a Generalized Likelihood Ratio:

Z (t) = ln ` (X1, . . . ,Xt ; µ(t))max

λ∈Alt(µ(t))`(X1, . . . ,Xt ; λ) = inf

λ∈Alt(µ(t))

K∑a=1

Na(t)kl(µa(t), λa)

Ü high value of Z (t) rejects the hypothesis “µ ∈ Alt(µ(t))”.

Stopping and recommendation rule

τδ = inf{

t ∈ N : Z (t) > β(t, δ)}

Bτ = argmaxa=1,...,K

µa(τ).

(can be traced back to [Chernoff, 1959])

Ü How to pick the threshold β(t, δ)?

.How to match the lower bound? Stopping rule.

I a Generalized Likelihood Ratio:

Z (t) = ln ` (X1, . . . ,Xt ; µ(t))max

λ∈Alt(µ(t))`(X1, . . . ,Xt ; λ) = inf

λ∈Alt(µ(t))

K∑a=1

Ü high value of Z (t) rejects the hypothesis “µ ∈ Alt(µ(t))”.

Stopping and recommendation rule

τδ = inf{

t ∈ N : Z (t) > β(t, δ)}

Bτ = argmaxa=1,...,K

µa(τ).

(can be traced back to [Chernoff, 1959])

Ü How to pick the threshold β(t, δ)?

.How to match the lower bound? Stopping rule.

Z (t) = infλ∈Alt(µ(t))

K∑a=1

= minb 6=Bt

inf{λ:λBt≤λb}

K∑a=1

Na(t)kl(µa(t), λa) bla

P (Bτδ 6= a?) ≤ P(∃t ∈ N?,∃a 6= a? : Bt = a, Z (t) > β(t, δ)

)≤ P

(∃t ∈ N?,∃a 6= a? : inf

λa≤λa?

∑i∈{a,a?}

Ni (t)kl(µi (t), λi ) > β(t, δ))

≤∑

a 6=a?P(∃t ∈ N? : Na(t)kl(µa(t), µa) + Na?(t)kl(µa?(t), µa?) > β(t, δ)

)︸︷︷︸

requires simultaneous deviations on the two arms

Bernoulli case: [Garivier and Kaufmann, 2016]Exponential families: [Kaufmann and Koolen, 2018]

.A δ-correct stopping rule

K∑a=1

= minb 6=Bt

inf{λ:λBt≤λb}

[NBt (t)kl(µBt (t), λBt ) + Nb(t)kl(µb(t), λb)

)≤ P

(∃t ∈ N?,∃a 6= a? : inf

λa≤λa?

∑i∈{a,a?}

≤∑

)︸︷︷︸

K∑a=1

= minb 6=Bt

inf{λ:λBt≤λb}

[NBt (t)kl(µBt (t), λBt ) + Nb(t)kl(µb(t), λb)

)≤ P

(∃t ∈ N?,∃a 6= a? : inf

λa≤λa?

∑i∈{a,a?}

≤∑

)︸︷︷︸

TheoremThe Track-and-Stop strategy, that usesI the Tracking sampling ruleI the GLRT stopping rule with

β(t, δ) ' ln(K − 1

)+ 2 ln ln

(K − 1δ

)+ 6 ln(ln(t))

I and recommends Bτ = argmaxa=1...K

µa(τ)

is δ-PAC for every δ ∈]0, 1[ and satisfies

lim supδ→0

Eµ[τδ]ln(1/δ) = T?(µ).

τδ = inf{

t ∈ N? : infλ∈Alt(µ(t))

K∑a=1

Na(t)kl (µa(t), λa) > β(t, δ)}

.An asymptotically optimal algorithm

β(t, δ) ' ln(K − 1

)+ 2 ln ln

(K − 1δ

)+ 6 ln(ln(t))

µa(τ)

lim supδ→0

τδ = inf{

t ∈ N? : infλ∈Alt(µ(t))

K∑a=1

Na(t)kl (µa(t), λa) > β(t, δ)}

β(t, δ) ' ln(K − 1

)+ 2 ln ln

(K − 1δ

)+ 6 ln(ln(t))

µa(τ)

lim supδ→0

τδ = inf{

t ∈ N? : t × infλ∈Alt(µ(t))

K∑a=1

Na(t)t kl (µa(t), λa) > β(t, δ)

β(t, δ) ' ln(K − 1

)+ 2 ln ln

(K − 1δ

)+ 6 ln(ln(t))

µa(τ)

lim supδ→0

τδ ' inf{

t ∈ N? : t × infλ∈Alt(µ)

K∑a=1

(w?(µ))akl (µa, λa) > β(t, δ)}

β(t, δ) ' ln(K − 1

)+ 2 ln ln

(K − 1δ

)+ 6 ln(ln(t))

µa(τ)

lim supδ→0

τδ ' inf{

t ∈ N? : t × T−1? (µ) > β(t, δ)

Experiments on two Bernoulli bandit models:I µ1 = [0.5 0.45 0.43 0.4], such that

w?(µ1) = [0.417 0.390 0.136 0.057]

I µ2 = [0.3 0.21 0.2 0.19 0.18], such that

w?(µ2) = [0.336 0.251 0.177 0.132 0.104]

In practice, set the threshold to β(t, δ) = ln(

ln(t)+1δ

Track-and-Stop GLRT-SE KL-LUCB KL-SEµ1 4052 4516 8437 9590µ2 1406 3078 2716 3334

Table: Expected number of draws Eµ[τδ] for δ = 0.1,averaged over N = 3000 experiments.

.Numerical experiments

I TaS: a lower-bound inspired algorithmI KL-UCB versus TaS: very different sampling rules!

947 312 91 58

600 558 210 40

I Two recent improvements of the Tracking sampling rule:Ü relax the need for computing the optimal weights at every round

[Menard, 2019]Ü get rid of the forced exploration by using Upper Confidence Bounds

[Degenne et al., 2019]Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 30

.Observations and improvements

I Unlike previously mentioned strategies, the sampling rule of TaS isanytime, i.e. does not depend on a budget T or a risk parameter δ.

Ü is it also a good strategy in the fixed budget setting?Ü can control E[rν(TaS, t)] for any t?

I Fixed-budget setting: no exactly matching upper and lower boundbest lower bounds: [Carpentier and Locatelli, 2016]

I Top-Two Thompson Sampling [Russo, 2016]: a Bayesian (anytime)strategy that is optimal in a different (Bayesian, asymptotic) sense

Ü can we obtain frequentist guarantees for this algorithm?

.Open problems

BeyondBest Arm Identification

.Outline

Context: exponential family bandit model

ν ↔ µ = (µ1, . . . , µK ) ∈ IK

Goal: Given M regions of IK , R1, . . . ,RM , the goal is to identify oneregion to which µ belongs.

Formalization: build aÜ sampling rule (At)Ü stopping rule τÜ recommendation rule ıτ ∈ {1, . . . ,M}

such that, for some risk parameter δ,

Pµ (µ /∈ Rıτ ) ≤ δ and Eµ[τ ] is small.

.Active Identification in Bandit Models

Context: exponential family bandit model

ν ↔ µ = (µ1, . . . , µK ) ∈ IK

Goal: Given M regions of IK , R1, . . . ,RM , the goal is to identify oneregion to which µ belongs.

Two cases:I R1, . . . ,RM form a partition : a lower-bound inspired sampling rule

+ a GLRT stopping rule essentially worksBcomputing the optimal allocation may be difficult[Juneja and Krishnasamy, 2019, Kaufmann and Koolen, 2018]

I R1, . . . ,RM are overlapping regions : a Track-and-Stop approachdoes not always work [Degenne and Koolen, 2019]

.Active Identification in Bandit Models

.Outline

Anomaly detection: given a threshold θ:I find all the arms whose mean is below θ [Locatelli et al., 2016]I find whether there is an arm with mean below θ

[Kaufmann et al., 2018]

Phase I clinical trial: find the arm with mean closest to the threshold...with increasing means. [Garivier et al., 2017]

λ ∈ Inc : a = argmini|θ − λi |

.Bandits and thresholds

Find the best move at the root of a game tree by actively sampling itsleaves. s? = argmax

s∈C(s0)Vs .

μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

µs if s ∈ L,

maxc∈C(s) Vc if s is a MAX node,minc∈C(s) Vc if s is a MIN node.

Ü more details laterEmilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 38

.Bandit and games

Bandit for Optimizationin a Larger Space

.Outline

f : {1, . . . ,K} −→ R maxa=1,...,K

f (a) ?

Sequential evaluations: at time t, choose At ∈ {1, . . . ,K}, observe

Xt ∼ νAt where νa has mean f (a).

After T observations,

Minimize the cumulative regret

minimize E[ T∑

t=1(f (a?)− f (At))

Minimize the simple regret (optimization error)If BT is a guess of the argmax

minimize E [f (a?)− f (BT )]

.Bandit problems from an optimization perspective

f : {1, . . . ,K} −→ R maxa=1,...,K

f (a) ?

Sequential evaluations: at time t, choose At ∈ {1, . . . ,K}, observe

Xt ∼ νAt where νa has mean f (a).

0 1 2 3 4 5 60

sequential optimization of a discrete functionbased on noisy observations

.Bandit problems from an optimization perspective

f : X −→ R maxx∈X

f (x) ?

Sequential evaluations: at time t, choose xt ∈ X , observe

yt = f (xt) + εt

sequential optimization of a black-box functionbased on noisy observations

.General Sequential (Noisy) Optimization

f : X −→ R maxx∈X

f (x) ?

Sequential evaluations: at time t, choose xt ∈ X , observe

yt = f (xt) + εt

After T observations,

Minimize the cumulative regret

minimize E[ T∑

t=1(f (x?)− f (xt))

Minimize the simple regret (optimization error)If zT is a guess of the argmax

minimize E [f (x?)− f (zT )]

.General Sequential (Noisy) Optimization

I learning based on (costly, noisy) function evaluations only!I no access to gradients of f

Examples of function fI a costly PDE solver (numerical experiments)I a simulator of the effect of a chemical compound (drug discovery)I training and evaluation a neural network

(hyper-parameter optimization)

.Black Box Optimization

.What we want to do

How to choose the next querry?

.What we want to do

How to choose the next querry?

.What we want to do

.Outline

Bandit in metric spaces [Kleinberg et al., 2008]X -armed bandits [Bubeck et al., 2011]

Idea: Partition the space, and adaptatively choose in which cell to sample

I For any depth h, X is partitioned in Kh cells (Ph,i )0≤Kh−1.

I K -ary tree T where depth h = 0 is the whole X .

.Hierarchical partitioning

Assumptions (given a metric `(x , y))I f (x?)− f (y) ≤ f (x?)− f (x) + max{f (x?)− f (x); `(x , y)}I sup(x ,y)∈Ph,i `(x , y) ≤ νρh.

Idea: Use Upper-Confidence Bounds on the maximum values of thefunction in each cell to guide exploration

U(h,i)(t) = µ(h,i)(t) +√

2 ln(t)N(h,i)(t) + νρh

B(h,i)(t) = min{

U(h,i)(t); B(h+1,2i)(t);

B(h+1,2i+1)(t)}

[Bubeck et al., 2011]

.Hierarchical Optimistic Optimization (HOO)

Cumulative regret of HOO

E[ T∑

t=1(f (x?)− f (xt))

]≤ CT

d+1d+2 (ln(T ))

for some near-optimality dimension d .

Ü how to turn this into an optimizer?

zT ∼ U(x1, . . . , xT ), then

E [f (x?)− f (zT )]=R(HOO,T )T ≤C

(ln(T )T

) 1d+2

a tree built by HOO

I Many variants!

DOO, SOO [Munos, 2011], StoSOO [Valko et al., 2013], POO[Grill et al., 2015], GPO [Shang et al., 2019]...

.Results

.Outline

Assumption. the function f is drawn from some Gaussian Process :

f ∼ GP(0, k(x , y)).

i.e. for any distinct points x1, . . . , x` in X ,f (x1)f (x2). . .

f (x`)

∼ N (0,K ) where K = (k(xi , xj))1≤i ,j≤`

Bayesian inferenceGiven some (possibly noisy) observations of f in x1, . . . , xt , the posterioron all the function value in any point is Gaussian

f (y)|x1, . . . , xt ∼ N(µt(y), σt(y)2

)[Rasmussen and Williams, 2005]

.Gaussian Process Regression

Ü use the current GP posterior to pick the new point to select

Example: [µt(y)± βσt(y)] is a kind of confidence interval on f (y).Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 51

.Bayesian Optimization

GP-UCB [Srinivas et al., 2012] selects at round t + 1

xt+1 = argmaxx∈X

µt(x) +√β(t, δ)σt(x)

Ü Bayesian and frequentist guarantees in terms of cumulative regret fordifferent β(t, δ): P

(RT (GP-UCB,T ) ≤ C

√Tβ(T , δ)γT

)≥ 1− δ.

.GP-UCB

More generally, many Bayesian Optimization algorithms optimize anacquisition function that depends on the posterior and select

xt+1 = argmaxx∈X

αt(x)

Many other acquisition functions: [Shahriari et al., 2016]I Expected improvementI Probability of improvementI Entropy Search ...

Remark: optimization the acquisition function is another (non-trivial)optimization problem!

I Thompson Sampling?

.BO Algorithms

Bandit Tools forPlanning in Games

.Outline

Goal: decide for the next move based on evaluation of possibletrajectories in the game, ending with a random evaluation.

A famous bandit approach: [Kocsis and Szepesvari, 2006]Ü use UCB in each node to decide the next children to explore

.Playout-Based Monte-Carlo Tree Search

N(s) : number of visits of node sS(s) : number of visits finishing ending with the root player winning

UCT in a MaxMin TreeIn a MAX node s (= root player move), go towards the children

argmaxc∈C(s)

S(c)N(c) + c

√ln(N(s))

2/6 2/3 4/2

N=19 visits

n3=6 visitsUCB3 = 4/6 + c√log(N)/n3

.The UCT algorithm

UCT in a MaxMin TreeIn a MIN node s (= adversary move), go towards the children

argminc∈C(s)

S(c)N(c) − c

√ln(N(s))

2/6 2/3 4/2

N=19 visits

.The UCT algorithm

argmaxc∈C(s)

S(c)N(c) + c

√ln(N(s))

2/6 2/3 4/2

N=19 visits

.The UCT algorithm

argmaxc∈C(s)

S(c)N(c) + c

√ln(N(s))

2/6 2/3 4/2

N=19 visits

+ first Go AI based on variants of UCT (+ heuristics)

.The UCT algorithm

argmaxc∈C(s)

S(c)N(c) + c

√ln(N(s))

2/6 2/3 4/2

N=19 visits

− UCT is not based on statistically-valid confidence intervals− no sample complexity guarantees− should we really minimize rewards?

.The UCT algorithm

.Outline

A fixed MAXMIN game tree T , with leaves L.

MAX node (= root player move)

MIN node (= adversary move)

Leaf `: stochastic oracle O` that evaluates the position

.A simple model for MCTS

μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

At round t a MCTS algorithm:I picks a path down to a leaf LtI get an evaluation of this leaf Xt ∼ OLt

Assumption: i.i.d. sucessive evaluations, EX∼O` [X ] = µ`

.A simple model for MCTS

μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

A MCTS algorithm should find the best move at the root:

µs if s ∈ L,

maxc∈C(s) Vc if s is a MAX node,minc∈C(s) Vc if s is a MIN node.

s∗ = argmaxs∈C(s0)

MCTS algorithm: (Lt , τ, sτ ), whereI Lt is the sampling ruleI τ is the stopping ruleI sτ ∈ C(s0) is the recommendation rule

μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

Goal: an (ε, δ)-PAC MCTS algorithm:

P(V sτ ≥ V s∗ − ε) ≥ 1 − δ

with a small sample complexity τ . [Teraoka et al., 2014]Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 62

.A structured BAI problem

MCTS algorithm: (Lt , τ, sτ ), whereI Lt is the sampling ruleI τ is the stopping ruleI sτ ∈ C(s0) is the recommendation rule

μ1 μ2 μ3 μ4 μ5 μ6 μ7 μ8

Idea: use LUCB on the depth-one nodesÜ requires confidence intervals on the values (Vs)s∈C0

Ü requires to identify a leaf to sample starting from s ∈ C0

.A structured BAI problem

Using the samples collected for the leaves, one can build, for ` ∈ L,[LCB`(t),UCB`(t)] a confidence interval on µ`

Idea: Propagate these confidence intervals up in the tree

.First tool: confidence intervals

MAX node:UCBs(t) = max

c∈C(s)UCBc(t) LCBs(t) = max

c∈C(s)LCBc(t)

MAX node:UCBs(t) = max

c∈C(s)UCBc(t) LCBs(t) = max

c∈C(s)LCBc(t)

MIN node:UCBs(t) = min

c∈C(s)UCBc(t) LCBs(t) = min

c∈C(s)LCBc(t)

`s(t): representative leaf of internal node s ∈ T .

Idea: alternate optimistic/pessimistic moves starting from s

.Second tool: representative leaves

I run a BAI algorithm on the depth-on nodes

→ selects Rt ∈ C0

I sample the representative leaf associated to that node:

Lt = `Rt (t)

(' from depth one, run UCT based on statistically valid CIs)I update the confidence intervalsI stop when the BAI algorithm tell us toI recommand the depth-one node chosen by the BAI algorithm

.The BAI-MCTS architecture

For some exploration function β, define

LCB`(t) = µ`(t)−√β(N`(t), δ)

2N`(t) ,

UCB`(t) = µ`(t) +√β(N`(t), δ)

2N`(t) .

Theorem [Kaufmann et al., 2018]Choosing

β(s, δ) ' ln( |L| ln(s)

LUCB-MCTS and UGapE-MCTS are (ε, δ)-PAC and

P(τ = O

(H∗ε (µ) ln

)))≥ 1− δ

for UGapE-MCTS.

.Theoretical guarantees

H∗ε (µ) :=∑`∈L

1∆2` ∨∆2

∗ ∨ ε2

where∆∗ := V (s∗)− V (s∗2 )∆` := max

s∈Ancestors(`)\{s0}

∣∣∣VParent(s) − Vs∣∣∣

0.5 0.3 0.2 0.4 0.3 0.5 0.6 0.1

0.5 0.4 0.3 0.6

0.4 0.3

(slightly improved complexity in the work of [Huang et al., 2017])

.The complexity term

I Optimal and efficient algorithms for solving best action identificationin a maxmin tree...

I ... and other generic active identification problemsI UCT versus BAI-MCTS on large-scale problem?

.Future work

I Best arm identification and regret minimization are two differentproblems that require different sampling rules

I Upper and Lower Confidence Bounds are useful in both settingsI Optimal algorithms for BAI are inspired by the lower bounds

(cf. structured bandits)I Tools for BAI → more general Active Identification problems

I Bandit tools inspire methods for sequential optimization in largespaces (games trees or continuous spaces)

.Take-home messages

That’s all!

now you’re ready to pull the right arm ;-)

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Carpentier, A. and Locatelli, A. (2016).Tight (lower) bounds for the fixed budget best arm identification bandit problem.In Conference on Learning Theory (COLT).

Chernoff, H. (1959).Sequential design of Experiments.The Annals of Mathematical Statistics, 30(3):755–770.

Degenne, R., Koolen, W., and Menard, P. (2019).Non asymptotic pure exploration by solving games.arXiv:1906.10431.

Degenne, R. and Koolen, W. M. (2019).Pure Exploration with Multiple Correct Answers.arXiv:1902.03475.

Even-Dar, E., Mannor, S., and Mansour, Y. (2006).Action Elimination and Stopping Conditions for the Multi-Armed Bandit andReinforcement Learning Problems.Journal of Machine Learning Research, 7:1079–1105.

Garivier, A. and Kaufmann, E. (2016).Optimal best arm identification with fixed confidence.In Proceedings of the 29th Conference On Learning Theory (to appear).

Garivier, A., Menard, P., and Rossi, L. (2017).Thresholding bandit for dose-ranging: The impact of monotonicity.arXiv:1711.04454.

Grill, J., Valko, M., and Munos, R. (2015).Black-box optimization of noisy functions with unknown smoothness.In Advances in Neural Information Processing Systems.

Huang, R., Ajallooeian, M. M., Szepesvari, C., and Muller, M. (2017).Structured best arm identification with fixed confidence.In International Conference on Algorithmic Learning Theory (ALT).

Jamieson, K., Malloy, M., Nowak, R., and Bubeck, S. (2014).lil’UCB: an Optimal Exploration Algorithm for Multi-Armed Bandits.In Proceedings of the 27th Conference on Learning Theory.

Jun, K.-W. and Nowak, R. (2016).Anytime exploration for multi-armed bandits using confidence information.In International Conference on Machine Learning (ICML).

Juneja, S. and Krishnasamy, S. (2019).Sample complexity of partition identification using multi-armed bandits.In Conference on Learning Theory (COLT).

Kalyanakrishnan, S., Tewari, A., Auer, P., and Stone, P. (2012).PAC subset selection in stochastic multi-armed bandits.In International Conference on Machine Learning (ICML).

Karnin, Z., Koren, T., and Somekh, O. (2013).Almost optimal Exploration in multi-armed bandits.In International Conference on Machine Learning (ICML).

Kaufmann, E., Cappe, O., and Garivier, A. (2016).On the Complexity of Best Arm Identification in Multi-Armed Bandit Models.Journal of Machine Learning Research, 17(1):1–42.

Kaufmann, E. and Koolen, W. (2018).Mixture martingales revisited with applications to sequential tests and confidenceintervals.arXiv:1811.11419.

Kaufmann, E., Koolen, W., and Garivier, A. (2018).Sequential test for the lowest mean: From Thompson to Murphy sampling.In Advances in Neural Information Processing Systems (NeurIPS).

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Menard, P. (2019).Emilie Kaufmann (CNRS) - Stochastic Bandits July, 2019 - 70

Gradient ascent for active exploration in bandit problems.arXiv:1905.08165.

Munos, R. (2011).Optimistic optimization of a deterministic function without the knowledge of itssmoothness.In Advances in Neural Information Processing Systems.

Rasmussen, C. E. and Williams, C. K. I. (2005).Gaussian Processes for Machine Learning.The MIT Press.

Russo, D. (2016).Simple bayesian algorithms for best arm identification.In Proceedings of the 29th Conference On Learning Theory.

Shahriari, B., Swersky, K., Wang, Z., Adams, R. P., and de Freitas, N. (2016).Taking the human out of the loop: A review of bayesian optimization.Proceedings of the IEEE, 104(1):148–175.

Shang, X., Kaufmann, E., and Valko, M. (2019).General parallel optimization a without metric.In Algorithmic Learning Theory (ALT).

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