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Efficient learning of smooth probability functions from Bernoulli tests with guarantees

Paul Rolland paul.rolland@epfl.ch Laboratory for Information and Inference Systems (LIONS) ´ Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL) Switzerland July 2019, Joint work with Ali Kavis, Alexander Immer, Adish Singla, Volkan Cevher @ LIONS

Introduction

• Setup f : X → [0, 1], X ⊂ Rd compact.
• Observations:

⊲ Static setting: yi ∼ Bernoulli(f(xi)) ⊲ Dynamic setting: yi ∼ Bernoulli(Aif(xi) + Bi), with 0 ≤ Ai + Bi ≤ 1

• Goal: Approximate f over X from observation set S = {(xi, yi)}i=1,...,n
• Need regularity assumption on f

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 2/ 8

Logistic Gaussian Process

• Regularity assumption:

f(x) = σ(h(x)), h ∼ GP(µ, κ) where σ(x) =

1 1+e−x .

• Observations: yi ∼ Bernoulli(σ(h((xi)))
• Issues:

⊲ No analytically tractable posterior ⊲ Requires costly Bayesian computations

Figure: Sample from GP prior Figure: Sample from LGP prior

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 3/ 8

Smooth Beta Processes: Static setting

• Regularity assumption: f is L-Lipschitz continuous, i.e.,

|f(x) − f(x′)| ≤ Lx − x′2 ∀x, x′ ∈ X

• Observations: yi ∼ Bernoulli(f(xi))
• Prior: p(y|x) = Beta(α(x), β(x))
• Update of ˜

f(x|X) after observing X = {(x1, y1), . . . , (xn, yn)} 1: p(y|X, x) = Beta

• α(x) +

n

• i=1

δyi=1κ(x, xi), β(x) +

n

• i=1

δyi=0κ(x, xi)

• Theorem (Informal - Convergence of static Beta process)

Using kernel κ(x, x′) = δx−x′2≤∆n,L where ∆t,L = L−

2 d+2 n− 1 d+2 ,

sup

x∈X

EX

• E

˜

f(x|X) − f(x)2 = O

• L

2d d+2 n− 2 d+2

• .

1”

Continuous Correlated Beta Processes” , Goetschalckx et al.

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 4/ 8

Smooth Beta Processes: Dynamic setting

• Regularity assumption: f is L-Lipschitz continuous, i.e.,

|f(x) − f(x′)| ≤ Lx − x′2 ∀x, x′ ∈ X

• Observations: yi ∼ Bernoulli(Aif(xi) + Bi), with 0 ≤ Ai + Bi ≤ 1.
• Prior: p(y|x) = Beta(α(x), β(x))
• Update of ˜

f(x|X) after observing X = {(x1, y1), . . . , (xn, yn)}: p(y|X, x) =

n

• i=1

Cn

i Beta (α(x) + i, β(x) + n − i)

where {Cn

i }i=1,...,n depend on {Ai, Bi}i=1,...,n and a kernel κ.

Theorem (Informal - Convergence of dynamic Beta process)

Using kernel κ(x, x′) = δx−x′2≤∆t,L where ∆n,L = L−

2 d+2 n− 1 d+2 , and under the

assumption Ai + Bi = 1, sup

x∈X

EX

• E

˜

f(x|X) − f(x)2 = O

• L

2d d+2 n− 2 d+2

• .

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 5/ 8

Numerical results in Dynamic setting

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 6/ 8

Benefits of SBP

• Fast computation of posterior update
• Can include contextual features directly influencing success probabilities
• Simple to implement

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 7/ 8

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Welcome to our poster #233!!

Efficient learning of smooth probability functions from Bernoulli tests | Paul Rolland, paul.rolland@epfl.ch Slide 8/ 8