Boosted Density Estimation Remastered Zac Cranko 1,2 and Richard Nock - - PowerPoint PPT Presentation

Boosted Density Estimation Remastered

Zac Cranko1,2 and Richard Nock2,1,3

1The Australian National University 2CSIRO Data61 3The University of Sydney

Quick Summary

• Learn a density function incrementally
• Use classifiers for the incremental updates (similar to GAN

discriminators)

• Unlike other state of the art attempts, achieve strong convergence

results (geometric) using a weak learning assumption on the classifiers (in the paper!)

1

sup

D:X→(0,1)

EQ0[log D] − EP [log(1 − D)]

2

Take f(t)

def

= t log t − (t + 1) log(t + 1) and ϕ(D)

def

=

D 1−D. Then

sup

D:X→(0,1)

EQ0[log D] − EP [log(1 − D)] = sup

D:X→(0,1)

EQ0[f ′ ◦ ϕ ◦ D] − EP [f ∗ ◦ f ′ ◦ ϕ ◦ D] = sup

d:X→(0,∞)

EQ0[f ′ ◦ d] − EP [f ∗ ◦ f ′ ◦ d] = EQ0

• f ′ ◦ dP

dQ0

• − EP
• f ∗ ◦ f ′ ◦ dP

dQ0

• 3

Take f(t)

def

= t log t − (t + 1) log(t + 1) and ϕ(D)

def

=

D 1−D. Then

sup

D:X→(0,1)

EQ0[log D] − EP [log(1 − D)] = sup

D:X→(0,1)

EQ0[f ′ ◦ ϕ ◦ D] − EP [f ∗ ◦ f ′ ◦ ϕ ◦ D] = sup

d:X→(0,∞)

EQ0[f ′ ◦ d] − EP [f ∗ ◦ f ′ ◦ d] = EQ0

• f ′ ◦ dP

dQ0

• − EP
• f ∗ ◦ f ′ ◦ dP

dQ0

• Recall:

∀f :

• f(x)P(dx) =
• f(x) dP

dQ0 (x)Q0(dx)

3

Main Idea

d1 ∈ argmax

d′:X→(0,∞)

EQ0[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′]

• 1. Find d1 as above

4

Main Idea

d1 ∈ argmax

d′:X→(0,∞)

EQ0[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′]

• 1. Find d1 as above
• 2. Multiply d1(x)Q0(dx) to find P(dx)

4

Main Idea

d1 ∈ argmax

d′:X→(0,∞)

EQ0[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′]

• 1. Find d1 as above
• 2. Multiply d1(x)Q0(dx) to find P(dx)
• 3. Finished. Get a job at a hedge fund next door

Unfortunately this is not so simple since in practice we can only approximately solve the maximisation.

4

Main Idea

d1 ∈ argmax

d′:X→(0,∞)

EQ0[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′]

• 1. Find d1 as above
• 2. Multiply d1(x)Q0(dx) to find P(dx)
• 3. Finished. Get a job at a hedge fund next door

Unfortunately this is not so simple since in practice we can only approximately solve the maximisation. Sadface.

4

Solution

dt ∈ argmax

d′:X→(0,∞)

EQt−1[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′] ˜ Qt(dx) = dαt

t (x) · ˜

Qt−1(dx), Qt = 1 Zt ˜ Qt, where Zt

def

=

• d ˜

Qt,

• 1. Some step size parameters αt ∈ (0, 1)
• 2. Treat the updates as classifiers dt = exp ◦ct

5

Solution

dt ∈ argmax

d′:X→(0,∞)

EQt−1[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′] ˜ Qt(dx) = dαt

t (x) · ˜

Qt−1(dx), Qt = 1 Zt ˜ Qt, where Zt

def

=

• d ˜

Qt,

• 1. Some step size parameters αt ∈ (0, 1)
• 2. Treat the updates as classifiers dt = exp ◦ct
• The classifiers are distinguishing between samples originating from

P and Qt−1 like in a GAN

• However unlike a GAN there is not necessarily a simple fast sampler

for Qt−1, but there is a closed-form density function

5

Solution

dt ∈ argmax

d′:X→(0,∞)

EQt−1[f ′ ◦ d′] − EP [f ∗ ◦ f ′ ◦ d′] ˜ Qt(dx) = dαt

t (x) · ˜

Qt−1(dx), Qt = 1 Zt ˜ Qt, where Zt

def

=

• d ˜

Qt,

• 1. Some step size parameters αt ∈ (0, 1)
• 2. Treat the updates as classifiers dt = exp ◦ct
• The classifiers are distinguishing between samples originating from

P and Qt−1 like in a GAN

• However unlike a GAN there is not necessarily a simple fast sampler

for Qt−1, but there is a closed-form density function

Convergence of Qt → P in KL-divergence with a weak learning assumption on the updates as classifiers. With additional minimal assumptions: geometric convergence.

5

Experiments

0.4 0.6 0.8 0.5 Accuracy (0.5 if Qt → P ) 1 2 3 4 5 1 2 3 4 KL(P, Qt) (lower is better) t = 0 t = 1 t = 2 t = 3

6

Thanks for listening, come chat to us at poster #161. (Bring beer!)

7