Adversarial Variational Optimization of Non-Differentiable Simulators

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Adversarial Variational Optimization of

Non-Differentiable Simulators

Gilles Louppe and Kyle Cranmer

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Likelihood-free assumptions

Operationally,

x ∼ p(x|θ) ≡ z ∼ p(z|θ), x = g(z; θ) where

• z provides a source of randomness;

• g is a non-differentiable deterministic function (e.g. a computer program).

Accordingly, the density p(x|θ) can be written as p(x|θ) =

Z

{z:g(z;θ)=x}

p(z|θ)µ(dz)

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Problem statement

Given observations x ∼ pr(x), we seek:

θ∗ = arg min

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Generative Adversarial Networks

LW = Ex∼p(x|θ)˜ [d(˜x; φ)] − Ex∼pr(x)[d(x; φ)] L_d= LW + λΩ(φ)

L_g= − LW

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Variational Optimization

min

θ f (θ) ≤ Eθ∼q(θ|ψ)[f (θ)] = U (ψ)

∇_ψ_{U (ψ) = E}_{θ∼q(θ|ψ)}[f (θ)∇ψlog q(θ|ψ)]

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Adversarial Variational Optimization

• Replace the generative network in a non-differentiable forward simulator g(z; θ).

• With VO, optimize upper bounds of the adversarial objectives:

Ud= Eθ∼q(θ|ψ)[Ld] (1)

Ug= Eθ∼q(θ|ψ)[Lg] (2)

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Operationally, we get the marginal model:

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41 42 43 Ebeam 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Gf 1 2 3 q( | ) = 0 q( | ) = 5 *_{= (42, 0.9)} 1.0 0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x pr(x) x p(x| ) = 0 x p(x| ) = 5 0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 Ud = 0 Ud = 5

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Summary

• We provide a new likelihood-free inference algorithm:

That works for non-differentiable forward simulators. Combines adversarial training with variational optimization.