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5.9 surrogate as an upper bound, and mm algorithms 78 Equation (40) requires that the integrand is differentiable, which is satisfied if all of the PDFs and c(depsc) are differentiable. Equation (39) follows by summing over all costs c ∈ C. Equation (38) follows from rearrangement of the terms in this equation. Theorem 2 It suffices to show that for a particular node v ∈ S, the following expectation (taken over all variables) vanishes ∂ E ∂θ log p(v | parentsv) b(NonInfluenced(v)) . Analogously to NonInfluenced(v), define Influenced(v) := {w | w ≻ v}. Note that the nodes can be ordered so that NonInfluenced(v) all come before v in the ordering. Thus, we can write ∂ ENonInfluenced(v) EInfluenced(v) ∂θ log p(v | parentsv) b(NonInfluenced(v)) ∂ = ENonInfluenced(v) EInfluenced(v) ∂θ log p(v | parentsv) b(NonInfluenced(v)) = ENonInfluenced(v) [0 · b(NonInfluenced(v))] =0 whereweusedEInfluenced(v)∂ logp(v|parentsv)=Ev∂ logp(v|parentsv)= ∂θ ∂θ 0. 5.9 surrogate as an upper bound, and mm algorithms L has additional significance besides allowing us to estimate the gradient of the expected sum of costs. Under certain conditions, L is a upper bound on on the true objective (plus a constant). We shall make two restrictions on the stochastic computation graph: (1) first, that all costs c ∈ C are negative. (2) the the costs are not deterministically influenced by the parameters Θ. First, let us use importance sampling to write down the expectation of a given cost node, when the sampling distribution is different from the distribution we are evaluating: for parameter θ ∈ Θ, θ = θold is used for sampling, but we are evaluating atPDF Image | OPTIMIZING EXPECTATIONS: FROM DEEP REINFORCEMENT LEARNING TO STOCHASTIC COMPUTATION GRAPHS
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