Join-graph Decomposition Bounds for Exponentiated IDs

Given the exponentiated ID M^e := hX, D, P, U^e, Oi of M := hX, D, P, U, Oi, a join-graph decomposition G_JG := hG(C, S), χ, ψi for M^e can be obtained in the usual way using the mini-bucket tree decomposition (see Example 3.1). Figure 3.5 and Example 3.5 show the

Figure 3.5: Join-graph Decomposition of the Exponentiated ID.

changes to the join-graph decomposition due to the exponentiated ID M^e.

Example 3.5. Figure 3.5 presents a join-graph decomposition G_JG:= hG(C, S), χ, ψi of the ID in Figure 3.1a. Here we do not use the valuation algebra over the Jensen’s potentials as was shown in Figure 3.1b. The join-graph in Figure 3.5 treats probability functions and exponentiated utility functions as unnormalized factors in a probabilistic graphical models.

Theorem 3.4, shows that the MEU of an ID M can be bounded by the log-partition function of the exponentiated ID M^e. Therefore, we can now apply any variational decomposition bounds in the mixed inference task for bounding the log-partition function to yield bounds to the MEU of M. Namely, we bound the MEU by first applying Theorem 3.4 to obtain M^e from M, and then by using variational decomposition bounds in the probabilistic graphical models to M^e.

3.3.2.1 Derivation of Upper Bounds

Given a join-graph decomposition G_JG := hG(C, S), χ, ψi of M^e, we denote a cost-shifting function over the separator (C_i, C_j) ∈ S by δ_Ci,Cj(X_Ci,Cj). The upper bounds on the MEU can be parameterized over G_JG as described in the following proposition.

Proposition 3.3. Given an ID M := hX, D, P, U, Oi, where U = {U₁, . . . , U_l} and its exponentiated ID M^e := hX, D, P, U^e, Oi, U^e = {e^U1, . . . , e^Ul}, a total elimination order O_elim := {X₁, X₂, . . . , X_N} over the variables X ∪ D, and given a join-graph decomposition G_JG := hG(C, S), χ, ψi of M^e, the upper bound on the MEU can be parameterized over G_JG as follows. U_MEU denotes as usual an upper bound over the MEU. We claim that for a given set of weights.

and we denote the expression on the right hand side as U_MEU where,

P_Ci(X_Ci) = Y

k is the weight for the variable X_k assigned at cluster C that satisfies w_Xk =P

C∈Cw^C_X

k.

Proof. We start our derivation from Eq. (3.61) in Theorem 3.4 bounding the MEU of M by the log-partition function of M^e repeated in Eq. (3.73). To get Eq. (3.74), we rearrange the probability functions and exponentiated utility functions at each cluster C_i according to G_JG, and introduce cost-shifting functions over the separators, which cancel each other. Namely,

δ_Ci,Cj(X_Ci,Cj)δ_Cj,Ci(X_Ci,Cj) = 0(X_Ci,Cj) Next, to derive Eq. (3.75), we rewrite functions in the log-space and replaces the exponentiated utility functions F_i(X_Fi) with the original utility functions U_i(X_Ui) Subsequently, we switch the order of the multiplication operation and the elimination operation using decomposition bounds on mini-buckets [Ping et al., 2015]

yielding Eq. (3.76), and obtain the desired expression by changing log product to sum log and rewriting the powered-sum operation explicitly, yielding Eq. (3.77).

MEU =

Proposition 3.3 introduces two kinds of optimization parameters: (1) cost-shifting functions δ_Ci,Cj(X_Ci,Cj), and (2) weight parameters w^Ci that satisfies the following condition, wX = P

Ci∈Cw^C_Xⁱ for all X ∈ χ(C_i). Compared with algorithm JGD-ID presented in Section 3.2.2, algorithm JGD-EXP has only two types of optimization parameters because JGD-EXP does not use the valuation algebra for IDs.

Algorithm 3.8 Join-graph GDD of Exponentiated IDs (JGD-EXP)

Require: Exponentiated ID M^e := hX, D, P, U^e, Oi of M := hX, D, P, U, Oi, weights w_Xi associated with each variable X_i ∈ X, i-bound, iteration limit M for the outer-loop of BCD.

Ensure: an upper bound of the MEU, U_MEU Initialization Steps

1: Generate a join-graph decomposition G_JG = hG(C, S), χ, ψi of M^e with given i-bound

2: Initialize weights for each chance variable X_i ∈ X by uniform weights, w_X^C

i = _|{C|X¹

i∈χ(C)}|, and for each decision variable D_i ∈ D by a constant close to zero, w_D^C

i ≈ 0.

Outer-loop of BCD

3: iter=0, U_MEU= inf

4: while iter < M or U_value is not converged do Inner-loop of BCD

5: for each variable X_i ∈ X do

6: U_MEU ← min(U_MEU, Update-Weights(GJG, X_i)) . use Algorithm 3.3 with the gradient expression shown in Eq. (3.78)

7: for each edge (C_i, C_j) ∈ S do

8: U_MEU ← min(U_MEU, Update-Costs(GJG, (C_i, C_j))) . see Algorithm 3.9

9: iter = iter + 1 return U_MEU

3.3.2.2 Optimizing the Paramterized Bounds

Next, we present algorithm JGD-EXP that solves a convex optimization problem for tighten-ing the upper bound U_MEUdefined in Eq. (3.68) by optimizing over the cost-shifting functions δ_Ci,Cj(X_Ci,Cj) and the weight parameters w^C_X

k parameterized over the join-graph G(C, S). As we will see, we will need to modify algorithm JGD-ID shown in Algorithm 3.2 in order to present JGD-EXP.

Message Passing Algorithm (JGD-EXP) Algorithm 3.8 outlines the overall procedure of BCD updates in JGD-EXP. In the initialization step of JGD-EXP, we generate a join-graph of M^e and initialize its weights as usual (line 1-2), The inner-loop of JGD-EXP alternates updating weights by UPDATE-WEIGHTS(GJG,Xi) and updating cost-shifting functions by UPDATE-COSTS(G_JG,(Ci,Cj)) (line 5-8). During the updating if the weight parameters, we optimize a subset of weight parameters w_X = {w_X^C|X ∈ χ(C)} for each variable X using

Algorithm 3.3, which was also used for updating the weight parameters in JGD-ID. Yet, we modify the gradient expression for the exponentiated utility bounds as will be shown in Eq. (3.78). For updating the cost-shifting functions, we use Algorithm 3.9 to be presented shortly which performs gradient descent updates for δ_Ci,Cj(X_Ci,Cj) over the separators S.

Since both JGD-ID and JGD-EXP are based on GDD bounds [Ping et al., 2015], they are quite similar. However, JGD-EXP does not involve the complicated non-convex optimization problems for updating the parameters, and the derived gradient expressions are much simpler than JGD-ID, as we will show in the following Section. The main difference between these two bounding schemes is in how each handles the additive utility functions. In JGD-ID, we generalize the powered-sum elimination for IDs and derived decomposition bounds by Minkowski’s inequality and Hölder’s inequality. In JGD-EXP, we bound the MEU of M by first using the exponentiated utility bound, which gives M^e, and then by applying the decomposition bounds to the log-partition function of M^e.

Next, we will present the optimization routines for optimzing the weight parametrs and the cost-shifting functions.

Updating Weights The UPDATE-WEIGHTS routine in Algorithm 3.8 updates the weights w^C_X^j_i for each variable Xi over clusters {C|Xi ∈ χ(C)} like in algorithm JGD-ID. The only modification required in Algorithm 3.3 to accommodate that is to provide the gradient expression due to the new objective function U_MEU defined in Eq. (3.68). Namely,

∂U_MEU

Algorithm 3.9 UPDATE-COSTS(G_JG, (C_i, C_j))

Require: a join-graph decomposition GJG, Xi) of an exponentiated ID M^e = hX, D, P, U^e, Oi of M = hX, D, P, U, Oi, separator (C_i, C_j) iteration limit M for the gradient descent

Ensure: an upper bound of the MEU, UMEU,

1: λ_Ci ←P

6: Compute gradient of U_MEU with respect to the cost-shifting function by Eq. (3.80)

7: Find cost-shifting functions δ_Ci,Cj(X_Ci,Cj) by Eq. (3.32).

8: λ_Ci(X_Ci) ← F_Ci(X_Ci) − δ_Ci,Cj(X_Ci,Cj)

9: λ_Cj(X_Cj) ← F_Cj(X_Cj) + δ_Ci,Cj(X_Ci,Cj)

10: Compute new U_MEU using Eq. (3.68) with the updated λ_Ci(X_Ci) and λ_Cj(X_Cj)

11: t = t + 1

Eq. (3.78) is a conditional entropy of the pseudo marginal at cluster C defined by Eq. (3.79).

For more details for deriving Eq. (3.78), see [Ping et al., 2015].

Updating Cost-shifting Functions We now derive the UPDATE-COSTS routine that is used in Algorithm 3.8 to update the cost-shifting functions δ_Ci,Cj(X_Ci,Cj) over separators (C_i, C_j) ∈ S. As usual, this is done via a gradient descent algorithm. We obtain a closed-form gradient expression for the cost-shifting paramters by evaluating the partial derivatives of UMEU in Eq. (3.68) with respect to δCi,Cj(X_Ci,Cj) as follows.

For the derivation of the gradients, see [Ping et al., 2015].

Algorithm 3.9 presents the gradient descent algorithm for optimizing the cost-shifting func-tions. Given a separator (Ci, C_j), we combine the probability functions and the utility functions assigned at cluster C_i and C_j to λ_Ci(X_Ci) and λ_Cj(X_Cj), respectively (line 1-2).

Note that this algorithm performs factor operations in the log-scale. In each iteration, we

set the cost-shifting function δ_Ci,Cj(X_Ci,Cj) to a neutral factor 0(X_Ci,Cj). Then we compute gradient using Eq. (3.80) and find the cost-shifting function by Eq. (3.32) (line 6-7), and then we reparameterize λ_Ci(X_Ci) and λ_Cj(X_Cj) (line 8-9). We repeat this until we reach the iteration limit M or until convergence.

Complexity of JGD-EXP We can analyze the time and space complexity of Algorithm 3.8 by following the same reasoning for analyzing the complexity of algorithm JGD-ID (Theorem 3.2).

Theorem 3.5. Given an ID M := hX, D, P, U, Oi, and its exponentiated ID M^e :=

hX, D, P, U^e, Oi, and given a join-graph decomposition G_JG := hG(C, S), χ, ψi of M^ewith an i-bound, the time complexity of JGD-EXP is O M1·M2·(|X∪D|·|C|·Gw·kⁱ⁺¹+·|S|·Gc·kⁱ⁺¹+)
and the space complexity is O (|C| · kⁱ⁺¹+ |S| · k^|sep|)
, where M1 is the iteration limit of the outer-loop of the BCD method, M₂ is the iteration limit of the gradient descent algorithms in the inner-loop of the BCD method, |X ∪ D| is the number of variables, k is the maximum domain size, |C| and |S| are the number of clusters and separators in the join-graph decom-position G_JG, and |sep| is the separator width. G_w, and G_c are constants that bounds the number of factor operations for evaluating gradients.

Summary Algorithm JGD-EXP generates upper bounds by adapting the GDD algorithm over the join-graph decomposition after exponentiating the utility functions. This formu-lation avoids the non-convexity issue in the previous algorithms JGD-ID and WMBE-ID.

However, the bounding scheme cannot recover the exact solution, even if the i bound ex-ceeds the constrained induced width.

3.3.3 Weighted Mini-bucket Moment Matching Bounds for Exponentiated