Deriving the EM-Based Update Rules in VARSAT University of Toronto Technical Report CSRG-580

Here we show how to derive the EM-Based update rules (EMBP-L, EMBP-G, EMSPL, EMSP-G) that are used in [4] as bias estimators for SAT. The main idea of the derivation can be viewed as recharacterizing the bias estimation task as a maximumlikelihood parameter estimation task that can be solved by variational methods like EM (Expecatation Maximization). The basic framework can actually be applied to arbitrary discrete constraint satisfaction problems, such as the quasigroup-completion problem considered in [2]. For a more in-depth explanation of the difference between BP (Belief Propagation–for SAT) and SP (Survey Propagation), please consult [3]. 1 Preliminaries We will derive four update rules, from the space spanned by choosing either the twovalued model of BP, or the three-valued model of SP, and then applying Expectation Maximization (EM) with either a local or global consistency approximation. Thus the four methods are labeled EMBP-L, EMBP-G, EMSP-L, and EMSP-G. Both BP and SP attempt to label a variable with ‘+’ or ‘-’. Importantly, this indicates that the variable is positively or negatively constrained, as opposed to merely appearing positively or negatively in some satisfying assignment. That is, when we examine a satisfying assignment there must exist some clause that is entirely dependent on the variable being positive in order for us to label it ‘+’. Under BP all variables are assumed to be constrained, while SP introduces the additional ‘*’ case indicating that a variable is unconstrained, i.e. all clauses are already satisfied by the other variables. The derivation will be written out in terms of SP; in the end it will be simple to handle BP just by assigning zero weight to the third ‘*’ state of SP. The algebra of how clauses are supported comes into play when we consider the local and global consistency approximations. The local consistency considered below corresponds to generalized arc-consistency: according to a specific clause c, variable v can hold a value like ‘+’ only if the other variables of c are consistent with v being positive. Note that this cannot capture whether v is truly constrained to be positive; each individual clause might report that they are fine with v being labeled ‘+’, but between them there is no way to determine whether v is constrained to be positive. When we move to the more global form of consistency in the following proof, this shortcoming is remedied at potentially greater computational cost. 2 The Basic EM Framework We first present the general EM framework [1], from a perspective that motivates its application to traditional message-passing tasks, and ultimately, to bias estimation. The result is an improved way to calculate surveys, one that always converges. At a high level, EM accepts a vector of observations Y , and seeks some model parametersΘ that maximize the log-likelihood of having seen Y . This likelihood consists of a posterior probability represented by some model designed for the domain at hand; we seek the best setting of Θ for fitting the model to the observations. Maximizing log P (Y |Θ) would ordinarily be straightforward, but for the additional complication that we posit some latent variables Z that contributed to the generation of Y , but that we did not get to observe. That is, we want to set Θ to maximize log P (Y, Z|Θ), but cannot marginalize on Z. So, we bootstrap by constructing an artificial probability distribution Q(Z) to estimate P (Z|Y,Θ) and then use this distribution to maximize the expected log-likelihood log P (Y, Z|Θ)with respect toΘ. In a sense we simulate the marginalization over Z by using Q(Z) as a surrogate weight: EQ[log P (Z, Y |Θ)] = ∑ Z Q(Z) log P (Z, Y |Θ). The first step of hypothesizing a Q() distribution is called the E-Step, and the second phase of setting Θ is the M-Step. The two are repeated until convergence to a local maximum in likelihood, which is guaranteed.