Probabilistic logic programming (PLP) is a powerful tool for reasoning in relational domains with uncertainty. However, both inference and learning from examples are computationally expensive tasks. We consider a restriction of PLP called hierarchical PLP whose clauses and predicates are hierarchically organized forming a deep neural network or arithmetic circuit. Inference in this language is much cheaper than for general PLP languages. In this work we present an algorithm called Deep Parameter learning for HIerarchical probabilistic Logic programs (DPHIL) that learns hierarchical PLP parameters using gradient descent and back-propagation.
Probabilistic logic programming (PLP) under the distribution semantics [
The paper is organized as follows: we describe HPLP in Section 2 and present parameter learning in Section 3. Related work is discussed in Section 4 and Section 5 concludes the paper. hi1; : : : ; hini are logical atoms, bi1; : : : ; bimi are logical literals and wih1e;r:e:: P;ninii are real numbers in the interval [0; 1] that sum up to 1. Clauses k=1 ik < 1 implicitly contain an extra atom null that does not appear in the body of any clause and whose annotation is 1 Pkn=i1 ik.
An HPLP is a restricted LPAD where clauses are of the form Cpi = ap :
pi : pi; bpi1; : : : ; bpimp . ap, an atom possibly about target predicate (the
atom we aim at predicting), is the single head atom annotated with probability
pi. The body of the clause has 2 parts: a conjunction of input predicates, pi, in
the sense that their de nition is given as input and is certain, and a conjunction
of literals for hidden predicates, bpik, that are de ned by clauses of the program.
Hidden predicates are disjoint from input and target predicates. Moreover, the
program is hierarchically organized so that it can be divided into layers forming
an Arithmetic circuit or a deep neural netwoks, see [
Let us consider the following example in the UW-CSE domain where the objective is to predict the \advised by" relation (advisedby/2 is the target predicate)
C1 = advisedby(A; B) : 0:3 : student(A); prof essor(B); project(C; A); project(C; B); r11(A; B; C): C2 = advisedby(A; B) : 0:6 :
student(A); prof essor(B); ta(C; A); taughtby(C; B): C111 = r11(A; B; C) : 0:2 :
publication(D; A; C); publication(D; B; C): where project(C; A) means that C is a project with participant A, ta(C; A) means that A is a teaching assistant (TA) for course C and taughtby(C; B) means that course C is taught by B. publication(A; B; C) means that A is a publication with author B produced in project C. student=1, prof essor=1, project=2, ta=2, taughtby=2 and publication=3 are input predicates and r11=3 is a hidden predicate.
The probability p = advisedby(harry; ben) depends on the number of joint courses and projects and on the number of joint publications from projects. The clause for the hidden predicate r11=3 computes an aggregation over publications of the same project and the clause of the level above aggregates over projects. The corresponding arithmetic circuit is shown in Fig 1 .
In order to perform inference with such a program, we can generate its grounding. Each ground probabilistic clause is associated with a random variable whose probability of being true is given by the parameter of the clause and that is independent of all the other clause random variables. Given a ground instance of Cpi, the probability that the body is true is computed by multiplying the probability of being true of each individual atom in positive literals and one minus the probability of being true of each individual atom in negative literals. Therefore the probability of the body of Cpi is P (bpi1; : : : ; bpimp ) = Qim=pk P (bpik) and P (bpik) = 1 P (apik) if bpik = :apik. If bpik = a is a literal for an input predicate, P (a) = 1 if it belongs to the example interpretation and P (a) = 0 otherwise. To compute the probability, P (ap), of an atom ap of a given hidden 0:3 0:2
L L
L
0:6
literal bp, we need to take into account the contribution of every ground clause
for the atom ap. For a single clause, P (ap) = p1 P (body(Cp1)). For two clauses,
we can compute the contribution of each clause as above and then combine them
with the formula P (api) = 1 (1 p1 P (body(Cp1)) (1 p2 P (body(Cp2))).
We de ned the operator that combines two probabilities as follows p q =
1 (1 p) (1 q). This operator is commutative and associative. For many
clauses, we can compute sequences of applications as Li pi = 1 Qi(1 pi). AC
are built by performing inference using tabling and answer subsumption using
PITA(IND,IND) [
We propose a back-propagation algorithm for learning HPLP parameters. Given
an HPLP T with parameters , an interpretation I de ning input predicates and
a set of positive and negative examples E = fe1; : : : ; eM ; not eM+1; : : : ; not eN g
where each ei is a ground atom for the target predicate r, we want to nd the
values of that minimize the sum of cross entropy errors, erri = yi log(pi)
(1 yi) log(1 pi), for all the examples. The algorithm, called DPHIL 1, starts by
building a set of ground ACs sharing parameters , line 2. Then weights (W [i]),
gradients (G[i]) and moments (adam's parameters, see [
node
i) if n= (Wi)
pan = not(n)
(1)
where pan is the parent of node n and i = (Wi) = 1+e1 Wi , lines 9{16.
Parameters are updated using Adam the optimizer [
HPLP is related to [
We have presented the algorithm DPHIL that performs parameter learning of
HPLP. DPHIL converts an HPLP into an AC and performs gradient descent
and back-propagation. In the future we plan to experiments with DPHIL and
investigate the Expectation Maximization algorithm as an alternative parameter
learning method. We also plan to design an algorithm for learning both the
structure and the parameters of HPLPs, taking inspiration from SLIPCOVER
[