: f 1; 1gn ! f 1; 1g where n is a natural number that we call the arity of . A classifier class is a (possibly infinite) family of classifiers. We denote by n the restriction of to the n-ary classifiers in . An n-ary training collection is a multiset T of pairs hx; yi where x 2 f 1; 1gn and y 2 f 1; 1g. We denote by Tn the set of all n-ary training collections. A cost function for a classifier class is a function of the form
In the design of machine-learning solutions, a critical and often the most resourceful
task is that of feature engineering [
Classifiers and Learning In this work, a classifier is a function of the form where R 0 is the set of nonnegative numbers. In the context of a classifier class and a cost function c, learning a classifier is the task of finding a classifier 2 n that minimizes c( ; T ), given a training collection T 2 Tn.
We illustrate the above definitions on the important class of linear classifiers. An n-ary linear classifier is parameterized by a vector w 2 Rn, is denoted by w, and is defined as follows for all a 2 f 1; 1gn.
w(a) =def where “ ” denotes the operation of dot product. By Lin we denote the class of linear classifiers. An example of a cost function is the least square cost lsq that is given by lsq ( w; T ) =def (x w
y)2
X hx;yi2T for the arguments
w 2 Linn and T 2 Tn. 2.2
Relational Formalism Our relational terminology is as follows. A schema is a pair (A; ), where A is a signature that consists of relation symbols, and is a set of logical integrity constraints over A. Each relation symbol R has an associated arity. We assume an infinite set Const of constants. An instance I over a schema S = (A; ) associates with every k-ary relation symbol R 2 A a finite subset RI of Constk, such that all the constraints of are satisfied. The active domain of an instance I, denoted adom(I), is the set of all the constants in Const that are mentioned in I.
Let S be schema. A query (over S) is a function Q that is associated with an arity k, and that maps every relation instance I over S into a finite subset Q(I) of Constk. A query Q0 contains a query Q if Q(I) Q0(I) for all instances I over S; if Q Q0 and Q0 Q then Q and Q0 are said to be equivalent. A query Q is additive if for every two instances I1 and I2, if adom(I1) and adom(I2) are disjoint, then
Q(I1 [ I2) = Q(I1) [ Q(I2) : A query class is a mapping that associates with every schema S a class of queries over S. An example of a query class is that of the conjunctive queries. A conjunctive query (CQ) is represented by the logical formula q(x) that has the form 9y[ 1(x; y; d) ^ ^ m(x; y; d)] where x and y are disjoint sequences of variables, d is a sequence of constants, and each i is an atomic query over S (i.e., a formula that consists of a single relation symbol and no logical operators). The result of applying the CQ Q = q(x) to the instance I consists of all the tuples a (of the same length as x) such that q(a) is true in I; we denote this result is denoted by Q(I). 2.3
Classification Framework We now present our formal framework. An entity schema is a triple (A; ; E), where (A; ) is a schema and E is a relation symbol in A that represents the entities (as tuples). An instance I over a schema (A; ; E) is simply an instance over (A; ). Intuitively, an instance I over an entity schema (A; ; E) represents a set of entities, namely EI (i.e., the set of tuples in E), along with information about the entities that is contained in the remaining relations RI . For example, E may be the relation Persons and A may include, besides Persons, relations such as PersonAddress, PersonCompany, CompanyCity, and so on. If S = (A; ; E) is an entity schema, then the elements A, and E are denoted by AS, S and ES, respectively.
Let S be an entity schema, and let I be an instance over S. A feature query (over S) is a query over the schema S, such that ES, where ES is viewed as the query that, given an instance I, copies the relation ESI. In other words, a feature query is a query that selects entities. For example, if E is Persons(ssn; name) then a feature can be the following CQ q(s; n) (selecting persons working in New York City).
9c Persons(s; n) ^ PersonCompany(s; c) ^ CompanyCity(c; ‘NYC’) If is a feature query, then I denotes the function f : ESI ! f 1; 1g where f (e) = (1
if e 2 1 otherwise.
(I); A statistic (over S) is a sequence = ( 1; : : : ; n) of feature queries. We denote by I the function ( 1I ; : : : ; nI) from ESI to f 1; 1gn.
A feature schema is a pair (S; ), where S is an entity schema, and is a statistic over S that produces a sequence of features for every entity of a given input instance. We say that and (S; ) are in a query class Q if every query in belongs to Q. A training instance over S is a pair (I; o), where I is an instance over S and o : ESI ! f 1; 1g is a function that partitions the entities into positive and negative examples. Given a feature schema (S; ) and a classifier class , the training instance (I; o) defines the training collection that consists of the tuple h I (e); o(e)i for every e 2 ESI. 3
In feature engineering, one devises feature queries for a classification task. Next, we discuss two computational challenges that naturally arise in feature engineering. 3.1
Separability Let (S; ) be a feature schema, and let be a classifier class. A training instance (I; o) is said to be -separable with respect to (w.r.t.) if there exists a classifier 2 that fully agrees with o; that is, and have the same arity, and (e) = o(e) for every e 2 ESI. We define the following core computational problem.
Problem 1 (Separability). Let S be an entity schema, let Q be a query class over S, and let be a classifier class. The separability problem is the following. Given a training instance (I; o) over S, determine whether there exists a statistic in Q such that (I; o) is -separable w.r.t. .
The separability problem, as defined, can be extended in various practical directions. The input can include a bound N on the length n of the statistic (hence, limiting the model complexity, which results in classifiers that are more efficient and less overfitting). One can allow for an approximate agreement with o (e.g., the classifier should agree with o on at least (1 ) of the entities, or at most k examples should be misclassified). And one can impose various constraints on common query classes Q (e.g., limit the size of queries, number of constants, etc., again to limit the model complexity and potential overfitting). The following theorem considers the complexity of testing for separability in the case where the class of queries is that of CQs without constants,3 which we denote by CQnc. It states that, in the absence of such extensions of the problem, it can very quickly get intractable.
Theorem 1. Let Q be the class CQnc, and let be the class Lin. For every entity schema S, separability is in NP. Moreover, there exists an entity schema S such that separability is NP-complete.
The proof of membership in NP is using the concept of a canonical database [
We note that a problem similar to separability has been studied in a recent paper by
Cohen and Weiss [
Statistic Identifiability We denote by 0m the vector of m zeroes. Let M be an n k real matrix. A linear column dependence in M is a weight vector w 2 Rk such that w 6= 0k and M w = 0n; if M does not have any linear column dependence, then we say that M is linearly column independent. Let (S; ) be a feature schema, and let I be an instance of S. We fix an arbitrary order over the entities in ESI, and denote by J I K the matrix that consists of the rows I (e) for every e 2 ESI in order. The second computational problem we define is the following.
Problem 2 (Identifiability). Let Q be a query class. Identifiability is the problem of testing, given a feature schema (S; ) in Q, whether there exists an instance I over S such that the matrix J I K is linearly column independent; in that case, we say that is identifiable.
Identifiability is an important property in the design of machine-learning solutions [
Next, we show that in the case of CQs. identifiability amounts to query equivalence. A statistic is said to have redundancy if it contains two distinct feature queries that are equivalent.
Theorem 2. Let Q be the query class of additive CQs, and let (S; ) be a feature schema such that is in Q. Then is identifiable if and only if has no redundancy.
We conclude with comments on Theorem 2. First, this theorem is proved again
by applying the concept of a canonical database of a CQ. Second, we can extend the
theorem to the class of all CQs, but the condition that characterizes identifiability is
significantly more complicated (and will be given in the extended version of this paper).
Third, this theorem generalizes to affine independence, which is important in different
cost functions such as maximum entropy [
Benny Kimelfeld is a Taub Fellow, supported by the Taub Foundation. The research of Christopher Re´ is supported by DARPA’s projects XDATA (FA8750-12-2-0335), DEFT (FA8750-13-2-0039), MEMEX, and SIMPLEX. His research is also supported by NSF Career Award IIS-1353606, ONR Awards N000141210041 and N000141310129, NIH Grant U54EB020405 (awarded by NIBIB through funds provided by the trans-NIH BD2K initiative), the Sloan Research Fellowship, the Moore Foundation, American Family Insurance, Google, and Toshiba. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA, AFRL, NSF, ONR, NIH, or the U.S. government.