Learning Binary Relations Presented by Alan Duan 1 / 97 Motivation - PowerPoint PPT Presentation

Learning Binary Relations Presented by Alan Duan 1 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T 2 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T and are related by some rule. S T 3 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T and are related by some rule. S T Consider one relation: Student presents topic . s t 4 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T and are related by some rule. S T Consider one relation: Student presents topic . s t For example, Alan presents the topic ' learning binary relations ', and Mark presented both ' tail inequalities ' and ' realizable selective sampling '. 5 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T and are related by some rule. S T Consider one relation: Student presents topic . s t For example, Alan presents the topic ' learning binary relations ', and Mark presented both ' tail inequalities ' and ' realizable selective sampling '. Clearly, student either presents topic , or does not. s t 6 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T and are related by some rule. S T Consider one relation: Student presents topic . s t For example, Alan presents the topic ' learning binary relations ', and Mark presented both ' tail inequalities ' and ' realizable selective sampling '. Clearly, student either presents topic , or does not. s t The predicate relating the two sets of variables is either true or false. 7 / 97

Motivation of Binary Relations Let's start by considering the set of all students (let's call it ), and the set of all topics in this course ( ). S T and are related by some rule. S T Consider one relation: Student presents topic . s t For example, Alan presents the topic ' learning binary relations ', and Mark presented both ' tail inequalities ' and ' realizable selective sampling '. Clearly, student either presents topic , or does not. s t The predicate relating the two sets of variables is either true or false. We call this a binary relation. 8 / 97

Formal De�nition of Binary Relations A binary relation between two sets and is a subset of . R A B A × B 9 / 97

Formal De�nition of Binary Relations A binary relation between two sets and is a subset of . R A B A × B Each binary relation is associated with a predicate : P : A × B ↦ {0, 1} 10 / 97

Formal De�nition of Binary Relations A binary relation between two sets and is a subset of . R A B A × B Each binary relation is associated with a predicate : P : A × B ↦ {0, 1} 1, if ( a , b ) ∈ R P ( a , b ) = { 0, otherwise 11 / 97

Formal De�nition of Binary Relations A binary relation between two sets and is a subset of . R A B A × B Each binary relation is associated with a predicate : P : A × B ↦ {0, 1} 1, if ( a , b ) ∈ R P ( a , b ) = { 0, otherwise Note : 1. Binary relations can be defined between different set (e.g.: Netflix user and movie), or the set with itself (e.g.: the relation 'divides' between and ). ℕ + ℕ + 12 / 97

Formal De�nition of Binary Relations A binary relation between two sets and is a subset of . R A B A × B Each binary relation is associated with a predicate : P : A × B ↦ {0, 1} 1, if ( a , b ) ∈ R P ( a , b ) = { 0, otherwise Note : 1. Binary relations can be defined between different set (e.g.: Netflix user and movie), or the set with itself (e.g.: the relation 'divides' between and ). ℕ + ℕ + 2. In binary relations, the order matters. 13 / 97

Representing a Binary Relations binary matrix n × m Topics in Learning Theory Machine Learning Operating System Alan 1 0 0 Bob 1 1 0 Cathy 0 0 1 David 0 0 0 14 / 97

Representing a Binary Relations binary matrix n × m Topics in Learning Theory Machine Learning Operating System Alan 1 0 0 Bob 1 1 0 Cathy 0 0 1 David 0 0 0 2-column table Student Course Alan Topics in Learning Theory Bob Topics in Learning Theory Bob Machine Learning Cathy Operating System 15 / 97

Representing a Binary Relations (cont'd) Bipartite graph 16 / 97

Learning Binary Relations Setting We are learning binary relations between two set and represented by predicate . Denote and A B P | A | = n . | B | = m In each trial : t learner is given an unlabeled pair of object , where x t = ( , a t b t ) a t ∈ A , b t ∈ B learner predicts 0 or 1 y ̂ = t reveals the answer y t if answer and prediction are different, record it as a mistake Goal: Minimize the number of incorrect predictions 17 / 97

Learning Binary Relations Question: Can we reduce the learning of binary relations to something we have seen? 18 / 97

Learning Binary Relations Question: Can we reduce the learning of binary relations to something we have seen? Yes! 19 / 97

Learning Binary Relations Question: Can we reduce the learning of binary relations to something we have seen? Yes! ,  = A × B  = {0, 1} Target hypothesis h = P This is an online concept learning (realizable) setting! 20 / 97

Learning Binary Relations Question: Can we reduce the learning of binary relations to something we have seen? Yes! ,  = A × B  = {0, 1} Target hypothesis h = P This is an online concept learning (realizable) setting! Note : 1. In this presentation, we will use these notation from concept learning interchangably from time to time. 2. We will see what is special about learning binary relations in a bit! 21 / 97

Learning Binary Relations A few more terms Let be a finite learning domain. Let be a concept class over .   C A learner is consistent if, on every trial, there exists some concept such that: c ∈ C y ̂ , if k = t t c ( x k ) = { y k , if k = 1, … , t − 1 A query sequence is a permutation of , where is the instance presented to the  x t ∈  π = ⟨ x 1 x 2 , , … , x |  | ⟩ learner at the trial. t th 22 / 97

Learning Binary Relations Who determines the query sequence? 23 / 97

Learning Binary Relations Who determines the query sequence? Director! 24 / 97

Learning Binary Relations Who determines the query sequence? Director! In this presentation, we will consider the following settings: Director Agnostic: we want some mistake bounds regardless of the director. 25 / 97

Learning Binary Relations Who determines the query sequence? Director! In this presentation, we will consider the following settings: Director Agnostic: we want some mistake bounds regardless of the director. Self-directed: the learner itself chooses . π 26 / 97

Learning Binary Relations Who determines the query sequence? Director! In this presentation, we will consider the following settings: Director Agnostic: we want some mistake bounds regardless of the director. Self-directed: the learner itself chooses . π Teacher-directed: A teacher who knows the target relation and wants to minimize the learner's mistakes by choosing ; Teacher can choose with the knowledge of 1) target relation, 2) , 3) x t π x 1 , … , x t − 1 . y ̂ y ̂ , … , 1 t − 1 27 / 97

Learning Binary Relations Who determines the query sequence? Director! In this presentation, we will consider the following settings: Director Agnostic: we want some mistake bounds regardless of the director. Self-directed: the learner itself chooses . π Teacher-directed: A teacher who knows the target relation and wants to minimize the learner's mistakes by choosing ; Teacher can choose with the knowledge of 1) target relation, 2) , 3) x t π x 1 , … , x t − 1 . y ̂ y ̂ , … , 1 t − 1 Adversary-directed: An adversary who tries to maximize the learner's mistakes, knows the learner's algorithm and has unlimited computing power, chooses . π 28 / 97

Learning Binary Relations Who determines the query sequence? Director! In this presentation, we will consider the following settings: Director Agnostic: we want some mistake bounds regardless of the director. Self-directed: the learner itself chooses . π Teacher-directed: A teacher who knows the target relation and wants to minimize the learner's mistakes by choosing ; Teacher can choose with the knowledge of 1) target relation, 2) , 3) x t π x 1 , … , x t − 1 . y ̂ y ̂ , … , 1 t − 1 Adversary-directed: An adversary who tries to maximize the learner's mistakes, knows the learner's algorithm and has unlimited computing power, chooses . π For teacher-directed setting, we want to consider worst case mistake bound over all consistent learners. (why?) 29 / 97

Motivation of k-binary-relations Now let's talk about what can be special about binary relations. 30 / 97