More about the Learning Algorithm More about the Class being learned Define the model Four examples More about the Model More about the Learning Algorithm More about the Class being learned More about the Teacher More about the Learning Complexity Back to the examples 11/30/2018 Content

Exact Learning Problem from Membership Queries Interpolation Exact Learning MQ Inferring from Q Active Learning Guessing Game Combinatorial Search Hitting Testing Verification Black Box PIT White Box PIT Find 𝑓 exactly Test whether 𝑓≡𝑔, 𝑓≡0 etc. 𝑓 𝑓(𝑥) 𝑓∈𝐶 𝑥 𝑓:𝐷→𝑅 Goal: Find 𝑓 exactly 11/30/2018 Exact learning from MQ

אוניית מלחמה - צוללות (משחק) Battleship game Type of ship Size aircraft carrier 5 battleship 4 submarine 3 destroyer patrol boat 2 1 2 3 4 5 6 7 8 9 10 A B C D E F G H I J Each player secretly arranges their ships אֳנִיָּה on their primary grid The game proceeds in a series of rounds. In each round, each player takes a turn to announce a target square in the opponent's grid which is to be shot יָרָה at. The opponent announces whether or not the square is occupied by a ship, and if it is a "hit" they mark this on their own primary grid. If all of a player's ships have been sunk שָׁקַע, the game is over and their opponent wins. 11/30/2018 Battleship game

𝒇: 𝟏,𝟐,⋯,𝟏𝟎 × 𝑨,𝑩,⋯,𝑱 →{𝟎,𝟏} 𝒇 𝟒,𝑫 =𝟏 a "hit" 𝒇 𝟕,𝑬 =𝟎 a “miss" 𝑪={𝒇} 1 2 3 4 5 6 7 8 9 10 A B C D E F G H I J 𝒇: 𝟏,𝟐,⋯,𝟏𝟎 × 𝑨,𝑩,⋯,𝑱 →{𝟎,𝟏} 𝒇 𝟒,𝑫 =𝟏 a "hit" 𝒇 𝟕,𝑬 =𝟎 a “miss" 𝑪={𝒇} 11/30/2018 Battleship game

1 2 3 4 5 6 7 8 9 10 A B C D E F G H I J Find 𝑓 exactly Test whether 𝑓≡0. Hitting set 11/30/2018 Battleship game

Group Testing Robert Dorfman's paper in 1943 introduced the field of (Combinatorial) Group Testing. The motivation arose during the Second World War when the United States Public Health Service and the Selective service embarked upon a large scale project. The objective was to weed out all syphilitic נדְבִּק men called up for induction. However, syphilis testing back then was expensive and testing every soldier individually would have been very cost heavy and inefficient. We can combine עִרְבּוּב blood samples and test a combined sample together to check if at least one soldier has syphilis. 11/30/2018 Group Testing

Group Testing & Learning 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 , 𝑥 5 , 𝑥 6 , 𝑥 7 , 𝑥 8 , 𝑥 9 , 𝑥 10 , 𝑥 11 , 𝑥 12 𝑥 13 , 𝑥 14 , 𝑥 15 , 𝑥 16 YES NO YES (1101011011010101) (0101010110100101) (1011100110010110) 𝑓= 𝑥 3 ∨ 𝑥 10 11/30/2018 Group Testing & Learning

Decision Tree Group Testing Decision Tree of Depth 𝑑 1 𝑥 2 1 1 1 1 𝑥 4 𝑥 3 2 𝑑 1 1 𝑥 3 1 𝑥 1 1 Group Testing 𝑑 1 1 𝑥 3 𝑓: 0,1 𝑛 →{0,1} 𝑓= 𝑥 3 ∨ 𝑥 10 𝑓 1011 =1 1 𝑥 10 1 𝑓 1000 =0 11/30/2018 Decision Tree

Battleship as Decision Tree 𝑥 1 1 2 3 4 5 6 7 8 9 𝑥 2 1 1 𝑥 2 =5 1 𝑥 1 <3 1 𝑥 2 >4 1 𝑥 2 >1 1 1 𝑥 1 =3 11/30/2018 Battleship as Decision Tree

The structure of part of a DNA 𝑦𝑒𝑠 1 𝑥 2 𝑦𝑒𝑠 1 1 𝑥 4 𝑥 3 𝑛𝑜 𝑦𝑒𝑠 𝑥 3 1 𝑛𝑜 1 𝑥 1 The structure of part of a DNA 𝑦𝑒𝑠 http://en.wikipedia.org/wiki/DNA 11/30/2018 DNA

Resistorנַגָּד Combinations Change the resistance הִתְנַגְּדוּת 𝑅= 1 1 𝑅 6 + 1 𝑅 4 + 𝑅 5 + 1 1 𝑅 1 + 1 𝑅 2 + 𝑅 3 11/30/2018 Resistor Combination

More about the Learning Algorithm More about the Class being learned Define the model Four examples More about the Model More about the Learning Algorithm More about the Class being learned More about the Teacher More about the Learning Complexity Back to the examples 11/30/2018 Content

𝑓 𝑓∈𝐶 𝑥 𝑓(𝑥) 𝑓:𝐷→𝑅 𝐴 𝐶 (𝑀 𝑄 𝑓 ,𝐼,𝑟) Learning Algorithm Goal: Find 𝑓 exactly Learning Algorithm 𝐴 𝐶 (𝑀 𝑄 𝑓 ,𝐼,𝑟) 11/30/2018 Algorithm

Learning Algorithm Sequential Parallel Deterministic Randomized Monte Carlo Alg. Pr 𝑟 𝐴 𝐶 𝑀 𝑄 𝑓 ,𝐼,𝑟 =𝑓 ≥1−𝛿 Deterministic Randomized Las Vegas Alg. Pr 𝑟 𝐴 𝐶 𝑀 𝑄 𝑓 ,𝐼,𝑟 =𝑓 =1 Adaptive Non-adaptive Rounds 11/30/2018 Learning Algorithm

Input Taget 𝒇 𝑪 𝑪 𝒕 Boolean 𝒇:𝑫→{𝟎,𝟏} Arithmetic 𝒇: 𝑹 𝒏 →𝑹 Discrete 𝒇:𝑫→{𝟎,𝟏,⋯,𝒎} 11/30/2018 Target

Output Interpolation Exact Learning MQ Inferring from Q Active Learning Guessing Game Combinatorial Search Hitting Testing Verification Black Box PIT Test if 𝑓≡𝑔, 𝑓≡0 𝒇∈𝑪 Find 𝑓 exactly NonProper computable 𝒉 Proper 𝒉∈𝑪 PAC, PAC+MQ Find 𝑓 approximately 11/30/2018 Summary

Teacher, Black Box Opponent player Honest הוֹגֵן Liar שַׁקְרָן Incomplete MQ Angluin Slonim 94 I DON’T KNOW wp 𝒑 (Limited) Malicious MQ Angluin Krikis Sloan Turan 95 m Incorrect Answers ? with probability p +persistent Malicious MQ Valiant 85- Kearns Li 88 Incorrect wp 𝒑 Limited MQ Angluin Krikis Sloan Turan 95 m I DON’T KNOW Answers incorrect to m + persistent Incorrect answers with probability p + persistent Answers ? with to m + persistent 11/30/2018 Teacher

Query Complexity=𝑂𝑃𝑇(𝐶) 𝑓 𝐶 𝑓(𝑥)∈𝐶 𝑥 =𝑛 𝑠𝑖𝑧𝑒 𝑓 ≤𝑠 Query Complexity=𝑂𝑃𝑇(𝐶) Query Complexity Time Complexity Learnable 𝑝𝑜𝑙𝑦(𝑂𝑃𝑇(𝐶),𝑛,𝑠) Efficient Learnable 𝑝𝑜𝑙𝑦(𝑂𝑃𝑇(𝐶)) 𝑝𝑜𝑙𝑦(𝑂𝑃𝑇(𝐶),𝑛,𝑠) Optimally Learnable 𝑂𝑃𝑇(𝐶) 1+𝑜 1 𝑝𝑜𝑙𝑦(𝑂𝑃𝑇(𝐶),𝑛,𝑠) Optimally Learn. in 𝑛 𝑂𝑃𝑇(𝐶) 1+𝑜 1 , 𝑠=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Randomized Deterministic Teacher Adaptive Non-adaptive Honest Liar 11/30/2018 Complexity

More about the Learning Algorithm More about the Class being learned Define the model Four examples More about the Model More about the Learning Algorithm More about the Class being learned More about the Teacher More about the Learning Complexity Back to the examples 11/30/2018 Content