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Massive Online Teaching to Bounded Learners Brendan Juba (Harvard) Ryan Williams (Stanford)

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Teaching

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Massive Online Teaching Arbitrary “consistent (proper) learner” [Goldman-Kearns, Shinohara-Miyano] f ∈ C f:{0,1} n →{0,1} x ∈ {0,1} n f(x) ∈ {0,1} y f(y) … Bounded complexity “consistent learner” (possibly improper) [this work] g?g? (g ∈ C ) g?g? f?f?

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THIS WORK We design strategies for teaching consistent learners of bounded computational complexity.

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A hard concept (1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1) (0,0,0,0) Requires all 2 n examples!! Prop’n: teaching this concept to the class of consistent learners with linear-size AC 0 circuits requires sending all 2 n examples. (learners’ initial hypothesis may be arbitrary) We primarily focus on the number of mistakes made during learning, not the number of examples sent. 00011011

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WE SHOW 1)The state complexity of the learners controls the optimal number of mistakes in teaching 2)It also controls the optimal length of an individually tailored sequence of examples 3)The strategy establishing (1) can be partially derandomized (to a polynomial-size seed), but full derandomization implies strong circuit lower bounds.

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I.The model (cont’d) II.Theorems: Teaching state- bounded learners III.Theorems: Derandomizing teaching

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Bounded consistent learners Learner given by pair of bounded functions, EVAL and UPDATE Consistent: learner correct on all seen examples – …for f ∈ C f(x) ∈ {0,1} σ (g?) σ’ (f?) x ∈ {0,1} n EVAL(σ,x) = g(x) (σ ∈ {0,1} s(n) ) UPDATE(σ,x,f(x)) = σ’ (Require EVAL(σ’,x) = f(x)) We consider all bounded & C -consistent (EVAL,UPDATE) f∈Cf∈C

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I.The model II.Theorems: Teaching state- bounded learners III.Theorems: Derandomizing teaching

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Uniform random examples are good Theorem: under uniform random examples, any consistent s(n)-state bounded learner identifies the concept with probability 1-δ after O(s(n)(n+log 1/δ)) mistakes. Corollary: Every consistent learner with S(n)- size (bdd. fan-in) circuits identifies the concept after O(S(n) 2 log S(n)) mistakes on an example sequence drawn from the uniform dist. (whp).

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A lower bound Theorem: There is a consistent learner for singleton/empty with s-bit states (and O(sn)- size AC 0 circuits) that makes s-1 mistakes on the empty concept. (2 n ≥ s ≥ n) Idea: Divide {0,1} n into s-1 intervals; the learner’s state initially indicates whether each interval should be labeled 0 or (by default) 1. It switches to a singleton hypothesis on a positive example, and switches off the corresponding interval on negative examples. vs. O(sn)

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Main Lemma After s(n) mistakes, the fraction of {0,1} n that the learner could ever label incorrectly is reduced by a ¾ factor whp. Suppose not: then since the learner is consistent, the mistakes are on ¼ of this initial set of examples W Uniform dist: hit S w.p. < ¼ conditioned on hitting W {0,1} n W S

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Main Lemma Each mistake must come from W (by def.); to reach the given state, they all must hit S. The ≥s(n) draws from W all fall into S w.p. < ¼ s(n) ⇒ we reach the state with ≥¾ of W remaining w.p. < ½ 2s(n) Union bound over the (≤2 s(n) ) states with ≥¾ of W remaining Custom sequences: sample from W directly. {0,1} n W S

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RECAP Theorem: under uniform random examples, any consistent s(n)-state bounded learner identifies the concept with probability 1-δ after O(s(n)(n+log 1/δ)) mistakes. Theorem: Every deterministic consistent s(n)-state bounded learner has a sequence of examples of length O(s(n) n) after which it identifies the concept.

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I.The model II.Theorems: Teaching state- bounded learners III.Theorems: Derandomizing teaching

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Derandomization Our strategy uses ≈n 2 2 n random bits The learner takes examples given by blocks of n uniform-random bits at a time and stores only s(n) bits between examples ☞ Nisan’s pseudorandom generator should apply!

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Using Nisan’s generator Theorem: Nisan’s generator produces a sequence of O(n2 n ) examples from a seed of length O((n+log 1/δ)(s(n)+n+log 1/δ)) s.t. w.p. 1-δ, a s(n)-state bounded learner identifies the concept and makes at most O(s(n)(n+log 1/δ)) mistakes. Idea: consider A’ that simulates A and counts its mistakes using n more bits. The “good event” for A can be defined in terms of the states of A’.

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Using Nisan’s generator still requires poly(n) random bits. Can we construct a low-mistake sequence deterministically? Theorem: Suppose there is an EXP algorithm that for every polynomial S(n) and suff. large n, produces a sequence s.t. every S(n)-size circuit learner makes less than 2 n mistakes to learn the empty concept. Then EXP ⊄ P/poly. Idea: there is then an EXP learner that switches to the next singleton in the sequence as long as the examples remain on the sequence. This learner makes 2 n mistakes, and so can’t be in P/poly.

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WE SHOWED 1)The state complexity of the learners controls the optimal number of mistakes in teaching 2)It also controls the optimal length of an individually tailored sequence of examples 3)The strategy establishing (1) can be partially derandomized (to a polynomial-size seed), but full derandomization implies strong circuit lower bounds. {0,1} n W S

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Open problems O(sn) mistake upper bound vs. Ω(s) mistake lower bound—what’s the right bound? Can we weaken the consistency requirement? Can we establish EXP ⊄ ACC by constructing deterministic teaching sequences for ACC? Does EXP ⊄ P/poly conversely imply deterministic algorithms for teaching?

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WE SHOWED 1)The state complexity of the learners controls the optimal number of mistakes in teaching 2)It also controls the optimal length of an individually tailored sequence of examples 3)The strategy establishing (1) can be partially derandomized (to a polynomial-size seed), but full derandomization implies strong circuit lower bounds. {0,1} n W S

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