Massive Online Teaching to
Bounded Learners
Brendan Juba (Harvard)
Ryan Williams (Stanford)
Teaching
Massive Online Teaching
f∈C
f:{0,1}n→{0,1}
Bounded
Arbitrary “consistent
complexity
“consistent
(proper) learner”
learner”
(possibly
[Goldman-Kearns,
improper)
Shinohara-Miyano]
[this work]
g? (g∈C)
f?
THIS WORK
We design strategies for teaching
consistent learners of bounded
computational complexity.
A hard concept
(1,0,0,0)
(0,1,0,0)
We primarily focus
on the number of
(0,0,1,0)
mistakes made (0,0,0,1)
during learning,
Requires all 2
not the number(0,0,0,0)
of examples sent.
examples!!
n
Prop’n: teaching this concept to the class of
consistent learners with linear-size AC0 circuits
requires sending all 2n examples.
(learners’ initial hypothesis may be arbitrary)
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.
I. The model (cont’d)
II. Theorems: Teaching statebounded learners
III.Theorems: Derandomizing
teaching
Bounded consistent learners
• Learner given by pair of bounded functions,
EVAL and UPDATE
• Consistent: learner correct on all seen examples
– …for f∈C
f∈C
σ (g?)
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)
σ’ (f?)
I. The model
II. Theorems: Teaching statebounded learners
III.Theorems: Derandomizing
teaching
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)2log S(n)) mistakes on an example
sequence drawn from the uniform dist. (whp).
A lower bound
vs. O(sn)
• Theorem: There is a consistent learner for
singleton/empty with s-bit states (and O(sn)size AC0 circuits) that makes s-1 mistakes on
the empty concept. (2n ≥ 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.
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.
n
{0,1}
• Suppose not: then since the
learner is consistent, the
mistakes are on ¼ of this
W
initial set of examples W
• Uniform dist: hit S w.p. < ¼
S
conditioned on hitting W
Main Lemma
• Each mistake must come from W
{0,1}n
(by def.); to reach the given
state, they all must hit S.
• The ≥s(n) draws from W all
W
fall into S w.p. < ¼s(n)
⇒we reach the state with ≥¾
of W remaining w.p. < ½2s(n)
S
• Union bound over the (≤2s(n))
states with ≥¾ of W remaining
• Custom sequences: sample from W directly.
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.
I. The model
II. Theorems: Teaching statebounded learners
III.Theorems: Derandomizing
teaching
Derandomization
• Our strategy uses ≈n22n 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!
Using Nisan’s generator
• Theorem: Nisan’s generator produces a
sequence of O(n2n) 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’.
• 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 2n 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 2n mistakes, and so can’t be in P/poly.
{0,1}n
WE SHOWED
W
S
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.
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?
{0,1}n
WE SHOWED
W
S
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.