p samples Test/Learn What if your samples arent quite right? What - - PowerPoint PPT Presentation

▶
p
SMART_READER_LITE
LIVE PREVIEW

p samples Test/Learn What if your samples arent quite right? What - - PowerPoint PPT Presentation

Nov 05, 2022 545 likes •1.07k views

Classy sample correctors 1 Ronitt Rubinfeld MIT and Tel Aviv University joint work with Clement Canonne (Columbia) and Themis Gouleakis (MIT) 1 thanks to Clement and G for inspiring this classy title Our usual model: p samples

slide-1
SLIDE 1

“Classy” sample correctors1

Ronitt Rubinfeld MIT and Tel Aviv University joint work with Clement Canonne (Columbia) and Themis Gouleakis (MIT)

1thanks to Clement and G for inspiring this classy title

slide-2
SLIDE 2

Our usual model:

p

Test/Learn samples

slide-3
SLIDE 3

What if your samples aren’t quite right?

slide-4
SLIDE 4

Some sensors lost power, others went crazy!

What are the traffic patterns?

slide-5
SLIDE 5

A meteor shower confused some of the measurements

Astronomical data

slide-6
SLIDE 6

Never received data from three of the community centers!

Teen drug addiction recovery rates

slide-7
SLIDE 7

Correction of location errors for presence-only species distribution models

[Hefley, Baasch, Tyre, Blankenship 2013]

Whooping cranes

slide-8
SLIDE 8

What is correct?

slide-9
SLIDE 9

What is correct?

slide-10
SLIDE 10
  • Outlier detection/removal
  • Imputation
  • Missingness
  • Robust statistics
  • …

What to do?

What if don’t know that the distribution (and even noise) is normal, Gaussian, …?

Weaker assumption?

slide-11
SLIDE 11

A suggestion for a methodology

slide-12
SLIDE 12

Sample corrector assumes that original distribution in class P (e.g., P is class of Lipshitz, monotone, k-modal, or k-histogram distributions)

What is correct?

slide-13
SLIDE 13
  • Classy Sample Correctors

P

q q’

slide-14
SLIDE 14
  • Classy Sample Correctors
  • 1. Sample complexity per output

sample of q’?

  • 2. Randomness complexity per
  • utput sample of q’?
slide-15
SLIDE 15

P’

  • Classy “non-Proper” Sample Correctors

P

q q’

slide-16
SLIDE 16
  • A very simple (nonproper) example
slide-17
SLIDE 17

k-histogram distribution

1

n

slide-18
SLIDE 18

Close to k-histogram distribution

1

n

slide-19
SLIDE 19

A generic way to get a sample corrector:

slide-20
SLIDE 20

Agnostic learner

An observation

Sample corrector

What is an agnostic learner? Or even a learner?

slide-21
SLIDE 21
  • What is a ``classy’’ learner?
slide-22
SLIDE 22
  • What is a ``classy’’ agnostic learner?
slide-23
SLIDE 23

Agnostic learner

An observation

Sample corrector

Corollaries: Sample correctors for

  • monotone distributions
  • histogram distributions
  • histogram distributions under promises (e.g.,

distribution is MHR or monotone)

slide-24
SLIDE 24
  • Learning monotone distributions
slide-25
SLIDE 25
  • Birge Buckets

You know the boundaries! Enough to learn the marginals

  • f each bucket
slide-26
SLIDE 26
  • A very special kind of error
  • 1. Pick sample x from p
  • 2. Output y chosen UNIFORMLY

from x’s Birge Bucket

“Birge Bucket Correction”

slide-27
SLIDE 27

When can sample correctors be more efficient than agnostic learners?

Some answers for monotone distributions:

  • Error is REALLY small
  • Have access to powerful queries
  • Missing consecutive data errors
  • Unfortunately, not likely in general case (constant

arbitrary error, no extra queries) [P. Valiant]

The big open question:

slide-28
SLIDE 28
  • Learning monotone distributions

Proof Idea: Mix Birge Bucket correction with slightly decreasing distribution (flat on buckets with some space between buckets)

OBLIVIOUS CORRECTION!!

slide-29
SLIDE 29
  • A lower bound [P. Valiant]
slide-30
SLIDE 30
  • What about stronger queries?
slide-31
SLIDE 31

Use Birge bucketing to reduce p to an O(log n)-histogram distribution

First step

slide-32
SLIDE 32
  • Fixing with CDF queries

superbuckets

slide-33
SLIDE 33
  • Fixing with CDF queries

Add some weight Remove some weight

slide-34
SLIDE 34
  • Fixing with CDF queries
slide-35
SLIDE 35

Reweighting within a superbucket

slide-36
SLIDE 36

“Water pouring” to fix superbucket boundaries

Extra “water”

What if there is not enough pink water? What if there is too much pink water? Could it cascade arbitrarily far?

slide-37
SLIDE 37
  • Missing data segment errors – p is a member
  • f P with a segment of the domain removed
  • E.g. power failure for a whole block in traffic data

Special error classes

More efficient sample correctors via “learning” missing part

slide-38
SLIDE 38

Sample correctors provide power!

slide-39
SLIDE 39
  • Sample correctors provide more

powerful learners:

slide-40
SLIDE 40

Sample correctors provide more powerful property testers:

  • Often much

harder

slide-41
SLIDE 41
  • Sample correctors provide more

powerful testers:

slide-42
SLIDE 42
  • Sample correctors provide more

powerful testers:

Estimates distance between two distributions

slide-43
SLIDE 43
  • Use sample corrector on p to output p’
  • Test that p’ in D
  • Ensure that p’ close to p using distance

approximator

Proof: Modifying Brakerski’s idea to get tolerant tester

If p close to D, then p’ close to p and in D If p not close to D, we know nothing about p’: (1) may not be in D (2) may not be close to p

slide-44
SLIDE 44
  • Can we correct using little randomness of our
  • wn?
  • Note that agnostic learning method relies on using
  • ur own random source
  • Compare to extractors (not the same)

Randomness Scarcity

slide-45
SLIDE 45
  • Can we correct using little randomness of our
  • wn?
  • Generalization of Von Neumann corrector of

biased coin

  • For monotone distributions, YES!

Randomness Scarcity

slide-46
SLIDE 46
  • Correcting to uniform distribution
  • Output convolution of a few samples

Randomness scarcity: a simple case

slide-47
SLIDE 47

Yet another new model!

In conclusion…

slide-48
SLIDE 48

What classes can we correct?

What next for correction?

slide-49
SLIDE 49

When is correction easier than agnostic learning?

What next for correction?

When is correction easier than (non-agnostic) learning?

slide-50
SLIDE 50
  • Estimating averages of survey/experimental

data

  • Learning

How good is the corrected data?

slide-51
SLIDE 51

Thank you