Active Learning and the Importance of Feedback in Sampling

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Active Learningand the Importance of Feedback
in Sampling

• Rui Castro
• Rebecca Willett and Robert Nowak

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Motivation twenty questions
Goal Accurately learn a concept, as fast as
possible, by strategically focusing in regions of
interest
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Active Sampling in Regression
Learning by asking carefully chosen questions,
constructed using information gleaned from
previous observations
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Passive Sampling
Sample locations are chosen a priori, before any
observations are made
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Active Sampling
Sample locations are chosen as a function of
previous observations
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Problem Formulation
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Passive vs. Active
Passive Sampling
Active Sampling
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Estimation and Sampling Strategies
Goal
The estimator
The sampling strategy
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Classical Smoothness Spaces
Functions with homogeneous complexity over the
entire domain

• Hölder smooth function class

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Smooth Functions minimax lower bound
Theorem (Castro, RW, Nowak 05)
The performance one can achieve with active
learning is the same achievable with passive
learning!!!
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Inhomogeneous Functions
Homogenous functions spread-out complexity
Inhomogeneous functions localized complexity
The relevant features of inhomogeneous functions
are very localized in space, making active
sampling promising
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Piecewise Constant Functions d2
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Passive Learning in the PC Class
Estimation using Recursive Dyadic Partitions (RDP)

Prune the partition, adapting to the data
Recursively divide the domain into hypercubes
Decorate each partition set with a constant
Distribute sample points uniformly over 0,1d
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RDP-based Algorithm
Choose an RDP that fits the data well, but it is
not overly complicated

This estimator can be computed efficiently using
a tree-pruning algorithm.
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Error Bounds
Oracle bounding techniques, akin to the work of
Barron91, can be used to upper bound the
performance of our estimator
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Active Sampling in the PC class
Active Sampling Key learn the location of the
boundary
Use Recursive Dyadic Partitions to find the
boundary
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Active Sampling in the PC Class
Stage 1 Oversample at coarse resolution

• n/2 samples uniformly distributed

• Limit the resolution many more samples than
  cells

• biased, but very low variance result
• (high approximation error, but low estimation
  error)
• boundary zone is
• reliably detected

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Active Sampling in the PC Class
Stage 2 Critically sample in boundary zone

• n/2 samples uniformly distributed
• within boundary zone

• construct fine partition around boundary

• prune partition according to
• standard multiscale methods
• high resolution
• estimate of boundary

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Main Theorem
Main Theorem (Castro 05)

Cusp-free boundaries cannot behave like the
graph of x1/2 at the origin, but milder kinks
like x at 0 are allowable.
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Sketch of the Proof - Approach
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Controlling the Bias
Cells intersecting the boundary may be pruned if
aligned with cell edge

• Solution
• Repeat Stage 1 d-times, using d slightly offset
  partitions
• Small cells remaining in any of the d1
  partitions are passed on to Stage 2

Potential Problem Area
Not a problem after shift
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Multi-Stage Approach
Iterating the approach yields a L-step method
Compare with minimax lower bound
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Learning PC Functions - Summary
Passive Sampling
Active Sampling
This rates are nearly achieved using RDP-based
estimators, that are easily implemented and have
low computational complexity.
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Spatial Adaptivity and Active Learning
Spatially adaptive estimators based on sparse
model selection (e.g., wavelet thresholding) may
provide automatic mechanisms for guiding active
learning processes
Instead of choosing where-to-sample one can
also choose where-to-compute to actively reduce
computation.
Can active learning provably work in even more
realistic situations and under little or no prior
assumptions ?
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Piecewise Constant Functions d 1
Consider first the simplest non-homogenous
function class
step function
This is a parametric class
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Passive Sampling
Distribute sample points uniformly over 0,1 and
use a maximum likelihood estimator
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Active Sampling
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Learning Rates d 1
Passive Sampling
Active Sampling
(Burnashev Zigangirov 74)
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Sketch of the Proof - Stage 1
Error due to approximation of the boundary regions
estimation error
Intuition tells us that this should be the error
we experience away from the boundary
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Sketch of the Proof - Stage 1
Key Limit the resolution of the RDPs
1/k
1/k
This is the performance away from the boundary
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Sketch of the Proof - Stage 1
Are we finding more than the boundary?
Lemma
At least we are not detecting too many areas
outside the boundary.
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Sketch of the Proof - Stage 2
n/2 more samples distributed uniformly over the
boundary
Total error contribution from boundary zone
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Sketch of the Proof Overall Error
Error away from the boundary
Error in the boundary region
Balancing the two errors yields