Active Learning and the Importance of Feedback in Sampling Rui Castro Rebecca Willett and Robert Nowak

Motivation – “twenty questions” Goal: Accurately “learn” a concept, as fast as possible, by strategically focusing in regions of interest

Learning by asking carefully chosen questions, constructed using information gleaned from previous observations Active Sampling in Regression

Sample locations are chosen a priori, before any observations are made Passive Sampling

Sample locations are chosen as a function of previous observations Active Sampling

Problem Formulation

Passive vs. Active Passive Sampling: Active Sampling:

Estimation and Sampling Strategies Goal: The estimator : The sampling strategy :

Classical Smoothness Spaces Functions with homogeneous complexity over the entire domain - Hölder smooth function class

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!!!

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

Piecewise Constant Functions – d ≥ 2 best possible rate

Passive Learning in the PC Class Estimation using Recursive Dyadic Partitions (RDP) Prune the partition, adapting to the dataRecursively divide the domain into hypercubesDecorate each partition set with a constantDistribute sample points uniformly over [0,1] d

RDP-based Algorithm Choose an RDP that fits the data well, but it is not overly complicated empirical risk measures fit of the data Complexity penalty This estimator can be computed efficiently using a tree-pruning algorithm.

Error Bounds Oracle bounding techniques, akin to the work of Barron’91, can be used to upper bound the performance of our estimator approximation errorcomplexity penalty balancing the two terms

Active Sampling in the PC class Active Sampling Key: learn the location of the boundary Use Recursive Dyadic Partitions to find the boundary

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

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

Main Theorem * Cusp-free boundaries cannot behave like the graph of |x| 1/2 at the origin, but milder “kinks” like |x| at 0 are allowable. Main Theorem (Castro ’05): *

Sketch of the Proof - Approach

Controlling the Bias Not a problem after shift Potential Problem Area 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 d+1 partitions are passed on to Stage 2

Iterating the approach yields a L-step method Compare with minimax lower bound: Multi-Stage Approach

Passive Sampling: Learning PC Functions - Summary Active Sampling: This rates are nearly achieved using RDP-based estimators, that are easily implemented and have low computational complexity.

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 ? Spatial Adaptivity and Active Learning

Piecewise Constant Functions – d =1 Consider first the simplest non-homogenous function class step function This is a parametric class

Passive Sampling Distribute sample points uniformly over [0,1] and use a maximum likelihood estimator

Active Sampling

Learning Rates – d =1 Passive Sampling: Active Sampling: (Burnashev & Zigangirov ’74)

Sketch of the Proof - Stage 1 Intuition tells us that this should be the error we experience away from the boundary Error due to approximation of the boundary regions estimation error

Sketch of the Proof - Stage 1 Key: Limit the resolution of the RDPs 1/k This is the performance away from the boundary 1/k

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.

Sketch of the Proof - Stage 2 n/2 more samples distributed uniformly over the boundary Total error contribution from boundary zone:

Sketch of the Proof – Overall Error Error away from the boundary Balancing the two errors yields Error in the boundary region