Predictive models from interpolation

Daniel Hsu

Computer Science Department & Data Science InstituteColumbia University

Simons Institute for the Theory of ComputingJune 3, 2019

Spoilers

"A model with zero training error is overfit to the training data and will typically generalize poorly."

– Hastie, Tibshirani, & Friedman, The Elements of Statistical Learning

2

Springer Series in Statistics

Trevor HastieRobert TibshiraniJerome Friedman

Springer Series in Statistics

The Elements ofStatistical LearningData Mining, Inference, and Prediction

The Elements of Statistical Learning

During the past decade there has been an explosion in computation and information tech-nology. With it have come vast amounts of data in a variety of fields such as medicine, biolo-gy, finance, and marketing. The challenge of understanding these data has led to the devel-opment of new tools in the field of statistics, and spawned new areas such as data mining,machine learning, and bioinformatics. Many of these tools have common underpinnings butare often expressed with different terminology. This book describes the important ideas inthese areas in a common conceptual framework. While the approach is statistical, theemphasis is on concepts rather than mathematics. Many examples are given, with a liberaluse of color graphics. It should be a valuable resource for statisticians and anyone interestedin data mining in science or industry. The book’s coverage is broad, from supervised learning(prediction) to unsupervised learning. The many topics include neural networks, supportvector machines, classification trees and boosting—the first comprehensive treatment of thistopic in any book.

This major new edition features many topics not covered in the original, including graphicalmodels, random forests, ensemble methods, least angle regression & path algorithms for thelasso, non-negative matrix factorization, and spectral clustering. There is also a chapter onmethods for “wide” data (p bigger than n), including multiple testing and false discovery rates.

Trevor Hastie, Robert Tibshirani, and Jerome Friedman are professors of statistics atStanford University. They are prominent researchers in this area: Hastie and Tibshiranideveloped generalized additive models and wrote a popular book of that title. Hastie co-developed much of the statistical modeling software and environment in R/S-PLUS andinvented principal curves and surfaces. Tibshirani proposed the lasso and is co-author of thevery successful An Introduction to the Bootstrap. Friedman is the co-inventor of many data-mining tools including CART, MARS, projection pursuit and gradient boosting.

› springer.com

S T A T I S T I C S

ISBN 978-0-387-84857-0

Trevor Hastie • Robert Tibshirani • Jerome FriedmanThe Elements of Statictical Learning

Hastie • Tibshirani • Friedman

Second Edition

We'll give empirical and theoretical evidence against this conventional wisdom, at least in "modern" settings of machine learning.

Outline

1. Observations that counter the conventional wisdom2. Risk bounds for prediction rules that interpolate3. Speculation about some neural networks

3

Supervised learning

4

Learning algorithm

Training data (labeled examples)!", $" , … , (!', $') from )×+

Prediction function,-:) → +

Test point!′ ∈ )

Predicted label,- !′ ∈ +

/t/

/k/ /a/…

2 ← 2 − 5∇ 7ℛ(2)

(IID from 9)

Risk: ℛ - ≔ ; ℓ - != , $=where !′, $′ ∼ 9

Modern machine learning algorithms

• Choose (parameterized) function class ℱ ⊂ #$• E.g., linear functions, polynomials, neural networks with certain architecture

• Use optimization algorithm to (attempt to) minimize empirical risk

%ℛ ' ≔ 1*+,-.

/ℓ ' 1, , 3,

(a.k.a. training error).

• But how "big" or "complex" should this function class be?(Degree of polynomial, size of neural network architecture, …)

5

Overfitting

6

True risk

Empirical risk

Model complexity

Generalization theory

• Generalization theory explains how overfitting can be avoided• Most basic form:

! max%∈ℱ

ℛ(*) − -ℛ(*) ≲ Complexity(ℱ)7

• Complexity of 8 can be measured in many ways:• Combinatorial parameter (e.g., Vapnik-Chervonenkis dimension)• Log-covering number in 9: ; metric• Rademacher complexity (supremum of Rademacher process)• Functional / parameter norms (e.g., Reproducing Kernel Hilbert Space norm)• …

7

Observations from the field(Belkin, Ma, & Mandal, 2018)

8

Neural nets & kernel machines:• Can fit any training data.• Can generalize even when

training data has substantial amount of label noise.

Overfitting or perfect fitting?

• Training produces a function !" that perfectly fits noisy training data.• !" is likely a very complex function!

• Yet, test error of !" is non-trivial: e.g., noise rate + 5%.

9

Existing generalization bounds appear uninformative for function classes that can interpolate noisy data.• !" chosen from class rich enough to express all possible

ways to label Ω(%) training examples.• Instead, must exploit properties of how !" is chosen.

Existing theory about local interpolation

Nearest neighbor (Cover & Hart, 1967)

• Predict with label of nearest training example• Interpolates training data• Risk → 2 ⋅ ℛ(&∗) (sort of)

Hilbert kernel (Devroye, Györfi, & Krzyżak, 1998)

• Special kind of smoothing kernel regression (like Shepard's method)

• Interpolates training data• Consistent, but no convergence rates

) * − *, = 1* − *, /

10

Our goals

• Further counter the "conventional wisdom" re: interpolationShow interpolation methods can be consistent (or almost consistent) for classification & regression problems• Simplicial interpolation• Weighted & interpolated nearest neighbor

• Identify some useful properties of good interpolation methods• Suggest connections to practical methods

11

Simplicial interpolation

12

Basic idea

• Construct estimate "̂ of the regression function" # = % &' #' = #

• Regression function " is minimizer of risk for squared lossℓ )&, & = )& − & ,

• For binary classification - = {0,1}:• " # = Pr(&' = 1 ∣ #' = #)• Optimal classifier: 7∗ # = 9: ; <=>• We'll construct plug-in classifier ?@ # = 9A: ; <=>

based on "̂

13

Consistency and convergence rates

Questions of interest:• What is the (expected) risk of !" as # → ∞? Is it near optimal (ℛ((∗))?• What what rate (as function of #) does + ℛ !" approach ℛ((∗)?

14

Interpolation via multivariate triangulation

• IID training examples !", $" , … , !&, $& ∈ ℝ)×[0,1]• Partition / ≔ conv !", … , !& into simplices with !5 as vertices via Delaunay.• Define 7̂(!) on each simplex by affine interpolation of vertices' labels.• Result is piecewise linear on /. (Punt on what happens outside of /.)

• For classification ($ ∈ {0,1}), let <= be plug-in classifier based on 7̂.

15

!"

!#

!$

What happens on a single simplex

• Simplex on !", … , !'(" with corresponding labels )", … , )'("• Test point ! in simplex, with barycentric coordinates (+", … ,+'(").• Linear interpolation at ! (i.e., least squares fit, evaluated at !):

.̂ ! = 012"

'("+1)1

!

16

Key idea: aggregates information from all vertices to make prediction.(C.f. nearest neighbor rule.)

Comparison to nearest neighbor rule

• Suppose ! " = Pr(' = 1 ∣ ") < 1/2 for all points in a simplex• Optimal prediction of .∗ is 0 for all points in simplex.

• Suppose '0 = ⋯ = '2 = 0, but '240 = 1 (due to "label noise")

x1

x3x2

0

0 1

Nearest neighbor rule

x1

x3x2

0

0 1

Simplicial interpolation

56 " = 1 hereEffect is exponentially more pronounced in high dimensions!

17

Asymptotic risk (binary classification)Theorem: Assume distribution of !′ is uniform on some convex set, and # is bounded away from 1/2. Then simplicial interpolation's plug-in classifier '( satisfies

limsup/

0 ℛ( '() ≤ 1 + 678 9 ⋅ ℛ ;∗

18

• Near-consistency in high-dimension• C.f. nearest neighbor classifier: limsup

/0 ℛ( '() ≈ 2 ⋅ ℛ ;∗

• "Blessing" of dimensionality (with caveat about convergence rate).• Also have analysis for regression + classification w/o condition on #

[Belkin, H., Mitra, '18]

Weighted & interpolated NN

19

Weighted & interpolated NN scheme

• For given test point !, let !(#), … , ! ' be ( nearest neighbors in training data, and let )(#), … , ) ' be corresponding labels.

20

!(#)

!(*)!(')

!

Define

,̂ ! = ∑/0#' 1(!, ! / ) ) /∑/0#' 1(!, ! / )

where1 !, ! / = ! − ! /

34, 5 > 0

Interpolation: ,̂ ! → )/ as ! → !/

Comparison to Hilbert kernel estimate

Weighted & interpolated NN Hilbert kernel (Devroye, Györfi, & Krzyżak, 1998)

"̂ # = ∑&'() *(#, # & ) . &∑&'() *(#, # & )

*(#, # & ) = ‖# − # & ‖12

Our analysis allows 0 < 5 < 6/2

"̂ # = ∑&'(9 *(#, #&) .&∑&'(9 *(#, #&)

* #, #& = # − #& 12

MUST have 5 = 6 for consistency

21

Localization is essential to get non-asymptotic rate.

Convergence rates (regression)Theorem: Assume distribution of !′ is uniform on some compact set satisfying regularity condition, and # is $-Holder smooth.

For appropriate setting of %, weighted & interpolated NN estimate #̂satisfies

' ℛ #̂ ≤ ℛ # + + ,-.//(./23)

22

• Consistency + optimal rates of convergence for interpolating method.• Also get consistency and rates for classification.• Follow-up work by Belkin, Rakhlin, Tsybakov '19: also for Nadaraya-Watson

with compact & singular kernel.

[Belkin, H., Mitra, '18]

What about the other NN?

23

Two layer fully-connected neural networks

24

[Belkin, H., Ma, Mandal, '18]

Random first layer Trained first layer

"Double descent" risk curve

25

[Belkin, H., Ma, Mandal, '18]

Linear regression w/ weak features

26

Number of features chosen

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0.0 0.2 0.4 0.6 0.8 1.0

05

1015

p/N

risk

E ErrorR(alpha)

0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

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risk

[Belkin, H., Xu, '19; Xu, H., '19]

Gaussian design linear model with ! featuresAll features are "relevant" but equally weak

Only use " of the features (1 ≤ " ≤ !)Least squares (" ≤ %) or least norm (" ≥ %) fit

Concurrent work by Hastie, Montanari, Rosset, Tibshirani '19 (also for some non-linear models!),and by Muthukumar, Vodrahalli, Sahai, '19; Bartlett, Long, Lugosi, Tsigler, '19.

What happens in the limit ! → ∞?

! → ∞ --- infinite width neural net.

(Note: number of data points $ is constant.)

ℎ&,( = lim-→(ℎ&,-

What is the classifier corresponding to this neural net?

Kernel machine!

ℎ&,( = ./012$3∈ℋ,3 67 897 ||ℎ||ℋ

Min norm interpolation in Gaussian RKHS ℋ.

RFF approximates Kernel Machine

Min RFF norm convergences to min ℋ norm

Kernel machine

Norm of the solution

Test error

Kernel machine

Is infinite width optimal?

ℎ",$ appears to be near-optimal empirically.

Suppose ∀& '& = ℎ∗ *& for some ℎ∗ ∈ ℋ (Gaussian RKHS).

Theorem:

|ℎ∗(*) − ℎ",$(*)| = 1234 "/ 678 " 9/: ||ℎ∗||ℋ

Compare to ; <" for classical bias-variance

analyses.

[B., Hsu, Ma, Mandal, 18]

Occams’s razor

Occam’s razor based on inductive bias:

Choose the smoothest function subject to interpolating the data.

Three ways to increase smoothness:

Ø Explicit: minimum functional norm solutions (RFF, ReLUfeatures).

Ø Implicit: SGD/optimization (Neural networks)

(All coincide for kernel machines)

Conclusions and open problems

1. Interpolation is compatible with good statistical properties2. Need good inductive bias: e.g.,

1. Functions that do local averaging in high-dimensions.2. Low function space norm

Open problems• Formally characterize inductive bias of interpolation with existing

methods (e.g., neural nets, kernel machines, random forests)• Srebro: Simplicial interpolation = GD on infinite width ReLU network (d=1)

• Benefits of interpolation?

30

Acknowledgements

• Collaborators:Misha Belkin, Siyuan Ma, Soumik Mandal, Partha Mitra, Ji Xu• Some slides borrowed from Misha!• National Science Foundation• Sloan Foundation• Simons Institute for the Theory of Computing

31