Unbiased Risk Estimators Can Mislead: A Case Study of Learning with Complementary Labels

Unbiased Risk Estimators Can Mislead: A Case Study of Learning with Complementary Labels

Yu-Ting Chou, Gang Niu, Hsuan-Tien Lin, Masashi Sugiyama

ICML 2020 work done during Chou’s internship at RIKEN AIP, Japan;

resulting M.S. thesis of Chou won the 2020 thesis award of TAAI

October 8, 2021, AI Forum, Taipei, Taiwan

Introduction

Supervised Learning

(Slide Modified from My ML Foundations MOOC)

25

5 1

Mass

Size 10

unknown target function f : X → Y

training examples D : (x1,y1), · · · , (xN,yN)

learning algorithm

A

final hypothesis g≈f

hypothesis set H

supervised learning:

every input vectorxn with its (possibly expensive) label y_n,

Introduction

Weakly-supervised: Learning without True Labels y

(a) positive-unlabeled learning [EN08] (b) learning with complementary labels [Ish+17] (c) learning with noisy labels [Nat+13]

• positive-unlabeled: some of true y_n= +1 revealed

• complementary: ‘not label’ y_ninstead of true yn

• noisy: noisy label y_n⁰ instead of true yn

weakly-supervised: arealisticandhot research direction to reduce labeling burden

[EN08] Learning classifiers from only positive and unlabeled data, KDD’08.

[Ish+17] Learning from complementary labels, NeurIPS’17.

Introduction

Motivation

popular weakly-supervised models [DNS15; Ish+19; Pat+17]

• deriveUnbiased Risk Estimators (URE) as new loss

• theoretically, nice properties (unbiased, consistent, etc.) [Ish+17]

• practically,sometimes bad performance(overfitting) our contributions: on Learning with Complementary Labels (LCL)

• analysis:identify weaknessof URE framework

• algorithm: propose animproved framework

• experiment: demonstratestronger performance

next: introduction to LCL

[DNS15] Convex formulation for learning from positive and unlabeled data, ICML’15.

[Ish+19] Complementary-Label Learning for Arbitrary Losses and Models, ICML’19.

[Pat+17] Making deep neural networks robust to label noise: A loss correction approach, CVPR’17.

Introduction

Motivation behind Learning with Complementary Label

complementary label y_n instead of true yn

Figure 1 of [Yu+18]

complementary label: easier/cheaperto obtain for some applications

Introduction

Fruit Labeling Task (Image from AICup in 2020)

hard: true label

• orange?

• mango?

• cherry

• banana

easy: complementary label

• orange

• mango

• cherry

• banana 7

complementary:less labeling cost/expertiserequired

Introduction

Comparison

Ordinary (Supervised) Learning

training: {(xn= ,yn =mango)} →classifier Complementary Learning

training: {(xn= ,y_n=banana)} →classifier

testing goal: classifier( ) →cherry

ordinary versus complementary: same goal via different training data

Introduction

Learning with Complementary Labels Setup

Given

N examples (inputxn,complementary label y_n) ∈ X × {1, 2, · · · K } in data set D such that y_n6= ynfor some hidden ordinary label

y_n∈ {1, 2, · · · K }.

Goal

a multi-class classifier g(x) thatclosely predicts(0/1 error) the ordinary label y associated with someunseeninputs x

LCL model design: connecting complementary & ordinary

Introduction

Unbiased Risk Estimation for LCL

Ordinary Learning

• empirical risk minimization (ERM) on training data

risk: E(x,y )[`(y , g(x))] empirical risk: E(xn,yn)∈D[`(y_n,g(x_n))]

• loss `: usuallysurrogateof 0/1 error

LCL [Ish+19]

• rewrite the loss ` to `, such that

unbiased risk estimator: E(x,y )[`(y , g(x))] = E(x,y )[`(y , g(x))]

• LCL by minimizingURE

URE:pioneer modelsfor LCL

Introduction

Example of URE

Cross Entropy Loss

for g(x) = argmaxk ∈{1,2,...,K }p(k | x),

• `<sub>CE</sub>: derived by maximum likelihood as surrogate of 0/1 risk: R(g; `_CE) = E(x,y )(− log(p(y | x)))

| {z }

`_CE

Complementary Learning [Ish+19]

URE: R(g; `) = E(x,y )



`

z }| {

(K − 1) log(p(y | x))

| {z }

negative

−

K

X

k =1

log(p(k | x))

under uniform y assumption

ERM with URE: minpR with E taken on D

Problems of URE

URE overfits on single label

` = − log(p(y | x)

` = (K − 1) log(p(y | x)) −

K

X

k =1

log(p(k | x))

ordinary risk and URE very different

• ` >0 → ordinary risk non-negative

• smallp(y | x) (often) → possibly very negative `

empiricalURE can be negative: only observingsome but not all y

• negative empirical UREdrags minimizationtowards overfitting practical remedy: [Ish+19]

NN-URE: constrain emprical URE to be non-negative how can we avoid negative empirical URE?

Proposed Framework

Proposed Framework

Minimize Complementary 0/1

• Recall the goal: We minimize 0-1 loss instead of `

• The unbiased estimator of R₀₁

R₀₁ : Ey[`<sub>01</sub>(y , g(x))] = `₀₁(y , g(x))

• We denote `₀₁ as the complementary 0-1 loss:

`₀₁(y , g(x)) =Jy = g (x)K Surrogate Complementary Loss (SCL)

• Surrogate loss to optimize `₀₁

• Unify previous work as surrogates of `₀₁[Yu+18; Kim+19]

[Yu+18] Learning with biased complementary labels, ECCV’18.

[Kim+19] Nlnl: Negative learning for noisy labels, ICCV’19.

Proposed Framework

Negative Risk Avoided

Unbiased Risk Estimator (URE)

URE loss `<sub>CE</sub> [Ish+19] from cross-entropy `_CE,

`_CE(y , g(x)) = (K − 1) log(p(y | x))

| {z }

negative loss term

−

K

X

j=1

log(p(j | x))

can go negative.

Surrogate Complementary Loss (SCL) another loss [Kim+19], a surrogate `₀₁

φ_NL(y , g(x)) = − log(1 − p(y | x))) remains non-negative.

Proposed Framework

Illustrative Difference between URE and SCE

URE

SCL

R

R

R

ˆ R

R

R

ˆ R

E

E

A URE: Ripple effect of errors

• Theoretical motivation [Ish+17]

• Estimation step (E) amplifies approximation error (A) in ` SCL: ‘Directly’ minimize complementary likelihood

• Non-negative loss φ

• Practically prevents ripple effect

Proposed Framework

Classification Accuracy

Methods

1 Unbiased risk estimator (URE) [Ish+19]

2 Non-negative correction methods on URE (NN) [Ish+19]

3 Surrogate complementary loss (SCL)

Table:URE and NN are based on ` rewritten from cross-entropy loss, while SCL is based on exponential loss φEXP(y , g(x)) = exp(p_y).

Data set + Model URE NN SCL

MNIST + Linear 0.850 0.818 0.902

MNIST + MLP 0.801 0.867 0.925

CIFAR10 + ResNet 0.109 0.308 0.492 CIFAR10 + DenseNet 0.291 0.338 0.544

Gradient Analysis

Gradient Analysis

Gradient Direction of URE

• Very diverged directions on each y to maintain unbiasedness

• Low correlation to the target `₀₁

∇`(y , g(x))

∇`(y , g(x))

Figure:Illustration of URE

Gradient Direction of SCL

• Targets to minimum likelihood objective

• High correlation to the target

`01

Gradient Analysis

Gradient Estimation Error

Bias-Variance Decomposition

MSE = E(f − c)²

= E(f − h)²

| {z }

Bias²

+ E(h − c)²

| {z }

Variance

Gradient Estimation

1 Ordinary gradient f = ∇`(y , g(x))

2 Complementary gradient c = ∇`(y , g(x))

3 Expected complementary gradient h

Gradient Analysis

Bias-Variance Tradeoff

(a) MSE (b) Bias² (c) Variance

Findings

• SCL reduces variance by introducing small bias (towards y ) Bias Variance MSE

URE 0 Big Big

SCL Small Small Small

Conclusion

Conclusion

Explain Overfitting of URE

• Unbiased method only do well in expectation

• Single fixed complementary label cause overfitting Surrogate Complementary Loss (SCL)

• Minimum likelihood approach

• Avoids negative risk problem Experiment Results

• SCL significantly outperforms other methods

• Introduce small bias for lower gradient variance

Conclusion

References

Marthinus Du Plessis, Gang Niu, and Masashi Sugiyama.“Convex formulation for learning from positive and unlabeled data”.In: International Conference on Machine Learning. 2015, pp. 1386–1394.

Charles Elkan and Keith Noto.“Learning classifiers from only positive and unlabeled data”.In: Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining. 2008, pp. 213–220.

Takashi Ishida et al.“Learning from complementary labels”.In: Advances in neural information processing systems. 2017, pp. 5639–5649.

Takashi Ishida et al.“Complementary-Label Learning for Arbitrary Losses and Models”.In: International Conference on Machine Learning. 2019, pp. 2971–2980.

Youngdong Kim et al.“Nlnl: Negative learning for noisy labels”.In: Proceedings of the IEEE International Conference on Computer Vision. 2019, pp. 101–110.

Nagarajan Natarajan et al.“Learning with noisy labels”.In: Advances in neural information processing systems. 2013, pp. 1196–1204.

Giorgio Patrini et al.“Making deep neural networks robust to label noise: A loss correction approach”.In:

Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2017, pp. 1944–1952.

Xiyu Yu et al.“Learning with biased complementary labels”.In: Proceedings of the European Conference on Computer Vision (ECCV). 2018, pp. 68–83.