Learning Noise Transition Matrix from Only Noisy Labels via Total Variation Regularization
Yivan Zhang1, 21

Gang Niu2

The University of Tokyo2

Masashi Sugiyama2, 1
RIKEN AIP

Introduction

Overconﬁdence

ProblemNoise transition matrix is important in learning from noisy labels.
However, it is usually unavailable or hard to obtain.
Existing methods often depend on unreliable noisy class-posterior estimation.

Data

Implementation
Empirical distribution

Estimated distribution

Histogram of entropy
102

Empirical
Estimated

overconfidence

101

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overconfidenceContributionWe characterized the class-conditional label corruption process.
We proposed a conceptually novel method for transition matrix estimation.

Total Variation Regularization
We sample a ﬁxed number of pairs to reduce the additional computational cost.

102

101dTV (p, q) := kp − qk1
2R(W ) :=
E
E

1

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X1∼p(X) X2∼p(X)101

where pi := p(Y |Xi; W )

100
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[dTV (p1, p2)]

i = 1, 2Methodology
Make probabilities more distinguishable: total variation regularization
Capture uncertainties during training: Dirichlet posterior update

Problem
The estimation of the noisy class-posterior could be unreliable due to the overconﬁdence ofdeep neural networks [Guo et al., 2017, Hein et al., 2019].

p = model (x) # probability [ batch_size , num_classes ]idx_1 , idx_2 = randint (0 , batch_size , (2 , num_pairs ))tv = 0.5 * l1_norm (p[ idx_1 ] - p[ idx_2 ], dim =1). mean ()Learning from Noisy Labels

Solution
Do not estimate the noisy class-posterior directly using neural networks.

Dirichlet Posterior Update
Inspired by the closed-form posterior update rule for the Dirichlet-multinomial conjugate, weupdate the concentration parameters A during training using the confusion matrix C,where (β1, β2) are ﬁxed hyperparameters.

Notation
X: input features
Y : true labels
Ye : noisy labels

Assumption
XY

Motivation
Transition Matrix as a Contraction Mapping

YeClass-Conditional Noise (CCN) assumes that the noisy label Ye is independent of the input
feature X given the true label Y : p(Ye |Y, X) = p(Ye |Y ).
Noise Transition Matrix Tij = p(Ye = j|Y = i) p(Y = 1|X)
p(Ye = 1|Y = 1) . . . p(Ye = 1|Y = K)p(Ye = 1|X) 
..
.
.
.
.
.
.
.
.
.

=

p(Y = K|X)
p(Ye = K|Y = 1) . . . p(Ye = K|Y = K)p(Ye = K|X)
⇓p(Ye |X) = T Tp(Y |X)

The mapping ∆ → Conv(U ) deﬁned by p 7→ U Tp is acontraction mapping over the simplex ∆ relative to thetotal variation distance [Del Moral et al., 2003]:∀U ∈ T , ∀p, q ∈ ∆,dTV (U Tp, U Tq) ≤ dTV (p, q)
Clean class-posteriors are always more distinguishable
from each other than noisy class-posteriors.

A(posterior) = A(prior) + C (observation)A ← β1A + β2C
y = Categorical (p ). sample ()C = confusion_matrix (y , y_ )A = beta_1 * A + beta_2 * C

Optimization
For each batch of data, we sample a transition matrix from the Dirichlet posterior.
Ti ∼ Dirichlet(Ai) (i = 1, . . . , K)
ihDKL p(Ye |X) T Tp(Y |X; W )L0(W, T ) :=E

Transition Matrix Estimation

Noise Transition Matrix

X∼p(X)

Symmetric

PairwiseL(W, T ) := L0(W, T ) − γR(W )GeneralIn addition to the gradient information, the confusion
matrix can be used to estimate the transition matrix.
To capture uncertainties during training, we could use
Dirichlet distributions to accumulate information ofconfusion matrices, which leads us to a derivative-free
approach for transition matrix estimation.

1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0


0.8 0.1 0.1
0.1 0.8 0.1
0.1 0.1 0.8


0.7 0.3 0.0
0.0 0.7 0.3
0.3 0.0 0.7


0.7 0.2 0.1
0.2 0.5 0.3
0.1 0.2 0.7

Proposed Method

Class-conditional label corruption maps the probability simplex ∆K−1 to a convex hullConv(T ) of the rows of the noise transition matrix T .
Outer black triangle: probability simplex ∆2
Inner colored triangle: convex hull Conv(T )Good news: if the ground-truth noise transition matrix T is known, p(Y |X) is identiﬁable
based on observations of p(Ye |X) [Patrini et al., 2017].

Anchor Points

Solution
Do not rely on a separate set of anchor points.

Improved classiﬁcation performance, measured by accuracy.

CIFAR-100

Solution
Learn the noise transition matrix from only noisy labels.

ProblemAnchor points are hard to obtain [Xia et al., 2019, Yao et al., 2020].

Experiments(a) Clean (b) Symm.
(c) Pair(d) Pair2(e) Trid.
(f) Rand.
MAE
11.23(1.02) 7.89(0.67)
6.94(1.11)
6.60(0.74)
7.45(0.55)
7.15(0.98)
CCE
70.58(0.29) 42.94(0.47) 44.00(0.71) 41.37(0.27) 46.55(0.54) 42.41(0.48)
GCE
57.10(0.85) 48.66(0.58) 45.27(0.85) 43.67(0.94) 50.98(0.33) 48.66(0.63)
Forward 70.58(0.28) 44.32(0.64) 44.17(0.57) 42.07(0.55) 47.48(0.40) 43.15(0.53)
T-Revision 70.47(0.26) 46.52(0.57) 44.08(0.42) 42.01(0.52) 47.59(0.60) 45.33(0.40)
Dual-T 70.56(0.28) 55.92(0.60) 46.22(0.72) 44.74(0.65) 61.68(0.51) 57.92(0.50)
TVG
70.02(0.30) 57.33(0.42) 45.68(0.85) 44.38(0.72) 54.23(0.53) 59.85(0.61)
TVD
69.93(0.21) 52.54(0.45) 56.02(0.82) 49.18(0.53) 62.45(0.44) 53.95(0.47)

Improved transition matrix estimation, measured by average total variation.

ProblemNoise transition matrix is usually not available [Patrini et al., 2017].

An instance x is called an anchor point for class i if p(Y = i|X = x) = 1.
Based on anchor points, we can estimate p(Ye |X) to obtain an estimate of T .
p(Ye |X = x) = T Tp(Y |X = x) = Ti

T = Dirichlet (A ). sample () # transition matrix
loss = cross_entropy (p @ T , y_ ) - gamma * tv

CIFAR100Clean# predicted labels# confusion matrix# update(a) Clean (b) Symm.
(c) Pair(d) Pair2(e) Trid.
(f) Rand.
Forward 0.00(0.00) 48.62(0.11) 39.81(0.03) 43.57(0.04) 40.92(0.07) 49.06(0.10)
T-Revision 0.46(0.05) 31.58(0.46) 39.45(0.03) 42.77(0.06) 40.01(0.09) 39.49(0.26)
Dual-T
3.10(0.08) 17.10(0.18) 33.26(0.20) 33.79(0.26) 23.56(0.43) 22.59(0.23)
TVG
1.59(0.02) 13.11(0.10) 37.79(0.30) 38.83(0.34) 30.80(0.51) 16.47(0.18)
TVD
21.98(0.11) 26.46(0.15) 29.47(0.26) 31.34(0.30) 23.86(0.22) 35.37(0.30)

References
Our model has two modules:(a) a neural network for predicting p(Y |X)(b) a Dirichlet posterior for the noise transition matrix TThe learning objective also contains two parts:(i) the usual cross-entropy loss for classiﬁcation from noisy labels(ii) a total variation regularization term for the predicted probability

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