Causally regularized machine learning

CAUSALLY REGULARIZED
MACHINE LEARNING
Peng Cui, Tsinghua University
Kun Kuang, Zhejiang University
Bo Li, Tsinghua University

ML techniques are impacting our life
2
• A day in our life with ML techniques
8:30 am
8:00 am 10:00 am
4:00 pm
6:00 pm
8:00 pm

Now we are stepping into risk-sensitive areas
3
Shifting from Performance Driven to Risk Sensitive

Problems of today’s ML - Explainability
4
Most machine learning models are black-box models

5
Most ML methods are developed under I.I.D hypothesis
Problems of today’s ML - Stability

6
Yes
Maybe
No

7
• Cancer survival rate prediction
Training Data
Predictive Model
Testing Data
City Hospital
University HospitalHigher income, higher survival rate.
City Hospital
Survival rate is not so correlated with income.

8
A plausible reason: Correlation
Correlation is the very basics of machine learning.

9
Correlation is not explainable

10
Correlation is ‘unstable’

11
It’s not the fault of correlation, but the way we use it
• Three sources of correlation:
• Causation
• Causal mechanism
• Stable and explainable
• Confounding
• Ignoring X
• Spurious Correlation
• Sample Selection Bias
• Conditional on S
• Spurious Correlation
T Y
T Y
X
T Y
S
Accepted
Income
Financial
product offer
DogGrass
Sample
Selection
Ice Cream
Sales
Summer

A Practical Definition of Causality
Definition: T causes Y if and only if
changing T leads to a change in Y,
while keeping everything else constant.
Causal effect is defined as the magnitude by which Y is
changed by a unit change in T.
Called the “interventionist” interpretation of causality.
12
http://plato.stanford.edu/entries/causation-mani/
X
T Y

13
The benefits of bringing causality into learning
Causal Framework
T grass
X dog nose
Y label
Grass—Label: Strong correlation
Weak causation
Dog nose—Label: Strong correlation
Strong causation
X
T Y
More Explainable and More Stable

14
The gap between causality and learning
pHow to evaluate the outcome?
pWild environments
p High-dimensional
p Highly noisy
p Little prior knowledge (model specification, confounding structures)
p Targeting problems
p Understanding v.s. Prediction
p Depth v.s. Scale and Performance
How to bridge the gap between causality and (stable) learning?

Outline
ØCorrelation v.s. Causality
ØCausal Inference
ØStable Learning
ØNICO: An Image Dataset for Stable Learning
ØConclusions
15

16
T Y
U Z W• Causal Identification with back
door criterion
• Causal Estimation with do
calculus
Paradigms - Structural Causal Model
A graphical model to describe the causal mechanisms of a system
How to discover the causal structure?

17
• Causal Discovery
• Constraint-based: conditional independence
• Functional causal model based
Paradigms – Structural Causal Model
A generative model with strong expressive power.
But it induces high complexity.

Paradigms - Potential Outcome Framework
• A simpler setting
• Suppose the confounders of T are known a priori
• The computational complexity is affordable
• Under stronger assumptions
• E.g. all confounders need to be observed
18
More like a discriminative way to estimate treatment’s
partial effect on outcome.

Causal Effect Estimation
• Treatment Variable: ! = 1 or ! = 0
• Treated Group (! = 1) and Control Group (! = 0)
• Potential Outcome: &(! = 1) and &(! = 0)
• Average Causal Effect of Treatment (ATE):
19
(!) = )[& ! = 1 − & ! = 0 ]

Counterfactual Problem
• Two key points for causal effect
estimation
• Changing T
• Keeping everything else constant
• For each person, observe only one:
either !"#$or !"#%
• For different group (T=1 and T=0),
something else are not constant
20
Person T &'#( &'#)
P1 1 0.4 ?
P2 0 ? 0.6
P3 1 0.3 ?
P4 0 ? 0.1
P5 1 0.5 ?
P6 0 ? 0.5
P7 0 ? 0.1

Ideal Solution: Counterfactual World
• Reason about a world that does not exist
• Everything in the counterfactual world is the same as the
real world, except the treatment
21
! " = 1 ! " = 0

Randomized Experiments are the “Gold Standard”
• Drawbacks of randomized experiments:
• Cost
• Unethical
• Unrealistic
22
What can we do when an experiment is
not possible?
Observational Studies!

Recap: Causal Effect and Potential Outcome
• Two key points for causal effect estimation
• Changing T
• Keeping everything else (X) constant
• Counterfactual Problem
• Ideal Solution: Counterfactual World
• “Gold Standard”: Randomized Experiments
• We will discuss other solutions in next Section.
23
! " = 1 or ! " = 0

Outline
ØCausal Inference
ØMethods for Causal Inference
ØStable Learning
ØConclusions
24

Causal Inference with Observational Data
• Average Treatment Effect (ATE) represents the mean
(average) difference between the potential outcome of
units under treated (T=1) and control (T=0) status.
• Treated (T=1): taking a particular medication
• Control (T=0): not taking any medications
• ATE: the causal effect of the particular medication
25
!"# = #[& " = 1 − & " = 0 ]

• Counterfactual Problem:
• Can we estimate ATE by directly comparing the average
outcome between treated and control groups?
• Yes with randomized experiments (X are the same)
• No with observational data (X might be different)
• Two key points:
• Changing T (T=1 and T=0)
• Keeping everything else (Confounder X) constant
26
! " = 1 or ! " = 0
Balancing Confounders’ Distribution

Methods for Causal Inference
• Matching
• Propensity Score Based Methods
• Propensity Score Matching
• Inverse of Propensity Weighting (IPW)
• Doubly Robust
• Data-Driven Variable Decomposition (D2VD)
• Directly Confounder Balancing
• Entropy Balancing
• Approximate Residual Balancing
• Differentiated Confounder Balancing (DCB)
27

Assumptions of Causal Inference
• A1: Stable Unit Treatment Value (SUTV): The effect of treatment on
a unit is independent of the treatment assignment of other units
! "# $#, $&, '# = ! "# $#, '#
• A2: Unconfounderness: The distribution of treatment is independent
of potential outcome when given the observed variables
$ ⊥ " 0 , " 1 | '
No unmeasured confounders
• A3: Overlap: Each unit has nonzero probability to receive either
treatment status when given the observed variables
0 < ! $ = 1 ' = . < 1
28

• Matching
• Doubly Robust
• Differentiated Confounder Balancing
29

Matching
• Identify pairs of treated (T=1) and control (T=0)
units whose confounders X are similar or even
identical to each other
• Paired units provide the everything else
(Confounders) approximate constant
• Estimating average causal effect by comparing
average outcome in the paired dataset
• Smaller !: less bias, but higher variance
32
" #$%&'()*+ ,-, ,/ ≤ !

Matching
• Exactly Matching:
• Easy to implement, but limited to low-
dimensional settings
• Since in high-dimensional settings, there
will be few exact matches
33
!"#$%&'( )*, ), ≤ .
!"#$%&'( )*, ), = 0
0, )* = ),
∞, )* ≠ ), 4 5

• Matching
• Doubly Robust
34

Propensity Score Based Methods
• Propensity score !(#) is the probability of a unit to be treated
• Then, Rubin shows that the propensity score is sufficient to
control or summarize the information of confounders
• Propensity score are rarely observed, need to be estimated
35
! # = &(' = 1|#)
' ⫫ # | !(#) ' ⫫ (+ 1 , +(0)) | !(#)

Propensity Score Matching
• Estimating propensity score:
• Supervised learning: predicting a known
label T based on observed covariates X.
• Conventionally, use logistic regression
• Matching pairs by distance between
propensity score:
• High dimensional challenge:
36
!"#$%&'( )*, ), ≤ .
̂( ) = 1(3 = 1|))
!"#$%&'( )*, ), = | ̂( )* − ̂( ), |
transferred from matching to PS estimation

• Matching
• Doubly Robust
37

Inverse of Propensity Weighting (IPW)
• Why weighting with inverse of propensity score is helpful?
• Propensity score induces the distribution bias on confounders X
38
Unit !(#) % − !(#) #units #units
(T=1)
#units
(T=0)
A 0.7 0.3 10 7 3
B 0.6 0.4 50 30 20
C 0.2 0.8 40 8 32
' ( = *(+ = 1|()
Reweighting by inverse of propensity score:
Unit #units
(T=1)
#units
(T=0)
A
B
C
./ =
+/
'/
+
1 − +/
1 − '/
Confounders
are the same!
10 10
50 50
40 40
Distribution Bias

• Estimating ATE by IPW [1]:
• Interpretation: IPW creates a pseudo-population where the
confounders are the same between treated and control groups.
• Why does this work? Consider
39
!" =
$"
%"
+
1 − $"
1 − %"

• If: , the true propensity score
• Similarly:
40
! = # ∗ !% + % − # ∗ !(
# ⊥ !%, !( | ,
- , = .(#|,)
123 = 3[5 1 − 5 0 ]

• If: , the true propensity score, the IPW
estimator is unbiased
• Wildly used in many applications
• But requires the propensity score model is correct
• High variance when ! is close to 0 or 1
41

• Matching
• Doubly Robust
42

Doubly Robust
• Recap:
• Simple outcome regression:
• Unbiased if the regression models are correct
• IPW estimator:
• Unbiased if the propensity score model is correct
• Doubly Robust [2]: combine both approaches
43
!"# = #[& " = 1 − & " = 0 ]
and

Doubly Robust
• Estimating ATE with Doubly Robust estimator:
• Unbiased if either propensity score or regression model is correct
• This property is referred to as double robustness
44

Doubly Robust
• Theoretical Proof:
45

Doubly Robust
• Estimating ATE with Doubly Robust estimator:
• Unbiased if propensity score or regression model is correct
• This property is referred to as double robustness
• But may be very biased if both models are incorrect
46

Propensity Score based Methods
•Recap:
• Inverse of Propensity Weighting
• Doubly Robust
•Need to estimate propensity score
• Treat all observed variables as confounders
• In Big Data Era, High dimensional data
• But, not all variables are confounders
47

Propensity Score based Methods
•Recap:
• Inverse of Propensity Weighting
• Doubly Robust
•Need to estimate propensity score
• Treat all observed variables as confounders
• In Big Data Era, High dimensional data
• But, not all variables are confounders
48
How to automatically separate
the confounders?

• Matching
• Doubly Robust
49

50
• Treat all observed variables U as
confounders X
• Propensity Score Estimation:
• Adjusted Outcome:
• IPW ATE Estimator:

Data-Driven Variable Decomposition (D2VD)
51
• Separateness Assumption:
• All observed variables U can be decomposed into
three sets: Confounders X, Adjustment Variables Z,
and Irrelevant variables I (Omitted).
• Propensity Score Estimation:
• Adjusted Outcome:
• Our D2VD ATE Estimator:

• Confounders Separation & ATE Estimation.
• With our D2VD estimator:
• By minimizing following objective function:
• We can estimate the ATE as:
52

53
• Adjustment variables:
• Confounders:
• Treatment Effect:
!, #, $
Where
Where
Replace X, Z with U

54
Bias Analysis:
Our D2VD algorithm is unbiased to estimate causal effect
Variance Analysis:
The asymptotic variance of Our D2VD algorithm is smaller

•OUR: Data-Driven Variable Decomposition (D2VD)
•Baselines
•Directly Estimator (dir): ignores confounding bias
•IPW Estimator (IPW): treats all variables as confounders
•Doubly Robust Estimator (DR): IPW+regression
•Non-Separation Estimator (D2VD-): no variables separation
55

• Dataset generation:
• Sample size m={1000,5000}
• Dimension of observed variables n={50,100,200}
• Observed variables:
• Treatment: logistic and misspecified
• Outcome:
56

• Dataset generation:
• Sample size m={1000,5000}
• Dimension of observed variables n={50,100,200}
• Observed variables:
• Treatment: logistic and misspecified
• Outcome:
57
The true treatment effect in synthetic data is 1.

• Experimental Results on Synthetic Data:
58

59
1. The direct estimator is failed under all settings.
2. IPW and DR estimators are good when T=Tlogit, but poor when T=Tmissp.
3. D2VD(-) has no variables separation, get similar results with DR estimator.
4. D2VD can improve accuracy and reduce variance for ATE estimation.

60
TPR: true positive rate
TNR: true negative rate
Our D2VD algorithm
can precisely separate
the confounders and
adjustment variables.

Experiments on Real World Data
• Dataset Description:
• Online advertising campaign (LONGCHAMP)
• Users Feedback: 14,891 LIKE; 93,108 DISLIKE
• 56 Features for each user
• Age, gender, #friends, device, user setting on WeChat
• Experimental Setting:
• Outcome Y: users feedback
• Treatment T: one feature
• Observed Variables U: other features
61
2015
Y = 1, if LIKE
Y = 0, if DISLIKE

Experiments Results
• ATE Estimation.
62
1. Our D2VD estimator evaluate the ATE more accuracy.
2. Our D2VD estimator can reduce the variance of estimated ATE.
3. Younger Ladies are with higher probability to like the LONGCHAMP ads.

Experiments Results
• Variables Decomposition.
63
1. The confounders are many other ways for adding friends on WeChat.
2. The adjustment variables have significant effect on outcome.
3. Our D2VD algorithm can precisely separate the confounders and
adjustment variables.

Summary: Propensity Score based Methods
• Propensity Score Matching (PSM):
• Units matching by their propensity score
• Inverse of Propensity Weighting (IPW):
• Units reweighted by inverse of propensity score
• Doubly Robust (DR):
• Combing IPW and regression
• Data-Driven Variable Decomposition (D2VD):
• Automatically separate the confounders and adjustment variables
• Confounder: estimate propensity score for IPW
• Adjustment variables: regression on outcome for reducing variance
• Improving accuracy and reducing variance on treatment effect estimation
• But, these methods need propensity score model is correct
64
Treat all observed
variables as confounder,
ignoring non-confounders
! " = $(& = 1|")

• Matching
• Doubly Robust
65

• Average Treatment Effect (ATE):
• Average Treatment effect on the Treated (ATT):
• Two key points:
66
!"# = #[& " = 1 − & " = 0 ]
!"" = #[& 1 |" = 1] − #[& 0 |" = 1]

• Average Treatment Effect (ATE):
• Average Treatment effect on the Treated (ATT):
• Two key points:
67
!"# = #[& " = 1 − & " = 0 ]
!"" = #[& 1 |" = 1] − #[& 0 |" = 1]
Balancing Confounders’ Distribution

Directly Confounder Balancing
• Recap: Propensity score based methods
• Sample reweighting for confounder balancing
• But, need propensity score model is correct
• Weights would be very large if propensity score is close to 0 or 1
• Can we directly learn sample weight that can balance
confounders’ distribution between treated and control?
68
!" =
$"
%"
+
1 − $"
1 − %"
Yes!

• Motivation: The collection of all the moments of variables
uniquely determine their distributions.
• Methods: Learning sample weights by directly balancing
confounders’ moments as follows
69
The first moments of X
on the Control Group
on the Treated Group
With moments, the sample weights can be learned
without any model specification.

• Motivation: The collection of all the moments of variables
uniquely determine their distributions.
• Methods: Learning sample weights by directly balancing
confounders’ moments as follows
• Estimating ATT by:
70

• Matching
• Doubly Robust
71

Entropy Balancing
• Directly confounder balancing by sample weights W
• Maximize the entropy of sample weights W
• But, treat all variables as confounders and balance
them equally
72

Approximate Residual Balancing
• 1. compute approximate balancing weights W as
• 2. Fit !" in the linear model using a lasso or elastic net,
• 3. Estimate the ATT as
• Double Robustness: Exact confounder balancing or regression is correct.
• But, treats all variables as confounders and balance them equally
73

• Recap:
• Entropy Balancing, Approximate Residual Balancing etc.
• Moments uniquely determine variables’ distribution
• Learning sample weights by balancing confounders’ moments
• But, treat all variables as confounders, and balance them equally
• Different confounders make different confounding bias
74

• Recap:
• Entropy Balancing, Approximate Residual Balancing etc.
• Moments uniquely determine variables’ distribution
• Learning sample weights by balancing confounders’ moments
• But, treat all variables as confounders, and balance them equally
• Different confounders make different confounding bias
75
How to differentiated confounders and
their bias?

• Matching
• Doubly Robust
76

Differentiated Confounder Balancing
•Ideas: simultaneously learn confounder weights ! and
sample weighs ".
•Confounder weights determine which variable is
confounder and its contribution on confounding bias.
•Sample weights are designed for confounder balancing.
77
How to learn the confounder weights?

Confounder Weights Learning
• General relationship among !, ", and #:
78
Confounding biasConfounder weights
If $% = 0, then (% is not confounder, no need to balance.
Different confounders have different confounding weights.

Confounder Weights Learning
Propositions:
• In observational studies, not all observed variables are confounders, and
different confounders make unequal confounding bias on ATT with their
own weights.
• The confounder weights can be learned by regressing potential outcome
! 0 on augmented variables #.
79

• Objective Function
81
The ENT[3] and ARB[4] algorithms are special case of our DCB
algorithm by setting the confounder weights as unit vector.
Our DCB algorithm is more generalize for
treatment effect estimation.

• Algorithm
• Training Complexity: ! "#
• ": sample size, #: dimensions of variables
82
In each iteration, we first
update $ by fixing %, and
then update % by fixing $

Experiments
•Experimental Tasks:
ØRobustness Test (high-dimensional and noisy)
ØAccuracy Test (real world dataset)
ØPredictive Power Test (real ad application)
83

Experiments
• Baselines:
• Directly Estimator: comparing average outcome between treated and control units.
• IPW Estimator [1]: reweighting via inverse of propensity score
• Doubly Robust Estimator [2]: IPW + regression method
• Entropy Balancing Estimator [3]: directly confounder balancing with entropy loss
• Approximate Residual Balancing [4]: confounder balancing + regression
• Evaluation Metric:
84

Experiments - Robustness Test
• Dataset
ØSample size: ! = {2000, 5000}
ØVariables’ dimensions: ) = {50,100}
ØObserved Variables:
ØTreatment: from logistic function +,-./0 and misspecified function +1/223
• Confounding rate 45: the ratio of confounders to all observed variables.
• Confounding strength 65: the bias strength of confounders
ØOutcome: from linear function 7,/89:; and nonlinear function 78-8,/8
85

• Directly estimator fails in all settings, since it ignores confounding bias.
• IPW and DR estimators make huge error when facing high dimensional
variables or the model specifications are incorrect.
• ENT and ARB estimators have poor performance since they balance all
variables equally.
86
More results see our paper!

87
Our DCB estimator achieves significant improvements over
the baselines in different settings.
More results see our paper!
Our DCB estimator is very robust!

88
The MAE of our DCB estimator is consistent
stable and small.
• Sample Size
• Dimension of variables
• Confounding rate
• Confounding strength

89
Our DCB algorithm is very robust for
treatment effect estimation.

Experiments - Accuracy Test
• LaLonde Dataset [5]: Would the job training program increase people’s earnings
in the year of 1978?
• Randomized experiments: provide ground truth of treatment effect
• Observational studies: check the performance of all estimators
• V-RAW: variables set of 10 raw observed variables, including employment,
education, age ethnicity and married status.
• V-INTERACTION: variables set of raw variables, their pairwise one way
interaction and their squared terms.
90

Experiments - Accuracy Test
91
Results of ATT estimation
Our DCB estimator is more accurate than the baselines.
Our DCB estimator achieve a better confounder
balancing under V-INTERACTION setting.

Experiments - Predictive Power
• Dataset Description:
• Online advertising campaign (LONGCHAMP)
• Users Feedback: 14,891 LIKE; 93,108 DISLIKE
• 56 Features for each user
• Age, gender, #friends, device, user settings on WeChat
• Outcome Y: users feedback
• Treatment T: one feature
• Observed Variables X: other features
93
2015
Y = 1, if LIKE
Y = 0, if DISLIKE
Select the top k features with high causal effect for prediction

Experiments - Predictive Power
• Two correlation-based feature
selection baselines:
• MRel [6]: maximum relevance
• mRMR [7]: Maximum relevance
and minimum redundancy.
94
ØOur DCB estimator achieves the best prediction accuracy.
ØCorrelation based methods perform worse than causal methods.

Summary: Directly Confounder Balancing
• Motivation: Moments can uniquely determine distribution
• Confounder balancing with maximizing entropy of sample weights
• Combine confounder balancing and regression for doubly robust
• Treat all variables as confounders, and balance them equally
• But different confounders make different bias
• Theoretical proof on the necessary of differentiation on confounders
• Improving the accuracy and robust on treatment effect estimation
95

Sectional Summary: Methods for Causal Inference
• Matching
• Doubly Robust
96
Not all observed variables
are confounders
Different confounders
make different bias
Treat all observed
variables as confounder
Balance all confounder
equally
Limited to low-dimensional settings

Sectional Summary: Methods for Causal Inference
97
p Progress has been made to draw causality from
big data.
p From single to group
p From binary to continuous
p Weak assumptions
Ready for Learning?

Outline
ØCausal Inference
ØStable Learning
ØFuture Directions and Conclusions
98

Stability and Prediction
99
True Model
Learning Process
Prediction
Performance
TraditionalLearning
StableLearning
Bin Yu (2016), Three Principles of Data Science: predictability, computability, stability

Stable Learning
100
ModelDistribution 1
Distribution 1
Distribution 2
Distribution 3
Distribution n
…
Accuracy 1
Accuracy 2
Accuracy 3
Accuracy n
…
I.I.D. Learning
Transfer Learning
VAR (Acc)
Stable
Learning
Training
Testing

Stability and Robustness
• Robustness
• More on prediction performance over data perturbations
• Prediction performance-driven
• Stability
• More on the true model
• Lay more emphasis on Bias
• Sufficient for robustness
101
Stable learning is a (intrinsic?) way to realize robust prediction

Domain Generalization / Invariant Learning
105
• Given data from different
observed environments :
• The task is to predict Y given X
such that the prediction works
well (is “robust”) for “all possible”
(including unseen) environments

Domain Generalization
• Assumption: the conditional probability P(Y|X) is stable or
invariant across different environments.
• Idea: taking knowledge acquired from a number of related domains
and applying it to previously unseen domains
• Theorem: Under reasonable technical assumptions. Then with
probability at least
106
Muandet K, Balduzzi D, Schölkopf B. Domain generalization via invariant feature. ICML 2013.

Invariant Prediction
• Invariant Assumption: There exists a subset ! ∈ # is causal for the prediction
of $, and the conditional distribution P(Y|S) is stable across all environments.
• Idea: Linking to causality
• Structural Causal Model (Pearl 2009):
• The parent variables of Y in SCM satisfies Invariant Assumption
• The causal variables lead to invariance w.r.t. “all” possible environments
107
Peters, J., Bühlmann, P., & Meinshausen, N. (2016). Causal inference by using invariant prediction: identification and
confidence intervals. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2016

From Variable Selection to Sample Reweighting
108
X
T Y
Typical Causal Framework
Sample reweighting can make a variable independent of other
variables.
Given a feature T
Assign different weights to samples so that
the samples with T and the samples without
T have similar distributions in X
Calculate the difference of Y distribution in
treated and controlled groups. (correlation
between T and Y)

Global Balancing: Decorrelating Variables
109
X
T Y
Typical Causal Framework
Partial effect can be regarded as causal effect. Predicting with causal
variables is stable across different environments.
Given ANY feature T
Assign different weights to samples so that the
samples with T and the samples without T have
similar distributions in X
Calculate the difference of Y distribution in
treated and controlled groups. (correlation
between T and Y)
Kun Kuang, Peng Cui, Susan Athey, Ruoxuan Li, Bo Li. Stable Prediction across Unknown Environments. KDD, 2018.

Theoretical Guarantee
110
à
0

Causal Regularizer
111
Set feature j as treatment variable
Zheyan Shen, Peng Cui, Kun Kuang, Bo Li. Causally Regularized Learning on Data with Agnostic Bias. ACM MM, 2018.

Causally Regularized Logistic Regression
112
Zheyan Shen, Peng Cui, Kun Kuang, Bo Li. Causally Regularized Learning on Data with Agnostic Bias. ACM MM, 2018.

From Shallow to Deep - DGBR
113

Experiment 1 – non-i.i.d. image classification
• )
• - 0 1 0 -
• ) 0 ) - ) : 0 1 -
• 0 ) : ) , 5 0 ) )) :
) - 0 ( 0
114

Experimental Result - insights

Experimental Result - insights
116

From Causal problem to Learning problem
118
• Previous logic:
• More direct logic:
Sample
Reweighting
Independent
Variables
Causal
Variable
Stable
Prediction
Sample
Reweighting
Independent
Variables
Stable
Prediction

Thinking from the Learning end
119
!"#$%&(() !"*+"(()
,-.// 01121
/.130 01121
Zheyan Shen, Peng Cui, Tong Zhang. Stable Learning of Linear Models via Sample Reweighting. (under review)

Stable Learning of Linear Models
• Consider the linear regression with misspecification bias
• By accurately estimating with the property that ! " is uniformly
small for all ", we can achieve stable learning.
• However, the estimation error caused by misspecification term can
be as bad as , where #$
is the smallest
eigenvalue of centered covariance matrix.
120
Bias term with bound ! " ≤ &Goes to infinity when perfect collinearity exists!

Toy Example
• Assume the design matrix ! consists of two variables !", !$,
generated from a multivariate normal distribution:
• By changing %, we can simulate different extent of collinearity.
• To induce bias related to collinearity, we generate bias term & !
with & ! = !(, where ( is the eigenvector of centered covariance
matrix corresponding to its smallest eigenvalue )$
.
• The bias term is sensitive to collinearity.
121

Simulation Results
122
!"#$% %##&# (%()*+")*&, -*"()
!"#$% /"#*",0% *, 1*22%#%,) 1*()#*-3)*&,(
*,0#%"(% 0&!!*,%"#*)4

Reducing collinearity by sample reweighting
123
Idea: Learn a new set of sample weights !(#) to decorrelate the
input variables and increase the smallest eigenvalue
• Weighted Least Square Estimation
which is equivalent to
So, how to find an “oracle” distribution which holds the desired
property?

Sample Reweighted Decorrelation Operator (cont.)
124
Decorrelation
where !, #, $, %, &, ' are drawn from 1 … * at random
• By treating the different columns independently while performing
random resampling, we can obtain a column-decorrelated design
matrix with the same marginal as before.
• Then we can use density ratio estimation to get +(-).

Experimental Results
• Simulation Study
125

Experimental Results
• Regression
• Classification
126
• Regression
• Classification

Disentanglement Representation Learning
• Learning Multiple Levels of Abstraction
• The big payoff of deep learning is to allow learning higher levels of
abstraction
• Higher-level abstractions disentangle the factor of variation,
which allows much easier generalization and transfer
127
Yoshua Bengio, From Deep Learning of Disentangled Representations to Higher-level Cognition. (2019). YouTube. Retrieved 22 February 2019.
From decorrelating input variables to learning
disentangled representation

Disentanglement for Causality
• Causal / mechanism independence
• Independently Controllable Factors (Thomas, Bengio et al., 2017)
• Optimize both !" and #" to minimize
128
A policy !" A representation #"
selectively change correspond to value
Require subtle design on the
policy set to guarantee causality.

Sectional Summary
129
p Causal inference provide valuable insights for stable learning
p Complete causal structure means data generation process,
necessarily leading to stable prediction
p Stable learning can also help to advance causal inference
p Performance driven and practical applications
Benchmark is important!

Outline
ØCausal Inference
ØStable Learning
ØFuture Directions and Conclusions
130

Non-I.I.D. Image Classification
• Non I.I.D. Image Classification
• Two tasks
• Targeted Non-I.I.D. Image Classification
• Have prior knowledge on testing data
• e.g. transfer learning, domain adaptation
• General Non-I.I.D. Image Classification
• Testing is unknown, no prior
• more practical & realistic
131
!(#$%&'( = *$%&'(, ,$%&'( ) ≠ !(#$/0$ = *$/0$, ,$/0$ )
unknown
known
#$%&'( #$/0$

Existence of Non-I.I.Dness
• One metric (NI) for Non-I.I.Dness
• Existence of Non-I.I.Dness on Dataset consisted of 10 subclasses from ImageNet
• For each class
• Training data
• Testing data
• CNN for prediction
132
ubiquitous
strong correlation
Distribution shift
For normalization

Related Datasets
• DatasetA & DatasetB & DatasetC
• NI is ubiquitous, but small on these datasets
• NI is Uncontrollable, not friendly for Non IID setting
133
Small NI
A dataset for Non-I.I.D. image classification is demanded.
ImageNet
PASCAL
VOC MSCOCO
Uncontrollable NI
Average NI: 2.7

NICO - Non-I.I.D. Image Dataset with Contexts
• NICO Datasets:
• Object label: e.g. dog
• Contextual labels (Contexts)
• the background or scene of a object, e.g. grass/water
• Structure of NICO
134
Animal Vehicle
Dog …
…
Train
Grass on bridge…
…
2 Superclasses
10 Classes
10 Contexts
per
per Diverse &
Meaningful
Overlapping

• Data size of each class in NICO
• Sample size: thousands for each class
• Each superclass: 10,000 images
• Sufficient for some basic neural networks (CNN)
• Samples with contexts in NICO
135

Controlling NI on NICO Dataset
•Minimum Bias (comparing with ImageNet)
•Proportional Bias (controllable)
• Number of samples in each context
•Compositional Bias (controllable)
• Number of contexts that observed
136

Minimum Bias
• In this setting, the way of random sampling leads to minimum distribution shift between
training and testing distributions in dataset, which simulates a nearly i.i.d. scenario.
• 8000 samples for training and 2000 samples for testing in each superclass (ConvNet)
137
Average NI Testing Accuracy
Animal 3.85 49.6%
Vehicle 3.20 63.0%
Images in NICO
are with rich contextual
information
more challenging for
image classification
Average NI on ImageNet: 2.7
Our NICO data is more Non-iid, more challenging

Proportional Bias
• Given a class, when sampling positive samples, we use all contexts for both training and
testing, but the percentage of each context is different between training and testing dataset.
138
4
4.1
4.2
4.3
4.4
4.5
1:1 2:1 3:1 4:1 5:1 6:1
NI
Dominant Ratio in Training Data
Testing
1 : 1
Dominate
Context (55%)
(5%) (5%) (5%) (5%) (5%) (5%) (5%) (5%) (5%)
We can control NI by varying dominate ratio

Compositional Bias
• Given a class, the observed contexts are different between training and testing data.
139
Moderate setting
(Overlap)
Radical setting
(No Overlap &
Dominant ratio)
4.44
4.0
4.2
4.4
4.6
4.8
5.0
1:1 2:1 3:1 4:1 5:1
NI
Dominant Ratio in Training data
4.34
4.0
4.1
4.2
4.3
4.4
7 6 5 4 3
NI
Number of Contexts in Training Data
Training:
Testing:
Training:
Testing:
Testing
1 : 1

• Large and controllable NI
140
Controllable NILarge NI
small NI
large NI

• The dataset can be downloaded from (temporary address):
• https://www.dropbox.com/sh/8mouawi5guaupyb/AAD4fdySrA6fn3P
gSmhKwFgva?dl=0
• Please refer to the following paper for details:
• Yue He, Zheyan Shen, Peng Cui. NICO: A Dataset Towards Non-
I.I.D. Image Classification. https://arxiv.org/pdf/1906.02899.pdf
141

Outline
ØCausal Inference
ØStable Learning
ØConclusions
142

Conclusions
• Predictive modeling is not only about Accuracy.
• Stability is critical for us to trust a predictive model.
• Causality has been demonstrated to be useful in stable prediction.
• How to marry causality with predictive modeling effectively and
efficiently is still an open problem.
143

Conclusions
144
Debiasing
Prediction
Causal Inference
Stable Learning
Propensity
Score
Direct Confounder
Balancing
Global
Balancing
Linear Stable
Learning
Disentangled
Learning

Reference
• Shen Z, Cui P, Kuang K, et al. Causally regularized learning with agnostic data
selection bias[C]//2018 ACM Multimedia Conference on Multimedia Conference.
ACM, 2018: 411-419.
• Kuang K, Cui P, Athey S, et al. Stable prediction across unknown
environments[C]//Proceedings of the 24th ACM SIGKDD International Conference on
Knowledge Discovery & Data Mining. ACM, 2018: 1617-1626.
• Kuang K, Cui P, Li B, et al. Estimating treatment effect in the wild via differentiated
confounder balancing[C]//Proceedings of the 23rd ACM SIGKDD International
Conference on Knowledge Discovery and Data Mining. ACM, 2017: 265-274.
• Kuang K, Cui P, Li B, et al. Treatment effect estimation with data-driven variable
decomposition[C]//Thirty-First AAAI Conference on Artificial Intelligence. 2017.
• Kuang K, Jiang M, Cui P, et al. Steering social media promotions with effective
strategies[C]//2016 IEEE 16th International Conference on Data Mining (ICDM).
IEEE, 2016: 985-990.
145

Reference
• Pearl J. Causality[M]. Cambridge university press, 2009.
• Austin P C. An introduction to propensity score methods for reducing the effects of
confounding in observational studies[J]. Multivariate behavioral research, 2011, 46(3): 399-
424.
• Johansson F, Shalit U, Sontag D. Learning representations for counterfactual
inference[C]//International conference on machine learning. 2016: 3020-3029.
• Shalit U, Johansson F D, Sontag D. Estimating individual treatment effect: generalization
bounds and algorithms[C]//Proceedings of the 34th International Conference on Machine
Learning-Volume 70. JMLR. org, 2017: 3076-3085.
• Johansson F D, Kallus N, Shalit U, et al. Learning weighted representations for generalization
across designs[J]. arXiv preprint arXiv:1802.08598, 2018.
• Louizos C, Shalit U, Mooij J M, et al. Causal effect inference with deep latent-variable
models[C]//Advances in Neural Information Processing Systems. 2017: 6446-6456.
• Thomas V, Bengio E, Fedus W, et al. Disentangling the independently controllable factors of
variation by interacting with the world[J]. arXiv preprint arXiv:1802.09484, 2018.
• Bengio Y, Deleu T, Rahaman N, et al. A Meta-Transfer Objective for Learning to Disentangle
Causal Mechanisms[J]. arXiv preprint arXiv:1901.10912, 2019.
146

Reference
• Yu B. Stability[J]. Bernoulli, 2013, 19(4): 1484-1500.
• Szegedy C, Zaremba W, Sutskever I, et al. Intriguing properties of neural networks[J]. arXiv
preprint arXiv:1312.6199, 2013.
• Volpi R, Namkoong H, Sener O, et al. Generalizing to unseen domains via adversarial data
augmentation[C]//Advances in Neural Information Processing Systems. 2018: 5334-5344.
• Ye N, Zhu Z. Bayesian adversarial learning[C]//Proceedings of the 32nd International
Conference on Neural Information Processing Systems. Curran Associates Inc., 2018: 6892-
6901.
• Muandet K, Balduzzi D, Schölkopf B. Domain generalization via invariant feature
representation[C]//International Conference on Machine Learning. 2013: 10-18.
• Peters J, Bühlmann P, Meinshausen N. Causal inference by using invariant prediction:
identification and confidence intervals[J]. Journal of the Royal Statistical Society: Series B
(Statistical Methodology), 2016, 78(5): 947-1012.
• Rojas-Carulla M, Schölkopf B, Turner R, et al. Invariant models for causal transfer learning[J].
The Journal of Machine Learning Research, 2018, 19(1): 1309-1342.
• Rothenhäusler D, Meinshausen N, Bühlmann P, et al. Anchor regression: heterogeneous data
meets causality[J]. arXiv preprint arXiv:1801.06229, 2018.
147

Thanks!
Peng Cui
cuip@tsinghua.edu.cn
http://pengcui.thumedialab.com
149
Kun Kuang
kunkuang@zju.edu.cn
https://kunkuang.github.io/

Causally regularized machine learning

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