Doing Fast Adaptation Fast: Conditionally Independent Deep Ensembles for Distribution Shifts
Authors: Wanqian Yang
Aahlad Manas Puli
Andrew Gordon Wilson
Rajesh Ranganath
Venue: Submitted to ICLR 2023 (2023)
Keywords: deep ensemble
diverse ensemble
shortcut learning
spurious correlations
conditional mutual information

Abstract: Classifiers in a diverse ensemble capture distinct predictive signals, which is valuable for datasets containing multiple strongly predictive signals. Performing fast adaptation at test time allows us to generalize to distributions where certain signals are no longer predictive, or to avoid relying on sensitive or protected attributes. However, ensemble learning is often expensive, even more so when we need to enforce diversity constraints between the high-dimensional representations of the classifiers. Instead, we propose an efficient and fast method for learning ensemble diversity. We minimize conditional mutual information of the output distributions between classifiers, a quantity which can be cheaply and exactly computed from empirical data. The resulting ensemble contains individually strong predictors that are only dependent because they predict the label. We demonstrate the efficacy of our method on shortcut learning tasks. Performing fast adaptation on our ensemble selects shortcut-invariant models that generalize well to test distributions where the shortcuts are uncorrelated with the label.


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DOING FAST ADAPTATION FAST: CONDITIONALLY INDEPENDENT DEEP ENSEMBLES FOR DISTRIBUTION SHIFTS
Anonymous authors Paper under doublc-blind review
ABSTRACT
Classifiers in diverse ensemble capture distinct predictive signals, which is valuable for datasets containing multiple strongly predictive signals. Performing fast adaptation test time allows to generalize distributions where certain signals are no longer predictive, or to avoid relying on sensitive or protected attributes. However, ensemble learning is often expensive, even more so when we need to enforce diversity constraints between the high-dimensional representations of the classifiers. Instead, we propose an efficient and fast method for learning ensemble diversity. We minimize conditional mutual information of the output distributions between classifiers, quantity which can be cheaply and exactly computed from empirical data. The resulting ensemble contains individually strong predictors that are only dependent because they predict the label. We demonstrate the efficacy of our method on shortcut learning tasks. Performing fast adaptation on our ensemble selects mhorteut-invarian models that generalize well to test distributions where the shorteuts uncorrelated with the label.
INTRODUCTION
Some of the strongest scientific theories are supported by multiple sources of evidence, principle described by 19th century philosopher William Whewell as "consilience". Evolution is one such example, having been firmly corroborated by fields ranging from paleontology genetics. In many real-world applications of machine learning, datasets can similarly contain multiple predictive signals that explain the label well. In these settings, standard model typically learns from combination of predictive features (Ross et al. 2018 Kirichenko et al.) 2022). Such a model will fail to generalize to distribution shifts that break the correlation between certain signals and the label (Hovy & Søgaard 2013 Hashimoto et al. 2018: Puli et al. 2022).
This shortcoming be addressed by learning diverse set ensemble of classifiers. Such methods typically exploit some notion of independence to learn multiple classifiers that rely on different predictive signals. We then perform fast adaptation, using small amount of duitr--distribution (OOD) validation data to select the model that generalizes best. Learning diversity is also beneficial in and of itself: these classifiers are empirically shown to be more numan-interpretaa than if were to fit single model (Ross et al. 20188), possibly because they learn disentangled representations that correspond natural factors of variation (Shu al. 2019).
The key challenge is quantifying the right notion of diversity. Existing work has exploited concepts like input gradient parameter orthogonality as a proxy for statistical independence (Teney et al.) 2021; Xu et al., 2021). To tackle OOD generalization, which fundamentally requires additional assumptions or data beyond the observed training data (Bareinboim et al., 2022; Schölkopf et al.. 2021), previous work have also assumed access to unlabelled test data and measured disagreement on those examples (Lee al. 2022 Pagliardini et al. 2022). However, these objectives or assumptions are often prohibitive or unrealistic in real-world settings. For example, group-balanced test data is not always obtainable, e.g. when deploying a pneumonia model to multiple new hospitals whose patient profiles may change over time. Another costly example is enforcing input gradient orthogonality high-dimension covariates like images or text, where it can be challenging to avoid learning from orthogonal covariates of the same underlying feature, such neighboring pixels.

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To avoid the pitfalls of operating in high- dighnsimensional input or parameter space, promising line of work instead adopts the cntrmmation-thhrree perspective and tackles the problem as representation learning. These approaches apply the information bottleneck method and minimize mutual informa- tion between the representations learnt by each classifier. Such an objective forces the classifiers to rely on distinctly meaningful features for prediction. Most notably, |aace et al.) (2020) and [Rame & Cord (202)) minimize mutual information between the classifier representations conditioned on the label. Since any pair of predictors cannot both be accurate while remaining unconditionally inde- pendent, the extra conditioning prevents learning weak classifiers. The resulting ensemble contains accurate classifiers that nevertheless rely on distinct predictive signals. The only core assumption is that the underlying predictive signals are themselves conditionally independent.
These approaches are conceptually appealing but practically challenging. Mutual information between high-dimensional representations is intractable and must be approximated, either via variational (e.g. Fischer 2020) or contrastive (e-g. Oord et al.) 2018) bounds. Furthermore, such approximations are computationally expensive, a problem that compounded in the ensemble setting where we wish to train multiple classifiers speedily.
We seek to learn ensemble diversity fast and effectively. Our key insight is that it suffices to enforce conditional independence on the output distributions of the classifiers. Our first contribution is proposing conditional mutual information (CMI) between output distributions the regulariz- ing objective. Assuming conditionally independent predictive signals, enforcing CMI between output distributions also guarantees that the ensemble where separate predictive signals are learnt by separate classifiers is minimizing solution. Since the output distribution is categorical, CMI can be cheaply and exactly computed from empirical data. In addition, our method avoids using additional sources of data that cannot be found in many real-world domains, such as unlabelled test data or "group" labels for each predictive signal in the dataset. We only permit a small amount of validation data from the test distribution for (1) hyperparameter tuning and (2) selecting the final predictor from our ensemble. We dub our approach as Conditionally Independent Deep Ensembles (CoDE).
Our second contribution is evaluating CoDE on benchmark datasets for shorteut learning (Geirhos jtt al. 2020). Shortcuts are signals that are (i) highly but spuriously correlated to the label in the training distribution, possibly due to biases in data collection other systematic pre-processing errors (Torralba & Efros| 2011). and (fi) preferentially learnt by a neural network, possibly due to simplicity biases (Shah al. 2020) or architectural biases (e.g. convolutional neural networks (CNNs) relying on texture over shape (Baker et al.] 2018)). An empirical risk minimizing (ERM) model will rely on shortcuts and fail to generalize test distributions where they are longer correlated to the label. This is a natural application for our method as the core assumption of conditional independence applies many such datasets- for example, in natural images, the foreground is typically the label and is thus conditionally independent from the background (shortcut). We show that CoDE effectively recovers an ensemble where the shortcut features and the true signal are learnt by separate classifiers.
2 PRELIMINARIES: SETUP AND NOTATION
In Section 3 we will fully motivate the assumptions behind our model of the data-generating process (DGP). However, we describe it here first to establish key terminology and concepts.
Generatng Process Let z denote the set of latent factors that generate the set of observed
features x E R.. Let y E {0, I.....KK K - 1} denote the label. The data (x y, z) is generated from family of distributions indexed by e, the environment. We only consider: (i) single training environment (e = tr), from which we have access to i.i.d. labelled training examples Dr, fx,, yi M and (ii) a test environment (e tc), from which we draw unlabelled test examples that our model should perform well on. We also allow access to a small set of labelled validation data Dral = {x, [x4, J from the test environment, which is used only for hyperparameter tuning and ensembling (i.c. constructing the final model from the set of learnt classifiers).
We make the following assumptions on the DGP:
(i all label information is encoded by z, i.e. Pe xv = (ylzz) ) for all (ii) Pa (x 2) pxx(z) is invariant across all e
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(iii Pe (z) >> 0 for all and
(iv) Pe (v) > 0 for all and
(V) [Latent Conditional Independence] 2; 11 2g ly for all e and j
Based on these assumptions, we can factorize Pe (x, y, z) as:
Pe(z, x, = po(v))
11 Po(czll)) p(x(2)
Example: ColoredMNIST' As introduced in Arjovsky al. (2019). y is binary label which determines color (21 E (red, green)) with probability Pe and digit (22 E {0-4, 5-9}) with probability
pa- Pe and pa are independently chosen. In the training distribution, pe 0.25 and pa 0.1, as such, an ERM model will primarily learn from color. and Pa can be arbitrary in the test distribution.
Example: Waterbirds As introduced in Sagawa et (2019), y is binary label determin- ing if the image represents water or land bird. It perfectly determines the foreground (21 E [water bird, land bird}) and highly but spuriously correlated to the background (22 (water, land)) in the training distribution. An ERM model will learn from background features.
Group Robustness When z discrete, each possible value that can take is known as group. Due to the spurious correlations created by (zi(zi)y), groups that are highly represented in the training set are called "majority groups". and poorly-represented groups are "minority groups". Group robustness refers to the goal of generalizing well all groups and is natural way of evaluating if model has been learning shortcuts. For example, both ColoredMNIST and Waterbirde admits four groups formed by the Cartesian product of 21 and 22-
Ensembles and Fast Adaptation class probabilities. We will use ý == po(y) to denote the unconditional output distribution. We use the term "ensemble" loosely to refer to a set of M classifiers {fm J that can be learnt jointly or sequentially. (Section 4 clarifies the relationship traditional ensemble methods.) After all M classifiers are learnt, the final model 0 is selected using validation data Deal:
classifier f(x) po(y!x) is parametrized by and outputs
N'
0" arg
min Om ME (I.....M) N
log po.. (U: (x:)
(2)
This process is referred to as fast adaptation.
3 CONDITIONALLY INDEPENDENT DEEP ENSEMBLES
To motivate our approach and the assumptions made in (T) we first define what it means to learn diverse ensemble and explain why conditional independence a sound measure of diversity.
3.1 DIVERSITY CONDITIONAL INDEPENDENCE
Diverse classifiers utilize separate predictive signals, intuitively, they predict the "same things for different reasons" (Rame & Cord 202I). Our setup in Section formalizes this notion of "different reasons" by explicitly defining the latent variable which models the total underlying set of predictive signals that relate x to y. A classifier that learns a mapping from x to y can then be interpreted as implicitly inferring from x and learning mapping from to y. We can thus define diverse classifiers that rely on separate predictive signals as learning from separate dimensions or subspaces of z.
To formalize the idea that classifier f learns using only a subspace of z, naive approach might be to define If as relying only on the subspace Z(n] if and only if (some distribution computed from) f is independent of its complement z)] This definition is convenient as suggests that the appropriate objective to learn diverse ensemble is simply enforce statistical independence between the classifiers. This follows because two classifiers that rely on overlapping subspaces will necessarily be dependent.
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However, the definition above assumes that distinct predictive signals (i.e. subspaces of z) themselves unconditionally independent. This is not always true when a dataset contains multiple strongly predictive signals. Dimensions of can be dependent by virtue of their correlation to y- Classifiers that learn from such signals will similarly he dependent. Shortcut learning is precisely a problem because meaningful and spurious features are highly correlated in the training environment.
This conundrum can be resolved by establishing independence of the latent factors with conditioning on y. Doing so is equivalent to assuming that upon knowing the true label, observing one set of features yields no additional information about other features. This is usually a realistic assumption to make. As the Waterbirds example in Section shows, backgrounds and foregrounds are often conditionally independent in the test distributions we care about. This motivates our assumption (v) of latent conditional independence in Section where the individual factors 2; are conditionally independent given y. We formalize this notion of "diversity as conditional independence" below.
Definition 3.1. Let [[a]
(Zor).... 201) denote some subspace of z. Let h(f) denote some distribution computed from f. We say f is invariant to [[a] if h -L (Za,...., Ray )
Ig.. Let Z be the maximal subset of z that f is invariant to. Then f is said to rely on [1] == z z| for prediction.
Definition 3.2. Let and f' be pair of classifiers that rely on 2|g] and [[r] respectively. f and f' are said to be diverse if xz O [1]] = 0. An ensemble (Sm J is diverse if every pair of classifiers fa, fu in the ensemble are diverse.
It follows immediately from Definition 3.2 that diverse classifiers must themselves be conditionally independent, i.e. f4 IL hy y. Our training objective for learning diverse ensemble should therefore enforce conditional independence on all pairs of classifiers:
N
M
arg max
Σ
log po.. (Ma (uilx;;
1=1
subject
11
1s,
We interpret (3) as follows: the main objective guarantees that the learnt ensemble contains indi- vidually strong predictors, whereas the constraint guarantees that each predictor is uninformative of the others when conditioned on the label. Put together, (3) learns classifiers that rely conditionally independent subspaces of z and thus provide no additional information about each other. As is typical in machine learning (Krogh & Hertz 199T Deb) 2014), optimize an unconstrained analogue of (3 by expressing the constraint as a regularization term.
3.2 ENFORCING CONDITIONAL INDEPENDENCE VIA OUTPUT DISTRIBUTIONS
remains for us decide on the distribution h that constrain, well as the (unconstrained) regularization objective from (3). These choices are crucial in many ways. Since independence with
respect to å underpins the notions of invariance and diversity in Definitions 3.1 and 3.2 it must be informative about the underlying predictive signals that classifier is relying on. Furthermore, ft and the regularization objective must be tractable.
Earlier work such as Pace et al. 2020) and Rame & Cord 2202 choose h to be the representations learnt by the classifiers, e.g. by constructing j = Ji Je as deep encoder network fe that is attached linear classifier and letting h [x(... As the regularization objective for conditional independence, /Rameeeeeeeeemmmemeeeeeeaeeeeee & Cord (2021) compute pairwise conditional mutual information C.MI fe.:) whereas Pace et al. (2020) compute total correlation TC.fe....... ... fe,rr) Since the encoder representations are high-dimensional, these terms must be approximated.
We propose far simpler and more efficient method. Instead of network representations, we choose to simply be the output distribution h = f(x) (o(y) of the classifier. Accordingly. our regular- ization objective is conditional mutual information (CMI) between the output distributions of the classifiers. For any pair of classifiers fj, fú. we have:
C.MI(fa By [DKL (p(f.. file) (I ((fll)))((((l)))
(4)
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CMI is zero iff fs TT fely for all values of y. Enforcing conditional independence on the classifiers' predicted output probabilities rather than underlying representations trades off granularity of the independence constraint for computational efficiency. We believe that this is valuable trade-off. Since ý has categorical support, 4 can be cheaply and exactly estimated from training data. As our experiments in Section show, even noisier signal like output distributions, enforcing conditional independence is sufficient to learn a diverse ensemble.
Even though a diverse ensemble implies pairwise conditionally independent classifiers, the converse is not necessarily true. Mutual information is also zero if one of the classifiers outputs random or constant class probabilities. particular, optimizing a weighted sum of the cross-entropy term and the CMI term can be challenging overly weak regularization produces an ensemble that is not diverse, whereas overly strong regularization tends towards solutions containing close-to-random classifiers. Instead, we propose adding another term to regularize for confident predictions:
R(f)
Σ
p(0ly = k) -
(5)
1=1
where 1g is the indicator function k. Put together, the overall loss objective
M
s-1 log pe.. (y1 x.) λ1 ΣΣ cM
M
C((0m)ml AM
Σ
42
R(fm)
(6)
where A and λ2 are hyperparameters controlling the strength of regularization. A solution that minimizes (6) contains an ensemble where: (i) each classifier is accurate (first term) and confident (third term), and (ii) different classifiers rely on different subspaces of for prediction (second term).
We name such an ensemble a Conditionally Independent Deep Ensemble (CoDE).
3.3 CODE: COMPUTATIONAL DETAILS
The hyperparameters of the method are M, X1 and Ag. Unlike traditional ensembles, M (ensemble size) will typically be small (M 2 for all our experiments) since A cannot be larger than the number of conditionally independent predictive signals inherent in the dataset. As is typical for OOD problems, we assume access to validation data from the test environment for hyperparameter tuning.
Objective (6) describes the situation where all M classifiers are jointly optimized. Since M is typically small, doing so is not difficult or computationally expensive (as might be with traditional ensembles). An alternative to joint optimization is to learn the classifiers a sequential fashion. The analogue to (6) becomes:
((0m)
log Pm... (1, + A Σ CMI + A R(fm)
Sequential optimization presents a natural way to determine M, as we can terminate the training process when no more predictive classifiers can be learnt. However, it will fail if earlier classifiers in the sequence learn multiple predictive signals. We discuss this further in Section
RELATED WORK
Ensemble Methods In statistics, ensembling traditionally refers to combining multiple predictors into a single model that outperforms the individual learners, typically by bagging (Breiman) 1996) or boosting (Schapire, 1990). Diversity in this context refers to minimizing correlation between individual learners, which reduces variance and improve generalization (Kuncheva Whitaker 2003). Deep ensembling (Lakshminarayanal et al.) 2017) is analogous approach in deep learning where multiple mandomly-initallinn networks are trained in parallel. however, they are generally used for the purpose of uncertainty estimation. Unlike these works, consider diversity specifically in the context of datasets with multiple predictive signals, and learning a diverse ensemble as recovering all such signals for the purpose of OOD generalization.
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Various Approaches For Learning Diversity As an unsupervised task, diversity refers to learning disentangled representations where natural factors of variation in the dataset are encoded into distinct latent dimensions (Bengio et al.) 2013; Higgins et al. 2018); however, recent work has proposed incorporating weak supervision in this process (Locatello et al. 2019; Shu et al. 2019 Brehmer al., 2022). As supervised problem without OOD shifts, diversity refers to learning functions that
disagree outside training points. Methods in this space have generally made use of input gradients
(Ross al. 2017 2018) and orthogonality (Mashhadi et al. |02|] Xu al.) 2021) Finally, diversity is considered in the context of distribution shifts either to improve robustness against adversarial attacks (Pang et al. 2019), to disambiguate between perfectly correlated signals (Lee et al. 2022). or
to evade the simplicity bias by learning more complex functions (Pagliardini et al. [202]] Teney et al.,
202T). Our work most closely aligned with this last category. Unlike the approaches above, we exploit cotemaatioo-theorette measures as our objective.
Shorteut Learning and Spurious Correlations Shortcut learning (Geirhos et al. 2020) involves distribution shifts arising from spurious correlations (Buolamwini & Gebru 2018 Xiao et al. 2020
Moayeri et al.; 2022) and neural network biases (architectural or simplicity biases) (Geirhos et al.
2018|[Shah al. 2220 Teney al. (021). Methods that tackle distribution shifts must use additional
data and/or assumptions. Examples of additional data include having multiple training environments (Arjovsky al. 2019), counterfactual examples (Teney et al., 2020), access to enough validation
data to fine-tune the model (Kirichenko et al. 2022), or group labels (Sagawa et al.) 2019 Puli et al.,
2022). Examples of additional assumptions include exploiting the lottery ticket hypothesis (Zhang ct al. 202T) or treating misclassified training examples by an initial model as proxy for minority groups (Liu et al., 202T, Zhang al., 2022). Unlike these methods, we aim learn all predictive signals in the dataset, rather than performing well on a single test distribution. Furthermore, we use validation data for hyperparameter tuning only, without additional sources of data (e.g. group labels).
Information Bottleneck and Conditional Independence The line of work most similar to ours also exploits the information bottleneck method to learn diversity. (Sinha et al. (2020) minimizes the mutual information Z(ž., & between learnt representations åm however, this term is unconditional
and will simply learn weak (biased) predictors, as noted in Section 3 Rame & Cord (20TT) introduce
DICE, which minimizes the conditional term C., &g). Pace et al. (2020) considers total correlation TC(2) 2M1) instead of pairwise terms. Unlike CODE, both of these approaches compute mutual information terms on the high-dimensional representations 2. Their objectives
intractable and must be approximated. For example, DICE requires both variational approximations and a jointly trained adversarial discriminator that learns to distinguish pairwise classifiers. Compared to these approaches, CoDE is by far computationally advantageous as mutual information for categorical output distributions can be computed faster and exactly.
5 EXPERIMENTS
Section 5.1 presents experiments on Co oredMNIST, which is used both demonstrate the viability of our approach and to highlight pivotal observations and ablations. Section 5.2 then evaluates CoDE on larger benchmark datasets for shorteut learning to show that it scales effectively.
5.1 COLOREDMNIST
Setup As described in Section 2 the original MNIST (LeCun et al. 1998) labels are binarized (0-4, 5-9) and used to generate true labels y with noise Pa- y then generates binary color labels with noise Pe, used to color the image (red or green). As per Arjovsky et al. (2019), we consider two test environments: the training distribution where pa = 0.25 and p. = 0.1, and the adversarial distribution where pa = 0.25 but P. = 0.9 (hence the shortcut-label correlation is reversed).
Evaluation Baselines and Metries As is standard in existing work, we evaluate predictive accuracy on the training and adversarial distributions. In choosing baselines, we considered the following desiderata for fairness and comprehensiveness: (i) comparing to both ensembling and non-ensenene non-ennsmbling methods, (ii) amongst ensembling methods, comparing both conditional independence based methods and those that do not, and (iii) comparing only to methods that do not require additional sources of data besides validation data for hyperparameter tuning. We chose the following baselines:

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Results ColoredMNIIST
Training (0.25, 0.1)
Adversarial
Random-Color
Random-Color + Perfect-Dight
(pd)Pe)
(0.25, 0.9)
(0.25, 0.5)
(0.0, 0.5)
Invariant
75
75
75
100
ERM
88.6
15.3
52.5
53,4
JTT
17.8
87.9
52.5
56.6
Ortho Ensemble
89.8
11.1
50.3
49.2
TC-Ensemble 89.1
69.8*
CoDE
70.7
70.0
70.8
91.2
Table 1: Results on ColoredMNIST. A theoretically ideal classifier relying only on digit (denoted as "Invariant") will be upper-bounded by the digit-label noise pa (75%), hence any result above 75% is relying on the color shortcut. CoDE has the strongest performance the adversarial distribution. We were unable to reproduce TC- ECCEmmemble on
and are citing their results in lieu.
1. ERM classifier (ERM): single, standard classifier trained with ERM
2. Just Train Twice (Liu et al. 2021) (JTT): initial classifier is trained for limited number epochs; classifidd examples are upweighted to train the final classifier
3. Ensembles using input gradient orthogonality (Teney et al. 2021 Onseblens:mmle): an ensemble where the regularizing term is the dot product of the two models' input gradients
4. Ensembles using conditional total correlation (CTC) (Pace al. 2020) (TC-Ensemble): ensemble learnt by minimizing CTC over the encoder network ' representation
Table 1 shows all results on Col ColoredMNIST. We discuss the most important findings below.
1. Enforcing conditional independence output distributions achieves diversity effectively.
Since ColoredMNIST is an artificially-created dataset whose DGP we know satisfy latent conditional independence (pe and pa are independently determined), it is the ideal dataset to evaluate our key claim. Indeed, the strong performance of CoDE shows that it is sufficient to enforce conditional independence output distributions. The final predictor selected via fast adap- tation achieves near-invariant results, suggesting that it has correctly learnt from digit rather than color.
2. CoDE generalizes to multiple OOD test distributions, without overfitting on any one specific distribution.
In Table D JTT achieved about 90% on the adversarial distribution, implying that it overfitted to the adversarial distribution by learning the opposite shorteut (color) correlation rather than the true signal (digit). This is further confirmed with additional results on two other test environments (Random-Colee and Random-Coll
+ Perfect-Digit) where Pe 0.5. JTT is close random these two environments, suggesting that it is still relying on color as the predictive feature. In contrast, CoDE achieves 91% 31% when Pa = 0.0, suggesting that it has learnt to predict using digit.
While concerning, these results are not entirely surprising. A method like JTT did exactly what it was designed to do, which is to minimize classification enrors on the adversarial test distribution. Since Pc. 0.1, the opposite color correlation is precisely this loss-minimizing function. In contrast, CoDE will not find such solution because two classifiers that return opposite predictions using the same feature (color) perfectly correlated, even when conditioned on y.
These results highlight the shortcomings single classifier methods like JTT. Such methods are designed to generalize to specific test distribution, in general, this does not imply that they have learnt the desired predictive signal merely that they have learnt an arbitrary function that does well on the test distribution. In contrast, methods that enforce diversity, such CoDE, explicitly recover meaningful predictive signals that can generalize to any test distribution where p(zly) changes.
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ColoredMNIST
CelebA
Training
Adversarial
Ave
Worst
CoDE (sequential fi)
90.0
10.2
95.2
31.1
CoDE (sequential f2)
70.1
70.0
95.0
33.3
CoDE (sequential fa)
63.2
49.0
CoDE (sequential fs)
64.4
42.2
CoDE (joint 2)
73.4
60.2
89.2
83.3
CoDE (joint = 3)
74.6
44.3
CoDE (joint M = 5)
71.9
43.1
Table 2: Additional results ColoredMNIST and CelebA.
3. Joint and sequential optimization suited to different datasets.
From our experiments, we found that there is no clear preference between either choice terms of generalization ability. Table T shows both joint and sequential results the ColoredMNIS IST and CelebA datasets. For ColoredMor ColoredMNIST, we found that sequential training performed better than joint training. For CelebA, joint training yielded a stronger classifier.
This might be explained by the biases of the ERM model. In Co 1 oredMNIST, as both latent factors (color and digit) noisy predictors and color presents particularly simple shortcut, the ERM model solely learns from color. As such, second classifier model that is trained sequentially can learn predict solely from the digit feature. In contrast, the ERM model in CelebA has likely picked up some combination of the spurious (gender) and true (hair color) features, possibly because gender gives rise to complex features that are not ncessarily simpler learn. This corroborates previous findings indicating that ERM models can learn an arbitrary combination of all predictive signals (Zhang et al. 2021] Kirichenko al. 2022). As such, when trained sequentially, the second model fails to learn from hair color alone.
The advantages of sequential optimization are: (i) cheaper computational costs as M increases, and (ii) providing a natural stopping point for training. The latter comes from the fact that can select for M by terminating the training process when the subsequent classifier is no longer predictive, which indicates that there are further predictive factors to be learnt. In contrast, joint optimization is advantageous as allows us to avoid the pathological sitation where earlier models learn combinations of predictive factors. As small values of M work well for CoDEs, note that the computational cost of CoDEs are not prohibitive.
5.2 BENCHMARK DATASETS
Setup
We consider the following benchmark datasets:
CelebA Liu et al. (2018): ) Sagawa et al. (2019: A dataset of celebrity faces with various
labelled attributes. We consider the benchmark task in (Sagawa al.) 2019) of predicting the binary hair color attribute (blond or not), with gender (female or male) the spurious attribute. There are therefore four groups.
Materbi rás (Wah et al. 2011 Sagawa et al. 2019) Setup described in Section 2 There are also four groups as both latent factors (background and foreground) are binary.
MF-Dome Domm i noes (MNIS ST Fash i inMMNSTT) (LeCun et al. 1998 Xiao et al. 2017 Shah et al. 2020 Pagliardini et al. 2022): Each input image concatenates an MNIST digit or 1) with FashionMNIST object (coat dress). The true label is the FashionMNIST object; the simpler MNIST feature is the shortcut. The minority groups represent 5% of the data.
Jshows all results on the benchmark datasets.
4. CoDE scales well to large datasets and retains effectiveness at preventing shortcut learning.
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CelebA
Waterbirds
MF-Dominoes
Method
Ave
Worst
Ave
Worst
Ave
Worst
ERM
94.8
46.7
90.4
78.3
88.9
76.9
JTT
88.0"
81.1"
93.3*
86.7"
89.5
76.1
CoDE
89.2
83.3
91.5
79.4
92.1
91.4
Table 3: Main results on all datasets. CoDE achieves better adversarial or wrost-group accuracy than the other methods on all datasets except Wat erbi rds.
Results from the JTT paper. We share the same model and training environment as their paper.
On CelebA and MF MF-Dominoes, CoDE achieves the best worst-group accuracy. Unlike the earlier ColoredMNIST dataset, we have no guarantees that the core assumption of latent conditional independence holds. However, the strong performance of CoDE on these datasets shows that such an assumption is generally valid and useful when scaled to more realistic datasets.
We note that CoDE performs poorly Waterbirds In experiments, selected M as the ensemble size. Even though there are no guarantees what will be the two conditionally independent classifiers that CoDE learns, in the other datasets, the results show that they do each correspond to the shortcut and true signal. This implies that in these datasets: (a) there are no features conditionally independent to both the shortcut and true signals and yet also strongly predictive of the label. and (b) the shortcut or true signal cannot be decomposed themselves into conditionally independent signals. Our hypothesis is that (b) is not for Wat waterbirds. As the dataset is varied and contains range of land and water backgrounds, there could be multiple spurious signals in the background that are somehow conditionally independent, resulting in these signals being learnt. Another possibility is that the ensemble could have learnt imperfect partial foreground signal.
5. Computational effectiveness is crucial to learn diverse ensembles at scale.
Beyond ColoredMNI ST, we found that it was computationally prohibitive to run as the size of ensembles required to work well (48 or 96) was too high. We also noted that could not implement TC-Ensembles successfully on larger datasets, noting that the original authors do not test datasets besides ColoredMNJ either. We believe that this further highlights the importance of computational efficiency in diverse ensembling.
DISCUSSION AND CONCLUSION
Appendix B discusses potential failure modes of method.
We introduce CoDE, a method for learning an ensemble of diverse classifiers that rely on different predictive signals in the dataset. The key assumption made by CoDE conditional independence between predictive signals, which it enforces on classifiers' output distributions. We find that CoDE works well in practice when applied to shortcut learning tasks, Future work includes: (a) evaluating CoDEs on other applications where multiple predictive signals exist, such as faimess--relttd tasks where we might want to learn classifiers that do not rely on sensitive attributes, and (b) considering other metrics for conditional independence that might provide more fine-gre tneeggained signals than output distributions (e.g. minimizing mutual information between latent representations).
ETHICS STATEMENT
Positive Impact Being robust to distribution shifts, CoDE will have positive impact when de- ployed to high-stakes domains, where learning shortcut signals can have harmful social consequences. One such notable example is pneumonia prediction models trained pneumonia labels from chest X-ray scans have been shown to learn machine-specific artifacts in the background. which is a shortcut as hospitals have differing positivity rates and different machines (Zech al. 2018).
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Negative Impact There notable negative impacts of using CoDE specifically, besides the general potential for all machine learning models to abused the wrong hands.
REPRODUCIBILITY STATEMENT
intend to release public code with a camera-ready version of the paper.
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A EXPERIMENTAL DETAILS
Architecture and Training Details For ColoreaMNIST, we use CNN as the classifier, con- taining two convolutional layers and two fully-connected layers. Adam (Kingma Ba 2014) is used for optimization, with learning rate of 0.001. For CelebA, Waterbirdse and MF - Domi noes, we use a ResNet-50 (He et al. 2016). SGD is used for optimization, with a learning rate of 0.001, momentum decay of 0.9, and weight decay of 0.001. Additionally, following previous work (e.g. Sagawa et al. 2019 Liu et al. 202T), the Waterbirg rds model is pre-trained on ImageNet (Deng et 2009) and includes data augmentation in the form of random horizontal flips and random resized cropping. For CelebA and Waterbirds, class reweighting is performed to ensure that there are roughly equal positive and negative labels. The random seed used for all experiments is 13.
Hyperparameters for CoDE and Baselines CoDE. For all four datasets, we used ensemble size, besides ablations for M as detailed in Appendix B The results in Table 3 were achieved with sequential training for moolorecMMISS and with joint training for the other three datasets. For CoLoredMNIST, Å 1200 and A2 = 10. For CelebA, A 500 and 02 0.1. For Waterbirds, A = 500 and X2 = 0.1. For -d-mmmnne,,, A = 300 and X = JTT. We performed hyperparameter sweep with T {1, 5, 10) (number of epochs for initial model training) and a E {2, 10, 100} (upweighting factor for mis-classified examples). Orthogonal Ensembles. All classifiers share the same feature extractor (i.e. convolutional output for Col ore and ResNet-50 feature representation for the other three datasets). We experimented with different values of M, however, values of M above 16 (for @oloredMNIST and above 4 (for the other three datasets) were prohitibively expensive. As such, we did not try M = 48 or M = 96 as used by Teney al. (202T). J. For these smaller values of M that we tried, we did not notice an improvement from the ERM model. Besides Co oredMNT ST, we did not report these results.
2 as the
B MODEL MIS-SPECIFICATION: POTENTIAL FAILURE MODES
The success of any method tackling distribution shifts depends on how well the assumptions made have been upheld. We discuss the potential implications when the model is mis-specified and these assumptions are no longer valid.
Conditional Dependence CoDE relies the assumption that predictive signals are conditionally independent. We using the synthetic Co coleorede e dMNIST dataset to generate a DGP where such an assumption does not hold true. Instead of the standard setup where color labels are generated from the true labels, we generate color labels from the original (binarized) MNIST labels instead, at the same noise level Pe 0.1. This means that the color and digit signals now highly correlated. Both are still predictive since the true labels themselves were generated from MNIST labels.
Table 4 shows the results of this experiment. As we expect, conditionally dependent features cannot be recovered by minimizing conditional mutual information. The ensemble either recovers one of the two features (when trained sequentially) or neither. This confirms our intuition that conditional independence must be correctly specified for CoDE to work. While these results demonstrate a failure mode of CoDE, conditional independence between predictive factors of interest does hold well in many natural image datasets, as shown in Table
The size of the ensemble M specifies how many predictive latent factors we believe generated the dataset. We can consider the mis- muissppecificatio of M in either direction: (i) if the true dimension of is smaller than M, and (ii) if the true dimension of is larger than M.
Latent Mis-specification
In case (i). since the number of conditional independent components has been over-specified, whether the ensemble has been jointly or sequentially trained makes a difference. Consider the results on the ColoredMNIS
dataset in Table 2lbiin. In the sequential regime, the first two classifiers fi and f2 correspond to the color and digit classifiers respectively, however, the subsequent few classifiers (/3 and fs) do not learn anything meaningful and perform poorly on both training and adversarial distributions. However, as noted in Section 5 this does not pose a serious problem since we can use validation data to naturally determine the stopping point. On the other hand, ver-specification of M is more worrying in the joint regime, as there is no guarantee that any of the true latent factors are learnt at all. As Table shows, for M 3 or 5, the besf-performing classifier does not generalize.
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Training (0.25, 0.1)
Adversarial (0.25, 0.9)
Random-Color (0.25, 0.5)
Perfect-Digh (0.0, 0.5)
(pa, Pe)
CoDE (sequential fi)
77.1
63.2
70.6
90.5
CoDE (sequential f2)
53.6
50.1
51.5
54.5
CoDE (joint fi)
84.9
25.8
56.0
60.4
CoDE (joint f2)
54.3
73.0
64.4
77.5
Table 4: Results on CoLoredMNIST with color-digit conditional dependence, on both joint and sequential training with M = 2 classifiers. When trained sequentially, the first classifier fi learns the digit correlation since digit is most predictive in this setup. However, as color no longer conditionally independent of digit, there is predictive feature that can be learnt by the second classifier /2, resulting close-to-random predictor. When trained jointly, neither of the classifiers correspond to the color digit feature.
In case (ii), where the number of conditional independent components is under-specified, the learnt ensemble may correspond to any subset of the true latent factors and individual classifiers could also learn arbitrary combinations of the latent factors. For example, the trivial case where M 1 underspecified simply returns the ERM model. In general, since M a hyperparameter, latent mis-specification does not pose serious problem we tune its value using the validation data.
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Reviews:
[{"review_id": "Oa67aHlFuPK", "reviewer": "Reviewer_q3rX", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper quantifies a notion of diversity for deep ensembles that facilitates efficient estimation. The authors show that it is sufficient to enforce conditional independence on the output distributions of the classifiers. This leads to their main contribution concerning the regularizing metric: conditional mutual information (CMI), efficiently computed in classification problems. The authors name this approach Conditionally Independent Deep Ensembles (CoDE).  The authors evaluate CoDE on benchmark datasets for shortcut learning.", "summary_of_the_review": "The authors' main contribution is the development of CoDE approach and the objective for efficient learning in CoDE.", "strength_and_weaknesses": "Strength:\n- The main strength of this paper lies in the clear exposition and development of CoDE.\n- The definitions of invariance and diversity specific to CoDE are novel.\n- The CoDE optimization problem is new, and its solution is efficient.  \n- The numerical results are promising. \n\nWeakness:\n- Novelty of the CoDE objective is unclear in light of the earlier work on conditional mutual information.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "AwiqRVOrSS", "reviewer": "Reviewer_aogU", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper discusses a method for learning diverse ensembles “Conditionally Independent Deep Ensembles (CoDE)” and benchmarking its performance with shortcut learning datasets (i.e. ColoredMNIST, Waterbirds). The main objective is to enforce variability of the signals picked  avoiding to rely always on the same and/or strongest signal. \n\nAuthors aim to enforce conditional independence  on the output distributions (there are good properties of doing so as stated by authors i.e. being computationally cheaper). Also authors focus on enforcing confident predictors. That takes the form of two loss components: \n\n(i) CMI - a conditional mutual information component that is computed on pairwise predictors/networks\n(ii) R - a “confident”-prediction regularization\n\nBoth are controlled by scalar factors to weight their importance.\n\nIn terms of literature counterparts, authors reference two main papers: Pace et al. (2020) and Rame & Cord (2021) ", "summary_of_the_review": "Overall, I initially recommend rejecting this submission as results are not very strong in showing performance or computational cost improvement. ", "strength_and_weaknesses": "Results on the ColoredMNIST dataset (table 1) raises some questions to me (i) does the train column refer to metrics of testing on the train set? and if so, does it show a tendency of ‘competing’ methods to overfit? -(ii) The difference of performance with TC-Ensemble Pace et al. is small, is there perhaps a computational benefit in comparison? (iii) the work by Pace et al. (2020) is included in the table but Rame & Cord (2021) not – being one of the closest references it would help to see its benchmark as well, or understand why it may not be applicable.\n\nThe model has a few possible variations: being sequential/joint, which is addressed by results in table 2. However, the two proposed loss components are not further studied beyond the hyperparameter comments on the appendices.\n\nIn terms of the data generation process or the impact of enforcing variability of the ensemble learnt it is not further linked to the results i.e. (i) is it a valid assumption or (ii) does the classifiers learnt show signs of being independent.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "84IJWIObfF", "reviewer": "Reviewer_iHMc", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper aims to learn a set of classifiers to take care of different predictive signals in the dataset. They propose that the \"diversity\" of the ensemble is important and the conditional independence is an effective way to realize this goal.", "summary_of_the_review": "Please refer to Strength And Weaknesses.", "strength_and_weaknesses": "Pros:\n* The proposed fast adaption method for ensemble learning can addresses distribution shift and shortcut learning simultaneously. These two issues are common and realistic in practice.\n\nCons:\n* Based on the description in this paper, I find the separate predictive signals are very similar to multi-view learning. Both of them are to use the different perspectives of a dataset to perform learning tasks. More discussions about the similarities and differences between these two settings are required. If the methods of multi-view learning could be extended to this problem.\n\n* The motivations are not clear. For example, they claim that considering the independence of latent factors conditioned on labels is better. However, I cannot get its advantages over unconditioned independence from this paper. \"Dimensions of z can be dependent by virtue of their correlation to y\" may be a reason. However, this explanation is too abstract, and more high-level analyses and practical examples will be better. As a way to realize the conditional independence, why do they choose the conditional mutual information? What are its advantages? If other metrics that are related to independence would make the method fail?\n\n* The experiment design is problematic. The title of this paper emphasize the \"distribution shift\". However, in experiments, they mainly verify the effectiveness for shortcut learning,  and the distribution shift is not verified specifically. They should perform additional experiments on the datasets of domain adaptation or OOD generalization. Besides, due to the similarity with multi-view learning, comparing CoDE with the corresponding methods is necessary.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject