Confounder-Free Continual Learning via Recursive Feature Normalization
Authors: Yash Shah
Camila Gonzalez
Mohammad H. Abbasi
Qingyu Zhao
Kilian M. Pohl
Ehsan Adeli
Venue: ICML 2025 poster (2025)
Keywords: deep neural networks
confounders
continual learning
invariant representations
statistical regression
Primary area: applications->health_medicine
TL;DR: We introduce the Recursive Metadata Normalization (R-MDN) layer to learn confounder-invariant feature representations under changing distributions of the data during continual learning.

Abstract: Confounders are extraneous variables that affect both the input and the target, resulting in spurious correlations and biased predictions. There are recent advances in dealing with or removing confounders in traditional models, such as metadata normalization (MDN), where the distribution of the learned features is adjusted based on the study confounders. However, in the context of continual learning, where a model learns continuously from new data over time without forgetting, learning feature representations that are invariant to confounders remains a significant challenge. To remove their influence from intermediate feature representations, we introduce the Recursive MDN (R-MDN) layer, which can be integrated into any deep learning architecture, including vision transformers, and at any model stage. R-MDN performs statistical regression via the recursive least squares algorithm to maintain and continually update an internal model state with respect to changing distributions of data and confounding variables. Our experiments demonstrate that R-MDN promotes equitable predictions across population groups, both within static learning and across different stages of continual learning, by reducing catastrophic forgetting caused by confounder effects changing over time.

Confounder-Free Continual Learning via Recursive Feature Normalization
Yash Shah Camila Gonzalez! Mohammad H. Abbasi
Qingyu Zhao Kilian M. Pohl Ehsan Adeli
Abstract
correlations introduce bias into learning algorithms, causing the feature representations learned by deep neural networks (DNNs) to be skewed (Buolamwini & Gebru, 2018; Ober- meyer et al., 2019; Oakden-Rayee et al., 2020; Chen et al., 2021; Seyyed-Kalantari Kalantari et al., 2020).
Confounders are extraneous variables that affect both the input and the target, resulting in spuri- ous correlations and biased predictions. There are recent advances in dealing with or remov- ing confounders in traditional models, such as metadata normalization (MDN), where the distri- bution of the learned features is adjusted based on the study confounders. However, in the con- text of continual learning, where a model Iearns continuously from new data over time without forgetting, learning feature representations that are invariant to confounders remains significant challenge. To remove their influence from inter- mediate feature representations, we introduce the Recursive MDN (R-MDN) layer, which can be integrated into any deep learning architecture, in- cluding vision transformers, and at any model stage. R-MDN performs statistical regression via the recursive least squares algorithm maintain and continually update an internal model state with respect to changing distributions of data and confounding variables. Our experiments demon- strate that R-MDN promotes equitable predictions across population groups, both within static learn- ing and across different stages of continual learn- ing, by reducing catastrophic forgetting caused by confounder effects changing over time.
This problem is particularly relevant in medical studies, such as those related to brain development (Casey al., 2018), bi- ological and behavioral health (Petersen al., 2010; Brown et al., 2015), and dermatoscopic images (Tschandl et al., 2018), 8), which are often confounded by demographic data such as age, sex, socio-economic background, and by fac- tors like acquisition protocols and discase comorbidities. For example, a DNN trained to diagnose neurodegenerative disorders from brain MRIs could disproportionattly rely on age instead of the underlying pathology. This may occur due to the disease causing accelerated aging selection bias, i.e., having different distributions in the diseased co- hort versus the control group. This can lead to models that are inequitable and inaccurate for certain populations (Rao et al., 2017: Seyyed-Kalantari Adeli al., 2020b; Lu al., 2021; Vento et al., 2022). Given these challenges, it is crucial to develop techniques that enable DNNs to focus on task-relevant features while re- maining invariant to confounders, which are often available as auxiliary information or metadata in the dataset.
Kalantari et al., 2020; Zhao et al., 2020;
Methods such as BR-Net (Adeli et al., 2020a), MDN (Lu et al., 2021), P-MDN (Vento et al., 2022), and RegBN (Ghahremani Boozandani & Wachinger, 2024) have pre- viously been proposed address the challenges posed by confounders when training DNNs. However, in continual learning, where data becomes available sequentially, tra- ditional methods fail. This is because previous adversar- ial (BR-Net) or statistical (MDN, PMDN, etc.) methods need to use the entire dataset (implemented in batch-level statistics together with global memories) estimate and remove the distribution of the features with respect the confounders. In addition, because they require estimating batch-level statistics, they are unsuitable for modern archi- tectures like vision transformers (Vaswani et al., 2017). A continuum of data may arise in various contexts. For exam- ple, in a cross-sectional study (Tschandl et al., 2018), the training process may be divided into distinct stages, each stage featuring different data distributions. In contrast, in
1. Introduction
Confounders study variables that influence both input and target, resulting in spurious correlations that distort the true underlying relationships within the data (Greenland & Morgenstern, 2001; Ferrari et al., 2020). These spurious
'Stanford University. Stanford, United States "Weill Cor- nell Medicine, New York, United States. Correspondence to: Yash Shah <ynshah@stttt aeate@sstaanatuuuee Ehsan Adeli <ead- eli @ stanford.eduu>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Confounder- Free Continual Learning via Recursive Feature Normalization
Continuum of data
ternal parameters--consisinng of regression coefficients and estimated inverse covariance matrix, which together form an internal model state-based previously computed val- ues whenever new data are received. This state represents the current understanding of the relationship between the learned features and the confounders, enabling the model to adapt dynamically as new data flows in.
Inputs
Frequriey
Frequency
Froquanc
Distribution of the confounder
R-MDN, therefore, applies to continual learning, where each training stage consists of data drawn from different stationary distributions. This continuum of data can be understood as a sequence of uninterrupted examples that a model learns from over time. Here, R-MDN does not need to train a stage-specifie network. Instead, the internal state can be continuously updated over time as the model progresses through successive training stages. As result, only single network equipped with R-MDN layers needs to be trained on the entire dataset, with the model being able to generalize across stages (data or confounder distributions) maintaining stable performance while also removing the effects of the confounders (see Figure 1).
Sequence non- internpted inputs
dg
ag
a2
ax Ng XN, a Ng
Confounder
information
a certain "time" stop of the training process,
Distriution of confounder
Confounded foature
Confounder-
R-MON
at Xr
Figure 1. Confounder-free continual representation learning. a. A continuum of data with varying distributions of the confounder across different training stages continually passes through a DNN. b. An R-MDN layer removes the influence of confounders from the intermediate feature representations.
In summary, we propose R-MDN- a flexible normalization layer that is able to residualize the effects of confounding variables from learned feature representations of a DNN by leveraging the recursive least squares closed-form solution. It can do so under varying data or confounder distributions, making it an effective algorithm for continual learning. We provide a theoretical foundation for our approach (Section 3) and empirically validate it in different experimental setups (Sections 4.1, 4.2, 4.3). Furthermore, we demonstrate the applicability of R-MDN static learning (Section C, 4.4) and architectures that prohibit the calculation of batch-level statistics (Sections 4.2, 4.3). We find that R-MDN helps to make equitable predictions for population groups not only within a single cross-sectional study (Section 4.4), but also across different stages of training during continual learning (Sections 4.1, 4.2, 4.3), by minimizing catastrophic forgetting of confounder effects over time.
longitudinal study (Petersen et al., 2010; Brown et al., 2015; Casey al., 2018), the system does not have access all data at the outset; instead, new data-such such patient vis- arrive over an extended
its in a clinical mudy--onninune period, often spanning several years. These limitations of existing methods create a gap, as there is a need for algo- rithms that effectively and explicitly remove the influence of confounding variables under changing data distributions.
To this end, we propose Recursive Metadata Normalization (R-MDN) to remove (normalize) the effects of the confound- ing variables from the learned features of a DNN through statistical regression. Specifically, R-MDN leverages the recursive least squares (RLS) algorithm (Albert & Sittler, 1965), which has been widely used in adaptive filtering, con- trol systems for reinforcement learning, and online learning scenarios (Xu et al., 2002; Gao al., 2020). R-MDN is layer that can be inserted at any stage within a DNN. The use of statistical linear regression is motivated by its suc- cess in de-confounding learned feature representations (Mc- Namee, 2005; Brookhart et al., 2010; Pourhoseingholi et al., 2012; Adeli et al., 2018). Similar to the original MDN for- mulation, R-MDN does not assume that confounders have to linearly influence the input image because the method does not directly operate images, but rather on the fea- ture embeddings, which are non-linear abstractions of the input. Morcover, R-MDN can be applied to feature em- beddings at multiple layers, so overall can effectively remove non- linear confounding between the input and the confounders. R-MDN operates by iteratively updating its in-
2. Related Work
Widely used techniques such as batch (loffe, 2015), layer (Ba et al., 2016), instance (Ulyanov, 2016), and group (Wu & He, 2018) normalization standardize intermediate feature representations of DNNs, i.e., they normalize them have mean and unit standard deviation across different repre- sentational axes. They do not explicitly remove the effects of confounding variables from these features.
Prior works have proposed methods for learning confounder- invariant feature representations based on domain- adversarial training (Liu et al., 2018; Wang et al., 2018;
'The implementation code available at htttp:://githhii com/ at a a 1ab/RMON, RMDN. git.
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Confounder-Frie Freee Continual Learning via Recursive Feature Normalization
Sadeghi et al., 2019; Adeli al., 2020a), closed-form sta- tistical linear regression analysis (Lu al., 2021), penalty approach to gradient descent (Vento et al., 2022), regulariza- tion (Ghahremani Boozandani & Wachinger, 2024), disen- tanglement (Liu et al., 2021; Tartaglione et al., 2021), coun- terfactual generative modeling (Neto, 2020; Lahiri et al., 2022), fair inference (Baharlouei et al., 2020), and distri- bution matching (Baktashmotlagh et al., 2016; Cao al., 2018). Among these, distribution matching techniques do not specifically remove the influence of individual con- founders from learned features. Adversarial training, on the other hand, typically involves a ceooonddr--reddcttin network applied to pre-logits feature representations, with an adversarial loss used to minimize the correlation between features and confounders. However, adversarial approaches struggle to scale effectively when faced with multiple con- founding variables. Likewise, disentanglement, fair infer- ence, and counterfactual generative modeling techniques only partially remove confounder effects from a single layer of the network (Zhao et al., 2020; Vento et al., 2022).
network after the current layer, y are the target labels, and Y is penalty parameter that trades off task learning with confounder-free feature learning. After these modifications, P-MDN is able to work with arbitrary batch sizes. However, as we see in the results of this work, 7 becomes very difficult to tune, and optimizing the proxy objective often leads to non-robust results with high variance across different runs.
For continual learning, methods based on regularization (Kirkpatrick et al., 2017), knowledge distillation (Li & Hoiem, 2017), and architectural changes (Rusu et al., 2016; Mallya & Lazebnik, 2018; Bayasi et al., 2024) have been proposed to overcome catastrophic forgetting- thetttnnnghhhhaae phe- nomenon where DNNs forget information learned in prior training stages when acquiring new knowledge. Some of these methods are motivated by dealing with task (domain) specific biases by learning task (domain) general features (Arjovsky et al., 2019; Zhao et al., 2019; Creager et al., 2021). Such approaches, however, do not leverage meta- data to remove effects due to specific confounders from the learned features. While Nomain-adversariin training and P-MDN still apply to the continual learning setting, we show in this paper that they do not perform well in many scenarios.
Among the methods listed earlier, Metadata Normalization (MDN) (Lu et al., 2021), which uses statistical regression analysis, is popular technique. MDN is layer that can be inserted into the DNN to residualize confounder effects from intermediate learned features. It does so through the ordinary least squares algorithm, wherein it computes a closed form solution for the expression = = X33 + r as 3- (X x) Xx where z is the intermediate learned feature vector, X is the confounder matrix, ,3 are regression coefficients, and r is the component in the learned features invariant to the confounder. To work with minibatches of data, MDN re-formulates the closed-form solution as 3 = NE -E- -((z), where Σ-1 = (xT x) is pre-computed with respect to training data, with the amount being how much is nceded to estimate its distribution (ideally, the entire dataset) at the start of training. and the expectation Exx2)) is computed using batch-level estimates during training. In the context of continual learning where we do not have all data at the outset of training, or where an infinite replay buffer is not assumed, this pre-computation step prohibits effective transfer of performance to training stages with different data distributions. Even in static learning, employing batch- level statistics during training precludes MDN from being used with architectures such as vision transformers, where computation is parallelized over individual examples.
3. Methodology
Consider that we have N training samples, where the in- put matrix A E Nxd for some dimension d, is associ- ated with target labels y E IR and information about the confounding variable ã E R.N. Let the output after a par- ticular layer of deep network be the features E R.N. Our goal is to obtain the residual r from the expression
= = z3, + y/s + X + r, where X = [ y] and B = [$_; SyI is set of learnable parameters. In other words, the learned features z are first projected onto the subspace spanned by the confounding variable and the labels, with the term 33 corresponding to the component in z explained by the confounder, and y, to that explained by the labels. We want to remove the influence of z from z while pre- serving the variance related to the labels. We thus compute the composite / as theorized below, but obtain the residual = . i.e., only with respect to 8r. This residual explains the components in the intermediate features that are irrelevant to the confounder but relevant to the labels and, thus, for the classification task.
To alleviate issues around the use of batch statistics, penalty approach to MDN (P-MDN) was proposed ( (Vento et al., 2022). The authors of P-MDN observe that MDN solves a bi-level nested optimization problem by having the network learn task-relevant features while also being invari- ant the confounder, and instead suggest to solve a proxy objective ming, w [(p(2 - X (8), y) +>< X), where s is the non-linear computation to be performed within the
To accomplish this, we use the recursive least squares ap- proach. We start from the closed-form solution found for the ordinary least squares (OLS) estimator:
N Σ
B XX.XX bi
(1)
where X is the 7th row of X. If we represent R(N)
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Confounder- FFee Continual Learning via Recursive Feature Normalization
X and Q(N) E 2,, this is equivalent to writing 3 = QQ().
Engan, 2010). From Figure 2, for a range of e values such 10, 100, and 1000, the RLS estimates approach that of OLS quite quickly.
Now, say that we have a new sample AN+ come in. The confounding variable and intermediate features for this sam- ple are XN+1, and 2 respectively. This means that we need to compute new parameters:
3.2. Mini-batch learning
So far, we have theorized R-MDN within an online learning setting, with the system adapting to new information as it comes in one at a time. However, we can extrapolate this method to work with mini-batches of data. Consider that the system receives new mini-batches of information X E RR Bxd, for some batch size B. We can adapt the R-MDN approach by now using the Woodbury formula (Woodbury, 1950):
8 = R(N + 1) + 1)
(2)
(R(N) + X Xx+1) (Q(N) + #N41)
Fortunately, R(N + 1) can be efficiently computed us- ing the Aoterman-Morrisee rank-l update rule (Sherman & Morrison, 1950):
P xx**)
(R(N) + XN- xx11.)
(4)
R(N) X N+1. R(N)- (3)
= -P-1-P-1
- P- (=*x(++Tp-'x)) x
XTp-1
= -R(N)- 1+ X N+1. R(N) IXN+1:
3.3. From batch to layer statistics
With training, Q(N) changes as the features z; learned by the model change. In continual learning setting, we cannot recompute Q(N) and instead take it to be an estimate of the true value. Empirically, we see that this is reliable estimate. The entire derivation is presented in Suppl. A. Importantly, applying R-MDN does not incur significant computational and memory overhead (a detailed analysis is presented in Suppl. B).
Remember that one of the drawbacks of MDN is that it has compute and store batch-level statistics Σ with re- spect to the entire training data prior to training (refer to Section 2, 3rd paragraph). This requirement makes MDN unsuitable for modern architectures like vision transformers, wherein computations happen in parallel over all examples in a mini-batch. Incorporating an MDN module will in- herently require aggregation step for the computation of batch-level statistics, resulting in significant computa- tional overhead. R-MDN, on the other hand, operates on the level of individual examples in a minibatch. Thus, it works in a purely online regime and can be inserted in vision transformers to residualize intermediate learned features.
3.1. Estimates produced by R-MDN
Stoica & Åhgren (2002) theorize that RLS (i.e., the linear model constructed by R-MDN) provides an estimate that coincides with the estimate from OLS (used by MDN) as the amount of data N - co. However, a careful initialization of the inverse covariance matrix can speed up this convergence. We thus initialize R(0)-1 = el, where e > 0 is a small scalar, as most commonly used by prior works (Haykin, 2002; Stoica& & Åhgren, 2002; Liu al., 2009; Skretting &
3.4. Regularization
R-MDN can adapt quickly to changing data distributions over time due to its iterative nature. However, this iterative nature of the method might sometimes make it too sensitive to small changes in the data. Random fluctuations, or data noise, can lead to unstable updates to R-MDN parameters. Therefore, we add regularization term XI to P(N + B). A is a hyperparameter that is tuned during training (ablation in Suppl. H). This has the effect of smoothing out the updates and stabilizing the residualization process, resulting in some robustness to noise. Additionally, it helps to ensure numerical stability by preventing the computation of an inverse for matrix that might be singular ill-conditioned.
E
2.5
0.001
[|oLS RRLSII
0.01
0.1
0
10
100
0.0
1000
1000
Number of samples
20000
4. Experimental Results
Figure 2. Ea between estimates produced from the linear model constructed by R-MDN and MDN on synthetically gener- ated data when varying e. Estimates converge fairly quickly and for a range of hyporparameter initializations.
We train on a continuum of data by slightly modifying the setting described by Lopez-Paz & Ranzato (2017): A 4- tuple (a;, 3z, Us, 8;) for E [N], where a, E A is the in-
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Confounder-Frie Free Continual Learning via Recursive Feature Normalization
Table 1. Quantifying metrics for the synthetic dataset in continual learning, ACCd, BWTd (x 10-0)) 9), and FWTd mean and standard deviation for different methods and datasets. A total of 5 runs were performed with different model initialization seeds. We compare against prior works-BR-Net (Adeli el al., 2020a), MDN (Lu et al., 2021), and P-MDN (Vento et al., 2022). MDN, a CNN was trained separately on each training stage and then evaluated against all stages. Best results in bold, second-to-best, underlined.
Method
Dataset
Dataset
Dataset 3 Both distributions chunge
Confounder dist. changes
Main effects chunge
ACCd
BWTd
FWTd
ACCa
BWTd
FWTd
ACCd
BWTd
FWTd
CNN Baseline
0.18 +: 0.00
0.03 0.04
0.19 + 0.00
0.2 = 0.00
-0.05 = 0.13
0.21 = 0.00
0.28 + 0.00
-0.37 d: 0.02
0.31 0.00
BR-Net
0.04 + 0.03
-1.28 1.37
0,05 + 0.04
0.04 + 0.02
-1.43 = 1.81
0.04 + 0.03
0.07 = 0.04
-0.55 + 0.56
0.08 + 0.04
Stage-specific MDN
0,25 + 0.00
2622
0.09 + 0.02
0.13 0.00
12.2 = 0.57
0.02 = 0.01
0.13 = 0.00
11.7 = 1.48
0.02 4: 0.01
P-MDN
+ 0.01
-0.61 1.37
0.04
0.04 0.00
-0.76 2.46
0.01
0.05
-1.37 + 2.21
0.06 0.01
R-MDN
0.02 + 0.01
0.15 + 0.14
0.02 + 0.01
0.03 0.00
0.07 = 2.17
0.03 + 0.00
0.02 = 0.01
-0,04 + 0.78
0.02 0.00
Baseline
BR-Net
Slage-Specffic MDN
P-MDN
R-MDN
Accuracy
group 2 group 1
Multiplier ö for distribution of confounding variable
Stage tested on:
3
5
Theoretical maximum accuracy:
0.75 0.6875
0.625
0.5625
0.5
Figure 3. Effects of the presence of the confounder for task generalization. After training the different methods on data with confounders, we evaluate them on test data with varying intensities the confounder, from (completely present) to 0 (completely absent), R-MDN does not get influenced by the change in confounder distribution during inference as it maintains its prediction accuracy (i.c., the lines are straight). This shows that it does not "cheat" by looking at the confounder and instcad learns task-relevant features.
put, 3y E X is the confounder, E Y is the label, and 8: E S is the training stage descriptor, satisfies local iid, i... (a,, 91) iid P.. (A,XYY) Y). Our goal is learn a classifier g: A x S Y, that is able to predict the label y associated with an unseen 2-tuple (a, s), where (a, v) ~ P. at any point during or after training on the S stages, in way such that it does not use confounder information in x.
and Forward Transfer distance (FWTd) metrics. These met- rics are adapted from ACC, BWT, and FWT defined by Lopez-Paz & Ranzato (2017) for the setting where a model is expected to achieve certain theoretical accuracies on data from both past and future stages of training. Say we have total of S stages. Let Ra, denote the classification accuracy of the model stage sj after learning stage 8;. And let A, denote the theoretical maximum accuracy for stage s, Then,
4.1. A Continuum of Synthetic Datasets
We first use a synthetic dataset with varying distributions of the confounding variable and main effects as playground to test how models behave under controlled variations in indi- vidual variables. Specifically, we design different datasets, each with stages of training that arrive sequentially. Over time, the distributions of the confounder, the main effects, or both change in way that emphasizes biased learning of a classifier that uses confounder information for discrimina- tion. A complete description is presented in Suppl. E.
S Rg A,b,
1
AcCd
=
5
(5)
i=1
S-1
BWTd
Rs. AAl
-RRA, Ai), (6)
5
S
FWTd
=
| - A,/.
(7)
S-1
To quantify knowledge transfer, we define the Average Accu- racy distance (ACCd), Backward Transfer distance (BWTd),
The smaller the metrics, the better the model. We use a con-
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Confounder-Frre -Fee Continual Learning via Recursive Feature Normalization
some resistance to the distribution shift but fail when the confounder is entirely absent. In contrast, R-MDN main- tains consistent performance across all distributions.
Table Training a 2D vision transformer on synthetic continual data. The setup is the same as in Section 4.1. We use dataset where the distributions of both the confounder and main effects change. Results are averaged over 5 runs of random model initialization seeds.
In addition to comparing with prior methods such as MDN, BR-Net, and P-MDN that have been proposed to remove the influence of confounders from learned features, we also want to compare against recently proposed methods in the continual learning setting. Analytic continual learning meth- ods such as ACIL (Zhuang et al., 2022) and F-OAL (Zhuang et al., 2024) share some similarities to our method in terms of using recursive least squares, although they address an inherently different problem that regardless is interesting to compare against. These methods address catastrophic forgetting and improve task accuracy, but do not explicitly remove the influence of confounders from learned features. R-MDN is normalization layer, and can be integrated with various continual learning frameworks, as we show later in Section 4.2. In Table 3, we see that both ACIL and F-OAL have excellent BWTd, which means that they effectively mitigate catastrophic forgetting, as their papers propose. However, they result in significantly worse ACCd and FWTd, which means that they make use of confounder information to make predictions (exhibiting large deviations from the theoretical maximum accuracy). On the other hand, R-MDN has better BWTd, ACCd, and FWTd, meaning that it learns confounder-fres features for making predictions, thus approaching the theoretical accuracy.
Method
ACCd
BWTd
FWTd
VIT Baseline
0.110 = 0.027
-0.173 = 0.024
0.306 0.009
R-MDN
0.061 = 0.013
-0.111 = 0.019
0.188 + 0.015
volutional neural network backbone for each method (details in Suppl. F). A stage-specific MDN does not perform well on the average accuracy metric, as indicated by high ACCd (Table and Figure 12b). While other methods are good at backward transfer, R-MDN also improves forward trans- fer. This means that even with changing distributions of the confounding variable, R-MDN only "looks at" the main effects for classification, learning features that transfer well to later tasks while remaining invariant to the confounder itself. Other methods make use of confounders to various degrees, pulling their classification accuracy away from A. This is also reflected in R-MDN consistently achieving an accuracy close the theoretical maximum for the test sets of each stage, while also showing the lowest correlation with the confounding variable (Figure 12c). This outcome is also reflected when starting with a vision transformer backbone (Table 2), and when constructing slightly more complex synthetic setup where we vary two different axes of possibly confounding information: position and intensity of variables (Suppl. D).
4.2. HAM10000 Skin Lesion Classification
Next, we classify 2D dermatoscopic images of pigmented skin lesions into seven distinct diagnostic categories with the HAM10000 dataset (Tschandl et al., 2018). The dataset consists of 10015 images, which we divide into five training stages. In each stage, the age distribution-the confounding variable (see Suppl. E with younger populations represented in the earlier stages and older populations in the later stages. For each stage, we randomly allocate 80% of the images for training and the remaining 20% for testing.
We further quantify how different methods generalize to unseen images where the confounder is absent (Figure 3, Suppl. K). This issue arises in situations where, for instance, a model is trained on data from multiple hospitals and then tested on data from a previously-unseen one (Zech et al., 2018). In this scenario, the base model experiences a sharp drop in performance when the distribution of the confounder changes in the test data. Both BR-Net and P-MDN show
In this experiment, we utilize a vision transformer as the base architecture and as the encoder for BR-Net. For R- MDN, we explore three different variants: (A) inserting the R-MDN layer after the self-altention layer in every trans- former block, as well as after the pre-logits layer; (B) in- serting it at the end of every transformer block and after the pre-logits layer; and (C) inserting it only after the pre-logits layer. For P-MDN, we place the P-MDN layer right after the pre-logits layer. Additionally, we compare to three continual learning methods as baselines: elastic weight consolidation (EWC) (Kirkpatrick et al., 2017) as regularization method, learning without forgetting (LwF) (Li & Hoiem, 2017) for knowledge distillation, and PackNet (Mallya & Lazebnik, 2018), an architectural method that applies iterative pruning.
Table 3. Comparing R-MDN against recent analytic continual learning frameworks. We compare R-MDN against ACIL (Zhuang et al., 2022) and F-OAL (Zhuang et al., 2024) under CNN backbone. The sctup is the same as in Section 4. We use dataset 3 w-eee the distributions of both the confounder and main effects change. Results are averaged over 3 runs of random model initialization seeds.
Method
ACCd
BWTd
FWTd
ACIL
0.29 + 0.00
-0.00 + 0.00
0.30 + 0.00
F-OAL
0.28 1 0.00
0.01
1 0.00
0.28 1: 0.00
R-MDN
0.02 + 0.01
-0.00 + 0.01
0.02 + 0.00
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Confounder-Frie Free Continual Learning via Recursive Feature Normalization
Table HAMIOK skin lesion classification results for continual learning, Results shown as the mean and standard deviation over test sets of different stages of training for the model after being trained on the last training stage. Best and second- to-best results shown in bold underlined respectively. Our metries interest are deor*. BWT, and FWT. A higher accuracy in our case does not equate a better model as it might be "cheating" by looking at the confounder for prediction.
Method
Accuracy
Average dcor
BWT
FWT
VIT Baseline
0.7095 1 0.0626
0.0864 i 0.0336
0.0278 + 0.0446
0,5125 1 0.0705
BR-Net (Adeli et al., 2020a)
0.7247 = 0.0627
0.0544 1 0,0534
-0.0277 + 0.0166
0.5592 + 0.0897
P-MDN (Vento et al., 2022)
0.6750 + 0.0945
0.2595 0.0620
0.0706 0.0622
0.4391 = 0.0372
R-MDN (A)
0,5503 + 0.0541
0.0928 1 0.0630
-0.0268 + 0.0248
0.4130 + 0.0709
R-MDN (B)
0,5288 = 0.0571
0.0739 + 0.0555
0.0571 + 0.0693
0.3362 + 0.0881
R-MDN (C)
0.6919 + 0.0723
0.0475 0.0247
0.1246 0.2123
0.3997 + O. 1.555
EWC (Kirkpatrick et al., 2017)
0.6437 = 0.0586
0.0938 + 0,0506
0.0698 +: 0.0238
0,4457 = 0,0620
EWC R-MDN (C)
0.6739 = 0.0686
0.0592 0.0488
0.0754 + 0.1614
0.4404 = 0.1305
LwF (Li & Hoiem, 2017)
0.7356 + 0.0757
0.0512 + 0.0407
0.0387 + 0.0390
0,5277 + 0.0605
LwF R-MDN (C)
0.7186 0.0736
0.0354 0.0210
0.1348 + 0.1994
0.4434 + O. 0.1403
PackNet (Mallya & Lazebnik, 2018)
0.6849 + 0.0745
0.0470 + 0.0304
0.0538 + 0.0670
0,4965 + 0.0611
a
Colored by distribution of the confounder
b
Colored by distribution of the labels
Without R-M R-MIN
With R-MDN
40
20
Stage
Stage 2
Stage 3
Stage 4
Stage 5
Stage
Stage 2
Stage
Stage
Stage
Figure 4. Visualizing learned features for HAMIOK skin lesion classification. t-SNE representation of the learned features after training a model with and without R-MDN on all stages of continual leamning Without R-MDN, the model drives fine-grained separation of feature clusters based category labels, especially for stage 5, which roughly overlap with confounder distributions for that training stage. By looking at the confounder for fine-grained discrimination, the bascline fails to generalize to previous data. R-MDN, on the other hand, does seem to be using confounder information drive fine-grained fcature separation, leading to better backward transfer.
Results are summarized in Table 4 and visualized in Fig- ure 14. While the base model performs decently on the classification task, it exhibits significantly lower backward transfer in earlier stages of training. In contrast, R-MDN not only effectively removes the confounder's influence from the features, as indicated by a low dcor? value, but also demonstrates significantly better backward transfer than the base model. We explore this effect by observing the 1-SNE visualizations of their features (Figure 4). When the base model is trained in the final stage, learns to clearly separate feature clusters from each of the seven diagnostic categories, especially after being trained stage 5 (Figure 4b). However, it is possible that this separation is influenced by the stage-specific distribution of the confounding vari- able (Figure 4a), leading to spurious correlations driving cluster separation and, thus, poor transfer to previous data. On the other hand, R-MDN- relies on task-relevant information also forms feature clusters for the different
categories but without introducing the same level of sepa- ration possible by looking at the confounder. R-MDN is able apply the knowledge learned in the current stage previous stages of training, improving its overall backward transfer performance.
Of the three R-MDN variants, R-MDN (C) performs the best. Moreover, applying R-MDN to classic continual learning frameworks such as EWC and LwF still drives the corre- lation with the confounder significantly down. In contrast, other methods, such as BR-Net and P-MDN, do not perform as well. BR-Net catastrophically forgets past information, and P-MDN fails to effectively remove confounder effects.
4.3. ADNI Diagnostic Classification
Finally, move from cross-sectional dataset to a lon- gitudinal study. The Alzheimer's Disease Neuroimaging Initiative is a multi-center observational study that collects
7

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Confounder-Frid Free Continual Learning via Recursive Feature Normalization
4.4. Static Learning
Table ADNI diagnostic classification for continual learning. Average squared distance correlation after training on the last training stage for different groups across different methods (Baseline ViT, P-MDN (Vento et al., 2022), and R-MDN). Note that we do not train BR-Net as adversarial training does not scale well to more than one confounder (Zhao et al., 2020).
R-MDN is not only helpful in continual settings, but also effective regularizer for static training. To demonstrate this, explore setting where the system only receives data from single stationary distribution. Experiments were conducted on a proof-of-prineipl synthetic dataset (Suppl. C) and a neuroimaging dataset.
Method
dcor? (MCI)
dcor? (CN)
dcor? (AD)
ABCD Sex Classification: We used Tlw MRIs from the ABCD (Adolescent Brain Cognitive Development) study (Casey al., 2018) for the task of binary sex classification. Within a cross-sectional study setting, we take 10686 base- line (i.e., first visit) MRIs, confounded by scores from the Pubertal Development Scale (PDS)-a validated measure of pubertal stage identified through self-assessment, PDS is a confounder because is significantly (Suppl. E) larger in girls (2.182 = 0.9) than in boys (1.372 0.6) (Adeli et al., 2020b). PDS categorizes participants as either (1) pre-pubertal, (2) early-pubertal, (3) mid-pubertal, (4) late- pubertal, or (5) post- post-pubertal (Carskadon & Acebo, 1993).
ViT Baseline
0.20 + 0.09
0.20 + 0.14
0.25 + 0.22
P-MDN
0.06 = 0.05
0.08 + 0.05
0.24 + 0.16
R-MDN
0.08 + 0.06
0.05 = 0.04
0.09 = 0.07
Acquisition Year: c=2010
Aug. Accurscy
2011
2012
2013
>-2014
<-2010
>-2014
Acquisition Year dcar' (AD)
b.
door' (MC1)
dcor? (CN)
Haseline P-MDN
150
125
R.DDN
c-2010
2-1014
We start with a 3D CNN as the base architecture consisting of three stacks of convolutional layers, each followed by ReL.U non-linearity and max pooling, and ending with two fully connected layers. We insert a residualization module after every layer except the last one. In addition to this approach, we establish two additional baselines- one where we use BR-Net, and adversarial training framework, with the same base model as the encoder, and another where we pre-process the input images prior to training by regressing out the influence of confounders directly from the pixel space (hereafter referred to as Pixel-Space MDN). We set the batch size to 128, which is the largest that can be fit in GPU memory (see additional details in Suppl. F).
Acquisition Year
Figure 5. Visualizations for ADNI diagnostic classification. Average accuracy and b. squared distance correlation for each participant group for different stages of training. The baseline and P-MDN have a poor forward transfer for the first acquisition year, as they use confounder information to overfit to that training stage. R-MDN has a significantly better performance on future training stages as it does not look at the confounder, indicated by lower dcor? for all diagnostic groups.
neuroimaging data from participants that fall into different diagnostic groups over several years (Mueller et al., 2005; Petersen et al., 2010). We pose a diagnostic classification task aask-Alzheimer's s Disease (AD), Mild Cognitive Impair- ment (MCI), and Control (CN) from TIw MRIs under changing distributions of age and sex, the two confounders for the experiment (additional details provided in Suppl. E). We train a 3D vision transformer as the base architecture.
We observe that the base model has high accuracy, but at the cost of being significantly biased towards girls-its feature representations have a larger dcor? with the confounder for girls than boys, and a higher true negative rate (Table 6). This is because the base model makes use of the pubertal development to drive its predictions. On the other hand, R-MDN incurs modest decrease in performance but sig- nificantly drives down the correlation between the learned features and the confounder for both boys and girls. More- over, it has the lowest mean difference between the true positive and true negative rates among all evaluated meth- ods (quantified in Table 6, visualized in Figure 1,c, signifying that it is not biased toward children of either sex. Other methods such as MDN and P-MDN have higher pre- diction accuracy but either fail to drive down the correlation between the features and the confounder due to requiring relatively larger batch sizes or remain biased towards girls despite driving the correlation down, As the authors of the MDN paper hypothesize, batch-level statistics could be ag- gregated over several batches to virtually increase the batch
Similar to previous experiments, applying R-MDN results in lower squared distance correlation, implying that it does not take into account confounders when making predictions (Table 5, Figure 5b). P-MDN does the same only to some extent, as it is not able to drive correlation down for all three participant groups. This leads to the baseline and P-MDN models "cheating" by using discriminatory cues from the confounders to better classify inputs and demonstrating slightly higher average accuracy (Figure 5a). This occurs at the expense of not being able to generalize to future stages of training, as, for example, seen by a roughly monoton- ically decreasing accuracy curve for the acquisition year <2010 (Figure 5a). R-MDN learns task-relevant cues, char- acterizing model performance in the absence of "cheating"
8

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Confounder-Free Free Continual Learning via Recursive Feature Normalization
Table ABCD sex classification results for static learning, Accuracy, true positive (TPR) and negative rates (TNR), difference between TPR and TNR, and squared distance correlation for both boys and girls for different methods. Results are shown as the mean and standard deviation over 5 folds 5-fold cross-vale vols-vaaii,tinn, with data split by subject and site ID. Our metrics interest are (TPR- TNR) and dcor". A higher accuracy in our case does not equate a better model as it might be "cheating" by looking at the confounder for prediction.
Method
Accuracy
TPR
TNR
TPR TNR
dcor (boys)
dcor (girls)
CNN Baseline
86.86 = 0.354
85.41 + 0.781
88.32 + 0.770
-0.029 + 0.016
0.0127 + 0.0022
0.0218 + 0.0029
Pixel-Space
84.91 = 0.447
83.04 + 2.900
86,77 + 2.352
-0.037 + 0.059
0.0168 + 0.0041
0.0239 + 0.0083
BR-Net (Adeli et al.. 2020a)
81.63 = 0.499
80.26 0.388
83.01 0.908
-0.027 0.011
0.0127 1 0.0006
0.0148 0.0002
MDN (Lu et al., 2021)
87.55 + 0.6630
87.43 +: 3,301
87.66 4.277
-0.002 0.084
0.0329 + 0.0140
0.0624 1 0.0283
P-MDN (Vento et al., 2022)
86.41 = 0.876
84.25 4: 1.651
88.57 + 1.540
-0.043 +: 0.030
0.0031 + 0.0009
0.0108 +: 0.0017
R-MDN
85.08 + 0.591
84.98 0.842
85.18 1.125
-0.002 + 0.018
0.0099 1 0.0029
0.0090 0.0027
a
Light Lateral Ventride
b
Light Lateral Ventricle Right Lsteral Ventricle
Planum Temporale
Supramarginal Gyrus, anterior div.
Middle Frontal Gyrus
CerebrA
Temporal Fusiform Cortex, anterior div.
Precuneous Cortex
Right Putamen
Left Hippocampus
Planum Temporale
Left Accumbens
Left Accumbens
Central Opercular Cortex
Brain-Stem
Occipital Fusiform Gyrus
Occipital Fusiform Gyrus
Left Pallidum
Left Thalamus
Figure 6. Visualizing ROIS for ABCD sex classification. The top 10 most relevant regions for distinguishing sex as determined by model trained a. without and b. with R-MDN, respectively. One of the ROIs identified by the baseline is the cerchellum, which is implicated in driving sex differences in preadolescence and is the region mostly affected by the confounder for this study. R-MDN does not use the cerebellum for classification.
size (Lu et al., 2021), and thus the performance of MDN, but this is research question orthogonal to our paper.
angles, or speaker accents might introduce spurious correla- tions in the data and bias the learning algorithm.
Further validation of R-MDN in learning confounder-fi feature representations is revealed when the model does not use the cerebellum-we which is the region mostly confounded by PDS (Adeli et al., 2020b)-for distinguishing sex, while the base model does (Figure ba, These regions are in line with findings in the neuroscience literature about sex differences in preadolescence (Chung et al., 2005; Tiemeier et al., 2010; Fan et al., 2010).
Acknowledgements
This research was supported in part by the NIH grants RO1 AGO89169,, AG184444,,, R01DA057567, and 75N95023C00013. This study was also supported by the Stanford Institute for Human-Centered AI (HAI) Hoffman- Yee award.
Impact Statement
5. Conclusion
Computer vision models taken out-of-the-box often gen- erate task predictions that are influenced by the presence of confounding variables in the dataset. These variables, which for instance take the form of patient demographic information, lead to biased predictions. In this paper, we control for confounder influences by proposing Recursive Metadata Normalization (R-MDN), a layer that can be easily integrated in existing models. Upon evaluation, the models trained with R-MDN learn to make equitable predictions across population groups in both continual and static learning settings. We helieve that this will encourage the community at large to look towards these models that learn confounder free feature representations when thinking about deploying systems for society-facing applications.
In this work, we presented Recursive Metadata Normal- ization (R-MDNJ-a flexible layer that can be inserted at any stage within deep neural networks to remove the influ- ence confounding variables from feature representations. R-MDN leverages the recursive least squares algorithm to operate the level of individual examples, enabling it to adapt changing data and confounder distributions in con- tinual learning. It also promotes equitable outcomes across population groups and mitigates catastrophic forgetting of confounder effects over time. As a direction for future work, R-MDN should be explored beyond medical contexts, such as video streams and audio signals, where confounding vari- ables like environmental noise, lighting conditions, camera
find that

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Confounder- Free Continual Learning via Recursive Feature Normalization
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Confounder-Frn Free Continual Learning via Recursive Feature Normalization
A. Deriving Parameter Updates for R-MDN
To derive parameter updates for R-MDN, which builds regression model based on the recursive least squares (RLS) method, begin from the closed-form solution of ordinary least squares (OLS)-
X
Σ
21 R(N)-IQ(N)
(8)
Firstly,
Q(N + 1)= Q(N) + 2N+1XN+1.
(9)
Additionally, using the Sherman-Morrison rank-1 update rule,
R(N 1 (R(N)+ X XN4.)) RRNN 1+ 14NN R(N)
(10)
Let
P(N + 1) R(N = K(N+ 1IX
(11)
where the Kalman Gain
P(N)XN+1:
K(N 1 R(N) -(N))-NN+1111
(12)
Rewriting eq. 12,
K(N + X P(N)XN+1. +N
-(NN)XN4I
(13)
K(N+ +1 = [[P(N) K(N + +1, XN+1.
K(N+1) = PPN + 1)Xx+1. [using eq. 11]
Finally,
8(N + 1) = P(N + 1)Q(N+ +1)
P(N + 1)Q(N) P(N + 1)2N+1 N+1, [using eq. 9]
= [P(N) - K(N P(NN Q(N) + P(N + 1)2N+1XN41: [using eq. 11]
[P(NN) K(N +1) Q(N) + K(N + 1)=N+1 [using eq. 12]
(14)
-P(N)Q(N) + K(N +1) [2N+1 N P(N)Q(()) [using eq. 12]
= B(N) + K(N 1) [aN+1 - (x+1,6(NN) [using eq. 8]
B(N) K(N 1)e(N +
where e(N +1) = EN+1 - x the a priori error computed before we update residual model parameters 3.
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Confounder-Fine Free Continual Learning via Recursive Feature Normalization
B. Computational and Memory Complexity
MDN, P-MDN, and R-MDN each have their tradeoffs in terms computational complexity, memory complexity, and the extent to which the influence of the confounder is removed from the learned feature representations. As demonstrated in this work, R-MDN empirically works better than both MDN and P-MDN. The asymptotic complexity of each is presented here.
Say there are N training examples, broken into batches of size B. Let the confounder matrix X have shape N p. where p is associated with the number of confounders, the target, and bias of 1, and the intermediate learned feature representations have a size of N x h.
Firstly, MDN internally uses the linear least squares estimator, which requires pre-computing the matrix Σ X in O(N)²) steps. Inverting this p x p matrix further requires O(p²) steps. Then, for every batch of information during training, batch level estimate X is produced, where the (+) operation refers batch instead of the entire training data. This takes O(Bph) steps. Post-multiplying this p X h matrix with E-1 requires (pp²h) steps. If computations over batches of information occur E times, the total computational complexity becomes Opp² + Np² + Elp²h
h + Bph)). In terms of memory complexity, p x E-1 needs to be stored, along with the residual model parameters / of size x h.
For R-MDN, computations only occur over batches of information. In memory, residual model parameters of size p x and an estimate of the inverse covariance matrix P of size p x p are required. For every processing iteration, computing the Kalman gain K requires O(Bp²) steps for PX O(B2p + Bp²) steps for XPX O(B)) for inverting this latter matrix, and O BB2p) steps for multiplying the matrices together. Updating P using KX requires ((B²p) steps. And finally, updating 3 requires computing Ke in ((Bhh) steps. The total computational complexity turns out to be O(E(B3 + B2p + Bp² + Bph)) steps. Empirically. R-MDN works best with small batch sizes B, showing very fast convergence rates, and having a computational complexity that is independent of the size of the training dataset. This becomes important for continual learning, especially longitudinal studies, where data collected over several years or decades can prohibit the use of MDN.
P-MDN does not use closed-form solution to linear statistical regression. Instead it uses gradient descent to optimize proxy objective. Thus, the only memory complexity stems from storing 3 parameters of size p x h. The computational complexity is dominated by the number of iterations required to navigate the proxy loss landscape, with results that are not often robust with high variance across runs.
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Confounder-Fre FFee Continual Learning via Recursive Feature Normalization
C. Synthetic Dataset for Static Learning
mf: 1.9, cf: 3.0
mf. 3.8, cf: 3.3
Group 1
Group 2
BB Metadata
Figure 7. A sample from the synthetic dataset used for static learning.
We construct this dataset, following Adeli et al. (2020a) and Lu et al. (2021), by generating 2048 images of size 32x32, equally divided between two groups (categories). Each image consists of Gaussian kernels: two on the main diagonal, i.c., quadrants and 4, whose magnitudes are controlled by parameter A,, and one on the off-diagonal, i.e., quadrant 3, whose magnitude is controlled by оg (see Figure 7). Differences in the distributions of σ between the two groups are associated with the main effects (true discrimination eues) that should be learned by the system, whereas d is confounding variable. An unbiased system will only use information from the main effects for categorization. Both A and σB are sampled from the distribution (4(1, 4) for group 1, and ((33 6) for group 2. Since there is an overlap in the sampling range of the main effects between the two groups, the theoretical maximum accuracy that the system can achieve, were it to not depend on the confounding variable to make discrimination choices, would be 1 (1) Plo E U(3 4)] = 1 - (1) (t) = 0.833.
We use 2D convolutional neural network comprising of 2 stacks of convolutions and ReLU non non-linea,ity. followed by 2 fully-connected layers. We apply residualization modules (either MDN, P-MDN, or R-MDN) after every convolution and pre-logits layers (other placement choices explored in Suppl. 1). During and after training, we quantify the high-dimensional non-linear correlation between the learned features from the pre-logits layer of the system and the confounding variable through the squared distance correlation (dcor?) metric (Székely et al., 2007). A dcor? = 0 implies statistical independence between the two distributions.
Our results are summarized in Table 7, We observe that R-MDN consistently reaches the theoretically optimal accuracy and
Table 7. Synthetic dataset results for static learning, Absolute deviation from the theoretical accuracy |bAcc- - A/ (L) and squared distance correlation (L) for various methods and batch sizes. Results are shown over 100 runs of random model initialization seeds with a 95% confidence interval. Best results for each batch size are in bold. There is significant difference in all metrics across all hatch sizes for different methods (one-way ANOVA p < 10 )8)) Our method has a significantly better squared distance correlation than MDN for batch sizes less than 128 (post-hoc Tukey's HSD p 0.05) and than P-MDN for batch sizes 2, 64, 256, and 1024 (p < 0.05).
Method
Metric
Batch size
64
256
1024
Baseline
|bAcc- - A|
10.49 1 0.037
10.49 1 0.025
10.50 1 0.023
10.52 1 0.022
10.55 1 0.029
dcor² (group 1)
0.408 + 0.002
0.420 : 0.001
0.421 1: 0.001
0.416 + 0.001
0.310 + 0.005
dcor? (group 2)
0.388 + 0.003
0.397 + 0.001
0.394 + 0.001
0.391 + 0.001
0.281 + 0.005
MDN
|bAcc - A|
8.13 + 1.203
4.93 = 0.424
3.21 + 0.532
0.52 : 0.335
0,95 + 0.335
dcor² (group I)
0.977 + 0.010
0.142 0.016
0.086 0.010
0.014 0.002
0.003 + 0.000
dcor² (group 2)
0.999 = 0.001
0.046 0.009
0.024 + 0.003
0.000 + 0.001
0.000 + 0.000
P-MDN
|bAcc A]
4.65 1. 0.448
3.49 1 0.373
1.85 + 1.151
0.23 1 1.361
1.58 1 1.983
d (group 1)
0.042 + 0.013
0.022 + 0.003
0.050 + 0.007
0.048 + 0.007
0.098 + 0.020
dcor2 (group 2)
0.060 + 0.021
0.013 0.005
0.015 0.002
0.027 + 0.004
0.091 + 0.026
R-MDN
|bAcc A
0.28 0.414
0.04 1 0.213
0.13 1 0.088
1.19 1 0.215
0.19 1 0.296
deor? (group I)
0.019 + 0.003
0.014 + 0.002
0.006 + 0.001
0.013 + 0.002
0.015 + 0.020
dcor² (group 2)
0.005 + 0.001
0.001 = 0.000
0.000 + 0.000
0.001 + 0.000
0.008 + 0.017
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Confounder- Free Continual Learning via Recursive Feature Normalization
lower dcor² across all batch sizes. The baseline "cheats" by making of information from the confounding variable, resulting in higher balanced accuracy and dcor?. MDN is successful only for large batch sizes like 1024, while being significantly worse for small ones (Lu et al., 2021). While P-MDN is also able remove the effects of the confounding variable from the learned features to large extent across all batch sizes (as shown through smaller dcor?), the large variance across different seed runs suggests that the results are not consistent or robust.
I-SNE visualization (Van der Maaten Hinton, 2008) of the learned feature representations shows that the distribution overlap 4) for R-MDN is not separable from the two groups for all batch sizes, which means that the system does not use information from the confounding variable for categorization (Figure 8a). In terms of convergence speed, R-MDN does significantly better than both MDN and P-MDN in removing the effects of the confounding variable from the learned features very quickly, especially for small batch sizes (Figure 8b). This effect is attributable to fast convergence properties of the underlying RLS algorithm (Hayes, 1996: Haykin, 2002), and will be advantageous in continual learning setting when we might not want to train a system until convergence on each training stage, but only for a single or few epochs (read Suppl. L).
Baseline
MDN
PAMDN
R-MDN (Ours)
Baseline
MDN
P-MON
R-MDN (Ours)
a
MN
2
ooof (group 1]
Betch
1024
dcor' (group
@
1024
Group
Group
Distribution overlap
Figure& 8. Learned features and squared distance correlation. t-SNE visualization of feature representations for different methods across batch sizes 2, 16, and 1024. The more separable the distribution overlap ((33, 4) is in the feature space, the more the method relied the confounder for discriminating between the groups b. Squared distance correlation across batch sizes for different methods. Each curve represents a different batch size (ranging from 2 to 1024, increments of powers of 2). Results are shown as the average over 100 runs of random model initialization seeds.
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Confounder-Frne Continual Learning via Recursive Feature Normalization
D. A Second Synthetic Continual Learning Dataset
pos: 0.879, int: 4.441
pos: 0.637, int: 3.052
pos: 0.395, 4.416
pos: 0.152, 4.094
pos: 0.879 int: 4.858
pos: 0.637, int: 4.648
pos: U.395, int: 5.917
pos: 0.152, int: 4./18
Figure Synthetic continual dataset #2: changing intensities and positions the confounding variable. Here we construct second synthetic dataset for validating our method. We show the change in the distribution of position and intensity of the confounding variable over 4 training stages (moving from left right on the plot). Similar to Section 4.1, main effects are present on the main diagonal.
While we explore a synthetically generated enviromment in Section 4.1 to theoretically quantify the influence of confounding variables and the deviation of the model from achieving an un-confounded performance, here we construct a slightly more complex setup wherein we vary two different axes of possibly confounding information: position and intensity. More specifically, we generate 1024 32x32 images that are implicitly broken down into 8x8 grids. The top left and bottom right grids contain Gaussian kernels of intensity оA denoting the main effects. The confounder represented by Gaussian kernel of intensity dB, whose position varies from the bottom left to top right of the image over 4 different training stages. Both A and ов are sampled from the distribution ((3,5) 5) for group 1, and (44, 6) for group 2. A fair classifier should remain unaffected by confounder information, irrespective of its position within the image. Such setup also allows us to compute theoretical maximum accuracy achievable for each training stage that we can validate methods against. Table presents the results of the experiment, where we can see that R-MDN again allows the model to learn confounder- features.
Table Results on synthetic continual dataset #2. The theoretical maximum accuracy for every stage of training is 0.75, which is what unbiased classifier should achieve. The off-diagonal kernel intensity taken the confounder. Results averaged over 3 runs of random model initialization seeds.
Method
ACCd
BWTd
FWTd
CNN Baseline
0.124
0.002
-0.001 + 0.001
0.250 + 0.000
R-MDN
0.046 0.031
0.0006 + 0.0009
0.238 + 0.014
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Confounder-Frid Free Continual Learning via Recursive Feature Normalization
E. Additional Details on Datasets
E.1. A Continuum of Synthetic Datasets
This dataset an extension of the dataset introduced Suppl. C. For stage 1 across the 3 datasets, we start with the parameter controlling the main effects a E ((33.) 5) for group and E (4, 6) for group 2. We use the same distributions for o (which controls the magnitude of the confounding variable). This means that prediction kernels (i.e., those associated with true discrimination cues) more "intense" (have higher magnitude) for images in group 2 than in group 1.
Dataset 1: Confounding variable distribution changes. We keep the distribution of main effects constant but vary that of the confounding variable across different stages. With every new stage, we decrease the entire range of og by 0.125 for group 1, and increase by the same amount for group 2. For an unbiased classifier that uses short-cut learning by focusing on the confounder distribution, the problem becomes easier with "time" and performance will likely increase. This what we observe with the baseline model, which has the same architecture as that in Section C (see Figure 12c).
Dataset 2: Distribution of main effects changes. Next, keep the distribution of the confounding variable constant and vary that of the main effects across different stages instead, In contrast to the above, with every new stage, we increase the entire range of A by 0.125 for group 1, and decrease it by the same amount for group 2. This results in the problem becoming more difficult with "time". Performance for an unbiased classifier should drop for later tasks during learning.
Dataset 3: Distributions of both the confounding variable and main effects change. This dataset a combination of the above two, with the difference in the distribution of the confounding variable across the two groups becoming more pronounced with "time", while that of the main effects starting to become more similar.
E.2. HAM10000 Dataset
10a
10
10a
109
mel
bkl
boc
akiec
Diagnostic Categary
Age groups:
0-20
20-40
40-50
50-50
60+
Figure 10. Distribution of dermatoscopic images across different diagnostic categories for various age brackets in the HAM10000 dataset.
The Human Against Machine with 10000 training images (HAM10000) dataset (Tschandl et al., 2018) is multi-source collection of 10015 dermatoscopic images for diagnosis of common pigmented skin lesions. These images have been collected from different populations through different modalities. Diagnostic categories include:
akiec: actinic keratoses and intraepithelial carcinoma or Bowen's disease
bee: basal cell carcinoma
bkl: benign keratosis-like lesions (solar lentigines or seborrheic keratoses and lichen-planus like keratoses)
df: dermatofibroma
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Confounder-Free Free Continual Learning via Recursive Feature Normalization
mel: melanoma
nv: melanocytic nevi
vasc:
vascular lesions (angiomas, angiokeratomas, pyogenic granulomas and hemorrhage)
Lesions were confirmed either through histopathology, follow-up examinations, expert consensus, in- iivivo confocal microscopy.
Figure 10 shows the distribution of age for various diagnostic categories. The change in the age distribution is significant. Age a confounder for this dataset because affects both the target categories (certain categories like melanoma mostly occur older patients) and the input images (skin appearance might change with age).
E.3. ADNI Dataset
The Alzheimer's s Disease Neuroimaging Initiative (ADNI) [adni.loni. loni. usc. edu) was launched 2003 with the goal to test whether serial MRI, PET, other biological markers, and clinical and neurophysiological assessments can be used to measure the progression of mild cognitive impairment (MCT) and early Alzheimer's disease (AD) (Mueller al., 2005; Petersen al., 2010). There a total of 3880 Tlw MRIs from 573 participants across years 2006 through 2017.
The distribution of age and sex across subjects diagnosed with MCI AD, and the control group (CN) is shown in Figure 11 and Table 9.
95
8
Group
MC
AD
75
55
<=2010
2011
2012
2013
>=2014
Acquisition Year
Figure 11. Distribution of age different participant groups in the ADNT dataset.
Table Sex distribution across different groups in the ADNI study. Differences are statistically significant, as measured via x 2tstt.
Group
Sex
Year
2010
2012
2013
2014
AD
Male
2
52
214
125
20
Femule
4
26
110
95
16
Male
71
238
410
208
206
Female
81
222
402
221
197
MCI
Male
117
145
140
81
98
Female
82
84
106
55
52
E.4. ABCD Study
The Adolescent Brain Cognitive Development (ABCD) study study. More than 10,000 boys and girls from the U.S. between the ages of 9-10 were recruited that were diverse in
org) is multisite, longitudinal
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Confounder- Free Continual Learning via Recursive Feature Normalization
terms of race/ethnicity, education and income levels, and living environments (Thompson al., 2019). Garavan al. (2018) provide detailed account of the population neuroscience approach recruitment and inclusion/excllsionn criteria. Appropriate consent was requested before participation in the ABCD study. Data is anonymized and curated, and is released annually to the research community through the NIMH Data Archive (see data sharing information at https:/ / abcdstudy orgg / sci ent.
shar i sharing/), The ABCD data repository grows and changes over time. The ABCD data used in this report came from release 5.0, with DOI 10. 15154 54/854/883333 /8873-877--- 2j65.
Table 10 shows the distribution participants (boys and girls) in the study with respect to age, pubertal development score (PDS), and race. PDS significantly larger for girls than boys, and thus serves as a confounder for this study.
Table 10. Variable distributions across boys and girls in the ABCD study. Mean and standard deviation for age and pubertal development scale (PDS), and the number subjects of each race in the study across boys and girls. PDS is an integer between 1-5. Differences significant across boys and girls for PDS (measured using two-sample t-test) but not age race. Girls have higher PDS than boys in the study. All values are for the first visit of each subject.
Boys
Girls
p-value
Age (in months)
119.18 + 7.562
118.80 + 7.525
>0.001
PDS
1.372 + 0.622
2.182 + 0.907
<0.001
White
2953
2659
Black
702
735
Race/ethnicity
Hispanic
1060
999
>0.01
Asian
103
114
Other
567
529
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Confounder-Fre Free Continual Learning via Recursive Feature Normalization
F. Additional Details on Methods
All experiments were run on a single NVIDIA GeForce RTX 2080 Ti with 1 IIGB memory size and workers on an internal cluster.
F.I Synthetic Dataset for Static Learning
The base model was a CNN consisting of two convolutional layers followed by two fully connected layers. The first convolutional layer had 16 output channels and kemel size of 5, the second had 32 output channels and a kernel size of 5, and the pre-logits fully connected layer had hidden size of 84. Models were trained for 100 epochs with different batch sizes. Parameters of the R-MDN model were optimized using Adam (Kingma & Ba, 2014), with learning rate initialization of 0.0001 that decayed by 0.8 times every 20 epochs. The regularization parameter for R-MDN was set to 0.0001.
E.2 ABCD Sex Classification
Raw MRI images were downloaded, skull-stripped, and affinely registered to the MNI 152 template (Mazziotta et al., 1995; 2001a;b). Data augmentation involved removing MRIs for all subjects that did not have associated PDS score recorded. We downscaled all MRIs to 64x64x66 64 volumes, performed random one voxel shift and one degree rotation in all three Cartesian directions, and random left-right flip (since sex affects the brain bilaterally (Hill et al., 2014; Hirnstein et al., 2019)) for training images. To evaluate the models, we perform 5 runs of 5-fold cross validation across different model initialization seeds, with images split by subject and site ID, and having approximately an equal number boys and girls in each fold.
The base model layers. The first convolutional layer had 8 output channels with kernel size of 3, the second had 16 output channels with a kernel size of 3, and the third had 32 output channels with kernel size of 3. The pre-logits fully connected layer had a hidden size of 32. For max pooling, the first and second layers used a kernel size of with a stride of 2, while the third layer had a kernel size of 4 with a stride of 4. Models were trained for 50 epochs with a batch size of 128. Parameters of the R-MDN model were optimized using Adam, with a learning rate initialization of 0.0005 that decayed by 0.7 times every epochs. The regularization parameter for R-MDN was set to O.
CNN consisting of three convolutional layers, each followed by max pooling, and two fully connected
F.3. A Continuum of Synthetic Datasets
Model architecture used the same as that for the static learning setting. Models were trained for 100 epochs with a batch size of 128. Parameters of the R-MDN model were optimized using Adam, with learning rate initialization of 0.0005 that decayed by 0.8 times every 20 epochs. The regularization parameter for R-MDN was set to 0.0001.
F.4 HAM10000 Skin Lesion Classification
The dataset was first downsampled to 64x64x64 and then divided into five training stages based on age groups: 20, [20, 40), [40, 50), [50, 60), and 2 60.
Stage 1: 50% of the images came from 20, 30% from [20, 40), 10% from 50). 5% from [50, 60). and 5% from
2 60.
Stage 2: 5% from < 20, 50% from [20, 40). 30% from [40, 50), 10% from [50, 60), and 5% from 2 60.
Stage 3: 5% from < 20, 5% from
[20, 40). 50% from
[40, 50), 30% from
[50, 60), and 10% from 2 60.
Stage 4: 10% from < 20, 5% from
[20, 40), 5% from
[40, 50). 50% from
[50, 60), and 30% from 2 60.
Stage 5: 30% from < 20, 10% from [20, 40). 5% from [40, 50). 5% from [50, 60). and 50% from 2 60.
The base model was a ViT with a patch size of 8, 12 hidden layers, 12 heads, a hidden dimension of 384, an MLP dimension of 1536, and the hidden size for the pre-logits layer as 96. Models were trained for 30 epochs with a batch size of 128. Parameters of the R-MDN model were optimized using AdamW (Loshchilov & Hutter, 2017), with a learning rate initialization of 0.0005 that decayed by 0.8 times every epochs. We imposed a weight decay of 0.001. The regularization parameter for R-MDN was set to 0. 0.00001.
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Confounder-Free Free Continual Learning via Recursive Feature Normalization
F.5. ADNI Diagnostic Classification
The dataset was first downsampled x4 and then divided into five training stages based on acquisition year bins: 22010, 2011, 2012, 2013, and 22014.
The base model was 3d VIT with patch size of 16, 6 hidden layers, 8 heads, hidden dimension of 384, an MLP dimension of 1536, and the hidden size for the pre-logits layer as 768. Models were trained for 50 epochs with a batch size of 128. Parameters of the R-MDN model were optimized using AdamW, with a learning rate initialization of 1e-05 that decayed according to a cosine annealing scheduler. We imposed a weight decay of lc-06. The regularization parameter for R-MDN was
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Confounder-Frie Free Continual Learning via Recursive Feature Normalization
G. Visualizing Results for Continual and Static Learning
Here we visualize the results we present through tables in Sections 4.1, 4.2, and 4.4.
Confounding variable distribution changes
Main affacts distribution changos
Both distributions change
b. 03-
Confounder dist. charges
group 24 43
proup 455 47
group re
group crich 12
0.-
11 4.43
0.1-
(3)
Stoge 2
affects dal. changes
Baseline
BR--Ww
P-MON
RIMON
IW/T (3)
FWTT ()
32
Both dat. change
0.2
0.1-
Stage
00-
BWTT
C.
Confounder dil changes
Main offects dist. changes
Both diats. change
Baseline
BR--Net
R-MON
Theorefical
stage
Stage
Stage
Figure 12. Synthetic dataset results for continual learning, a. Samples from the synthetic datasets used for continual learning. We annotate main effects confounders with boundaries different widths to visually aid distinguishing between their magnitudes. b. BWTd and FWTd mean and standard deviation for different methods and datasets. The closer the bar to 0, the better the model. A total of runs were performed with different model initialization seeds. post-hoc Conover's 's test with Bonferroni adjustment was performed hetween those groups of methods where Kruskal-Walli test showed significant differences (p 0.05). Accuracy and squared distance correlation for different methods and datasets. For each stage that the model is trained on, it is evaluated against the test sets of all 5 stages (shown through solid curves). Less opaque markers represent carlier stages, while more opaque markers represent later stages being evaluated on. Dotted red lines of various transparency values show the theoretical maximum accuracy that an unbiased model will get for each of the different stages.
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Confounder-Fre Free Continual Learning via Recursive Feature Normalization
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Figure Visualizing different metrics for the ABCD dataset. Accuracy, b. difference between True Positive Rate (TPR) and True Negative Rate (TNR), and c. dcor² between learned features and PDS for boys and girls for different methods. Results shown over folds 5-fold cross validation, data split by subject and ID. Statistically significant differences between R-MDN and other methods are measured first using Kruskal- Kruskal-Wallis Wallis and then post-hoc Conover's test with Bonferroni adjustment.
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Figure Visualizing different metries HAM10K skin lesion classification. Accuracy, squared distance correlation. backward transfer, and forward transfer for different methods. Results are shown after training each model on the final training stage.
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Confounder-Frne FFee Continual Learning via Recursive Feature Normalization
H. Effect of Regularization Hyperparameter
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Figure 15. Accuracy and squared distance correlation when the regularization hyperparameter for the R-MDN module varied. Results are computed for the synthetic dataset described in Section C, and we show the mean and 95% CI over 100 runs of random model initialization seeds.
In Figure 15, systematically vary the regularization hyperparameter feature representations to its value. We observe that model performance remains
how sensitive model performance and the
ability learn consistently robust across different values of X. However, we find that the capacity to residualize the confounder's effects improves with higher values of x, probably due of it stabilizing the residualization process.
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Confounder-Free Continual Learning via Recursive Feature Normalization
I. R-MDN Module Placement Choice
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Figure Accuracy and squared distance correlation when R-MDN module inserted at various locations in convolutional neural network. Results are computed for the synthetic dataset described in Section C, and we show the mean and 95% CI over 100 runs of random model initialization seeds.
In Figure I, vary the placement of the R-MDN layers within deep convolutional neural network observe the effects on model performance and correlation of the learned features with the confounder. We find that while model performance seems to be sensitive to the placement, the ability to remove the influence of the confounder from the feature representations is, overall, consistently high. For such an architecture, adding an R-MDN layer after every convolutional layer and the pre-logits layer seems to provide the best trade-off between model performance and residualization (as also observed by Lu al. (2021)).
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Confounder-Fue Free Continual Learning via Recursive Feature Normalization
J. Glass Brain Visualizations for ABCD Sex Classification
Light Lateral Ventricle
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Figure The top 10 regions identified as being relevant for distinguishing sex by base model, and b. the same model trained with R-MDN.
To identify the top 10 most relevant regions for distinguishing sex (Figure J), we first generate 3D saliency maps based on the test images, highlighting areas in the input image that most activate the model. A threshold of 0.05 is applied focus the most salient regions. A 5x5x5 smoothing filter is applied, replacing each voxel's value with the average of its neighboring voxels. These regions are then visualized using the Harvard atlas.
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Confounder-Fre Free Continual Learning via Recursive Feature Normalization
K. Quantifying Sensitivity To Confounders
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Figure 18. Accuracy for various methods get in continual learning setting when evaluated test sets with various distributions of the confounding variable. Each row represents a different method, each column the stage that the model is trained on after which it evaluated, and each hue of the curve the stage that the model evaluated on. A 8 implies that that input does not contain confounder. Results are evaluated on the synthetic datasct from Section 4.1 where we change the distributions for both the confounding variable and of random model initialization seeds.
main effets. We show the mean and 95% CI over
In Figure 18, provide additional plots for the experiment visualized in Figure 3d. We vary the intensity of the confounder by applying multiplier 5 E [0, 1].
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Confounder- Free Continual Learning via Recursive Feature Normalization
L. Effect of Training Protocol for Continual Learning
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Figure 19. Accuracy and squared distance correlation for different methods and number of training epochs per stage. We used the synthetic dataset where distributions both confounding variable and main effects change. For each stage that model trained on, it evaluated against the test sets of all 5 stages (shown through solid curves). Less opaque markers represent earlier stages, while more opaque markers represent later stages being evaluated on. Dotted red lines show the theoretical maximum accuracy that an unbiased model will get for each of the different stages. Results shown the mean and 95% CI over 5 runs.
Here, we quantify how task performance and the ability to learn confounder-free feature representations change with different number of training epochs per stage; i.e., with the number of times every example from the training data presented to the system. We observe that R-MDN is the only method that is able remove the influence the confounder from the learned features for smaller number of training epochs. This is perhaps because of R-MDN's fast convergence abilities (Hayes, 1996; Haykin, 2002) property that gradient- and adversarial-basee methods not able to demonstrate (Figure 19). This is further reinforced by R-MDN having a better forward transfer on future stages of training for both small and large numbers of training epochs (Figure 20). Both BR-Net and P-MDN are decent methods for continual learning, but they require the same training examples to be seen multiple times in order to drive high prediction scores and remove confounder influence from learned features.
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Confounder-Fre Free Continual Learning via Recursive Feature Normalization
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Figure 20. BWT and FWT mean and standard deviation for different methods and number of training epochs per task. We used the synthetic datasct where the distributions both the confounding variable and main effects change. The closer the bar the better the model. A total of 5 runs were performed with different model initialization seeds. A post-hoc Conover's 's test with Bonferroni adjustment was performed between those groups methods where Kruskal- Wallis test showed significant differences (p < 0.05).
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Reviews:
[{"review_id": "Q0k8q61k5R", "reviewer": "Reviewer_s7kX", "summary": "To remove the influence of confounding variables from intermediate feature representations, the authors introduce the Recursive MDN (R-MDN) layer. They note that such layer can be integrated into any deep learning architecture--including vision transformers--and at any model stage. \n\nR-MDN performs statistical regression via the recursive least squares algorithm to maintain and continually update an internal model state with respect to changing distributions of data and confounding variables. \n\nIn the empirical evaluation the authors claim that R-MDN promotes equitable predictions across population groups, for static learning and for different stages of continual learning, by reducing catastrophic forgetting caused by confounder effects changing over time.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your thorough review and thoughtful feedback - we respond below to the concerns you have raised.\n\n---\n\n> … marginal improvements or I was not able to see clear improvement separation \n\nIn this work, we introduce a method that allows DNNs to learn confounder-free features during continual learning. A consequence is that the model promotes equitable predictions across class categories. Thus, we must analyze results across multiple metrics, as a baseline model might perform well on specific individual metrics.\n\nFor example, in Exp. 4.1, a baseline CNN scores the best on the BWTd metric, but comparably on ACCd and FWTd. Here, lower is better all three metrics. Similarly, a stage-specific MDN model does poorly on ACCd. P-MDN, on the contrary, does well on ACCd and FWTd, but not on BWTd. With our approach, we observe that the method achieves competitive performance across all metrics, significantly outperforming other methods in metrics they perform poorly at (such as 1% improvement on FWTd but 10% on ACCd when compared to stage-specific MDN). In Exp. 4.3, while our approach leads to a modest 2% improvement on the dcor^2 (CN) metric, it has a significant 15% improvement on dcor^2 (AD) when compared to P-MDN. P-MDN is biased towards the AD category, while our approach has similar ranges of dcor^2 across all categories. Furthermore, significant improvements can be seen in Figure 3 (as evidenced by the five straight lines) and Figure 6 (as evidenced by the identification of the cerebellum).\n\nWe believe that these, along with other results from the paper, provide clear evidence of substantial improvement over prior works for the claims we make, as analyzed through a combination of qualitative and quantitative evaluations.\n\n---\n\n> I would have expected use of the methodology at 2 or three different NN architectures …\n\nPlease note that we tested our method on both CNN and ViT with a mix of datasets and different architectural setups; though, we agree that evaluating the method on more DNN architectures would add more value to the paper. We selected a CNN and ViT model backbone because they lend themselves well to the task of image categorization, which is what we focus on in this paper. Synthetic datasets (as presented in Exp. 4.1 and Suppl. C) are evaluated using a CNN backbone to enable direct comparison to prior de-biasing methods in this space such as MDN, P-MDN, and BR-Net, all of which adopt a similar backbone and model size. This helps us understand how well our method performs. Additionally, we include an experiment on the ABCD dataset (Exp. 4.4) using a 3D CNN model to extend its application to a real-world setup. Finally, we evaluate a 2D ViT (Exp. 4.2) and a 3D ViT (Exp. 4.3) on different real-world setups to analyze their performance. To the best of our knowledge, this is the first time transformers have been used for confounder-free feature learning in this context. We hope that the demonstration of both 2D and 3D CNNs and ViTs will add value to our claim. To complete the loop and further strengthen our claim, we are now running an experiment with a ViT as a model backbone on the synthetic dataset. Results are presented in Table 1 at https://docs.google.com/document/d/1VPSuH5B_XvGlGoSlg_jCikB8OPLfyI65oSgACuWxzco/edit?usp=sharing.\n\n---\n\n> On the metrics that they introduce it looks like the proposed approach does not have favorable scores.\n\nAcross the different metrics, our method favorably improves scores on dcor^2, which quantifies the influence of the confounder on learned features. Improvement is seen through a decrease in  dcor^2 in Tables 2, 3, and 4, often across all class categories to reduce inequitable predictions. On the accuracy metric, scores sometimes modestly drop to reflect the model not learning using shortcuts by looking at confounder information for prediction. Only for Exp. 4.2 we see that our approach leads to a slightly weaker score on FWT, possibly due to the corresponding influence from a slightly higher accuracy due to utilizing confounder information, which we seek to avoid. Since we can design carefully controlled environments through synthetic datasets, we can analyze the true theoretical accuracy, BWT, and FWT a fair model should get, and we see favorable scores across each in Exp. 4.1.\n\n---\n\n> Have the authors checked the performance over different NN architectures under one setup…\n\nYes, the placement of the R-MDN layer is an important hyperparameter that must be tuned to the specific DNN architecture being used. We analyze various placements across different architectures through Exp. 4.2 and Suppl. H. Prior work (Lu et al., 2021) has shown that applying MDN after both convolutional and fully connected layers results in the best performance in CNNs. We see similar performance improvement in CNNs with R-MDN in Suppl. H. We construct a similar setup for ViTs in Exp. 4.2 and find that adding R-MDN only after the pre-logits layer provides the best performance.", "claims_and_evidence": "Claims\n\n- The authors note that the introduced R-MDN layer can be integrated into any deep learning architecture--including vision transformers--and at any model stage. \n\n- The authors claim that R-MDN promotes equitable predictions across population groups, for static learning and for different stages of continual learning, by reducing catastrophic forgetting caused by confounder effects changing over time.\n\nEvidence\n\n- The authors claim that R-MDN layer can be integrated into any deep learning architecture, only on a small subset of architectures are this claim is evaluated.\n\n- The claim that R-MDN promotes equitable predictions across population groups, for static learning and for different stages of continual learning, by reducing catastrophic forgetting caused by confounder effects changing over time is evaluated on synthetic and real world datasets.", "methods_and_evaluation_criteria": "Methods\n\n- The methodology seems valid and clearly explained\n\nEvaluation Criteria\n\n- They define the Average Accuracy distance (ACCd), Backward Transfer distance (BWTd), and Forward Transfer distance (FWTd) metrics. As noted these metrics are adapted from ACC, BWT, and FWT defined by (Lopez-Paz & Ranzato, 2017) to work with the setting where a model is expected to achieve certain theoretical accuracies on data from both previous and future stages of training.", "theoretical_claims": "I have not cheeked the theoretical claims, the derivations seem right (related to equations 2, 3, and 4) but have not checked the detailed in the Appendix.", "experimental_designs_or_analyses": "The experimental design looks good.\n\nIn the experimental design they follow past works and use also exiting datasets from in this domain.\n\nThey evaluate on synthetic datasets and publicly available ones.", "relation_to_broader_scientific_literature": "The authors consider the residual r from the expression z = x˜β˜x + yβ˜y + r = Xβ + r, where X = [x y ˜ ] and β = [β˜x; β˜y] is a set of learnable parameters. They take learned features z that are first projected onto the subspace spanned by the confounding variable and the labels, with the term x˜β˜x corresponding to the component in z explained by the confounder, and yβ˜y to that explained by the labels.\n\nPractically they compute the residual r = z − x˜β˜x; i.e., only with respect to β˜x. They focus on this residual that explains the components in the intermediate features irrelevant to the confounder which seems relevant to the labels and, thus, for the classification task.", "essential_references_not_discussed": "I was not able to notice, although, I'm not an expert in the field.", "other_strengths_and_weaknesses": "Strengths\n\n- Very well introduced, written and nicely structured paper.\n- The methodology seems novel and it is clearly explained.\n- Empirical evaluations and results on synthetic and publicly available dataset, and comparison with existing work under different continual learning regimes.  \n\nWeaknesses\n- The claims somewhat not supported by the evaluation, marginal improvements or I was not able to see clear improvement separation \n- In the evaluation for the synthetic dataset they use CNN architecture, while in the evaluation on the publicly available dataset they use ViT architecture. I would have expected use of the methodology at 2 or three different NN architectures (to support the claim in the abstract and add more value to the paper).\n- On the metrics that they introduce it looks like the proposed approach does not have favorable scores.", "other_comments_or_suggestions": "/", "questions_for_authors": "I have seen that the proposed layer was applied and placed at different layers, Have the authors checked the performance over different NN architectures under one setup to check what kind of improvement we can get?", "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": "", "extra_scores": {}}, {"review_id": "AK0M8gmWKw", "reviewer": "Reviewer_Shhj", "summary": "The paper develops a method to debias intermediate representations in a continual learning setting. The idea is to regress the representation on the biased feature and the label. Then, only the residuals after removing the role biased feature are used as the representations. The experiments show the method improves over existing methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your thorough review and thoughtful feedback—we respond below to the concerns you have raised.\n\n---\n\n> Without the theory, the experiments should have explored more synthetic examples…\n\nWe agree that additional synthetic setups would strengthen the empirical results we observe. We selected the synthetic setup in the paper, which plays with the distribution of the intensity of main effects and the confounder across different training stages, as it is used extensively in prior work such as by Adeli et al. (2020a), Lu et al. (2021), and Vento et al. (2022). But complex setups seem important. We are now constructing a second synthetic setup which explores the influence of both the position of the confounder as well as the intensities of the main effects and the confounder. More specifically, we generate 1024 32$\\times$32 images that are implicitly broken down into 16 8$\\times$8 grids. The top left and bottom right grids contain Gaussian kernels of intensity $\\sigma_A$, denoting the main effects. The confounder is represented by a Gaussian kernel of intensity $\\sigma_B$, whose position varies from the bottom left to top right of the image over 4 different training stages. Both $\\sigma_A$ and $\\sigma_B$ are sampled from the distribution $\\mathcal{U}(3, 5)$ for group 1, and $\\mathcal{U}(4, 6)$ for group 2. A fair classifier should remain unaffected by confounder information, irrespective of its position within the image. Such a setup also allows us to compute theoretical maximum accuracy achievable for each training stage that we can validate methods against. Results are presented in Figure 1 and Table 2 at https://docs.google.com/document/d/1VPSuH5B_XvGlGoSlg_jCikB8OPLfyI65oSgACuWxzco/edit?usp=sharing. We will include the same in the final paper.\n\n---\n\n> The work does not discuss recent works like https://arxiv.org/pdf/2404.19132\n\nUnsupervised continual learning is an interesting framework to work in. We apologize for not discussing related works in the paper, but rest assured we will cite them in the final paper. In our work, we focus on supervised continual learning, where we train classification networks on labeled data. In UCL, the primary challenge is the lack of labeled confounders. Such a scenario is similar to when certain factors that bias learning are unobserved or hidden. Effectively ensuring plasticity, stability, and cross-task consolidation through the objective function, while making sure that intermediate features can be normalized by removing the influence of confounders (after identification) is an exciting opportunity for future work. In medical contexts, labeled information is often present as metadata with collected samples. Thus, techniques from UCL can be coupled with R-MDN to learn confounder-free representations that additionally exhibit minimized forgetting.\n\n---\n\n> It would be useful to understand why P-MDN works better in some settings.\n\nP-MDN minimizes a proxy to the bi-level optimization problem that MDN tries to minimize with a closed-form solution. P-MDN replaces the MDN’s closed-form solution, which requires a batch-level operation (not friendly for many architects and deep learning), with a proxy objective. Successful minimization of the P-MDN proxy objective depends on the careful tuning of the hyperparameter that controls the loss function computed over the features and the metadata. The success of P-MDN is then affected by this choice of hyperparameter as well as how close the proxy objective represents the true objective to be minimized. In our experiments, we extensively tuned this hyperparameter and analyzed the influence from the learning rate and optimizer and found that gradient updates due to this proxy objective led to large variance in performance across runs (different model seeds), showing improvements in some settings and weaker performance in others. In contrast, our approach is better since it is a recursive reformulation of MDN that theoretically approaches the solution from MDN under specific conditions we analyze in Section 3.1. \n\n---\n\n> I am not sure that I fully appreciate the reasons behind the linear de-confounding… Can the authors intuitively explain the type of relationships between confounder and input (as a thinking tool for representations) that the proposed method handles?\n\nSimilar to the original MDN formulation, R-MDN does not assume that confounders have to linearly influence the input image, because the method does not directly operate on images, but on the feature embeddings, which are non-linear abstractions of the input. Moreover, R-MDN can be applied to feature embeddings at multiple layers, so overall it can effectively remove non-linear confounding between the input and the confounders. However, we admit that neither R-MDN nor MDN has a theoretical guarantee to remove all the non-linear confounding relationships. We will make this discussion more explicit in the final paper.", "claims_and_evidence": "The claims made are not proved, which is okay because theory with neural networks is difficult. \n\nExperimentally, the method seems to do well across a verity of tasks.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "No theoretical claims. It would have been useful to see that type of biases that said method would get rid of but analyzing internal representations of neural networks is not a solved problem, so this can't be held as a big problem of the paper.\n\nOf course this does not mean no analysis is possible. It would have been very useful to see the type of  shifts over the continuum that the method can handle, maybe in a linear model? Does such analysis exist?", "experimental_designs_or_analyses": "The setup of the experiments vary the distributions over time and this does represent a few scenarios in the problem setup in the paper.\n\nWithout the theory, the experiments should have explored more synthetic examples, with varying features beyond position, to study the interaction of changing relationships between features and confounders. For example, the positions of the features affected by the confounder can switch positions of the features that are affected by the non-confounder features.", "relation_to_broader_scientific_literature": "Continual learning is important and shifts such as those induced by confounding (or spurious features) occur in real life (such as the age being correlated with certain diseases). The findings in the paper combine tools for debiasing with tools from continual learning and that improves accuracy on some real world data.", "essential_references_not_discussed": "The work does not discuss a recent works like https://arxiv.org/pdf/2404.19132", "other_strengths_and_weaknesses": "- The method is simple and the choices can be validated , as is done in the method.\n- The experiments done on real datasets are encouraging.\n\nMain weakness seems to be to understand what is missing with the method. See Questions.", "other_comments_or_suggestions": "It would be useful to understand why P-MDN works better in some settings.", "questions_for_authors": "1. I am not sure that I fully appreciate the reasons behind the linear de-confounding : \"The assumption of an underlying linear relationship between confounders and learned features arises from two key considerations: (1) decisions made by nonlinear models are often challenging to interpret, and (2) sufficiently powerful nonlinear models can extract almost any arbitrary variable from the information present in the features, even if those variables are not explicitly represented.\"\n\nYou build nonlinear models, so point 1 seems moot? Am I missing something? Second, linear regression is limited to not handle interactions between two coordinates in the representations. So any confounder that affects this interactions will not be removed with the proposed method. Is there reason to believe this is unlikely? I'm asking because it's not apriori clear which kinds of confounders can be debiased.\n\n2.  Can the authors intuitively explain the type of relationships between confounder and input (as a thinking tool for representations) that the proposed method handles?", "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": "", "extra_scores": {}}, {"review_id": "YVBhyeIpXE", "reviewer": "Reviewer_m5AR", "summary": "This paper studies how to remove confounder in continual learning process, and proposes the Recursive-MDN (R-MDN) layer. The R-MDN adopts  statistical regression via the recursive least squares for maintaining an internal state. By removing the confounder factors, the model will be less fitted to the irrelevant information in each task, and therefore the forgetting issue can be mitigated.\n\nThe proposed method is integrated with multiple baselines and evaluated over both synthetic and real datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your thorough review and thoughtful feedback—we respond below to the concerns you have raised.\n\n---\n\n> No theoretical analysis is provided to justify the proposed method. \n\nA more extensive exploration of the theoretical framework would surely be valuable. However, we would like to highlight that our work follows the definitions of MDN, which provides a closed-form solution to the generalized linear model removing the effect of the confounders (this is a common practice in statistics for controlling for confounders). Since the closed-form solution cannot be applied to newer architectures or continual learning settings (see the section on Related Works) because of its dependence on batch statistics and knowing all confounding values (even for future time points), we proposed a recursive reformulation of MDN. Theoretically, these two formulations are equivalent under specific conditions that we mention in Section 3.1 and Suppl. A. Additionally, we explore synthetic setups based on carefully controlled environments for various variables to identify theoretical maximum accuracies achievable that methods can be validated against. As an addition, we are now constructing a second, slightly more complex, synthetic setup to further justify the effectiveness of the proposed method---specifically, we vary both the position as well as the intensity of the confounding variable over various stages of continual learning. Please refer to our response to Reviewer Shhj above (first paragraph) for more details about the setup and results. Results are presented in Figure 1 and Table 2 at https://docs.google.com/document/d/1VPSuH5B_XvGlGoSlg_jCikB8OPLfyI65oSgACuWxzco/edit?usp=sharing, which we will add to the final paper. We hope that this addition can further support our findings.\n\n---\n\n> The performance improvement of the proposed method is not always significant.\n\nPlease see our response to Reviewer s7kX (first paragraph) about this point. Across multiple experiments, we observe that our method achieves substantial improvements over prior works when results are analyzed over multiple different metrics, since baseline models might perform well on specific individual metrics.\n\n---\n\n> The adopted baselines are pretty old, and recent baselines are not included. Discuss why baselines proposed in recent years (2023, 2024) are not included in experiments.\n\nBaselines for our work are of two different kinds: one that introduces methods for continual learning and two that introduces methods for confounder-free learning. Supervised continual learning is broadly partitioned into three categories: replay-based methods, regularization-based methods, and architecture-based methods. We include a baseline from each of these categories. While this might be non-exhaustive, these methods do not enforce any constraints on learning confounder-free features, so we think that any other recent baseline will perform poorly on our chosen metrics, specifically dcor$^2$. On the other hand, we test our method against the current state-of-the-art confounder-free learning methods such as BR-Net, MDN, and P-MDN, which encompass adversarial-, statistical regression-based, and gradient-based learning. We hope that this provides a comprehensive picture of the landscape.\n\n---\n\n> …whether the proposed method can be applied to structured data and integrated with models like graph neural networks?\n\nWhile the focus of our work in this paper has been on learning over Euclidean data (e.g., images), there are no reasons why the proposed method could not be applied to structured data. We used images because the prior work on confounder-free learning that we discussed in the paper also deals with image input, so this allowed for effective comparison. If the input data is of tabular or structured graph forms, as opposed to images, similar layers of neural network (e.g., fully connected for tabular data or graph convolution for graphs) could be applied to the input. This results in mid-level feature representations. Hence, the R-MDN (and all other prior work) could be simply applied on top of the feature representations for de-confounding.  \n\nIt is noteworthy that online continual graph learning is an emerging direction (Donghi et al., 2025). This is particularly interesting because we can construct causal graphs that capture the interactions of various observed variables (confounders, mediators, etc.) and use them for learning bias-free representations. Exploring the application of our proposed method, as well as other methods for confounder-free learning in static or continual graph learning frameworks, is an exciting direction for future work.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "There is no theoretical analysis in this work.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "This paper studies how to boost the performance of continual learning models by removing the confounder factors. From the continual learning perspective, this research would be relevant to different areas, e.g. the current LLMs also require the continual pre-training techniques for keeping an up-to-date knowledge base. From the confounder removal perspective, the contirbution is also likely to be beneficial to other machine learning areas.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n\n1. The studied problem is relevant to different machine learning research\n\n2. The experimental results are comprtehensive.\n\nWeakness:\n\n1. No theoretical analysis is provided to justify the proposed method. \n\n2. The performance improvement of the proposed method is not always significant.\n\n3. The adopted baselines are pretty old, and recent baselines are not included.", "other_comments_or_suggestions": "Discuss why baselines proposed in recent years (2023, 2024) are not included in experiments.", "questions_for_authors": "The proposed method is claimed to be able to applied with any deep learning architectures. However, the studied models are mostly for learning over Euclidean data, e.g. images. Therefore, a natural question is: whether the proposed method can be applied to structured data and integrated with models like graph neural networks? Then the setting would be continual graph learning, e.g. 'Online Continual Graph Learning', 'CGLB: Benchmark Tasks for Continual Graph Learning'. I would recommend the authors to discuss this to justify the applicability of the proposed technique.", "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": "", "extra_scores": {}}, {"review_id": "z7xylkzHRb", "reviewer": "Reviewer_97mW", "summary": "This paper introduces a Recursive metadata normalization (R-MDN) layer, in order to remove confounders, that are extraneous variables that affectboth the input and the target. The paper extends the confounder-removing activity to continual learning community. R-MDN performs statistical regression via the recursive least squares algorithm to maintain and continually update an internal model state\nwith respect to changing distributions of data and confounding variables.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your thorough review and thoughtful feedback—we respond below to the concerns you have raised.\n\n---\n\n> The paper only reports results in medical datasets. It seems that the method should cover various datasets (at least from the title and abstract). The reviewer concerns that the proposed method cannot handle large-scale problems, e.g., ImageNet-1000.\n\nLimitations of the methods on dataset size are definitely an important consideration. In this paper, we work within the framework of medical datasets for real-world setups (wherein issues with confounded learning is very prominent), and non-medical datasets for synthetic setups. We build on prior works from Zhao et al. (2020), Adeli et al. (2020a), Lu et al. (2021), and Vento et al. (2022), who have proposed methods within this framework that we can directly compare to. Several of these datasets are medium-sized, wherein our method successfully learns confounder-free representations, as demonstrated empirically. Larger problems manifest through either larger dataset sizes, longer pre-training, or both. We evaluate the effect of the number of training epochs on confounder-free learning in Suppl. K, where R-MDN’s quick convergence property becomes an advantage. For larger dataset sizes, catastrophic forgetting becomes an issue with task learning. We make preliminary advances in this direction in Exp. 4.2 by integrating R-MDN with existing continual learning frameworks such as LwF and EWC that deal with catastrophic forgetting. Memory replay is another technique that our method could be integrated with. Nevertheless, we acknowledge this issue, and implementing and analyzing plausible solutions is an exciting direction for future work.\n\n---\n\n> Please state the difference between your method and the analytic continual learning techniques. Could it be a trivial implementation? If not, please do make a thorough comparison.\n\nThank you for referencing the ACIL paper. ACIL works within the domain of class-incremental continual learning by breaking down training into two stages: (a) base training via backpropagation, which allows learning of model parameters for the task; and (b) analytic re-alignment, which maps intermediate model features onto the label matrix recursively. This recursive construction permits absolute memorization and effectively mitigates catastrophic forgetting. Our approach is different since we map intermediate features not onto the label space but onto the confounder space so that the residual component of the features can be extracted and passed to downstream layers. This effectively removes the influence of such confounding variables from learned feature representations. Additionally, everything happens in a single-stage end-to-end framework for R-MDN: no network parameters are frozen for the recursive updates. Thus, the network has to learn task-relevant features that are also confounder-free during base training. Overall, the implementation nuance (single-stage learning), the target for the recursive least squares estimator (confounding variables), as well as the input to downstream layers (the residual) differ between our method and ACIL. We will include these details in the final paper.", "claims_and_evidence": "The claims in this paper are well supported in both methology and experiment aspects.", "methods_and_evaluation_criteria": "In the main context, the paper only reports results in medical datasets. It seems that the method should cover various datasets (at least from the title and abstract). The reviewer concerns that the proposed method cannot handle large-scale problems, e.g., ImageNet-1000.", "theoretical_claims": "The proofs for theoretical claims seems valid.", "experimental_designs_or_analyses": "The experiment settings are overall quite good", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "This paper discusses the recursive learning method in continual learning. There is a new branch of continual learning, named analytic continual learning (starting from ACIL [1]), which also adopts recursive merhod, such as recursive least squares related techniques. The papers in this branch shares several similarities in methology (from derivation to result).\n\n[1] ACIL: analytic class-incremental learning with absolute memorization and privacy protection", "other_strengths_and_weaknesses": "See the comments marked above.", "other_comments_or_suggestions": "N.A", "questions_for_authors": "The use of RLS in updating the normalization layer seems interesting. Please state the difference between you method and the analytic continual learning techniques. Could it be a trivial implementation? If not, please do make a thorough comparison. Please note that my rating is not a final decision.\n\n##after rebuttal## according to the response, I am maintaining the rating.", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes Recursive Metadata Normalization (R-MDN), a novel and principled approach to removing confounding influences from intermediate feature representations in continual learning. Built upon recursive least squares (RLS), R-MDN operates online and can be flexibly inserted into deep networks, including vision transformers, without requiring batch-level statistics or task-specific architectures. The authors demonstrate its efficacy across synthetic and real-world datasets (e.g., HAM10000, ADNI, ABCD), showing improved fairness, backward and forward transfer, and confounder invariance.

While there has been some opinion that proposed method only gives marginal improvements (s7kX), lacks theoretical analysis (m5AR), the reviewers agreed that the flexibility of R-MDN, which can be plugged in any deep learning models, and positive experimental results on multiple datasets are worth publishing. Hence, the decision is Accept. But, AC strongly recommends the authors to incorporate the following crucial discussions / additional results in the final version:

  - Additional results for ViT architecture on synthetic data. 
  - More clear discussion on linear confounding.
  - Discussions and comparison with P-MDN. 
  - Clear definition on the metric $dcor^2$.
Decision: Accept (poster)