Interpret Your Decision: Logical Reasoning Regularization for Generalization in Visual Classification

We really appreciate your insightful comments, and we address your weaknesses and questions point-by-point.

W1. Thank you for your insightful comment. As discussed in Paper Lines 344-358, L-Reg relies on the precondition that each dimension of the semantic features represents independent semantics. When this condition is not met, applying L-Reg can lead to performance degradation. This is because the atomic formulas constructed for different classes could become sub-optimal if the minimal semantic supports for different classes correlate with each other. In cases where improper $z$ is used, especially between known and unknown classes, the model may struggle to effectively filter out irrelevant features for unknown classes, which can result in features that inadvertently overlap with the minimal semantic supports for known classes, resulting in degradation for them.

To address this issue, we hypothesize that enforcing independence between $z^i$ and $z^j$ could lead to further improvements. To test this hypothesis, we conducted experiments using orthogonality regularization (Ortho-Reg) to enforce feature independence in mDG and GCD tasks. As shown in PDF Tab.2,3&4, results indicate that while Ortho-Reg alone may not be very effective, combining L-Reg with Ortho-Reg leads to significant improvements.

Based on these findings, we propose that the performance drop observed could be mitigated further with a well-designed model architecture or additional regularization techniques that enhance independence between $z^i$ and $z^j$. We find this to be a very attractive and promising area of research, and we are eager to explore it further in future work.

W2. We appreciate your comment. To ensure a fair comparison and align with previous work, we use the same encoder, $g$, as employed by earlier studies to validate our L-Reg approach. Insights into the selection or design of $g$ can be found in our response to W1. As noted, L-Reg relies on the condition that the semantic features $z^i$ and $z^j$ (for $i \neq j$) are independent. Models that achieve an orthogonal semantic space are thus well-suited for applying L-Reg, as they naturally align with this requirement.

W3. hank you for highlighting this point; it has been very insightful. Currently, we use a plain strategy of selecting the regularization weights in the range of [0.01, 0.001, 0.0001], keeping them relatively small compared to the scale of other losses (approximately 1:10). We appreciate your comments and acknowledge that there may be deeper theoretical or empirical insights related to this issue. We plan to explore this further in future research.

Q1. Thank you for your recommendation. For a thorough evaluation, we compare our L-Reg to the aforementioned Ortho-Reg and a sparsity-based regularization approach. To validate this fairly, we re-implement the Ortho-Reg and Bernoulli sample of the latent features from the sparse linear concept discovery models [3] on the same PIM backbone that we used. The results, presented in PDF Tab.2&3, indicate that L-Reg consistently yields the most significant improvements when these regularization terms are applied alone. Additionally, as mentioned, combining L-Reg with Ortho-Reg further enhances performance since a more proper $z$ is obtained.

The authors' rebuttal dispelled some of my concerns, and I choose to maintain the score.

We really appreciate your insightful comments, and we address your weaknesses and questions point-by-point. Some points of weaknesses and questions are combined because they are very associated with each other.

W1&Q2. Thank you so much for your comments, and sorry for any confusion caused by our writing. We will follow your kind suggestions and add some intuitive explanations for key concepts in our paper. Please also kindly refer to Reply to All Reviewers for the theoretical analysis of atomic formulas and their relationship to the interpretability of L-Reg. To address your questions more concretely, we use Paper Fig. 5 as a typical example to provide more analysis on atomic formulas and the interpretability of L-Reg.

For the known classes, the efficacy of L-Reg can be intuitively understood as extracting the minimal semantic supports for a given class label. As examples shown here, the presence of a guitar's fingerboard, even in unseen domains, helps classify a sample as belonging to the guitar category, whose informal form can be denoted as $h(F_{(\gamma=\text{has the fingerboard}, y=\text{guitar})},d\in D)\rightarrow\text{True}$. For all known classes, samples with these minimal semantic supports are recognized accordingly.

In contrast, if a sample lacks these minimal supports for any known class, it is very likely categorized as an unknown class. This behavior stems from Paper Eq.10 which ensures $\mathcal{A}^{y_i*} \neq \mathcal{A}^{y_j*}$ through constraining $\gamma^{y_i} \neq \gamma^{y_j}$. L-Reg further enhances the model's ability to identify minimal supports for unknown classes by filtering out co-covariant features associated with other classes and thus generalizing to unseen domains. Therefore, the very interpretable features for unknown classes from unseen domains can be extracted using L-Reg. Paper Fig. 5 (right side) demonstrates that the model with L-Reg can even extract facial features for the unknown person class and can generalize this to the unseen domain. Similarly, here we obtain an (informal) atomic formula as $h(F_{(\gamma=\text{has a face}, y=\text{person})},d\in D)\rightarrow\text{True}$.

We will include a more detailed discussion on this topic in the final version of our paper.

W2. Thank you very much for this comment. Please refer to the Reply to All Reviewers for the experimental details on the ERM baseline. As shown in PDF Tab.4, under the same experimental settings and hyperparameters, incorporating L-Reg with ERM significantly enhances the overall mDG performance, improving it from 49.9% to 52.9%.

Q1. We apologize for any confusion caused and appreciate your constructive feedback. L-Reg is designed to focus on eliminating shared semantics across all classes rather than addressing frequent features within a single class. To clarify this, please refer to the PDF Fig.1. In this figure, we have included fixed images with the same coordinates and additional figures illustrating feature distributions of known and unknown classes. Note here we use the values of the first components of PCA results on the original features, denoted as $v1^{st}$.

Before applying L-Reg, the $v1^{st}$ feature predominantly falls within the range [-0.4, -0.2] across all classes, especially identical between known and unknown classes. This indicates that a few specific semantics overly influence many features. L-Reg mitigates this issue by focusing the model on disentangled minimal semantic supports for classifying each class, thereby reducing feature complexity and enhancing generalization.

Q2. Many thanks for this comment. The code for L-Reg is available in the supplementary materials we have provided, and we will release all code and hyperparameters to facilitate the reproduction of our experiments.

Regarding sensitivity, as discussed in Paper Line 344-358 and illustrated in Paper Tab.4, applying L-Reg to the semantic features from the deep layers improves performance for unknown classes without negatively impacting known classes. We hypothesize this is due to the fact that our L-Reg is derived based on the precondition that $z^i, z^j \in z, I \neq j$ is independent of each other. This condition holds for most deep-layer features but may not apply to shallow layers, and further regularizing the independence may lead to further improvements. To test this hypothesis, we conducted experiments with orthogonality regularization (Ortho-Reg) to enforce feature independence in mDG and GCD tasks. As shown in PDF Tab.2,3&4, while using Ortho-Reg alone may not be very effective, combining L-Reg with Ortho-Reg leads to further improvements. These findings support our hypothesis and suggest that L-Reg, particularly when applied to deep layers or in conjunction with Ortho-Reg, is beneficial.

We appreciate your insightful comments, and we address your weaknesses and questions point-by-point. Some points of weaknesses and questions are combined because they are very associated with each other.

W1. We really appreciate this insightful comment. In our study, we adopted the commonly used accuracy metric to align with the previous work we compared against. We agree that other entropy-based metrics could potentially provide a more effective evaluation. Additionally, it may be worthwhile to propose novel logic-based metrics for a more comprehensive assessment. We find this to be a very attractive and promising area of research, and we are eager to explore it further in future work.

W2&Q1. We appreciate these comments. For a more detailed explanation of L-Reg's interpretability, please refer to Reply to All Reviews 1. Notably, the CAM visualizations in our paper illustrate the model's learned atomic formulas. By using L-Reg, the model derives interpretable atomic formulas, which can also be understood as the most important features for predicting a given class. To address your questions more concretely, we use Paper Fig.5 as a typical example to analyze L-Reg’s interpretability further.

For the known classes, the efficacy of L-Reg can be intuitively understood as extracting the minimal semantic supports for a given class label. As examples shown here, the presence of a guitar's fingerboard, even in unseen domains, helps classify a sample as belonging to the guitar category, whose informal form can be denoted as $h(F_{(\gamma=\text{has the fingerboard}, y=\text{guitar})},d\in D)\rightarrow\text{True}$. For all known classes, samples with these minimal semantic supports are recognized accordingly.

In contrast, if a sample lacks these minimal supports for any known class, it is very likely categorized as an unknown class. This behavior stems from Paper Eq.10 which ensures $\mathcal{A}^{y_i*} \neq \mathcal{A}^{y_j*}$ through constraining $\gamma^{y_i} \neq \gamma^{y_j}$. L-Reg further enhances the model's ability to identify minimal supports for unknown classes by filtering out co-covariant features associated with other classes and thus generalizing to unseen domains. Therefore, the very interpretable features for unknown classes from unseen domains can be extracted using L-Reg. Paper Fig. 5 (right side) demonstrates that the model with L-Reg can even extract facial features for the unknown person class and can generalize this to the unseen domain. Similarly, here we obtain an (informal) atomic formula as $h(F_{(\gamma=\text{has a face}, y=\text{person})},d\in D)\rightarrow\text{True}$.

We will include a more detailed discussion on this topic in the final version of our paper.

W3&Q2. We really appreciate this inspiring comment. Following the Reply to All part 1, while a common sparse concept model may be able to achieve $\gamma^y\psi = z^y$ by filtering irrelevant features through the sparsity, it may not ensure $\gamma^{y_i} \neq \gamma^{y_j}$, which is crucial for disentangling features used for predicting different classes. This limitation can potentially lead to degradation in generalization performance for common sparse concept models.

To investigate this fairly, we re-implemented the Bernoulli Sample of the latent features from the Sparse Linear Concept Discovery Models [3] on the same PIM backbone that we used to achieve the sparsity. The results in PDF Tab.2&3 indicate that while L-Reg consistently achieves overall improvement, the sparse concept-based approach does not consistently improve generalization, validating the aforementioned difference.

These insightful comments are highly appreciated. We believe these two questions are very related; therefore, please allow us to address them together.

As discussed in Reply to All Reviews 1, the L-Reg's interpretability is rooted in learning the good general atomic formulas. Specifically, L-Reg encourages the model to identify the minimal semantic supports - the most important features - necessary for class recognition. Such an approach resembles humans' cognition process. Paper Fig.1,5 and visualizations included in the Paper appendix show that L-Reg enables the model to learn distinctive features such as facial features for recognizing the person class, the long-neck feature for the giraffe class, and so on.

To address your questions more concretely, we use Paper Fig.5 as a typical example to analyze L-Reg’s interpretability further.

For the known classes, the efficacy of L-Reg can be intuitively understood as extracting the minimal semantic supports for a given class label. As examples shown here, the presence of a guitar's fingerboard, even in unseen domains, helps classify a sample as belonging to the guitar category, whose informal form can be denoted as $h(F_{(\gamma=\text{has the fingerboard},y=\text{guitar})},d\in D)\rightarrow\text{True}$. For all known classes, samples with these minimal semantic supports are recognized accordingly.

In contrast, if a sample lacks these minimal supports for any known class, it is very likely categorized as an unknown class. This behavior stems from Paper Eq.10 which ensures $\mathcal{A}^{y_i*} \neq \mathcal{A}^{y_j*}$ through constraining $\gamma^{y_i} \neq \gamma^{y_j}$. L-Reg further enhances the model's ability to identify minimal supports for unknown classes by filtering out co-covariant features associated with other classes and thus generalizing to unseen domains. Therefore, the very interpretable features for unknown classes from unseen domains can be extracted using L-Reg. Paper Fig.5 (right side) demonstrates that the model with L-Reg can even extract facial features for the unknown person class and can generalize this to the unseen domain. Similarly, here we obtain an (informal) atomic formula as $h(F_{(\gamma=\text{has a human face},y=\text{person})},d\in D)\rightarrow\text{True}$.

However, as shown in Row 3, significant domain shifts, such as those between the sketch domain and other domains, pose challenges. Specifically, the differences between the stick-figure style of sketches of persons and figures from other domains can hinder the model's ability to cluster sketches with other domains' figures when the class label is unknown. Thus, under this circumstance, the model may fail to extract meaningful features from those sketches. We acknowledge this limitation and will explore solutions in future work.

Once again, we appreciate your thoughtful feedback. We will incorporate this analysis into the final version of our paper.

The rebuttal has well addressed my previous concerns about L_Reg and Fig.5, so I change my rating as WA.

We appreciate your insightful comments and we address your weaknesses and questions point-by-point.

W1. We humbly believe L-Reg delivers consistent and evident gains. Please refer to Reply to All for the highlighted improvements. Moreover, as you suggested, we further validate L-Reg's efficacy by applying it to the ERM for the mDG and CircuitFromer for additional Congestion prediction. PDF Tab.4 exhibits that L-Reg improves averaged performance with the ERM baseline from 49.9% to 52.9% and CircuitFormer from 0.6374 to 0.6553 in the Pearson metric, besides increases in other metrics.

W2. The sufficient and necessary conditions for achieving atomic formulas are the ultimate goal that remains a problem for the community. While we are working towards this, this paper proposes a first practical approach to approximate the most general atomic formulas. Please refer to the Reply to All, which provides more analysis about how the constraints in the paper are derived to achieve atomic formulas. Furthermore, according to the current limitation of L-Reg the paper discussed, we offer a future direction of obtaining proper $z$ to meet L-Reg's precondition. Additional experiments of ERM on mDG and GCD (cf. PDF Tab. 2,3&4) show that constraining proper $z$ leads to further improvement. We will discuss this grand topic in our revision and explore more aspects in the future.

W3. Sorry for any confusion. L-Reg is not designed to reduce the computational complexity but aims to reduce the complexity of the model parameters and the data features. As discussed in Paper Sec.3, L-Reg reduces the classifier's complexity by increasing the sparsity of its weights and features' complexity by removing over-dominated semantics across all classes (this comment is related to your Q3&4, where details of how L-Reg achieves these can be found). We will polish this part in the final version for better clarity.

Q1. Thanks. We will fix all these typos.

Q2. Paper Fig.1 shows CAM visualizations of the models trained under the GMDG with the RegNetY-16G backbone for mDG+GCD on the PACS with the unseen domain art painting, using DomainBed's protocols and codes. Both models share the same training parameters and seed 0. The only difference between them is that the latter uses L-Reg with its weight as an extra hyperparameter. Details such as specific parameter settings, as you requested, are in Paper Appendix E.1 and codes for our method are included in the supplementary materials. In short, we believe the comparison should be fair because of the same hyperparameters and the seed being used for both models except L-Reg. Note that the same models are used for Paper Fig.3&4, and all CAM visualizations in the Paper appendix.

Q3. Using the aforementioned models, Paper Fig.3(a) presents the heatmap of the classifiers' weights, where the x-axis is the index of the weights in the linear layers. Paper Fig.3(b) shows the distributions of the classifiers' weight values, with the x-axis representing the value of the normalized weights and the y-axis showing the count of weight values in bin intervals. Following your suggestion, the fixed figure with a denoted abscissa and more descriptive captions is shown in PDF Fig.2. PDF Fig.2 demonstrates that L-Reg reduces the classifier's complexity by alleviating extreme weight values. The heatmap further indicates that L-Reg increases the sparsity of the classifiers, thus better generalization. Please refer to the reply of W1 for details of more experimental results.

Q4. We apologize for any confusion caused by Paper Fig.4. We have re-drawn Paper Fig.4, now shown as PDF Fig.1, with distributions illustrated on the same coordinates and added features of known and unknown classes. Using the aforementioned models, PDF Fig.1 shows feature distributions based on the first component values after PCA (denoted as $v1^{st}$). Before using L-Reg, $v1^{st}$ mostly concentrates between [-0.4,-0.2] across all classes, indicating some specific semantics over-dominates the features. L-Reg alleviates this issue by forcing the model to obtain each class's minimal semantic supports, removing shared semantics across all classes to reduce feature complexity. PDF Fig.1 top row indicates the distance between the feature distributions of known and unknown classes is enlarged with L-Reg, making them more dividable for classification.

Q5. We provide more details about why $\vdash_{(h \circ g(X), Y)}=\models_{(g(X_s),Y_s)}$ in Paper Eq.6 can be safely omitted in the rest of the paper after the definition of logical framework. Consider the logic $L_{(X_s,Y_s)}=<F_{(X_s,Y_s)},D,\models_{(X_s,Y_s)},h,\vdash_{(h(X),Y)}>$, we want to study $\vdash$'s logic that is defined in the form of $L_{\vdash}\stackrel{\text{def}}{=}<F_{X_s,Y_s},D_{\vdash},h_{\vdash},\models_{\vdash}>$, where $D_{\vdash},h_{\vdash},\models_{\vdash}$ are pseudo-components associated with $\vdash$. Particularly, $D_{\vdash}$ is a subset of all possible world/domains from $F_{(X_s,Y_s)}:D_{\vdash}\stackrel{\text{def}}{=}\{T\subseteq F_{(X_s,Y_s)}:T\text{ is closed under }\vdash_{(h(X),Y)}\}$. For any $T\in D_{\vdash}$ and $a\in F_{(X_s,Y_s)}$, it has $h_{\vdash}(a,T)\stackrel{\text{def}}{=}\{b\in F:T\vdash(a\leftrightarrow b)\}$. Further, $\models_{\vdash}$ in $T\in D_{\vdash}$ is defined as $T\models_{\vdash}a\stackrel{\text{def}}{\Leftrightarrow}a\in T$. [1] points out that the following condition is almost always satisfied: (Cond) $\forall a, b\in F_{\vdash},d\in D_{\vdash}$, we have $(h_{\vdash}(a,d)=h_{\vdash}(b, d))\text{ and }d\models_{\vdash}a\Rightarrow d\models_{\vdash}b$. Hence, the semantical consequence relation induced by $\models_{\vdash}$ coincides with the original syntactical $\vdash_{(h\circ g(X),Y)}$ while Cond holds. Due to $D_{\vdash}\subseteq D$, $\models_{(g(X_s),Y_s)}$ coincides with $\models_{\vdash}$. Thus, $\vdash_{(h\circ g(X),Y)}=\models_{(g(X_s),Y_s)}$ can be safely omitted.