Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Second, we implement these concepts within two algorithms for architectural optimization of partially-equivariant networks.

The evidence to be evaluated is:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

Agrawal & Ostrowski (2022) presents theoretical contributions towards network morphisms for group-invariant shallow neural networks: in comparison, our work focuses on deep group convolutional architectures and implements the morphism in a NAS algorithm.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

The equivariance relaxation morphism generalizes this concept to any subgroup.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum. The equivariance relaxation morphism provides a mapping of group elements between a group and a subgroup. For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers. Towards finding the optimal full equivariance per layer, the equivariance relaxation morphism presented in Section 4.1 is applied as the genetic operator in an evolutionary hill-climbing algorithm. The Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) algorithm, given in Algorithm 1, is similar to other evolutionary NAS methods such as Elsken et al. (2017) with pareto selection as in Falanti et al. (2022). A population of networks, which starts with an individual with all layers equivariant to the largest possible group, undergoes mutation via equivariance relaxation and selection based on accuracy and parameter count to optimize neural architecture while learning network parameters.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance. We next study empirically the two EquiNAS methods on three datasets, one with known rotational symmetry and two with unknown but visually significant rotational and reflectional symmetry. We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation. The \(p4\) group consists of discrete translations and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes C_{4}\). The \(p4m\) group consists of discrete translations, reflections, and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes D_{4}\). As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively.

We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\). We present experiments on image classification for a variety of datasets. The Rotated MNIST dataset (Larochelle et al., 2007, RotMNIST) is a version of the MNIST handwritten digit dataset but with the images rotated by any angle. This task serves as a simple investigational study with known symmetry, while the following two tasks are more realistic and complex.

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

The advantages of equivariance search methods are most apparent in the Galaxy10 benchmark. While EquiNAS\({}_{E}\) outperforms most baselines on RotMNIST and all baselines on ISIC, it has similar performance and some final architectures to the \(D_{4}\) baseline for both tasks. However, the \(D_{4}\) baseline fails at the Galaxy10 task, demonstrating that the same equivariant architecture can not always be naively applied. Both search methods, EquiNAS\({}_{E}\) and RPP, outperform all baseline models on Galaxy10, and by a large margin for EquiNAS\({}_{E}\).

the selected population maintains a fully equivariant network in every generation.

These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

EquiNAS\({}_{E}\) may particularly work well on tasks that benefit from local equivariance, determined by analyzing the architecture weighting parameters from first applying the more efficient EquiNAS\({}_{D}\), as well as for finding good discrete architectures within which to retrain weights, based on the ablation and random comparisons. We thus recommend EquiNAS\({}_{D}\) for practical applications if the final model is not restricted to discrete equivariance, in which case it can be used to inform design decisions for applying EquiNAS\({}_{E}\).

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

Evolutionary algorithms are optimization methods inspired by evolution in biology, where individuals in a population compete with their phenotypic traits in order to pass on their genotypic traits to offspring. The _population_ is the current collection of individuals. Each _individual_ is an instance of the object to be optimized and has a genotype that is decoded into a phenotype. In this case, each individual is a neural network, with a genotype that encodes the parametrized equivariance group of each convolutional layer, represented as a vector of integers. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. Each parent itself is kept for the next population, as well as each valid child that is generated via the equivariance relaxation morphism, such that they are functionally equivalent to their parent at initialization (although with a different architecture) and thus have the same fitness before training. Each individual in this population is partially trained, such that these children diverge from their siblings and parent, so that the next set of parents may be selected and this process repeats.

To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion. The new filter tensor has a shape of \(C_{l}|R|\times C_{l-1}|R|\times S_{G_{l}}/|R|\times K_{l}\times K_{l}\). The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension. Every convolutional and linear layer except the output layer is immediately followed by a batchnorm then a ReLU.

For EquiNAS\({}_{E}\), the parent selection size is 5, the training time per generation is 0.5 epochs, and the number of generations is 50 for all tasks.

As ablation studies and comparisons, we implement two kinds of random search for each NAS method. The first ablates smart architecture search: EquiNAS\({}_{E}\) Random Select works as described in Algorithm 1 but with random parent selection (instead of pareto-front selection) and EquiNAS\({}_{D}\) Random \(Z\) works as described in Algorithm 2 but with random \(Z\) updates (instead of gradient descent) by shuffling \(Z\) gradients. The second is more akin to standard NAS random search: for the evolutionary paradigm, we train 30 randomly selected static architectures in the discrete architecture search space for the same training time and selecting the top 5 by validation accuracy, and for the differentiable paradigm, we train 25 randomly selected static architectures in the continuous architecture search space and selecting the top 5 by validation accuracy. 30 and 25 were respectively calculated to be approximately the same compute cost as the trials of EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\). These are labeled as "Random Static" for both the evolutionary and differentiable paradigms. Since Random Static trains static architectures while EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) dynamically search for both architectures and parameters, we take the best 5 architectures for each and retrain their param eters from scratch as in the standard NAS paradigm, labeled as EquiNAS\({}_{E}\) Retrain and EquiNAS\({}_{D}\) Retrain, respectively, for fair comparison to Random Static.

EquiNAS\({}_{D}\) finds competitive architectures on average and can find architectures which outperform baseline choices like architectures fully equivariant to \(C_{1}\) or \(D_{4}\). From comparing EquiNAS\({}_{D}\) Retrain to EquiNAS\({}_{D}\) results in Figure 8, retraining a resulting architecture is not consistently better or worse than using the final weights from EquiNAS\({}_{D}\), showing that there isn't a disadvantage to training weights during search and avoiding the additional cost of a post-search training step.

On this baseline, EquiNAS\({}_{D}\) greatly outperforms a search of random architectures, demonstrating that EquiNAS\({}_{D}\) can discover the appropriate equivariance for a specific dataset over fixed or randomly selected architectures.

Our methods enable searching for both architecture and parameters concurrently in a single training process. This approach is more efficient than NAS approaches that only search for architectures, requiring a retraining process within the resulting architecture for evaluation.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 8: Test errors for experiments of Section 6.2.

Input: Figure 2: Historical parameter counts of all selected individuals for EquiNASE on RotMNIST. The architectures of the final selected population are labeled. Each history is colored by the final test accuracy, measured of each individual upon removal. For other tasks, see Figure 7 in Appendix D.

Input: Figure 4: EquiNASD on classification of six augmentations of MNIST, where each row is a trial on MNIST augmented to exhibit symmetry to the labeled group.

Input: Figure 9: Test errors against stored parameter count for experiments of Section 6.2. Although EquiNASD and RPP train with a higher parameter count, the parameters may be stored in a single filter per layer using Equation 11 for testing.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 7: Historical parameter counts of each selected individual for EquiNASE (see Figure 2 for RotMNIST).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I carefully evaluated the claim "Second, we implement these concepts within two algorithms for architectural optimization of partially-equivariant networks" against the extensive evidence provided.

The evidence strongly supports this claim, showing detailed implementation of two distinct algorithms (EquiNAS_E and EquiNAS_D) that optimize network architectures for partial equivariance. The evidence includes:
- Comprehensive mathematical formulations and mechanisms (equivariance relaxation morphism and [G]-mixed equivariant layer)
- Detailed algorithmic descriptions
- Experimental results across multiple datasets demonstrating the algorithms' effectiveness
- Visualizations and comparisons showing how the algorithms optimize architectures

The claim is well-stated and accurately reflects what was implemented, without exaggeration. The evidence thoroughly documents the implementation and validation of these algorithms.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Second, we implement these concepts within two algorithms for architectural optimization of partially-equivariant networks.

The review comment to be evaluated is:

**Strength**
1. The paper is well motivated. Most G-CNN literature assume that the symmetry group of the learning task is explicitly known and perfect. The paper instead uses a equivariance-aware NAS procedure to search for the optimal architecture for (potentially) unknown and imperfect group transformations.
2. The paper is mostly well-written.
3. The experiments are detailed and thorough.

**Weakness**
1. The equivariance relaxation morphism, which relies on reparameterizing the group convolution on a subgroup, seems to be limited to discrete group. Can this be extended to continuous groups?
2. Moreover, the relaxation morphism is also restrictive in the sense that the number of the unstructured channel (c_out) cannot be adjusted after changing to a subgroup. This inevitably increases the model size as group size decreases, which is indeed the case for EquiNAS_D.
3. The clarity of the paper in section 5.1 can benefit from more explanation on the technical terms such as population, pareto selection, pareto front, etc.
4. In MNIST_rot, the review would like to see what happens when $C_8$ and $D_8$ are included in the picture.
5. If I understand correctly, the reported EquiNAS_E result on Galaxy10 in Table 1 might be misleading. Figure 4(a) shows that only one NAS output in the end achieves significantly smaller test error compared to all the remaining results.
6. Are the models involved in EquiNAS_D significantly larger compared to the baseline? After all, a linear combination of layers equivariant to different groups is taken. Does that mean the model is slow to train and (especially) test?

The evidence to be evaluated is:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

Agrawal & Ostrowski (2022) presents theoretical contributions towards network morphisms for group-invariant shallow neural networks: in comparison, our work focuses on deep group convolutional architectures and implements the morphism in a NAS algorithm.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

The equivariance relaxation morphism generalizes this concept to any subgroup.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum. The equivariance relaxation morphism provides a mapping of group elements between a group and a subgroup. For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers. Towards finding the optimal full equivariance per layer, the equivariance relaxation morphism presented in Section 4.1 is applied as the genetic operator in an evolutionary hill-climbing algorithm. The Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) algorithm, given in Algorithm 1, is similar to other evolutionary NAS methods such as Elsken et al. (2017) with pareto selection as in Falanti et al. (2022). A population of networks, which starts with an individual with all layers equivariant to the largest possible group, undergoes mutation via equivariance relaxation and selection based on accuracy and parameter count to optimize neural architecture while learning network parameters.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance. We next study empirically the two EquiNAS methods on three datasets, one with known rotational symmetry and two with unknown but visually significant rotational and reflectional symmetry. We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation. The \(p4\) group consists of discrete translations and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes C_{4}\). The \(p4m\) group consists of discrete translations, reflections, and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes D_{4}\). As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively.

We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\). We present experiments on image classification for a variety of datasets. The Rotated MNIST dataset (Larochelle et al., 2007, RotMNIST) is a version of the MNIST handwritten digit dataset but with the images rotated by any angle. This task serves as a simple investigational study with known symmetry, while the following two tasks are more realistic and complex.

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

The advantages of equivariance search methods are most apparent in the Galaxy10 benchmark. While EquiNAS\({}_{E}\) outperforms most baselines on RotMNIST and all baselines on ISIC, it has similar performance and some final architectures to the \(D_{4}\) baseline for both tasks. However, the \(D_{4}\) baseline fails at the Galaxy10 task, demonstrating that the same equivariant architecture can not always be naively applied. Both search methods, EquiNAS\({}_{E}\) and RPP, outperform all baseline models on Galaxy10, and by a large margin for EquiNAS\({}_{E}\).

the selected population maintains a fully equivariant network in every generation.

These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

EquiNAS\({}_{E}\) may particularly work well on tasks that benefit from local equivariance, determined by analyzing the architecture weighting parameters from first applying the more efficient EquiNAS\({}_{D}\), as well as for finding good discrete architectures within which to retrain weights, based on the ablation and random comparisons. We thus recommend EquiNAS\({}_{D}\) for practical applications if the final model is not restricted to discrete equivariance, in which case it can be used to inform design decisions for applying EquiNAS\({}_{E}\).

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

Evolutionary algorithms are optimization methods inspired by evolution in biology, where individuals in a population compete with their phenotypic traits in order to pass on their genotypic traits to offspring. The _population_ is the current collection of individuals. Each _individual_ is an instance of the object to be optimized and has a genotype that is decoded into a phenotype. In this case, each individual is a neural network, with a genotype that encodes the parametrized equivariance group of each convolutional layer, represented as a vector of integers. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. Each parent itself is kept for the next population, as well as each valid child that is generated via the equivariance relaxation morphism, such that they are functionally equivalent to their parent at initialization (although with a different architecture) and thus have the same fitness before training. Each individual in this population is partially trained, such that these children diverge from their siblings and parent, so that the next set of parents may be selected and this process repeats.

To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion. The new filter tensor has a shape of \(C_{l}|R|\times C_{l-1}|R|\times S_{G_{l}}/|R|\times K_{l}\times K_{l}\). The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension. Every convolutional and linear layer except the output layer is immediately followed by a batchnorm then a ReLU.

For EquiNAS\({}_{E}\), the parent selection size is 5, the training time per generation is 0.5 epochs, and the number of generations is 50 for all tasks.

As ablation studies and comparisons, we implement two kinds of random search for each NAS method. The first ablates smart architecture search: EquiNAS\({}_{E}\) Random Select works as described in Algorithm 1 but with random parent selection (instead of pareto-front selection) and EquiNAS\({}_{D}\) Random \(Z\) works as described in Algorithm 2 but with random \(Z\) updates (instead of gradient descent) by shuffling \(Z\) gradients. The second is more akin to standard NAS random search: for the evolutionary paradigm, we train 30 randomly selected static architectures in the discrete architecture search space for the same training time and selecting the top 5 by validation accuracy, and for the differentiable paradigm, we train 25 randomly selected static architectures in the continuous architecture search space and selecting the top 5 by validation accuracy. 30 and 25 were respectively calculated to be approximately the same compute cost as the trials of EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\). These are labeled as "Random Static" for both the evolutionary and differentiable paradigms. Since Random Static trains static architectures while EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) dynamically search for both architectures and parameters, we take the best 5 architectures for each and retrain their param eters from scratch as in the standard NAS paradigm, labeled as EquiNAS\({}_{E}\) Retrain and EquiNAS\({}_{D}\) Retrain, respectively, for fair comparison to Random Static.

EquiNAS\({}_{D}\) finds competitive architectures on average and can find architectures which outperform baseline choices like architectures fully equivariant to \(C_{1}\) or \(D_{4}\). From comparing EquiNAS\({}_{D}\) Retrain to EquiNAS\({}_{D}\) results in Figure 8, retraining a resulting architecture is not consistently better or worse than using the final weights from EquiNAS\({}_{D}\), showing that there isn't a disadvantage to training weights during search and avoiding the additional cost of a post-search training step.

On this baseline, EquiNAS\({}_{D}\) greatly outperforms a search of random architectures, demonstrating that EquiNAS\({}_{D}\) can discover the appropriate equivariance for a specific dataset over fixed or randomly selected architectures.

Our methods enable searching for both architecture and parameters concurrently in a single training process. This approach is more efficient than NAS approaches that only search for architectures, requiring a retraining process within the resulting architecture for evaluation.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 8: Test errors for experiments of Section 6.2.

Input: Figure 2: Historical parameter counts of all selected individuals for EquiNASE on RotMNIST. The architectures of the final selected population are labeled. Each history is colored by the final test accuracy, measured of each individual upon removal. For other tasks, see Figure 7 in Appendix D.

Input: Figure 4: EquiNASD on classification of six augmentations of MNIST, where each row is a trial on MNIST augmented to exhibit symmetry to the labeled group.

Input: Figure 9: Test errors against stored parameter count for experiments of Section 6.2. Although EquiNASD and RPP train with a higher parameter count, the parameters may be stored in a single filter per layer using Equation 11 for testing.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 7: Historical parameter counts of each selected individual for EquiNASE (see Figure 2 for RotMNIST).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim that "we implement these concepts within two algorithms for architectural optimization of partially-equivariant networks" appears well-stated. The evidence thoroughly demonstrates the implementation of two distinct algorithms (EquiNAS_E and EquiNAS_D) that use the equivariance relaxation morphism and [G]-mixed equivariant layer concepts for architectural optimization. The paper provides detailed mathematical formulations, algorithmic descriptions, experimental results across multiple datasets, and visualizations demonstrating how these concepts are implemented. The technical depth and empirical validation support this as an accurate representation without exaggeration.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Second, we implement these concepts within two algorithms for architectural optimization of partially-equivariant networks.

The review comment to be evaluated is:

## Strengths
- Equivariant networks have more hyperparameters to tune due to the choice of symmetry group.  For example, in E(2)-CNN (https://arxiv.org/abs/1911.08251), the authors experiment with symmetry groups $C_n$ and $D_n$ for many values of n before finding n=12 to work best.  This paper proposes a method for automatically determining the correct level of symmetry.
- Moreover, on many problems end-to-end strict symmetry is not desirable if the symmetry in the domain is only approximate or if symmetry constraints interfere with optimization.  The proposed method adds to the relatively few methods which learn relaxed symmetry constraints and adapt to the level of symmetry in the data.
- I found the equivariance relaxation morphism to be a simple and effective idea for relaxing symmetry constraints in networks on a per-layer basis.
- The [G]-mixed equivariant layer also seems like a reasonable idea for selecting the degree of equivariance by differentiably choosing weights over kernels with different levels of equivariance.  In theory this allows the model to learn a specific equivariance or an approximate equivariance which is in-between.  It also allows the network to reduce the symmetry constraint in later layers, which is what is done effectively by CNNs in practice by downsampling with stride.
- Experimental results on the learned subgroup weights (Fig. 2) reveal interesting trends showing that the networks do select for higher equivariance in early layers and lower equivariance in later layers which matches what has found to be effective in practice for equivariant networks.

## Weaknesses / Questions
- I am not sure of the framing of the method.  I am not an expert on NAS and am open to correction, but it seems to me both proposed methods find a trained network, not an architecture, since they are training the network weights and selecting an architecture at the same time.  The final architecture may not have great performance if trained from scratch.  In this sense, it seems the method is more a special training procedure than NAS method.   EquiNAS_D, in particular, is simply an architecture in which some weights are optimized alternatingly.
- Given the evolutionary algorithm starts with a single network, how do we know to what extent its evolution is guided by architecture versus the specific weights in the mutated networks?  In experiments, I think both population level variance and initialization variance should be accounted for.
- One of the biggest issues is the fact that the relaxation morphism increases the number of parameters making it difficult to distinguish performance increases are from relaxing constrains or increasing parameter count.  The authors do discuss this issue and suggest their optimization strategy over both parameter count and accuracy helps.  It would be better, however, if the number of parameters could be preserved by the relaxation.
- The experimental results are not completely convincing of when this method would add significant value over simply doing hyperparameter search over equivariant networks.  In Table 1, an equivariant baseline is best on rotMNIST and ISIC, and a non-rotationally equivariant network with equivariant initialization is best for Galaxy10.  In table 2, EquiNAS_D does do best for rotMNIST, but the score is much lower than the best values in E(2)-CNN after full tuning of the best equivariant method.  In Galaxy10, EquiNAS_D does outperform, but with very high variance.
- I have some issues with the clarity of the description of 4.2.  The condition that all groups in $[G]$ be subgroups or supergroups of all other groups in $[G]$ does not appear to be met by the expermental example since $D_2$ is not a subgroup or supergroup of $C_4$.  It's also not clear to me why this condition is necessary.  It is a fairly strict condition, although I think it would include useful cases.  I am also a bit unclear on the form of the input signal $f$.  In order to be convolved with $\tilde{\Psi}$ and have $G$-equivariance for each $G$ in $[G]$, $f$ should be defined over a supergroup of all $G \in [G]$.  Is this the case?   I am correct that if $z_G$ is a one-hot vector on the group $G$, then the operation is $G$-equivariant?

## Questions
- If the evolution algorithm can only mutate in the direction of symmetry constrain relaxation, doesn't this bias the algorithm towards relaxing all the symmetry constraints?
- Does this work have a relation to diffstride (https://arxiv.org/abs/2202.01653) in that stride is typically a hyperparameter that corresonds to a choice of the subgroup of the translation group and can be shown to differentiably optimized?
- After equivariance is broken in a given layer, there isn't much point in principle to persevering it downstream.  Are mutations automatically applied to all following layers or did your method discover this?  If so, I think that is interesting enough to be considered a strength.

## Minor
- In Eqn 1, if you reverse the order of $f$ and $\Psi$ and drop the sum over c and indices $c,d$, the equation can be simplified using matrix multiplication.
- 4.1, para 2, "neutral element" -> "identity element"
- 4.1 may benefit from writing as a proposition
- 4.1, Para 3, Line 5,  $h \in G'$ -> $h' \in G'$
- Sec 5., Line 4, typo "optimizffe"

The evidence to be evaluated is:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

Agrawal & Ostrowski (2022) presents theoretical contributions towards network morphisms for group-invariant shallow neural networks: in comparison, our work focuses on deep group convolutional architectures and implements the morphism in a NAS algorithm.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

The equivariance relaxation morphism generalizes this concept to any subgroup.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum. The equivariance relaxation morphism provides a mapping of group elements between a group and a subgroup. For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers. Towards finding the optimal full equivariance per layer, the equivariance relaxation morphism presented in Section 4.1 is applied as the genetic operator in an evolutionary hill-climbing algorithm. The Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) algorithm, given in Algorithm 1, is similar to other evolutionary NAS methods such as Elsken et al. (2017) with pareto selection as in Falanti et al. (2022). A population of networks, which starts with an individual with all layers equivariant to the largest possible group, undergoes mutation via equivariance relaxation and selection based on accuracy and parameter count to optimize neural architecture while learning network parameters.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance. We next study empirically the two EquiNAS methods on three datasets, one with known rotational symmetry and two with unknown but visually significant rotational and reflectional symmetry. We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation. The \(p4\) group consists of discrete translations and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes C_{4}\). The \(p4m\) group consists of discrete translations, reflections, and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes D_{4}\). As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively.

We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\). We present experiments on image classification for a variety of datasets. The Rotated MNIST dataset (Larochelle et al., 2007, RotMNIST) is a version of the MNIST handwritten digit dataset but with the images rotated by any angle. This task serves as a simple investigational study with known symmetry, while the following two tasks are more realistic and complex.

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

The advantages of equivariance search methods are most apparent in the Galaxy10 benchmark. While EquiNAS\({}_{E}\) outperforms most baselines on RotMNIST and all baselines on ISIC, it has similar performance and some final architectures to the \(D_{4}\) baseline for both tasks. However, the \(D_{4}\) baseline fails at the Galaxy10 task, demonstrating that the same equivariant architecture can not always be naively applied. Both search methods, EquiNAS\({}_{E}\) and RPP, outperform all baseline models on Galaxy10, and by a large margin for EquiNAS\({}_{E}\).

the selected population maintains a fully equivariant network in every generation.

These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

EquiNAS\({}_{E}\) may particularly work well on tasks that benefit from local equivariance, determined by analyzing the architecture weighting parameters from first applying the more efficient EquiNAS\({}_{D}\), as well as for finding good discrete architectures within which to retrain weights, based on the ablation and random comparisons. We thus recommend EquiNAS\({}_{D}\) for practical applications if the final model is not restricted to discrete equivariance, in which case it can be used to inform design decisions for applying EquiNAS\({}_{E}\).

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

Evolutionary algorithms are optimization methods inspired by evolution in biology, where individuals in a population compete with their phenotypic traits in order to pass on their genotypic traits to offspring. The _population_ is the current collection of individuals. Each _individual_ is an instance of the object to be optimized and has a genotype that is decoded into a phenotype. In this case, each individual is a neural network, with a genotype that encodes the parametrized equivariance group of each convolutional layer, represented as a vector of integers. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. Each parent itself is kept for the next population, as well as each valid child that is generated via the equivariance relaxation morphism, such that they are functionally equivalent to their parent at initialization (although with a different architecture) and thus have the same fitness before training. Each individual in this population is partially trained, such that these children diverge from their siblings and parent, so that the next set of parents may be selected and this process repeats.

To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion. The new filter tensor has a shape of \(C_{l}|R|\times C_{l-1}|R|\times S_{G_{l}}/|R|\times K_{l}\times K_{l}\). The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension. Every convolutional and linear layer except the output layer is immediately followed by a batchnorm then a ReLU.

For EquiNAS\({}_{E}\), the parent selection size is 5, the training time per generation is 0.5 epochs, and the number of generations is 50 for all tasks.

As ablation studies and comparisons, we implement two kinds of random search for each NAS method. The first ablates smart architecture search: EquiNAS\({}_{E}\) Random Select works as described in Algorithm 1 but with random parent selection (instead of pareto-front selection) and EquiNAS\({}_{D}\) Random \(Z\) works as described in Algorithm 2 but with random \(Z\) updates (instead of gradient descent) by shuffling \(Z\) gradients. The second is more akin to standard NAS random search: for the evolutionary paradigm, we train 30 randomly selected static architectures in the discrete architecture search space for the same training time and selecting the top 5 by validation accuracy, and for the differentiable paradigm, we train 25 randomly selected static architectures in the continuous architecture search space and selecting the top 5 by validation accuracy. 30 and 25 were respectively calculated to be approximately the same compute cost as the trials of EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\). These are labeled as "Random Static" for both the evolutionary and differentiable paradigms. Since Random Static trains static architectures while EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) dynamically search for both architectures and parameters, we take the best 5 architectures for each and retrain their param eters from scratch as in the standard NAS paradigm, labeled as EquiNAS\({}_{E}\) Retrain and EquiNAS\({}_{D}\) Retrain, respectively, for fair comparison to Random Static.

EquiNAS\({}_{D}\) finds competitive architectures on average and can find architectures which outperform baseline choices like architectures fully equivariant to \(C_{1}\) or \(D_{4}\). From comparing EquiNAS\({}_{D}\) Retrain to EquiNAS\({}_{D}\) results in Figure 8, retraining a resulting architecture is not consistently better or worse than using the final weights from EquiNAS\({}_{D}\), showing that there isn't a disadvantage to training weights during search and avoiding the additional cost of a post-search training step.

On this baseline, EquiNAS\({}_{D}\) greatly outperforms a search of random architectures, demonstrating that EquiNAS\({}_{D}\) can discover the appropriate equivariance for a specific dataset over fixed or randomly selected architectures.

Our methods enable searching for both architecture and parameters concurrently in a single training process. This approach is more efficient than NAS approaches that only search for architectures, requiring a retraining process within the resulting architecture for evaluation.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 8: Test errors for experiments of Section 6.2.

Input: Figure 2: Historical parameter counts of all selected individuals for EquiNASE on RotMNIST. The architectures of the final selected population are labeled. Each history is colored by the final test accuracy, measured of each individual upon removal. For other tasks, see Figure 7 in Appendix D.

Input: Figure 4: EquiNASD on classification of six augmentations of MNIST, where each row is a trial on MNIST augmented to exhibit symmetry to the labeled group.

Input: Figure 9: Test errors against stored parameter count for experiments of Section 6.2. Although EquiNASD and RPP train with a higher parameter count, the parameters may be stored in a single filter per layer using Equation 11 for testing.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 7: Historical parameter counts of each selected individual for EquiNASE (see Figure 2 for RotMNIST).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
The evidence strongly supports this claim. The authors clearly demonstrate implementation of two distinct algorithms:

1. EquiNAS_E: An evolutionary algorithm that uses the equivariance relaxation morphism to search for networks with different equivariance constraints per layer
2. EquiNAS_D: A differentiable algorithm that optimizes weighted mixtures of equivariant layers

The implementation details are thoroughly documented with pseudocode, architectures, and experimental results showing the algorithms in action (Figures 1-12). The reviewer's comments acknowledge these as "reasonable" approaches. The claim accurately reflects what was implemented without overstating capabilities or novelty.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Second, we implement these concepts within two algorithms for architectural optimization of partially-equivariant networks.

The review comment to be evaluated is:

### Strengths:
1. The problem of finding the correct equivariance that can exploit symmetries in the data is important for the development of general-purpose NAS methods.
2. The proposed approaches are simple and natural procedures.
3. The writing is fairly clear and the experimental results are interesting.
4. Code will be released upon acceptance.

### Weaknesses:
1. The experimental evaluation is somewhat limited. Given the fairly direct extension of prior work (morphisms and weight-sharing) to actually developing the search algorithms, I would expect more emphasis to be placed on experimental depth. This could either come in the form of evaluation on interesting benchmarks where we might expect finding symmetries to outperform regular CNNs (e.g. Tu et al. (2022)) and/or more detailed analysis on more involved search spaces of whether the proposed methods are recovering the “right” symmetry for the dataset, even if synthetic.
2. The authors do not consider the random search baseline for their method. More generally, it would be useful to follow the NAS reproducibility checklist here (Lindauer & Hutter, 2020).

### References:
- Lindauer, Hutter. Best practices for scientific research on neural architecture search. JMLR 2020.
- Tu, Roberts, Khodak, Shen, Sala, Talwalkar. NAS-Bench-360: Benchmarking neural architecture search on diverse tasks. NeurIPS 2022.

The evidence to be evaluated is:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

Agrawal & Ostrowski (2022) presents theoretical contributions towards network morphisms for group-invariant shallow neural networks: in comparison, our work focuses on deep group convolutional architectures and implements the morphism in a NAS algorithm.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

The equivariance relaxation morphism generalizes this concept to any subgroup.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum. The equivariance relaxation morphism provides a mapping of group elements between a group and a subgroup. For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers. Towards finding the optimal full equivariance per layer, the equivariance relaxation morphism presented in Section 4.1 is applied as the genetic operator in an evolutionary hill-climbing algorithm. The Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) algorithm, given in Algorithm 1, is similar to other evolutionary NAS methods such as Elsken et al. (2017) with pareto selection as in Falanti et al. (2022). A population of networks, which starts with an individual with all layers equivariant to the largest possible group, undergoes mutation via equivariance relaxation and selection based on accuracy and parameter count to optimize neural architecture while learning network parameters.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance. We next study empirically the two EquiNAS methods on three datasets, one with known rotational symmetry and two with unknown but visually significant rotational and reflectional symmetry. We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation. The \(p4\) group consists of discrete translations and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes C_{4}\). The \(p4m\) group consists of discrete translations, reflections, and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes D_{4}\). As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively.

We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\). We present experiments on image classification for a variety of datasets. The Rotated MNIST dataset (Larochelle et al., 2007, RotMNIST) is a version of the MNIST handwritten digit dataset but with the images rotated by any angle. This task serves as a simple investigational study with known symmetry, while the following two tasks are more realistic and complex.

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

The advantages of equivariance search methods are most apparent in the Galaxy10 benchmark. While EquiNAS\({}_{E}\) outperforms most baselines on RotMNIST and all baselines on ISIC, it has similar performance and some final architectures to the \(D_{4}\) baseline for both tasks. However, the \(D_{4}\) baseline fails at the Galaxy10 task, demonstrating that the same equivariant architecture can not always be naively applied. Both search methods, EquiNAS\({}_{E}\) and RPP, outperform all baseline models on Galaxy10, and by a large margin for EquiNAS\({}_{E}\).

the selected population maintains a fully equivariant network in every generation.

These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

EquiNAS\({}_{E}\) may particularly work well on tasks that benefit from local equivariance, determined by analyzing the architecture weighting parameters from first applying the more efficient EquiNAS\({}_{D}\), as well as for finding good discrete architectures within which to retrain weights, based on the ablation and random comparisons. We thus recommend EquiNAS\({}_{D}\) for practical applications if the final model is not restricted to discrete equivariance, in which case it can be used to inform design decisions for applying EquiNAS\({}_{E}\).

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

Evolutionary algorithms are optimization methods inspired by evolution in biology, where individuals in a population compete with their phenotypic traits in order to pass on their genotypic traits to offspring. The _population_ is the current collection of individuals. Each _individual_ is an instance of the object to be optimized and has a genotype that is decoded into a phenotype. In this case, each individual is a neural network, with a genotype that encodes the parametrized equivariance group of each convolutional layer, represented as a vector of integers. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. Each parent itself is kept for the next population, as well as each valid child that is generated via the equivariance relaxation morphism, such that they are functionally equivalent to their parent at initialization (although with a different architecture) and thus have the same fitness before training. Each individual in this population is partially trained, such that these children diverge from their siblings and parent, so that the next set of parents may be selected and this process repeats.

To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion. The new filter tensor has a shape of \(C_{l}|R|\times C_{l-1}|R|\times S_{G_{l}}/|R|\times K_{l}\times K_{l}\). The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension. Every convolutional and linear layer except the output layer is immediately followed by a batchnorm then a ReLU.

For EquiNAS\({}_{E}\), the parent selection size is 5, the training time per generation is 0.5 epochs, and the number of generations is 50 for all tasks.

As ablation studies and comparisons, we implement two kinds of random search for each NAS method. The first ablates smart architecture search: EquiNAS\({}_{E}\) Random Select works as described in Algorithm 1 but with random parent selection (instead of pareto-front selection) and EquiNAS\({}_{D}\) Random \(Z\) works as described in Algorithm 2 but with random \(Z\) updates (instead of gradient descent) by shuffling \(Z\) gradients. The second is more akin to standard NAS random search: for the evolutionary paradigm, we train 30 randomly selected static architectures in the discrete architecture search space for the same training time and selecting the top 5 by validation accuracy, and for the differentiable paradigm, we train 25 randomly selected static architectures in the continuous architecture search space and selecting the top 5 by validation accuracy. 30 and 25 were respectively calculated to be approximately the same compute cost as the trials of EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\). These are labeled as "Random Static" for both the evolutionary and differentiable paradigms. Since Random Static trains static architectures while EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) dynamically search for both architectures and parameters, we take the best 5 architectures for each and retrain their param eters from scratch as in the standard NAS paradigm, labeled as EquiNAS\({}_{E}\) Retrain and EquiNAS\({}_{D}\) Retrain, respectively, for fair comparison to Random Static.

EquiNAS\({}_{D}\) finds competitive architectures on average and can find architectures which outperform baseline choices like architectures fully equivariant to \(C_{1}\) or \(D_{4}\). From comparing EquiNAS\({}_{D}\) Retrain to EquiNAS\({}_{D}\) results in Figure 8, retraining a resulting architecture is not consistently better or worse than using the final weights from EquiNAS\({}_{D}\), showing that there isn't a disadvantage to training weights during search and avoiding the additional cost of a post-search training step.

On this baseline, EquiNAS\({}_{D}\) greatly outperforms a search of random architectures, demonstrating that EquiNAS\({}_{D}\) can discover the appropriate equivariance for a specific dataset over fixed or randomly selected architectures.

Our methods enable searching for both architecture and parameters concurrently in a single training process. This approach is more efficient than NAS approaches that only search for architectures, requiring a retraining process within the resulting architecture for evaluation.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 8: Test errors for experiments of Section 6.2.

Input: Figure 2: Historical parameter counts of all selected individuals for EquiNASE on RotMNIST. The architectures of the final selected population are labeled. Each history is colored by the final test accuracy, measured of each individual upon removal. For other tasks, see Figure 7 in Appendix D.

Input: Figure 4: EquiNASD on classification of six augmentations of MNIST, where each row is a trial on MNIST augmented to exhibit symmetry to the labeled group.

Input: Figure 9: Test errors against stored parameter count for experiments of Section 6.2. Although EquiNASD and RPP train with a higher parameter count, the parameters may be stored in a single filter per layer using Equation 11 for testing.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 7: Historical parameter counts of each selected individual for EquiNASE (see Figure 2 for RotMNIST).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Based on my review, the claim that the authors implement the concepts within two algorithms for architectural optimization of partially-equivariant networks appears well-stated. The evidence thoroughly demonstrates the implementation of:

1. EquiNAS_E: An evolutionary algorithm using the equivariance relaxation morphism
2. EquiNAS_D: A differentiable algorithm using the [G]-mixed equivariant layer

The evidence provides detailed descriptions of both algorithms, their implementations, and experimental results across multiple datasets. The visualizations in Figures 1-12 demonstrate the practical application of these algorithms. The review comments also acknowledge the approaches as "simple and natural procedures" with clear writing and interesting results.

<score>1</score>