ISTA-NAS: Efficient and Consistent Neural Architecture Search by Sparse Coding

Introduction

Current NAS studies can be mainly categorized into reinforcement learning-based , evolution-based , Bayesian optimization-based , and gradient-based methods . Most reinforcement learning and evolution-based algorithms suffer from huge computational cost, while gradient-based methods are simple to implement.

In gradient-based methods, an over-parameterized super-net is constructed to cover all candidate connections. Different architectures are sub-graphs of the super-net by weight sharing . Thus the aim of search is to determine a sparse solution from the high-dimensional architecture space. Liu et al. propose a differentiable search framework, DARTS , by introducing a set of architecture variables jointly optimized with the network weights. After search, the architecture variables are projected to be sparse by only keeping the top-2 strongest connections for each intermediate node as the target-net for evaluation. This method is the basis of many follow-up studies .

However, we note that the architecture variables do not satisfy the sparsity constraint during search. There is a gap between the dense super-net in search and the sparse target-net in evaluation (the term “evaluation” in this paper denotes re-training on the full train set and validation on the test set). From the perspective of optimization, solutions should be constrained in the feasible region at each iteration, as adopted in the Projected Gradient Descent (PGD) and proximal algorithms . Otherwise, the converged solution may be far from the optimal one in the feasible region. It explains the fact that there is a poor correlation between the performances of super-net for search and target-net for evaluation in DARTS . A high-performance super-net does not necessarily produce a good architecture after projection onto the sparsity constraint. Besides, the dense super-net covering all candidate connections is inefficient to train due to its huge computational and memory cost.

In some follow-up studies , the Gumbel Softmax strategy is adopted to enforce sparsification. Nevertheless, the super-net still contains the whole graph which is different from the derived architecture in target-net and does not reduce the computational and memory consumption. In ProxylessNAS and NASP , only sampled or projected connections are active for each edge to reduce the memory consumption and search cost. But the active paths of super-net still deviate from the target-net as there is no connection for some edges in the target-net. For single-path architecture search, DSNAS proposes a one-stage method to eliminate the inconsistency between super-net and target-net brought by two-stage methods, while there is yet no single-stage solution to DARTS-based architecture where the dimension of candidate connections is relatively higher.

Inspired by the observation that architecture variables in DARTS should have a sparse structure that can be well-represented by a compact space, in this paper, we propose to formulate NAS as a sparse coding problem, named ISTA-NAS. We construct an equivalent compressed search space where each point corresponds to a sparse solution in the original space. We perform gradient-based search in the compressed space with the sparsity constraint inherently satisfied, and then recover a new architecture by the sparse coding problem, which can be efficiently solved by well-developed methods, such as the iterative shrinkage thresholding algorithm (ISTA) . The differentiable search and architecture recovery are conducted in an alternate way, so at each update, the network for search is sparse and efficient to train. After convergence, there is no need of projection onto sparsity constraint by post-processing and the searched architecture is directly available for evaluation.

In previous studies , the super-net in search is dense, and thus has a smaller depth and width than the target-net setting due to limited memory. However, the number of cells and channels has a significant effect on the search result . In order to also eliminate these gaps between super-net in search and target-net in evaluation, we develop a one-stage framework where search and evaluation share the same super-net under the target-net settings, such as depth, width and batchsize. One-stage ISTA-NAS begins to search with all batch normalization (BN) layers frozen. When a termination condition is satisfied, architecture variables cease to change and one sparse architecture is searched and fixed. BN layers are now trainable and other network weights continue to be optimized. After training, architecture variables are absorbed into the parameters of BN layers, and we get the searched architecture and all optimized parameters in a single run with only evaluation cost.

We list the contributions of this paper as follows:

We formulate neural architecture search as a sparse coding problem and propose ISTA-NAS, which performs the differentiable search on an equivalent compressed space and recovers its corresponding sparse architecture in an alternate manner. The sparsity constraint is inherently satisfied at each update so the search is more efficient and consistent with evaluation.

We further develop a one-stage ISTA-NAS that incorporates search and evaluation in a single framework under the target-net settings. The network for search has no gap in terms of depth, width and even training batchsize with the derived architecture for evaluation.

In experiments, the two-stage ISTA-NAS finishes search on CIFAR-10 in only 0.05 GPU-day. And the one-stage version produces state-of-the-art performances in a single run on CIFAR-10 or directly on ImageNet at the cost of only evaluation time.

Related Work

Neural architecture search (NAS) has been proposed to spare the manual efforts in traditional networks and benefit application tasks, such as object detection , and semantic segmentation . Current gradient-based search methods widely adopt a two-stage strategy. In the first stage for search, the dense super-net covers all candidate connections and has to decrease the depth and width. As a result, it has a gap with the target-net for evaluation in the second stage, and the two stages correlate poorly. Several studies have been proposed for this problem.

Reducing the gap. To derive an architecture that is close to the one during search, Gumbel Softmax and sparsity or entropy regularization are adopted to enforce sparse architecture variables. But the whole graph still needs to be stored and computed, which is inefficient to train for the dense super-net. Some studies use a sparse super-net by only keeping sampled or projected connections active . P-DARTS reduces the depth gap by gradually dropping connections and increasing depths. Even if these studies improve the correlation between the two stages, their super-nets only approach to but do not strictly satisfy the sparsity constraint. A post-processing is still needed to derive the target-net. Our method differs from theses studies in that we keep the sparsity constraint inherently satisfied at each update, and do not need the projection as post-processing.

One-stage NAS. Some studies have proposed one-stage search methods to circumvent the gap brought by two-stage methods. The architecture variables and network weights are simultaneously optimized. However, they are all proposed for the chain-based search space . We show that ISTA-NAS also enables one-stage search. As a comparison, our method is developed for the cell-based search space in DARTS, which has a higher dimension because candidate connections for each intermediate node come from various operations of multiple previous nodes.

We note that a study introduces compressed sensing into NAS. But the method is based on a meta-learning algorithm, instead of our differentiable architecture search. Besides, tries to combine optimization algorithms with neural architecture design, instead of search.

ISTA-NAS

In this section, we first make a brief review of the differentiable architecture search and sparse coding. Then we build their relation and formulate NAS as a sparse coding problem. Finally, we introduce our two-stage and one-stage ISTA-NAS algorithms, respectively.

The search space of a cell in DARTS-based method is formed as a directed acyclic graph (DAG) consisting of $n$ ordered nodes $\{x_{1},x_{2},\cdots,x_{n}\}$ and their edges $\mathcal{E}=\{e^{(i,j)}|1\leq i<j\leq n\}$. Each edge has $K$ candidate operations from $\mathcal{O}=\{o_{1},o_{2},\cdots,o_{K}\}$, such as identity, max-pooling, and $3\times 3$ separable convolution. With a binary variable $z^{(i,j)}_{k}\in\{0,1\}$ to indicate whether the corresponding connection is active, we have the intermediate node $x_{j}$ as:

where ${Z}=\{\mathbf{z}_{j}\}_{j=2}^{n}$, ${W}$ is the weights of super-net $\mathcal{N}$, and $s_{j}$ denotes the sparseness of node $j$. Since the architecture variables $\mathbf{z}_{j}$ are binary and thus stubborn to optimize in a differentiable way, current studies adopt the continuous relaxation by $z^{(i,j)}_{k}=\exp(\alpha^{(i,j)}_{k})/\sum_{k}\exp(\alpha^{(i,j)}_{k})$ and optimize $\alpha^{(i,j)}_{k}$ as trainable variables . In the search phase, the sparsity constraint Eq. (4) is not considered. $\alpha^{(i,j)}_{k}$ and $W$ are optimized in an alternate manner by solving Eq. (2) and Eq. (3), respectively. After search, $\mathbf{z}_{j}$ is projected onto the sparsity constraint Eq. (4) by only keeping the top-2 strongest dimensions for node $j$, i.e., $s_{j}=2,\forall\ 1<j\leq n$.

It is shown that current studies widely ignore the sparsity constraint Eq. (4) during search, and the sparse architecture variables $\mathbf{z}_{j}$ are derived by post-processing. This may cause an invalid search process because there is few correlation between the optimal solutions inside and outside the feasible region. Constraints should be satisfied at each step of optimization . In addition, the dense super-net $\mathcal{N}({W},{Z})$ covers all candidate connections in search. As a result, it is inefficient to train, and has to adopt a small depth, width and training batchsize due to limited memory, which introduces the gap of depth, width and batchsize with the target-net in evaluation.

2 Preliminaries on Sparse Coding

where $\lambda$ is the regularization parameter. Many well-developed algorithms are proposed to solve the problem Eq. (6), such as the iterative shrinkage thresholding algorithm (ISTA) :

where $L$ is taken as the largest eigenvalue of $\mathbf{A}^{T}\mathbf{A}$, and $\eta_{\theta}$ is the component-wise soft-thresholding operator defined as: $\eta_{\theta}(x)={\rm sign}(x)\max(|x|-\theta,0)$. ISTA is a proximal gradient method applied to Eq. (6) and has an $O(1/t)$ convergence rate with a proper $\lambda$ .

3 Formulate Differentiable NAS as Sparse Coding

where $x_{j}$ and $\mathbf{o}_{j}$ are defined in Eq. (1), $B$ denotes the architecture variables in the network $\mathcal{N}({W},{B})$, i.e., $B=\{\mathbf{b}_{j}\}_{j=2}^{n}$, and $\mathbf{z}_{j}(\mathbf{b}_{j})$ is deemed as a function of $\mathbf{b}_{j}$ in $\mathcal{N}({W},{B})$. The two arrows denote how $x_{j}$ is produced in the networks $\mathcal{N}({W},{Z})$ and $\mathcal{N}({W},{B})$, respectively. We note that $\mathcal{N}({W},{B})$ and $\mathcal{N}({W},{Z})$ have the same propagation for node $x_{j},\forall\ 1<j\leq n$ when $\mathbf{b}_{j}=\mathbf{A}_{j}\mathbf{z}_{j}$.

Assume that $\mathbf{A}$ satisfies the RIP with its constant $\delta_{2s}$ and the exact $s$-sparse solution $\mathbf{z}^{*}$ can be recovered by $\operatorname*{argmin}_{\mathbf{z}}\frac{1}{2}\|\mathbf{A}\mathbf{z}-\mathbf{b}\|_{2}^{2}+\lambda\|\mathbf{z}\|_{1}$ and satisfies $\mathbf{A}\mathbf{z}^{*}=\mathbf{b}$. Then we have that $\mathbf{z}^{*}$ is the optimal solution of the network $\mathcal{N}({W},\mathbf{z})$ if and only if $\mathbf{b}^{*}=\mathbf{A}\mathbf{z}^{*}$ is the optimal solution of the network $\mathcal{N}({W},\mathbf{b})$.

For sufficiency, suppose that there exists another $s$-sparse $\mathbf{z}^{\prime}\neq\mathbf{z}^{*}$, such that $\mathcal{L}(\mathcal{N}({W},\mathbf{z}^{\prime}))<\mathcal{L}(\mathcal{N}({W},\mathbf{z}^{*}))$. Then $\mathbf{b}^{\prime}=\mathbf{A}\mathbf{z}^{\prime}\neq\mathbf{b}^{*}$ due to the RIP of $\mathbf{A}$. Because the two networks $\mathcal{N}({W},\mathbf{b})$ and $\mathcal{N}({W},\mathbf{z})$ have the same propagation by Eq. (8), we have $\mathcal{L}(\mathcal{N}({W},\mathbf{b}^{\prime}))=\mathcal{L}(\mathcal{N}({W},\mathbf{z}^{\prime}))<\mathcal{L}(\mathcal{N}({W},\mathbf{z}^{*}))=\mathcal{L}(\mathcal{N}({W},\mathbf{b}^{*}))$, which is in conflict with the fact that $\mathbf{b}^{*}=\mathbf{A}\mathbf{z}^{*}$ is the optimal solution of $\mathcal{N}({W},\mathbf{b})$. The necessity can be derived in the same way. Suppose that there exists another $\mathbf{b}^{\prime}\neq\mathbf{b}^{*}$, such that $\mathcal{L}(\mathcal{N}({W},\mathbf{b}^{\prime}))<\mathcal{L}(\mathcal{N}({W},\mathbf{b}^{*}))$. Then $\mathbf{z}^{\prime}=\operatorname*{argmin}_{\mathbf{z}}\frac{1}{2}\|\mathbf{A}\mathbf{z}-\mathbf{b}^{\prime}\|_{2}^{2}+\lambda\|\mathbf{z}\|_{1}\neq\mathbf{z}^{*}$. We have $\mathcal{L}(\mathcal{N}({W},\mathbf{z}^{\prime}))=\mathcal{L}(\mathcal{N}({W},\mathbf{b}^{\prime}))<\mathcal{L}(\mathcal{N}({W},\mathbf{b}^{*}))=\mathcal{L}(\mathcal{N}({W},\mathbf{z}^{*}))$, which is in conflict with the fact that $\mathbf{z}^{*}$ is the optimal solution of $\mathcal{N}({W},\mathbf{z})$, and concludes the proof. ∎

It is shown that a proper measurement matrix $\mathbf{A}$ builds the relation between the original and the compressed spaces. The optimal solution in $\Omega(\mathbf{z})$ can be searched by optimization in $\Omega(\mathbf{b})$. Thus we have our problem formulated as:

where $B=\{\mathbf{b}_{j}\}_{j=2}^{n}$ is the trainable variables in the network $\mathcal{N}({W},{B})$. $B$ and $W$ are optimized by the differentiable NAS without the sparsity constraint Eq. (4). $Z=\{\mathbf{z}_{j}\}_{j=2}^{n}$ is recovered by the sparse coding problem in Eq. (9), which inherently produces a sparse architecture.

4 Two-stage ISTA-NAS

The benefit of introducing sparse coding into NAS is that we can utilize the sparsity for more efficient search. In implementation, the solution of Eq. (9) may not lie in $\Omega(\mathbf{z})$ and strictly satisfy the sparsity constraint. We keep the top-$s$ strongest magnitudes and set other dimensions as zeros. Let $\mathcal{S}(\mathbf{z})$ denote the support set of $\mathbf{z}$, i.e., $\mathcal{S}(\mathbf{z})=\{i|\mathbf{z}(i)\neq 0\}$ and $|\mathcal{S}|=s$, where $i$ is the $i$-th dimension of $\mathbf{z}$ and $s$ is the sparseness. Then the network propagation in Eq. (8) is equivalent to:

where $\mathbf{z}_{(\mathcal{S})}$ denotes the elements of $\mathbf{z}$ indexed by $\mathcal{S}$, $\mathbf{A}_{(\mathcal{S})}$ denotes the columns of $\mathbf{A}$ indexed by ${\mathcal{S}}$, and $\mathbf{E}_{(\mathcal{S},\mathcal{S})}$ denotes the rows and columns of $\mathbf{E}$ indexed by ${\mathcal{S}}$.

We now introduce our two-stage ISTA-NAS outlined in Algorithm 1. The two-stage pipeline is consistent with current studies . A super-net with the whole DAG is constructed for search, after which the searched sparse architecture is for evaluation with a larger depth and width. We first initialize the network weights $W$ and architecture variables $B$ of the super-net. The measurement matrices $\mathbf{A}_{j}$ used to construct the relations between the original and the compressed spaces for each node are sampled and fixed. At each iteration of the algorithm, $\mathbf{b}_{j}$ corresponds to a sparse $\mathbf{z}_{j}$ by solving Eq. (9) with ISTA. Then we keep the top-$s$ strongest magnitudes of $\mathbf{z}_{j}$ and derive its support set $\mathcal{S}$, which indicates a sparse architecture. We only propagate the connections in $\mathcal{S}$ by Eq. (11), and then updates $W_{\mathcal{(S)}}$ and $\mathbf{b}$, respectively. The optimization of Eq. (10) is performed in an alternate way, which is the same as the widely-used bi-level scheme in current differentiable NAS methods . The difference lies in that $\mathcal{S}$ dynamically changes in our search, so the sparsity constraint is inherently satisfied at each update. We do not need a projection onto the target-net constraint by post-processing. Besides, the network $\mathcal{N}({W_{\mathcal{(S)}}},B)$ in our search is sparse and more efficient to train. An illustration of how we perform our search process is shown in Figure 1.

5 One-stage ISTA-NAS

Since the super-net of ISTA-NAS is as sparse as the target-net, our method is friendly to memory and is able to adopt the larger depth and width in target-net. If we can use one network for both search and evaluation under the target-net settings, the gap of depth, width, and even batch size in two-stage methods will be eliminated. To this end, we develop a one-stage method, where search and evaluation are finished in a single run, after which we get both architecture and its optimized parameters.

In Eq. (11), the architecture variables $\mathbf{b}$ and matrix $\mathbf{A}$ decide the coefficient of each connection for differentiable search. We need to combine these coefficients with network weights $W$ as the final optimized parameters. Considering the parameterized operations for search, such as $3\times 3$ separable convolution, $5\times 5$ dilation convolution, end up with a batch normalization (BN) layer, we also append BN layers after non-parametric operations, including identity and all pooling operations.

Experiments

We analyze the improved efficiency and correlation of the two-stage and one-stage ISTA-NAS, and then compare our search results on CIFAR-10 and ImageNet with state-of-the-art methods. All searched architectures are visualized in the supplementary material.

Our setting of search and evaluation is consistent with the convention in current studies . Please see the full description of our implementation details in the supplementary material. For our two-stage ISTA-NAS, the super-net is composed of 6 normals cells and 2 reduction cells. Each cell has 6 nodes. The first two nodes are input nodes output from the previous two cells. As convention, each intermediate node keeps two connections after search, so the sparseness $s_{j}=2$ in our method. We adopt the Adam optimizer for $\mathbf{b}_{j}$ and SGD for network weights $W$. We use the released tool MOSEK with CVX to efficiently solve Eq. (9). For our single-stage ISTA-NAS, we adopt the target-net settings of the evaluation stage. The network is stacked by 18 normal cells and 2 reduction cells. Other details are described in the supplementary material.

2 Efficiency and Correlation Improved by ISTA-NAS

ISTA-NAS inherently satisfies the sparsity constraint at each update, so significantly improves the search efficiency. As shown in Table 2, we re-implement DARTS and PC-DARTS using their released codes on CIFAR-10. When our two-stage ISTA-NAS uses a batchsize of 64, the search cost is on par with PC-DARTS, which uses a batchsize of 256 and consumes much more GPU memory. Two-stage ISTA-NAS supports a batchsize of 512, which leads to a search cost of only 0.03 GPU-day, about 20 times faster than DARTS. Their search processes are depicted in Figure 2 (left). The learning rates are also enlarged by the same times as batchsize. It is shown that ISTA-NAS with a batchsize of 256 has the most stable search accuracy on validation. Thus we adopt the 256 batchsize and 0.1 learning rate of $W$ for our two-stage search result on CIFAR-10. As for the one-stage ISTA-NAS, its accuracy on test is shown in Figure 2 (right). We also visualize the distance of $\mathbf{z}_{j}$ of two neighboring iterations averaged by all intermediate nodes $j$. It is observed that the distance of neighboring $\mathbf{z}$ decays fast. In about 10 GPU-hour, the termination condition is satisfied for all intermediate nodes, after which only network weights in the searched architecture get optimized. We have the architecture searched and optimized in a single run, which saves the search cost compared with the two-stage DARTS.

We also test the correlation between search and evaluation by the Kendall $\tau$ metric , which ranges from $-1$ to $1$ and evaluates the rank correlation of data pairs. If $\tau=-1$, the ranking order is reversed, while if the ranking is identical, $\tau$ will be 1. The full introduction of this metric is appended in the supplementary material. We implement DARTS, PC-DARTS and two-stage ISTA-NAS for 8 times on CIFAR-10 with different seeds and calculate their Kendall $\tau$ metrics based on the super-net accuracies in search and the target-net accuracies in evaluation. We measure the metric of one-stage ISTA-NAS using its converged accuracy and the one retrained for the same epochs without optimizing $\mathbf{b}$. As shown in Table 2, both two-stage and one-stage ISTA-NAS have positive correlation scores, and the one-stage ISTA-NAS has the best correlation. The Kendall $\tau$ metric is still not close to $1$, because accuracies on CIFAR-10 are not so distinguishable, and are easily affected by random noise.

3 Search Results on CIFAR-10

The search results of two-stage and one-stage ISTA-NAS on CIFAR-10 are shown in Table 3. We see that two-stage ISTA-NAS achieves a state-of-the-art error rate of 2.54% within only 0.05 GPU-day. The error rate is on par with P-DARTS, while the search cost is reduced to one sixth of the number in P-DARTS, and a half of that in PC-DARTS and NASP, which are the fastest approaches for DARTS-based search space before ISTA-NAS. The one-stage ISTA-NAS unifies the search and evaluation in a single run of 2.3 GPU-day, which is smaller than the total cost of most two-stage methods. In our implementation of one-stage ISTA-NAS, after the termination condition is satisfied, the same epochs of training as the evaluation stage in two-stage ISTA-NAS is performed. So its search cost is usually larger than that of two-stage ISTA-NAS because of its direct search in the evaluation settings that have a larger depth and width. But the performance gets better due to its improved consistency brought by further reducing the gaps apart from sparseness, such as depth, width and training batchsize.

4 Search Results on ImageNet

We use one-stage ISTA-NAS for experiments on ImageNet. As shown in Table 4, the architecture searched on CIFAR-10 has a top-1 error rate of 25.1%, which is better than DARTS by more than 1.5% top-1 error rate. Its search cost on CIFAR-10 is larger than that of SNAS and P-DARTS because our search is unified with evaluation in one stage. When one-stage ISTA-NAS is directly performed on ImageNet, we have a top-1/5 error rates of 24.0%/7.1%, which slightly surpasses PC-DARTS, the best performance to date of DARTS-based search space on ImageNet as we know. Still, the total cost is lower than most two-stage methods. PC-DARTS and our method both directly search on ImageNet. However, the difference is that our search is performed on the full training set instead of a sampled subset as adopted in PC-DARTS. Consequently, our search results may enjoy a better generalization ability due to the sufficient access to the data space.

Conclusion

In this paper, we formulate the NAS problem as optimizing a sparse solution from a high-dimensional space spanned by all candidate connections, and introduce a more efficient and consistent search method, named ISTA-NAS. The differentiable search is performed on a compressed space and the sparse solution is recovered by ISTA. The sparsity constraint is inherently satisfied at each update, which makes the search more efficient and consistent with the architecture for evaluation. We further develop a one-stage method that unifies the search and evaluation in a single run. Experiments verify the improved efficiency and correlation of two-stage and one-stage ISTA-NAS. The searched architectures surpass the state-of-the-art performances on both CIFAR-10 and ImageNet.

Acknowledgment

Z. Lin is supported by NSF China (grant no.s 61625301 and 61731018), Major Scientific Research Project of Zhejiang Lab (grant no.s 2019KB0AC01 and 2019KB0AB02), Beijing Academy of Artificial Intelligence, and Qualcomm.

Broader Impact

Our work proposes a new perspective to formulate the NAS problem and develops two algorithms that have better efficiency and correlation. The positive impacts are obvious. First, better architectures may be searched for some problems, so the practical application of neural network to the corresponding area can be fostered. It benefits industry because the products or services with more satisfactory performance can be employed. Second, NAS has been a resource demanding task. Our method saves much memory and time cost, which is friendly to environment and easy to use for practitioners. Finally, automation is believed as a complementary tool to human experts. We think that a good NAS method could bring researchers expertise in understanding neural architectures.

References

Supplementary Material

Appendix A Implementation Details

We perform our experiments on both CIFAR-10 and ImageNet. The CIFAR-10 dataset has 60,000 colored images in 10 classes, with 50,000 images for training and 10,000 images for testing. The images are normalized by mean and standard deviation. As convention, we perform the data augmentation by padding each image 4 pixels filled with 0 on each side and then randomly cropping a $32\times 32$ patch from each image or its horizontal flip. The ImageNet dataset contains 1.2 million training images, 50,000 validation images, and 100,000 test images in 1,000 classes. We adopt the standard data augmentation for training. A $224\times 224$ crop is randomly sampled from the images or its horizontal flip. The images are normalized by mean and standard deviation. We report the single-crop top-1/5 error rates on the validation set in our experiments.

The candidate operations are in accordance with current studies . They are $3\times 3$ and $5\times 5$ separable convolution, $3\times 3$ and $5\times 5$ dilated separable convolution, $3\times 3$ max and average pooling, and skip-connect. We do not use the zero operation since our methods do not rely on a post-processing process to derive the searched architecture.

A.2 Two-stage ISTA-NAS

The pipeline of our two-stage ISTA-NAS on CIFAR-10 is consistent with current two-stage methods for fair comparison. Concretely, the super-net for search is composed of 6 normal cells and 2 reduction cells, and has an initial number of channels of 16. Each cell has 6 nodes. The first 2 nodes are input nodes output from the previous two cells. The output of each cell is all intermediate nodes concatenated along the channel dimension. As convention, each intermediate node keeps two connections after search, so the sparseness $s_{j}=2$ in our method. The training set is split into two equal parts, with one for network weights $W$, and the other as the validation set for architecture variables. We train the super-net for 50 epochs with a batchsize of 256 on a single GPU. We use SGD to optimize the network weights $W$ with a momentum of 0.9, a weight decay of $3\times 10^{-4}$, and an initial learning rate of $0.2$ annealed down to zero by a cosine scheduler. The architecture variables $\mathbf{b}_{j}$ are optimized by Adam on the validation set with a learning rate of $6\times 10^{-4}$, a momentum of (0.5, 0.999), and a weight decay of $1\times 10^{-3}$. We adopt the released tool MOSEK with CVX to solve the sparse coding problem. The $\lambda$ in Eq. (9) is set as $1\times 10^{-5}$. We run our method for 5 times and choose the architecture that has the best performance on validation as the searched one.

In evaluation, the target-net is composed of 18 normal cells and 2 reduction cells, and has 36 initial channels. We train the target-net for 600 epochs on the full training set. The batchsize is 96 and the commonly used enhancements, such as cutout, dropout and auxiliary head are used. We use the SGD optimizer with a momentum of 0.9, a weight decay of $3\times 10^{-4}$, and an initial learning rate of 0.025 that is annealed down to zero by a cosine scheduler. We run the evaluation for 5 times with different seeds and report the mean error rate with its standard deviation on test set.

A.3 One-stage ISTA-NAS

The one-stage ISTA-NAS only uses one network and its implementation settings are in accordance with the evaluation stage of the two-stage ISTA-NAS.

For experiments on CIFAR-10, the network is stacked by 18 normals cells and 2 reduction cells with each cell covering all candidate connections. The initial number of channel is 36 and training batchsize is 96. The enhancements are also used accordingly. We run the Algorithm 2 with a fixed learning rate of 0.025 using the SGD optimizer. When the termination condition is satisfied for all intermediate nodes, the algorithm continues to run for 600 epochs with the learning rate annealed down to zero by a cosine scheduler. Finally, we re-evaluate the searched architecture for 4 times by running without the optimization of architecture variables $\mathbf{b}_{j}$, and report the mean and standard deviation of the error rates of these 5 results.

For experiments on ImageNet, the network starts with three convolution layers with a stride of 2 to reduce the resolution from $224\times 224$ to $28\times 28$. Then 12 normal cells and 2 reduction cells are stacked with the initial number of channels as 48. The training batchsize is 1,024. Enhancements including label smoothing and auxiliary head are used. The Adam optimizer for architecture variables is used with a learning rate of $6\times 10^{-3}$, a momentum of (0.5, 0.999), and a weight decay of $1\times 10^{-3}$. The SGD optimizer for network weights adopts a momentum of 0.9 and a weight decay of $3\times 10^{-5}$. Its initial learning rate is 0.5. When the termination condition is satisfied for all intermediate nodes, we continue to train for 250 epochs with the learning rate annealed down to zero linearly. Different from , our search is performed on the full training set instead of a sampled subset. When training finishes, we report the converged top-1/5 error rates on the validation set.

Appendix B Kendall Correlation

The Kendall correlation metric is proposed to measure the ranking correlation of pairwise data. For data pairs $(x_{i},y_{i})$ and $(x_{j},y_{j})$, if $x_{i}<x_{j}$ and $y_{i}<y_{j}$ (or $x_{i}>x_{j}$ and $y_{i}>y_{j}$), then we call the pair $(i,j)$ is concordant. Otherwise it is disconcordant. Assuming that there are $N$ samples, we have the Kendall correlation metric calculated as:

Appendix C Visualization of Architectures

We visualize the searched architectures of our methods. The two-stage ISTA-NAS on CIFAR-10 is shown in Figure 3. The one-stage ISTA-NAS on CIFAR-10 is shown in Figure 4. The one-stage ISTA-NAS on ImageNet is shown in Figure 5.