R ETHINKING A RCHITECTURE S ELECTION IN D IFFERENTIABLE NAS
Anonymous authors
Paper under double-blind review
A BSTRACT
Differentiable Neural Architecture Search is one of the most popular Neural Architecture Search (NAS) methods for its search efﬁciency and simplicity, accomplished by jointly optimizing the model weight and architecture parameters in a weight-sharing supernet via gradient-based algorithms.
At the end of the search phrase, the operations with the largest architecture parameters will be selected to form the ﬁnal architecture, with the implicit assumption that the values of architecture parameters reﬂect the operation strength.
While much has been discussed about the supernet’s optimization, the architecture selection process has received little attention.
We provide empirical and theoretical analysis to show that the magnitude of architecture parameters does not necessarily indicate how much the operation contributes to the supernet’s performance.
We propose an alternative perturbation-based architecture selection that directly measures each operation’s inﬂuence on the supernet.
We re-evaluate several differentiable NAS methods with the proposed architecture selection and ﬁnd that it is able to extract signiﬁcantly improved architectures from the underlying supernets consistently.
Furthermore, we ﬁnd that several failure modes of Darts can be greatly alleviated with the proposed selection method, indicating that much of the poor generalization observed in Darts can be attributed to the failure of magnitude-based architecture selection rather than entirely the optimization of its supernet.
Our code will be made publicly available shortly.
I NTRODUCTION
Neural Architecture Search (NAS) has been drawing increasing attention in both academia and industry for its potential to automatize the process of discovering high-performance architectures, which have long been handcrafted.
Early works on NAS deploy Evolutionary Algorithm (Stanley & Miikkulainen, 2002;
Real et al., 2017; Liu et al., 2017) and Reinforcement Learning (Zoph & Le, 2017; Pham et al., 2018; Zhong et al., 2018) to guide the architecture discovery process.
Recently, several one-shot methods have been proposed that signiﬁcantly improve the search efﬁciency (Brock et al., 2018; Guo et al., 2019; Bender et al., 2018).
As a particularly popular instance of one-shot methods, Darts (Liu et al., 2019) enables the search process to be performed with a gradient-based optimizer in an end-to-end manner.
It applies continuous relaxation that transforms the categorical choice of architectures into continuous architecture parameters α .
The resulting supernet can be optimized via gradient-based methods, and the operations associated with the largest architecture parameters are selected to form the ﬁnal architecture.
Despite its simplicity, several works cast doubt on the effectiveness of Darts.
For example, a simple randomized search (Li & Talwalkar, 2019) outperforms the original Darts; Zela et al.
(2020) observes that Darts degenerates to networks ﬁlled with parametric-free operations such as the skip connection or even random noise, leading to the poor performance of the selected architecture.
While the majority of previous research attributes the failure of Darts to its supernet optimization (Zela et al., 2020; Chen & Hsieh, 2020; Chen et al., 2020), little has been discussed about the validity of another important assumption: the value of α reﬂects the strength of the underlying operations.
In this paper, we conduct an in-depth analysis of this problem.
Surprisingly, we ﬁnd that in many cases, α does not really indicate the operation importance in a supernet.
Firstly, the operation associated with the largest α often does not result in the highest validation accuracy after discretization.
Secondly, as an important example, we show mathematically that the domination of skip connection observed in Darts (i.e. α skip becomes larger than other operations.)
is in fact a reasonable outcome of the supernet’s optimization but becomes problematic when we rely on α to select the best operation.
If α is not a good indicator of operation strength, how should we select the ﬁnal architecture from a pretrained supernet?
Our analysis indicates that the strength of each operation should be evaluated based on its contribution to the supernet performance instead.
To this end, we propose an alternative perturbation-based architecture selection method.
Given a pretrained supernet, the best operation on an edge is selected and discretized based on how much it perturbs the supernet accuracy; The ﬁnal architecture is derived edge by edge, with ﬁne-tuning in between so that the supernet remains converged for every operation decision.
We re-evaluate several differentiable NAS methods (Darts (Liu et al., 2019), SDarts (Chen & Hsieh, 2020), SGAS (Li et al., 2020)) and show that the proposed selection method is able to consistently extract signiﬁcantly improved architectures from the supernets than magnitude-based counterparts.
Furthermore, we ﬁnd that the robustness issues of Darts can be greatly alleviated by replacing the magnitude-based selection with our selection method.
B ACKGROUND AND RELATED WORK
Preliminaries of Differentiable Architecture Search (Darts)
We start by reviewing the formulation of Darts.
Darts’ search space consists of repetitions of cell-based microstructures.
Every cell can be viewed as a DAG with N nodes and E edges, where each node represents a latent feature map x i , and each edge is associated with an operation o
(e.g. skip connect , sep conv 3x3 ) from the search space O .
Continuous relaxation is then applied to this search space.
Concretely, every operation on an edge is activated during the search phase, with their outputs mixed by the architecture parameter α to form the ﬁnal mixed output of that edge ¯ m ( x i ) =
(cid:80) o ∈O [Equation] o (cid:48) exp α
o (cid:48) o ( x i ) .
This particular formulation allows the architecture search to be performed in a differentiable manner: Darts jointly optimizes α and model weight w with the following bilevel objective via alternative gradient updates:
[Equation]
We refer to the continuous relaxed network used in the search phase as the supernet of Darts.
At the end of the search phase, the operation associated with the largest α o on each edge will be selected from the supernet to form the ﬁnal architecture.
Failure mode analysis of Darts Several works cast doubt on the robustness of Darts.
Zela et al.
(2020) tests Darts on four different search spaces and observes signiﬁcantly degenerated performance.
They empirically ﬁnd that the selected architectures perform poorly when Darts’ supernet falls into high curvature areas of validation loss (captured by large dominant eigenvalues of the Hessian ∇ 2 α,α L val ( w, α ) ).
While Zela et al.
(2020) relates this problem to the failure of supernet training in Darts, we examine it from the architecture selection aspects of Darts, and show that much of Darts’ robustness issue can be alleviated by a better architecture selection method than the magnitude-based selection based on α .
Progressive search space shrinking There is a line of research on NAS that focuses on reducing the search cost and aligning the model sizes of the search and evaluation phases via progressive search space shrinking (Liu et al., 2018;
Li et al., 2019; Chen et al., 2020; Li et al., 2020).
The general scheme of these methods is to prune out weak operations and edges sequentially during the search phase, based on the magnitude of α following Darts.
Our method is orthogonal to them in this respect, since we select operations based on how much it contributes to the supernet’s performance rather than the α value.
Although we also discretize edges greedily and ﬁne-tune the network in between, the purpose is to let the supernet recover from the loss of accuracy after discretization to accurately evaluate operation strength on the next edge, rather than to reduce the search cost.
T HE PITFALL OF MAGNITUDE - BASED ARCHITECTURE SELECTION IN D ARTS
In this section
[Figure]
we put forward the opinion that the architecture parameter α does not necessarily represent the strength of the underlying operation in general, backed by both empirical and theoretical evidence.
As an important example [Figure] we mathematically justify that the skip connection domination phenomenon observed in Darts is reasonable by itself, and becomes problematic when combined with the magnitude-based architecture selection.
3. α MAY NOT REPRESENT THE OPERATION STRENGTH
[Figure]
Following Darts, existing differentiable NAS methods use the value of architecture parameters α to select the ﬁnal architecture from the supernet, with the implicit assumption that α represents the strength of the underlying operations.
In this section, we study the validity of this assumption in detail.
Consider one edge on a pretrained supernet; the strength of an operation on the edge can be naturally deﬁned as the supernet accuracy after we discretize to this operation and ﬁne-tune the remaining network until it converges again; we refer to it as ”discretization accuracy at convergence” for short.
The operation that achieves the best discretization accuracy at convergence can be considered as the best operation for the given edge.
Figure 1 shows the comparison of α (blue) and operation strength (orange) of randomly select edges on Darts supernet.
As we can see, the operations associated with the largest α s do not necessarily match the one that achieves the highest discretization validation accuracy at convergence.
Moreover, operations assigned with small α s are sometimes strong ones that lead to high discretization accuracy at convergence.
To further verify the mismatch, we investigate the operation strength on search space S2, where Darts fails dramatically due to excessive skip connections (Zela et al., 2020).
S2 is a variant of Darts search space that only contains two operations per edge ( skip connect, sep conv 3x3 ).
Figure 2 shows the selected operations based on α (left) and operation strength (right) on all edges on S2.
From Figure 2a, we can see that α skip connect > α sep conv 3 x 3 on 12 of 14 edges.
Consequently, the derived child architecture will lack representation ability and perform poorly due to too many skip connections.
However, as shown in Figure 2b, the supernet beneﬁts more from discretizing to sep conv 3x3 than skip connect on half of the edges.
3. A CASE STUDY : SKIP CONNECTION
Several works point out that Darts tends to assign large α to skip connections, resulting in shallow architectures with poor generability (Zela et al., 2020;
Liang et al., 2019; Bi et al., 2019).
This ”skip connection domination” issue is generally attributed to the failure of Darts’ supernet optimization.
In contrast, we draw inspiration from research on ResNet (He et al., 2016) and show that this phenomenon by itself is a reasonable outcome while Darts reﬁnes its estimation of the optimal feature map, rendering α skip ineffective in the architecture selection.
randomly swap two layers in each stage (deﬁned as successive pling
[Table]
In vanilla networks (e.g., VGG), each layer computes a new level of feature map from the output feature map of the predecessor layer; thus, reordering layers at test time would dramatically hurt the performance (Veit et al., 2016).
Unlike vanilla networks, Greff et al.
(2017) and Veit et al.
(2016) discover that successive layers in ResNet with compatible channel sizes are in fact estimating the same optimal feature map so that the outputs of these layers stay relatively close to each other at convergence; As a result, ResNet’s test accuracy remains robust under layer reordering.
Greff et al.
(2017) refers to this unique way of feature map estimation in ResNet as the ”unrolled estimation.”
Darts’ supernet resembles ResNet, rather than vanilla networks like VGG, in both appearance and behavior.
Appearance-wise, within a cell of Darts’ supernet, edges with skip connection are in direct correspondence with the successive residual layers in ResNet.
Behavior-wise, Darts’ supernet also exhibits a high degree of robustness under edge shufﬂing.
As shown in Table 1, randomly reordering edges on a pretrained Darts’ supernet at test time also has little effect on its performance.
This evidence indicates that Darts performs unrolled estimation like ResNet as well, i.e., edges within a cell share the same optimal feature map that they try to estimate.
In the following proposition, we apply this ﬁnding and provide the optimal solution of α in the sense of minimizing the variance of feature map estimation.
Proposition 1. 1 Without loss of generality, consider one cell from a simpliﬁed search space consists of two operations: (skip, conv).
Let m ∗ denotes the optimal feature map, which is shared across all edges according to the unrolled estimation view (Greff et al., 2017).
Let o e ( x e ) be the output of convolution operation, and let x e be the skip connection (i.e., the input feature map of edge e ).
Assume m ∗ , o e ( x e ) and x e are normalized to the same scale.
The current estimation of m ∗ can then be written as:
[Equation]
where α conv and α skip are the architecture parameters deﬁned in Darts.
The optimal α ∗ conv and α ∗ skip minimizing var ( m e ( x e ) − m ∗ ) , the variance of the difference between the optimal feature map m ∗ and its current estimation m e ( x e ) , are given by:
[Equation]
We refer the reader to Appendix A.4 for detailed proof.
From eq.
(3) and eq.
(4), we can see that the relative magnitudes of α skip and α conv come down to which one of x e and o e ( x e ) is closer to m ∗ in variance:
• x e (input of edge e ) comes from the mixed output of the previous edge.
Since the goal of every edge is to estimate m ∗ (unrolled estimation), x e is also directly estimating m ∗ .
• o e ( x e ) is the output of a single convolution operation instead of the complete mixed output of edge e , so it will deviate from m ∗ even at convergence.
Therefore, in a well-optimized supernet, x e will naturally be closer to m ∗ than o e ( x e ) , causing α skip to be greater than α conv .
Our analysis above indicates that the better the supernet, the larger the ( α skip − α conv ) gap (softmaxed) will become since x e gets closer and closer to m ∗ as the supernet is optimized.
This result is evidenced in Figure 3, where mean ( α skip − α conv ) continues to grow as the supernet gets better.
In this case, although α skip > α conv is reasonable by itself, it becomes an inductive bias to NAS if we were to select the ﬁnal architecture based on α .
P ERTURBATION - BASED ARCHITECTURE SELECTION
Instead of relying on the α value to select the best operation, we propose to directly evaluate operation strength in terms of its contribution to the supernet’s performance.
The operation selection criterion is laid out in section 4.1.
In section 4.2, we describe the entire architecture selection process.
4. E VALUATING THE STRENGTH OF EACH OPERATION
In section 3.1, we deﬁne the strength of each operation on a given edge as how much it contributes to the performance of the supernet, measured by discretization accuracy.
To avoid inaccurate evaluation due to large disturbance of the supernet during discretization, we ﬁne-tune the remaining supernet until it converges again, and then compute its validation accuracy (discretization accuracy at convergence).
The ﬁne-tuning process needs to be carried out for evaluating each operation on an edge, leading to substantial computation costs.
To alleviate the computational overhead, we consider a more practical measure of operation strength: for each operation on a given edge, we mask it out while keeping all other operations, and re-evaluate the supernet.
The one that results in the largest drop in the supernet’s validation accuracy will be considered as the most important operation on that edge.
This alternative criterion incurs much less perturbation to the supernet than discretization since it only deletes one operation from the supernet at a time.
As a result, the supernet’s validation accuracy after deletion stays close to the unmodiﬁed supernet, and thus it alleviates the requirement of tuning the remaining supernet to convergence.
Therefore, we implement this measurement for the operation selection in this work. Algorithm
Input: A pretrained supernet S , Set of edges E from S , Set of nodes N from S
Result: Set of selected operations { o ∗ e } e ∈E while |E| > 0 do randomly select an edge e ∈ E (and remove it from E ); forall operation o on edge e do evaluate the validation accuracy of S when o is removed ( ACC \ o ); end select the best operation for e : o ∗ e ← arg min o ACC \ o ; discretize edge e to o ∗ e and tune the remaining supernet for a few epochs; end
4. T HE COMPLETE ARCHITECTURE SELECTION PROCESS
Our method operates directly on top of Darts’ pretrained supernet.
Given a supernet, we randomly iterate over all of its edges. We evaluate each operation on an edge, and select the best one to be discretized based on the measurement described in section 4.1.
After that, we tune the supernet for a few epochs to recover the lost accuracy during discretization.
The above steps are repeated until all edges are decided.
Algorithm 1 summarizes the operation selection process.
The cell topology is decided in a similar fashion. We refer the reader to Appendix A.3 for the full algorithm, including deciding the cell topology.
This simple method is termed ”perturbation-based architecture selection (PT)” in the following section.
E XPERIMENTAL R ESULTS
In this section, we demonstrate that the perturbation-based architecture selection method is able to consistently ﬁnd better architectures than those selected based on the values of α .
The evaluation is based on the search space of Darts and NAS-Bench-201 (Dong & Yang, 2020), and we show that the perturbation-based architecture selection method can be applied to several variants of Darts.
5. R ESULTS ON D ARTS ’ CNN SEARCH SPACE
We keep all the search and retrain settings identical to Darts since our method only modiﬁes the architecture selection part.
After the search phase, we perform perturbation-based architecture selection following Algorithm 1 on the pretrained supernet.
We tune the supernet for 5 epochs between two selections as it is enough for the supernet to recover from the drop of accuracy after discretization.
We run the search and architecture selection phase with four random seeds and report both the best and average test errors of the obtained architectures.
As shown in Table 2, the proposed method (Darts+PT) improves Darts’ test error from 3.00% to 2.61%, with manageable search cost (0.8 GPU days).
Note that by only changing the architecture selection method, Darts performs signiﬁcantly better than many other differentiable NAS methods that enjoy carefully designed optimization process of the supernet, such as GDAS (Dong & Yang, 2019) and SNAS (Xie et al., 2019).
This empirical result suggests that architecture selection is crucial to Darts: with the proper selection algorithm, Darts remains a very competitive method.
Our method is also able to improve the performance of other variants of Darts.
To show this, we evaluate our method on SDarts(rs) and SGAS (Chen & Hsieh, 2020; Li et al., 2020).
SDarts(rs) is a variant of Darts that regularizes the search phase by applying Gaussian perturbation to α .
Unlike
Darts and SDarts, SGAS performs progressive search space shrinking.
Concretely, SGAS progressively discretizes its edges with the order from most to least important, based on a novel edge importance score.
For a fair comparison, we keep its unique search space shrinking process unmodiﬁed and only replace its magnitude-based operation selection with ours.
As we can see from Table 2, our method consistently achieves better average test errors than its magnitude-based counterpart.
Concretely, the proposed method improves SDarts’ test error from 2.67% to 2.54% and SGAS’ test error from 2.66% to 2.56%.
Moreover, the best architecture discovered in our experiments achieves a test error of 2.44%, ranked top among other NAS methods.
5. P ERFORMANCE ON NAS-B ENCH -201 SEARCH SPACE
To further verify the effectiveness of the proposed perturbation-based architecture selection, we conduct experiments on NAS-Bench-201.
NAS-Bench-201 provides a uniﬁed cell-based search space similar to Darts.
Every architecture in the search space is trained under the same protocol on three datasets (cifar10, cifar100, and imagenet16-120), and their performance can be obtained by querying the database.
As in section 5.1, we take the pretrained supernet from Darts and apply our method on top of it.
All other settings are kept unmodiﬁed.
Figure 4 shows the performance trajectory of
Darts+PT compared with Darts.
While the architectures found by magnitude-based selection degenerates over time, the perturbation-based method is able to extract better architectures from the same underlying supernets stably.
The result implies that the Darts’ degenerated performance comes from the failure of magnitude based architecture selection. [Table]
A NALYSIS 6. I SSUE WITH THE ROBUSTNESS OF D ARTS
Zela et al.
(2020) observes that Darts tends to yield degenerate architectures with abysmal performance.
We conjecture that this robustness issue of Darts can be explained by the failure of magnitude-based architecture selection. [Table] [Table] To show this
[Table]
we test Darts’ performance with perturbation-based architecture selection on four spaces proposed by Zela et al.
(2020) (S1-S4).
The complete speciﬁcations of these spaces can be found in
Appendix A.2.
Given a supernet, the architecture selected based on α performs poorly across spaces and datasets (column 3 in Table 3).
However, our method is able to consistently extract meaningful architectures with signiﬁcantly improved performance (Column 4 in Table 3).
Notably, Darts+PT is able to ﬁnd meaningful architecture on S2 ( skip connect, sep conv 3x3 ) and
S4 ( noise, sep conv 3x3 ), where Darts failed dramatically.
As shown in Figure 5, on S2, while magnitude-based selection degenerates to architectures ﬁlled with skip connections, Darts+PT is able to ﬁnd architecture with 4 convolutions; On S4, Darts+PT consistently favors sep conv 3x3 on edges where α selects noise .
[Figure]
-based method (Darts) fails dramatically.
The complete architecture can be found in Appendix A.9.
Nontrivial operations are marked red.
6. P ROGRESSIVE TUNING
In addition to operation selection, we also tune the supernet after an edge is discretized so that the supernet could regain the lost accuracy.
To measure the effectiveness of our operation selection criterion alone, we conduct an ablation study on the progressive tuning part.
Concretely, we test a baseline by combining progressive tuning with magnitude-based operation selection instead of our selection criterion, which we code-named Darts+PT-Mag.
Figure 6 plots the change of validation accuracy of
Darts+PT and Darts+PT-Mag during the operation selection phase.
As we can see, Darts+PT is able to identify better operations that lead to higher validation accuracy than the magnitudebased alternative, revealing the effectiveness of our operation selection criteria. Moreover,
Darts+PT-Mag is only able to obtain a test error of 2.85% on Darts space on cifar10, much worse than Darts+PT (2.61%), indicating that the operation selection part plays a crucial role in our method. 6. F IXING α TO U NIFORM [Table]
Since the proposed method does not rely on α for architecture selection, a natural question is whether it is necessary to optimize a stand-alone α . We ﬁnd that by ﬁxing α = 0 (uniform weights for all the operations) while training supernet and applying perturbation-based architecture selection, the resulting method performs onpar with Darts+PT, and in some cases even better. For example, Darts+PT (ﬁx α ) achieves better performance than Darts+PT on NAS-Bench-201. On Darts’ search space and its variants S1-S4,
Darts+PT (ﬁx α ) performs similarly to Darts+PT.
The results can be found in Table 3 and Table 4.
This surprising ﬁnding suggests that even the most naive approach, simply training a supernet without α , will be a competitive method when combining with the proposed perturbation-based architecture selection.
C ONCLUSION AND DISCUSSION
This paper attempts to understand Differentiable NAS methods from the architecture selection perspective.
We re-examine the magnitude-based architecture selection process of Darts and provide empirical and theoretical evidence on why it does not indicate the underlying operation strength.
We introduce an alternative perturbation-based architecture selection method that directly measures the operation strength via its contribution to the supernet performance.
The proposed selection method is able to consistently extract improved architecture from supernets trained identically to the respective base methods on several spaces and datasets.
Our method brings more freedom in supernet training as it does not rely on α to derive the ﬁnal architecture.
We hope the perturbation-based architecture selection can bring a new perspective to the NAS community to rethink the role of α in Differential NAS.
A A PPENDIX A. D ESCRIPTION ABOUT OUR BASELINE MODELS A.1. D ARTS
Darts (Liu et al., 2019) is a pioneering work that introduces the general differentiable NAS framework, which we reviewed in section 2.
In Darts, the topology and operation are searched together. Concretely, at the end of the search, it selects one operation for every edge in the normal (reduction) cell based on the architecture parameter α . Then it selects two input edges for each node in the cell by comparing the largest α of every input edge. The ﬁnal architecture consists of one operation on each of the eight edges in the normal (reduction) cell.
A.1. SD ARTS
SDarts (Chen & Hsieh, 2020) is a variant of Darts aiming at regularizing the bilevel optimization process in Darts via random Gaussian perturbation, inspired by the recent ﬁnding that regularizing Darts’ supernet leads to improved performance. While the optimization of architecture parameter α in SDarts is identical to Darts, it distorts the architecture parameters α with random Gaussian noise while training the model weights w . This simple yet effective regularizer is able to consistently improve the robustness and performance of Darts.
A.1. SGAS
SGAS (Li et al., 2020) represents a line of research on improving the search efﬁciency of differentiable NAS by progressively shrinking the search space.
It ﬁrst trains the model weights w alone for 10 epochs.
After that, it selects one edge from the supernet and then selects the best operation on that edge based on α to be discretized.
The edge selection is based on the ranking of the proposed edge importance score.
The process stops after all eight edges of the ﬁnal architecture are decided.
Zela et al. (2020) introduces four variants of the Darts’ space (S1, S2, S3, and S4) to study the robustness of Darts. These spaces differ from Darts’ original space only in the number and types of operations on each edge.
Apart from that, everything else is the same.
• S1 is a pre-optimized search space consisting of top2 operations selected by Darts.
As a result, each edge contains a different set of operations to be searched from.
• S2 consists of two operations: skip connect and sep conv 3x3 .
• S3 consists of three operations: none , skip connect and sep conv 3x3 .
• S4 consists of two operations: noise and sep conv 3x3 .
The noise operation outputs a random Gaussian noise N (0 , 1) regardless of the input.
This operation generally hurts the performance of discretized architecture, and should be avoided by NAS algorithm.
A. T HE COMPLETE ALGORITHM OF P ERTURBATION - BASED A RCHITECTURE S ELECTION
Input: A pretrained Supernet S , Set of Edges E from S , Set of Nodes N from S
Result: Set of selected operations { o ∗ e } e ∈E , and top2 input edges for each node { ( e (1) ∗ n , e (2) ∗ n } n ∈N while |E| > 0 do // operation selection phase randomly select an edge e ∈ E (and remove it from E ); forall operation o on edge e do evaluate the validation accuracy of S when o is removed ( ACC \ o ); end select the best operation on e : o ∗ e ← arg min o ACC \ o ; discretize edge e to o ∗ e and train the remaining supernet until it converges again; endwhile |N| > 0 do // topology selection phase randomly select a node n ∈ N (and remove it from N ); forall input edge e of node n do evaluate the validation accuracy of S when e is removed ( ACC \ e ); end set top2 edges on n ( e (1) ∗ n , e (2) ∗ n ) to be the ones with lowest and second lowest ACC \ e ; prune out all other edges of n and train the remaining supernet until it converges again; end
A. P ROOF OF P ROPOSITION
Proof.
Let θ skip =
Softmax ( α skip ) and θ conv = Softmax ( α conv ) .
Then the mixed operation can be written as m e ( x e ) = θ conv o e ( x e ) + θ skip x e .
We formally formulate the objective to be:
[Equation]
[Equation]
This constraint optimization problem can be solved with Lagrangian multipliers:
[Equation]
[Equation]
Setting:
[Equation]
Solving the above equations will give us:
[Equation]
[Equation]
Where Z = V ar ( o e ( x e ) − m ∗ ) − Cov ( o e ( x e ) − m ∗ , x
e − m ∗ )
+ V ar ( x e − m ∗ ) − Cov ( o e ( x e ) − m ∗ , x e − m ∗ ) .
Aligning basis with Darts, we get:
[Equation]
The only term that differentiates α skip from α conv is the ﬁrst term inside the logarithm, therefore:
[Equation]
[Equation]
A. M ORE F IGURES ON α AND D ISCRETIZATION A CCURACY AT C ONVERGENCE
We provide extra ﬁgures similar to Figure 1 to take into account the randomness of supernet’s training.
We ﬁrst train 6 supernets with different seeds, and then randomly select 1 edge from each of them.
We can see that the results are consistent with Figure 1.
As shown in Figure 7, operation associated with the largest α constantly deviates from the one that leads to the highest discretization accuracy at convergence.
supernet trained with different seeds (one subﬁgure for each edge).
Operations associated with the largest α constantly deviate from the one that lead to highest discretization accuracy at convergence.
A. P ERFORMANCE T RAJECTORY OF D ARTS +PT ( FIX α ) ON NAS-B ENCH -
We plot the performance trajectory of Darts+PT (ﬁx α ) on NAS-Bench-201 similar to Figure 4.
As shown in Figure 8, it consistently achieves strong performance without training α at all, indicating that the extra freedom of supernet training without α can be explored to develop improved search algorithm in the future.
[Table]
A. A BLATION S TUDY ON THE N UMBER OF F INE - TUNING E POCHS
As described in section 4.2, we perform ﬁne-tuning between two edge decisions to recover supernet from the accuracy drop after discretization.
The number of ﬁne-tuning epochs is set to 5 for all experiments because empirically we ﬁnd that it is enough for the supernet to converge again.
In this section, we conduct an ablation study on the effect of the number of ﬁne-tuning epochs.
As shown in Figure 9, the gain from tuning the supernet longer than 5 epochs is marginal.
A. A RCHITECTURE T RANSFERABILITY E VALUATION O N I MAGE N ET
We further evaluate the performance of the derived architecture on ImageNet.
We strictly follow the training protocals as well as the hyperparameter settings of Darts (Liu et al., 2019) for this experiment.
As shown in Table 5, the proposed method improves the top1 performance of Darts by 1.2%.
A. S EARCHED A RCHITECTURES
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