| Prompt Source | File Ranking |
| Current Parent Class | file with the target hole.Returns a single file. file that contains the parent class that occurs closest to the target hole. |
| Import | Returns a single file. files with the corresponding import usage ranked based on the proximity |
| Sibling | to the hole.Returns a ranked list of files. files with import usage common to the current file and the sibling file, |
| Similar Name | ranked based on the proximity to the hole.The total number of common imports between the current and the sibling file is used as a tie-breaker. Returnsa ranked list of files. files with import usage common to the current file and the similar name |
| Child Class | file,ranked based on the proximity to the hole.The total number of common imports between the current and the similar name file is used as a tie-breaker.Returns a ranked list of files. files with import usage common to the current file and the child file, |
| ranked based on the proximity to the hole.The total number of common imports between the current and the child class file is used as a tie-breaker. Returns a ranked list of files. |
| Import of Sibling | import files ranked based on the frequency of usage in all the sibling files. Returns a ranked list of files. |
| Import of Similar Name | import file ranked on the basis of frequency of usage in all the similar name files.Returns a ranked list of files. |
| Import of Parent Class | import file ranked on the basis of frequency of usage in all the parent |
| Importof Child Class | class files.Returns a ranked list of fles. import file ranked on the basis of frequency of usage in all the child class |
+
+# B.2 EXAMPLES OF PROMPT CONTEXT TYPE
+
+We provide examples of each of our prompt context type below:
+
+1. Post Lines (PL) : For the example shown in Figure 1 of the main paper, post lines will take all the lines after the line mg.InitializeToAssignment(CurrentAssignments()) till we reach the end of the file (AffinityPropagation.java).
+
+2. Identifiers (I): Identifiers are the names of variables used in the code. For example, for the prompt proposal context taken from the imported file shown in Figure 1 in the main paper (highlighted in violet), identifiers are InitializeToAssignment (line 1), a (line 1), currentAssignment_ (line 2), a ( line 2), clone (line 2), alreadyInitialized_ (line 3), justOneRound_ (line 4).
+
+3. Type Identifiers (TI): Type Identifiers define the type of an identifier. For example, in the code snippet class DPAffinityPropagation extends AffinityPropagation [ AffinityPropagation is labeled as a type identifier. Similarly in the snippet DPAPParameters parameters_;, DPAPParameters is a type identifier.
+
+4. Field Declarations (FD): The variables of a class type are introduced by field declarations. For example, double[][] mHijMujT_; and MessageValuePair[][] sortedMHijMujTs_; are examples of field declarations.
+
+5. String Literals (SL): A string literal is the sequence of characters enclosed in doublequotes. For example, in the code snippet, System.err.println("DPAP load Warning: unknown parameter " $^ +$ entries[0] + ", value $\begin{array} { r l r } { \mathbf { \Sigma } } & { { } = } & { \mathbf { \Sigma } ^ { \mathsf { ~ \pm ~ } } } \end{array} \quad \Sigma \left( \begin{array} { l l } { \mathbf { \Sigma } } & { \mathbf { \Sigma } } \\ { \mathbf { \Sigma } } & { \mathbf { \Sigma } } \end{array} \right. + \mathbf { \Sigma }$ entries[1]);, we have two string literals: (a) "DPAP load Warning: unknown parameter " ; (b) ", value = "
+
+6. Method Names (MN): For the example shown in Figure 1 of the main paper, public void InitializeToAssignment(int[] a) is the method name prompt context type. 7. Method Names and Bodies (MNB): For the example shown in Figure 1 of the main paper, the part highlighted in violet represents the method names and bodies.
+
+B.3 TRUNCATION STRATEGIES FOR PROMPT PROPOSAL CONTEXT
+
+If the prompt proposal context is greater than the context length allocated to it, then we need to truncate the prompt proposal context. We followed the below two schemes for truncating context:
+
+• front: We truncate the context from the front. This is used for all prompt sources except Parent Class and when we take PL from Current.
+• back: We truncate the context from the back. This is used when the prompt source is Parent Class and when we take prompt context types other than PL from Current.
+
+The truncation strategies for each case were selected based on results on a small validation set. For the prompt source Current, except when the prompt context type is PL, we always start by taking code of prompt context type from after the hole position. This makes sense as the default Codex context will anyways contain code before the hole. Only if this turns out to be blank, we will use the code of context type from before the hole.
+
+# B.4 LIST OF PROMPT PROPOSALS
+
+Table 3: List of our proposed repo-level prompt proposals
+
+| Dataset | Prefix | Categorydescription | Suffix |
| CIFAR10 | “a photo of a",“a jpeg photo ofa”,"a painting ofa”,"itap of a”,“graffiti of a”,“a cartoon”,“adoodle” | None | None,“It's common in daily life”，“It's cute"，“It's ugly”,“It's weird”,"Hope you like it” |
| CIFAR100 | “a jpeg photo of a”,“a painting of a”,“a good photo of a”,"abad photo of a”,“a photo of a”,“itap of a”,“a rendering of a” | None | None,“It's common in dailylife",“It'sbeautiful",“It'sugly”,"I like it”,“T take it to-day” |
| Caltech101 | “a photo of a”，“a cropped photo of a”,“a good photo ofa","a bad photo of a” | None | None,“I like it”，“I hate it","It's ugly”,"It's cute" |
| Stanford-Car | “a photo of a”，“a close-upphoto of a",“a good photo ofa”,“a bad photo of a” | “a type of car”,“a type of auto-mobile” | “I like it”,“It belongs to myfriend’,“It's brand new”,“It’'s popular recently”,“It's impor-tant to me”,“I take it today” |
| Flowers102 | “a photo of a (many)”,“a ren-dering of a (many)”, “itap of a(many）” | “a type offlower","a typeof bloom” | “It's beautiful”,“It's from mybest friend”,“It gives out a sweet perfume/fragrance |
| ImageNet | “a photo of a","a good photo ofa”,"a bad photo of a”,“a close-up photo of a”,“itap of a” | None | “I like it”,“It's common indaily life",“It's not common in daily life”,“It's ugly”,“It'scute","It's beautiful" |
| Food101 | “a photo of my”,“a close-up photo of my”,“itap of my” | “atypeoffood”，“a typeof nourish-ment” | “I made it today”,“I like it”,“I hate it”,"It's delicious","It'swith nice flavour”,“It's withterrible flavour"，“It's popularrecently” |
| SUN397 | “a photo of a”,“a good photo ofa",“a bad photo of a”,“a bright photo of a",a dark photo of a”,“a black and white photo of a",“a nice scene of a”,“a terriblescene of a” | None | None,“I like it”,“I hate it”,“It's beautiful”,“It's commonin daily life","It's important tome” |
| DTD | “itap of a”,“a close-up photo ofa” | “texture”,“surface",“material” | None,“It's out of style”,“It's popular in old days”,“It'sugly”,"It's beautiful” |
| Aircrafts | “a photo of the”，“a close-up photo of the","a good photo ofthe”,"a pixelated photo of the” | “a typeof plane”，“a typeof aircraft”,“atype of airliner” | None,"I like it”,“It's important to me”,“I take it today”,“Hopeyou like it” |
| Oxford Pet | “a photo of my”,“a low reso-lution photo of my”，“a goodphoto of my” | “a type of pet”,“a type of dogor cat” | None,“It's cute”,“It's impor-tant to me”，“I like it”,“It’sbeautiful" |
| EuroSAT | “a photo of a”,“apainting of a”,"a cropped photo of a”,"a good photo of a”,“a blurry photo ofa” | None，“an ex-ample of aerialor satellite im-ages” | None,“I like it”,“It's takenfrom an aircraft or some flying object”,"It's collected by imag-ing satellites” |
+
+Table 12: Prompt templates used for zero-shot image-text retrieval on Flickr30K and MSCOCO datasets.
+
+| Text | kNN Results | kNN Focus Score Score | |
| PG-19 | |
| S HE BA takes the gold-rimmed pince-nez which hangs upon THE DEAN'S waistcoat and places it before his eyes | Then if we're here with the closed carriage at ten-![They go together into the library. DARBEY.[To SHE BA.] | | 0.690.99 | |
| Oh![He sinks on to the settee with a vacant stare,his arms hanging helplessly. DAR- BEY. [To SHE BA is a burden to him! | .] There-now his career | 0.230.92 | |
| Papa! SHE BA. Papsey！[THE DEAN rouses himself, discovers his children and removes his hat. | | 0.025 0.92 | |
| Then if we're here with the closed carriage at ten-！[They go together into the library. DARBE Y [To SHEBA.] Oh! [He sinks on to the settee with a | 0.71 | | 0.99 |
| THE DEAN gives DAR BE Ya severe look,and with an important cough walks into the Library. The men and the girls speak in undertones. | vacant stare,his arms hanging helplessly. DARBE Y career is a burden to him! Oh,Salome!Papa!Papa! TARVER.The | [To SHEBA.] There-now his The Dean! | 0.190.99 0.080.99 | |
| Dean?DARBE Y Dictionary Lookup Task |
| <q> 14 42 23 38 <v> 40 41 05 56 <q> 30 55 07 23 <v> 36 17 26 63 <q> 10 41 26 39 <v> 48 38 11 24 | <k> 14 42 23 38 <v> 40 41 05 56 <k> 30 55 07 23 <v> 36 17 26 63 <k> 10 41 26 39 <v> 48 38 11 24 | 0.96 0.88 0.87 | 0.99 1.0 0.99 |
+
+# C Ablations
+
+In this section, we focus on two key properties of crossbatch training procedure: differentiability and the inclusion of negatives. We also discuss the relation to Memorizing Transformer in terms of the training protocol and memory integration. We refer to Appendix B.5 for a detailed technical description of differences between FOT and Memorizing Transformer.
+
+# C.1 Impact of differentiable keys and values
+
+We compare FOT to Memorizing Transformer, which uses a non-differentiable memory of keys and values during training. In the multi-doc experiment presented in Figure 5, both MT and FOT are trained with local context of 512. We observe that FOT is significantly better when the context is expanded during inference, which confirms that differentiable keys and values are beneficial.
+
+Table 8: Perplexity on PG-19 in the single-doc setting for various local context lengths during training. In these experiments, we used the same context length both during training and evaluation.
+
+| Outputs: The best robot design D* and controller C*. | Inputs:Task specification T, number of generations n, population size p. | | | |
| S←0 | | //Dataset of robot designs,controllers and reward | | |
| D1,..,Dp ← SAMPLEDESIGNS(p) fori←1tondo | | // Sample an initial population of robot designs | | |
| | | | |
| for j ←1 to p do | | | | |
| Cj ←OPTIMIZECONTROL(T,Dj) | / Optimize the controller of given robot design | | |
| | rj ←EvALUATEREWARD(T,Dj, Cj) // Evaluate the reward of given design and controller | | |
| S←SU{(Dj,Cj,rj)} | | // Update the evaluation result to the dataset | | |
| | D1,,Dp ← OPTIMiZEDESIGNs(S,p) // Optimize a population of robot designs to evaluate | | |
| | | | |
| Find the best design D* and controller C* in dataset S with the maximum reward r*. | | | | |
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+# PRUNING FILTERS FOR EFFICIENT CONVNETS
+
+Hao Li∗ University of Maryland haoli@cs.umd.edu
+
+Asim Kadav NEC Labs America asim@nec-labs.com
+
+Igor Durdanovic NEC Labs America igord@nec-labs.com
+
+Hanan Samet† University of Maryland hjs@cs.umd.edu
+
+Hans Peter Graf NEC Labs America hpg@nec-labs.com
+
+# ABSTRACT
+
+The success of CNNs in various applications is accompanied by a significant increase in the computation and parameter storage costs. Recent efforts toward reducing these overheads involve pruning and compressing the weights of various layers without hurting original accuracy. However, magnitude-based pruning of weights reduces a significant number of parameters from the fully connected layers and may not adequately reduce the computation costs in the convolutional layers due to irregular sparsity in the pruned networks. We present an acceleration method for CNNs, where we prune filters from CNNs that are identified as having a small effect on the output accuracy. By removing whole filters in the network together with their connecting feature maps, the computation costs are reduced significantly. In contrast to pruning weights, this approach does not result in sparse connectivity patterns. Hence, it does not need the support of sparse convolution libraries and can work with existing efficient BLAS libraries for dense matrix multiplications. We show that even simple filter pruning techniques can reduce inference costs for VGG-16 by up to $34 \%$ and ResNet-110 by up to $38 \%$ on CIFAR10 while regaining close to the original accuracy by retraining the networks.
+
+# 1 INTRODUCTION
+
+The ImageNet challenge has led to significant advancements in exploring various architectural choices in CNNs (Russakovsky et al. (2015); Krizhevsky et al. (2012); Simonyan & Zisserman (2015); Szegedy et al. (2015a); He et al. (2016)). The general trend since the past few years has been that the networks have grown deeper, with an overall increase in the number of parameters and convolution operations. These high capacity networks have significant inference costs especially when used with embedded sensors or mobile devices where computational and power resources may be limited. For these applications, in addition to accuracy, computational efficiency and small network sizes are crucial enabling factors (Szegedy et al. (2015b)). In addition, for web services that provide image search and image classification APIs that operate on a time budget often serving hundreds of thousands of images per second, benefit significantly from lower inference times.
+
+There has been a significant amount of work on reducing the storage and computation costs by model compression (Le Cun et al. (1989); Hassibi & Stork (1993); Srinivas & Babu (2015); Han et al. (2015); Mariet & Sra (2016)). Recently Han et al. (2015; 2016b) report impressive compression rates on AlexNet (Krizhevsky et al. (2012)) and VGGNet (Simonyan & Zisserman (2015)) by pruning weights with small magnitudes and then retraining without hurting the overall accuracy. However, pruning parameters does not necessarily reduce the computation time since the majority of the parameters removed are from the fully connected layers where the computation cost is low, e.g., the fully connected layers of VGG-16 occupy $90 \%$ of the total parameters but only contribute less than $1 \%$ of the overall floating point operations (FLOP). They also demonstrate that the convolutional layers can be compressed and accelerated (Iandola et al. (2016)), but additionally require sparse
+
+BLAS libraries or even specialized hardware (Han et al. (2016a)). Modern libraries that provide speedup using sparse operations over CNNs are often limited (Szegedy et al. (2015a); Liu et al. (2015)) and maintaining sparse data structures also creates an additional storage overhead which can be significant for low-precision weights.
+
+Recent work on CNNs have yielded deep architectures with more efficient design (Szegedy et al. (2015a;b); He & Sun (2015); He et al. (2016)), in which the fully connected layers are replaced with average pooling layers (Lin et al. (2013); He et al. (2016)), which reduces the number of parameters significantly. The computation cost is also reduced by downsampling the image at an early stage to reduce the size of feature maps (He & Sun (2015)). Nevertheless, as the networks continue to become deeper, the computation costs of convolutional layers continue to dominate.
+
+CNNs with large capacity usually have significant redundancy among different filters and feature channels. In this work, we focus on reducing the computation cost of well-trained CNNs by pruning filters. Compared to pruning weights across the network, filter pruning is a naturally structured way of pruning without introducing sparsity and therefore does not require using sparse libraries or any specialized hardware. The number of pruned filters correlates directly with acceleration by reducing the number of matrix multiplications, which is easy to tune for a target speedup. In addition, instead of layer-wise iterative fine-tuning (retraining), we adopt a one-shot pruning and retraining strategy to save retraining time for pruning filters across multiple layers, which is critical for pruning very deep networks. Finally, we observe that even for ResNets, which have significantly fewer parameters and inference costs than AlexNet or VGGNet, still have about $30 \%$ of FLOP reduction without sacrificing too much accuracy. We conduct sensitivity analysis for convolutional layers in ResNets that improves the understanding of ResNets.
+
+# 2 RELATED WORK
+
+The early work by Le Cun et al. (1989) introduces Optimal Brain Damage, which prunes weights with a theoretically justified saliency measure. Later, Hassibi & Stork (1993) propose Optimal Brain Surgeon to remove unimportant weights determined by the second-order derivative information. Mariet & Sra (2016) reduce the network redundancy by identifying a subset of diverse neurons that does not require retraining. However, this method only operates on the fully-connected layers and introduce sparse connections.
+
+To reduce the computation costs of the convolutional layers, past work have proposed to approximate convolutional operations by representing the weight matrix as a low rank product of two smaller matrices without changing the original number of filters (Denil et al. (2013); Jaderberg et al. (2014); Zhang et al. (2015b;a); Tai et al. (2016); Ioannou et al. (2016)). Other approaches to reduce the convolutional overheads include using FFT based convolutions (Mathieu et al. (2013)) and fast convolution using the Winograd algorithm (Lavin & Gray (2016)). Additionally, quantization (Han et al. (2016b)) and binarization (Rastegari et al. (2016); Courbariaux & Bengio (2016)) can be used to reduce the model size and lower the computation overheads. Our method can be used in addition to these techniques to reduce computation costs without incurring additional overheads.
+
+Several work have studied removing redundant feature maps from a well trained network (Anwar et al. (2015); Polyak & Wolf (2015)). Anwar et al. (2015) introduce a three-level pruning of the weights and locate the pruning candidates using particle filtering, which selects the best combination from a number of random generated masks. Polyak & Wolf (2015) detect the less frequently activated feature maps with sample input data for face detection applications. We choose to analyze the filter weights and prune filters with their corresponding feature maps using a simple magnitude based measure, without examining possible combinations. We also introduce network-wide holistic approaches to prune filters for simple and complex convolutional network architectures.
+
+Concurrently with our work, there is a growing interest in training compact CNNs with sparse constraints (Lebedev & Lempitsky (2016); Zhou et al. (2016); Wen et al. (2016)). Lebedev & Lempitsky (2016) leverage group-sparsity on the convolutional filters to achieve structured brain damage, i.e., prune the entries of the convolution kernel in a group-wise fashion. Zhou et al. (2016) add group-sparse regularization on neurons during training to learn compact CNNs with reduced filters. Wen et al. (2016) add structured sparsity regularizer on each layer to reduce trivial filters, channels or even layers. In the filter-level pruning, all above work use $\ell _ { 2 , 1 }$ -norm as a regularizer.
+
+Similar to the above work, we use $\ell _ { 1 }$ -norm to select unimportant filters and physically prune them. Our fine-tuning process is the same as the conventional training procedure, without introducing additional regularization. Our approach does not introduce extra layer-wise meta-parameters for the regularizer except for the percentage of filters to be pruned, which is directly related to the desired speedup. By employing stage-wise pruning, we can set a single pruning rate for all layers in one stage.
+
+# 3 PRUNING FILTERS AND FEATURE MAPS
+
+Let $n _ { i }$ denote the number of input channels for the $i$ th convolutional layer and $h _ { i } / w _ { i }$ be the height/width of the input feature maps. The convolutional layer transforms the input feature maps $\mathbf { x } _ { i } \ \in \ \mathbb { R } ^ { n _ { i } \times h _ { i } \times w _ { i } }$ into the output feature maps $\mathbf { x } _ { i + 1 } \in \mathbb { R } ^ { n _ { i + 1 } \times h _ { i + 1 } \times w _ { i + 1 } }$ , which are used as input feature maps for the next convolutional layer. This is achieved by applying $n _ { i + 1 }$ 3D filters $\dot { \mathcal { F } } _ { i , j } \in \mathbb { R } ^ { n _ { i } \times k \times k }$ on the $n _ { i }$ input channels, in which one filter generates one feature map. Each filter is composed by $n _ { i }$ 2D kernels $\mathcal { K } \in \mathbb { R } ^ { k \times k }$ (e.g., $3 \times 3 ,$ ). All the filters, together, constitute the kernel matrix $\bar { \mathcal { F } _ { i } } \in \mathbb { R } ^ { n _ { i } \times n _ { i + 1 } \times k \times k }$ . The number of operations of the convolutional layer is $n _ { i + 1 } n _ { i } k ^ { 2 } h _ { i + 1 } w _ { i + 1 }$ . As shown in Figure 1, when a filter $\mathcal { F } _ { i , j }$ is pruned, its corresponding feature map $\mathbf { x } _ { i + 1 , j }$ is removed, which reduces $n _ { i } k ^ { 2 } h _ { i + 1 } w _ { i + 1 }$ operations. The kernels that apply on the removed feature maps from the filters of the next convolutional layer are also removed, which saves an additional $n _ { i + 2 } k ^ { 2 } h _ { i + 2 } w _ { i + 2 }$ operations. Pruning $m$ filters of layer $i$ will reduce $m / n _ { i + 1 }$ of the computation cost for both layers $i$ and $i + 1$ .
+
+
+Figure 1: Pruning a filter results in removal of its corresponding feature map and related kernels in the next layer.
+
+# 3.1 DETERMINING WHICH FILTERS TO PRUNE WITHIN A SINGLE LAYER
+
+Our method prunes the less useful filters from a well-trained model for computational efficiency while minimizing the accuracy drop. We measure the relative importance of a filter in each layer by calculating the sum of its absolute weights $\sum | \mathcal { F } _ { i , j } |$ , i.e., its $\ell _ { 1 }$ -norm $\| \mathcal { F } _ { i , j } \| _ { 1 }$ . Since the number of input channels, $n _ { i }$ , is the same across filters, $\sum \lvert \mathcal { F } _ { i , j } \rvert$ also represents the average magnitude of its kernel weights. This value gives an expectation of the magnitude of the output feature map. Filters with smaller kernel weights tend to produce feature maps with weak activations as compared to the other filters in that layer. Figure 2(a) illustrates the distribution of filters’ absolute weights sum for each convolutional layer in a VGG-16 network trained on the CIFAR-10 dataset, where the distribution varies significantly across layers. We find that pruning the smallest filters works better in comparison with pruning the same number of random or largest filters (Section 4.4). Compared to other criteria for activation-based feature map pruning (Section 4.5), we find $\ell _ { 1 }$ -norm is a good criterion for data-free filter selection.
+
+The procedure of pruning $m$ filters from the ith convolutional layer is as follows:
+
+1. For each filter $\mathcal { F } _ { i , j }$ , calculate the sum of its absolute kernel weights $\begin{array} { r } { s _ { j } = \sum _ { l = 1 } ^ { n _ { i } } \sum | \mathcal { K } _ { l } | } \end{array}$ .
+2. Sort the filters by $s _ { j }$ .
+3. Prune $m$ filters with the smallest sum values and their corresponding feature maps. The kernels in the next convolutional layer corresponding to the pruned feature maps are also removed.
+4. A new kernel matrix is created for both the $i$ th and $i + 1$ th layers, and the remaining kernel weights are copied to the new model.
+
+
+Figure 2: (a) Sorting filters by absolute weights sum for each layer of VGG-16 on CIFAR-10. The $\mathbf { X }$ -axis is the filter index divided by the total number of filters. The y-axis is the filter weight sum divided by the max sum value among filters in that layer. (b) Pruning filters with the lowest absolute weights sum and their corresponding test accuracies on CIFAR-10. (c) Prune and retrain for each single layer of VGG-16 on CIFAR-10. Some layers are sensitive and it can be harder to recover accuracy after pruning them.
+
+Relationship to pruning weights Pruning filters with low absolute weights sum is similar to pruning low magnitude weights (Han et al. (2015)). Magnitude-based weight pruning may prune away whole filters when all the kernel weights of a filter are lower than a given threshold. However, it requires a careful tuning of the threshold and it is difficult to predict the exact number of filters that will eventually be pruned. Furthermore, it generates sparse convolutional kernels which can be hard to accelerate given the lack of efficient sparse libraries, especially for the case of low-sparsity.
+
+Relationship to group-sparse regularization on filters Recent work (Zhou et al. (2016); Wen et al. (2016)) apply group-sparse regularization $( \sum _ { j = 1 } ^ { n _ { i } } \| \mathcal { F } _ { i , j } \| _ { 2 }$ or $\ell _ { 2 , 1 }$ -norm) on convolutional filters, which also favor to zero-out filters with small $l _ { 2 }$ -norms, i.e. $\mathcal { F } _ { i , j } = \mathbf { 0 }$ . In practice, we do not observe noticeable difference between the $\ell _ { 2 }$ -norm and the $\ell _ { 1 }$ -norm for filter selection, as the important filters tend to have large values for both measures (Appendix 6.1). Zeroing out weights of multiple filters during training has a similar effect to pruning filters with the strategy of iterative pruning and retraining as introduced in Section 3.4.
+
+# 3.2 DETERMINING SINGLE LAYER’S SENSITIVITY TO PRUNING
+
+To understand the sensitivity of each layer, we prune each layer independently and evaluate the resulting pruned network’s accuracy on the validation set. Figure 2(b) shows that layers that maintain their accuracy as filters are pruned away correspond to layers with larger slopes in Figure 2(a). On the contrary, layers with relatively flat slopes are more sensitive to pruning. We empirically determine the number of filters to prune for each layer based on their sensitivity to pruning. For deep networks such as VGG-16 or ResNets, we observe that layers in the same stage (with the same feature map size) have a similar sensitivity to pruning. To avoid introducing layer-wise meta-parameters, we use the same pruning ratio for all layers in the same stage. For layers that are sensitive to pruning, we prune a smaller percentage of these layers or completely skip pruning them.
+
+# 3.3 PRUNING FILTERS ACROSS MULTIPLE LAYERS
+
+We now discuss how to prune filters across the network. Previous work prunes the weights on a layer by layer basis, followed by iteratively retraining and compensating for any loss of accuracy (Han et al. (2015)). However, understanding how to prune filters of multiple layers at once can be useful: 1) For deep networks, pruning and retraining on a layer by layer basis can be extremely time-consuming 2) Pruning layers across the network gives a holistic view of the robustness of the network resulting in a smaller network 3) For complex networks, a holistic approach may be necessary. For example, for the ResNet, pruning the identity feature maps or the second layer of each residual block results in additional pruning of other layers.
+
+To prune filters across multiple layers, we consider two strategies for layer-wise filter selection:
+
+• Independent pruning determines which filters should be pruned at each layer independent of other layers.
+
+• Greedy pruning accounts for the filters that have been removed in the previous layers. This strategy does not consider the kernels for the previously pruned feature maps while calculating the sum of absolute weights.
+
+Figure 3 illustrates the difference between two approaches in calculating the sum of absolute weights. The greedy approach, though not globally optimal, is holistic and results in pruned networks with higher accuracy especially when many filters are pruned.
+
+
+Figure 3: Pruning filters across consecutive layers. The independent pruning strategy calculates the filter sum (columns marked in green) without considering feature maps removed in previous layer (shown in blue), so the kernel weights marked in yellow are still included. The greedy pruning strategy does not count kernels for the already pruned feature maps. Both approaches result in a $( n _ { i + 1 } - 1 ) \times ( n _ { i + 2 } - 1 )$ kernel matrix.
+
+
+Figure 4: Pruning residual blocks with the projection shortcut. The filters to be pruned for the second layer of the residual block (marked as green) are determined by the pruning result of the shortcut projection. The first layer of the residual block can be pruned without restrictions.
+
+For simpler CNNs like VGGNet or AlexNet, we can easily prune any of the filters in any convolutional layer. However, for complex network architectures such as Residual networks (He et al. (2016)), pruning filters may not be straightforward. The architecture of ResNet imposes restrictions and the filters need to be pruned carefully. We show the filter pruning for residual blocks with projection mapping in Figure 4. Here, the filters of the first layer in the residual block can be arbitrarily pruned, as it does not change the number of output feature maps of the block. However, the correspondence between the output feature maps of the second convolutional layer and the identity feature maps makes it difficult to prune. Hence, to prune the second convolutional layer of the residual block, the corresponding projected feature maps must also be pruned. Since the identical feature maps are more important than the added residual maps, the feature maps to be pruned should be determined by the pruning results of the shortcut layer. To determine which identity feature maps are to be pruned, we use the same selection criterion based on the filters of the shortcut convolutional layers (with $1 \times 1$ kernels). The second layer of the residual block is pruned with the same filter index as selected by the pruning of the shortcut layer.
+
+# 3.4 RETRAINING PRUNED NETWORKS TO REGAIN ACCURACY
+
+After pruning the filters, the performance degradation should be compensated by retraining the network. There are two strategies to prune the filters across multiple layers:
+
+1. Prune once and retrain: Prune filters of multiple layers at once and retrain them until the original accuracy is restored.
+
+2. Prune and retrain iteratively: Prune filters layer by layer or filter by filter and then retrain iteratively. The model is retrained before pruning the next layer for the weights to adapt to the changes from the pruning process.
+
+We find that for the layers that are resilient to pruning, the prune and retrain once strategy can be used to prune away significant portions of the network and any loss in accuracy can be regained by retraining for a short period of time (less than the original training time). However, when some filters from the sensitive layers are pruned away or large portions of the networks are pruned away, it may not be possible to recover the original accuracy. Iterative pruning and retraining may yield better results, but the iterative process requires many more epochs especially for very deep networks.
+
+# 4 EXPERIMENTS
+
+We prune two types of networks: simple CNNs (VGG-16 on CIFAR-10) and Residual networks (ResNet-56/110 on CIFAR-10 and ResNet-34 on ImageNet). Unlike AlexNet or VGG (on ImageNet) that are often used to demonstrate model compression, both VGG (on CIFAR-10) and Residual networks have fewer parameters in the fully connected layers. Hence, pruning a large percentage of parameters from these networks is challenging. We implement our filter pruning method in Torch7 (Collobert et al. (2011)). When filters are pruned, a new model with fewer filters is created and the remaining parameters of the modified layers as well as the unaffected layers are copied into the new model. Furthermore, if a convolutional layer is pruned, the weights of the subsequent batch normalization layer are also removed. To get the baseline accuracies for each network, we train each model from scratch and follow the same pre-processing and hyper-parameters as ResNet (He et al. (2016)). For retraining, we use a constant learning rate 0.001 and retrain 40 epochs for CIFAR-10 and 20 epochs for ImageNet, which represents one-fourth of the original training epochs. Past work has reported up to $3 \times$ original training times to retrain pruned networks (Han et al. (2015)).
+
+Table 1: Overall results. The best test/validation accuracy during the retraining process is reported. Training a pruned model from scratch performs worse than retraining a pruned model, which may indicate the difficulty of training a network with a small capacity.
+
+