Question: The authors should explain a bit more the intuition behind these graph signatures. What type of dynamics do they capture? What do they reveal about the architecture? What is the advantage in comparison to other graph features (local or global)?
Evidence: A key objective of our work is to test whether the introduced graph representation of NNs is informative for predicting the performance of NNs. Next, we describe the steps of our proposed framework for solving Problem 1, namely performance prediction from the NN training dynamics.
As shown in Fig., our method consists of four steps: (S1) generation of a temporal graph for the training phase of each NN; (S2) extraction of node features; (S3) construction of a feature-based graph signature that captures the NN dynamics; (S4) prediction of NN performance by training a classifier or regressor on the constructed signatures. We describe these steps in more detail next.
(S1) Graph generation. The first step involves converting the training process of each input NN into a time-evolving graph. Each NN N tri ∈ N , is represented as a series of checkpoints saved for t training epochs. Based on these checkpoints and the graph representation approaches for fc and conv layers presented in 2.2, 3.1 and 2.3, we first convert the τ th NN checkpoint into weighted graph G (τ ) i at timestamp/epoch τ . We note that although our proposed graph representation involves node features, our framework does not leverage them, and thus we consider the generated graphs unattributed. Therefore, each NN N tri is mapped to a time-evolving graph
The output of this step is a set of n time-evolving graphs {G tr1 , G tr2 , ..., G trn } corresponding to the original n NNs in N .
(S2) Feature extraction. Next, the goal is to capture the structural dynamics of the NN training process. We aim to select graph measures that can capture changes during the training process, take into account the edge weight of graphs and can be calculated efficiently. In order to do that in an interpretable way, we extract two well-known node centralities from each snapshot of each generated time-evolving graph G i : weighted degree centrality and eigenvector centrality ( § 2.1). The weighted degree is a simple function of the learnable weight matrix W during the training phase of NN, therefore it gives us insights into the training dynamics at the node/neuron/filter level. The eigenvector centrality is an extension of the degree centrality, which captures the highly influential nodes, and has been successfully used in neuroscience to capture the dynamic changes of real neural networks (or connectomes). Eigenvector centrality can be used to capture importance and connectivity of filters/neurons (i.e., the nodes in our graph representation). Also, eigenvector centrality has been used for detecting communities or clusters, and thus provides structural information about the clusterability of the NN, which is complementary to that provided by the simpler and more efficient-to-compute degree centrality.
Our choice of features is also guided by the inherent k-partite structure of our proposed graph representation, which cannot be represented well by several other commonly-used graph features. For example, clustering-based features (e.g., number of triangles, transitivity, clustering coefficient) and cycle-based metrics which account for closed paths in a graph are always equal to 0 for k-partite graphs. Moreover, connected component-related features (i.e., strong/weak connectivity) do not capture the learned edge weights, which are important for modeling the training dynamics of NNs. Other features (e.g., betweenness centrality) tend to be computationally expensive, and would add significant overhead compared to early stopping methods.
(S3) Graph signature construction. In order to be able to compare NN-based graphs (with different number of nodes and edges), we summarize the structural changes in the generated time-evolving graphs at the graph level (rather than the node level, as in (S2)), and construct a statistical summary of the extracted node centralities (signature) per time-evolving graph G i . For each snapshot G (τ ) i of G i , we create a signature vector using five node feature aggregators, which were introduced in for graph similarity: median, mean, standard deviation, skewness, and kurtosis, where all but the median are moments of the corresponding distribution. Thus, G (τ ) i is mapped to a (static) signature vector s (τ ) i ∈ R 5 , representing the statistical summary of its node features (i.e., degree or eigenvector centrality) at time τ . To put more emphasis on the most recent timestamp, we can redefine the signature at time τ as the linear weighted average of the signatures up to that point,
, or an exponential function of the previous signatures, s i ← αs
. To obtain the temporal signature of the evolving graph G i , we aggregate the (static) signatures up to timestamp/epoch t:
, where ⊕ denotes concatenation. We note that global features such as algebraic connectivity, modularity, and average shortest paths may be seen as alternative ways for constructing global graph signatures while circumventing the local feature extraction step (S2); however, these features fail to capture the structural changes in our proposed graph representations (they remain (near-)constant over time) and lead to poor performance.
(S4) Performance prediction. For the last step of performance prediction, we consider two tasks: • Classification: We train a classifier (e.g., SVM, MLP) using the training graphs {G tr1 , G tr2 , ..., G trn } represented by their temporal signatures {s t tr1 , s t tr2 , ..., s t trn }, and their corresponding accuracies A = {α 1 , α 2 , . . . , α n } mapped to labels L = {l 1 , l 2 , . . . , l n } (e.g., high/low accuracy) based on some threshold. Any test NN instance is then classified using the trained classifier.
• Linear regression: We perform linear regression to estimate the actual accuracy value α tst of a new test instance N tst based on its signature obtained through steps (S1)-(S3).
(S2) Feature extraction. Next, the goal is to capture the structural dynamics of the NN training process. We aim to select graph measures that can capture changes during the training process, take into account the edge weight of graphs and can be calculated efficiently. In order to do that in an interpretable way, we extract two well-known node centralities from each snapshot of each generated time-evolving graph Gi : weighted degree centrality and eigenvector centrality (§ 2.1). The weighted degree is a simple function of the learnable weight matrix W during the training phase of NN, therefore it gives us insights into the training dynamics at the node/neuron/filter level. The eigenvector centrality is an extension of the degree centrality, which captures the highly influential nodes, and has been successfully used in neuroscience to capture the dynamic changes of real neural networks (or connectomes) (Lohmann et al., 2010). Eigenvector centrality can be used to capture importance and connectivity of filters/neurons (i.e., the nodes in our graph representation). Also, eigenvector centrality has been used for detecting communities (Newman, 2006) or clusters (Wu et al., 2013), and thus provides structural information about the clusterability of the NN, which is complementary to that provided by the simpler and more efficient-to-compute degree centrality. Our choice of features is also guided by the inherent k-partite structure of our proposed graph representation, which cannot be represented well by several other commonly-used graph features. For example, clustering-based features (e.g., number of triangles, transitivity, clustering coefficient) and cycle-based metrics which account for closed paths in a graph are always equal to 0 for k-partite graphs. Moreover, connected component-related features (i.e., strong/weak connectivity) do not capture the learned edge weights, which are important for modeling the training dynamics of NNs. Other features (e.g., betweenness centrality) tend to be computationally expensive, and would add significant overhead compared to early stopping methods.
Answer: Based on your feedback, we have updated our paper (Sec. 3.2 (S2) and (S3)) with additional justifications and intuition behind the graph features that we leverage during the signature construction step (degree and eigenvector centrality). We chose weighted degree centrality to efficiently capture (1) the weights that are learned in each layer of the NN at the node level (filter or neuron), which are the most important aspect of the training process; and (2) the change of the learned weights during the training process of NN over time (in the proposed time-evolving graph representation). Moreover, we use the node degrees to model the NN structure since they have been found to be effective in capturing the structural roles of nodes and a key notion for structural node embedding [1]. We note that we show the difference in how the degree-based signatures change during the training process for low accuracy NNs vs. high accuracy ones in Figure-7-9 in the appendix.
Eigenvector centrality is used to capture the highly influential nodes in a graph; in our case, the nodes correspond to filters of convolutional layers and neurons of fully connected layers. Additionally, eigenvector centrality has been used for detecting communities [2] or clusters [3], and thus provides structural information about the clusterability of the NN. Clusters or communities are central for network analysis and provide complementary information compared to the simpler and more efficient-to-compute degree centrality. Our hypothesis was that tracking changes in the eigenvector centrality (or node influence or clustering-related information) over time could provide complementary insights into the structural changes of the NN during training that may correlate with high or low performance. Global vs. local properties: Our proposed framework requires global graph summaries in order to be able to compare different NN-based graphs (with different number of nodes and edges) in the training set; we achieved this by computing statistical summaries (signatures) of graph properties that capture the structure locally at the node level and are efficient to compute (so that our approach adds little overhead). We also experimented with other global features such as algebraic connectivity, modularity, and average shortest paths, but those fail to capture the structural changes in our graphs and lead to poor classification performance (they remain constant at different timestamps of the represented graphs).