Deep neural networks (DNNs) have the same structure as the neocognitron proposed in 1979 but have much better performance, which is because DNNs include many heuristic techniques such as pre-training, dropout, skip connections, batch normalization (BN), and stochastic depth. However, the reason why these techniques improve the performance is not fully understood. Recently, two tools for theoretical analyses have been proposed. One is to evaluate the generalization gap, defined as the difference

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... between the expected loss and empirical loss, by calculating the algorithmic stability, and the other is to evaluate the convergence rate by calculating the eigenvalues of the Fisher information matrix of DNNs. This overview paper briefly introduces the tools and shows their usefulness by showing why the skip connections and BN improve the performance.