We propose an optimistic estimate to evaluate the best possible fitting performance of nonlinear models. It yields an optimistic sample size that quantifies the smallest possible sample size to fit/recover a target function using a nonlinear model. We estimate the optimistic sample sizes for matrix factorization models, deep models, and deep neural networks (DNNs) with fully-connected or convolutional architecture. For each nonlinear model, our estimates predict a specific subset of targets that can be fitted at overparameterization, which are confirmed by our experiments. Our optimistic estimate reveals two special properties of the DNN models -- free expressiveness in width and costly expressiveness in connection. These properties suggest the following architecture design principles of DNNs: (i) feel free to add neurons/kernels; (ii) restrain from connecting neurons. Overall, our optimistic estimate theoretically unveils the vast potential of nonlinear models in fitting at overparameterization. Based on this framework, we anticipate gaining a deeper understanding of how and why numerous nonlinear models such as DNNs can effectively realize their potential in practice in the near future.

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