Grunnleggende konsepter
Analyzing convergence rates for learning with convolutional neural networks.
Sammendrag
The article discusses approximation, learning capacities, and convergence rates of CNNs in various learning problems. It covers regression and classification scenarios, providing theoretical insights into the performance of estimators based on CNNs.
Introduction:
- Deep learning successes across applications.
- Theoretical research on deep neural networks' empirical successes.
- Optimal approximations by fully connected neural networks.
Approximation and Learning Capacities:
- CNNs' universal approximation properties.
- Bounds for smooth function approximation by CNNs.
- Covering number analysis for feed-forward neural networks.
Regression:
- Estimating minimax optimal rates for least squares with CNNs.
- Smooth function approximation bounds by CNNs.
Convolutional Neural Networks Architecture:
- Definition and properties of CNN layers.
- Weight constraint introduction to control network complexity.
Approximation Capacity Analysis:
- Error estimation for approximating Hölder functions by CNNs.
- Comparison with existing results for ResNet-type CNNs.
Covering Number Estimation:
- Framework to estimate covering numbers of feed-forward neural networks.
- Application to derive bounds for the covering number of CNNs.
Binary Classification:
Hinge Loss:
- Convergence rate analysis under the Tsybakov noise condition.
Logistic Loss:
- Convergence rate analysis under the SVB condition and Tsybakov noise condition.
Statistikk
Our first result proves a new approximation bound for CNNs with certain constraint on the weights. Our second result gives a new analysis on the covering number of feed-forward neural networks, which include CNNs as special cases. Using these two results, we are able to derive rates of convergence for estimators based on CNNs in many learning problems. In particular, we establish minimax optimal convergence rates of the least squares based on CNNs for learning smooth functions in the nonparametric regression setting. For binary classification, we derive convergence rates for CNN classifiers with hinge loss and logistic loss. It is also shown that the obtained rates are minimax optimal in several settings.
Sitater
"It has been shown that CNNs are universal for approximation" - [Zhou, 2020b]
"Our result is based on the approximation bound" - [Yang and Zhou, 2024]