Title: Calibration Regularized Training of Deep Neural Networks using Kernel Density Estimation. Abstract: Calibrated probabilistic classifiers are models whose predicted probabilities can directly be interpreted as uncertainty estimates. This property is particularly important in safety-critical applications such as medical diagnosis or autonomous driving. However, it has been shown recently that deep neural networks are poorly calibrated and tend to output overconfident predictions. As a remedy, we propose a trainable calibration error estimator based on Dirichlet kernel density estimates, which asymptotically converges to the true Lp calibration error. This novel estimator enables us to achieve the strongest notion of multiclass calibration, called canonical calibration, while other common calibration methods only allow for top-label and marginal calibration. The empirical results show that our estimator is competitive with the state-of-the-art, consistently yielding tradeoffs between calibration error and accuracy that are (near) Pareto optimal across a range of network architectures.  The computational complexity of our estimator is O(n^2), matching that of the kernel maximum mean discrepancy, used in a previously considered trainable calibration estimator. By contrast, the proposed method has a natural choice of kernel, and can be used to generate consistent estimates of other quantities based on conditional expectation, such as the sharpness of an estimator.