  Ensembling is now recognized as an effective approach for increasing the
predictive performance and calibration of deep networks. We introduce a new
approach, Parameter Ensembling by Perturbation (PEP), that constructs an
ensemble of parameter values as random perturbations of the optimal parameter
set from training by a Gaussian with a single variance parameter. The variance
is chosen to maximize the log-likelihood of the ensemble average ($\mathbb{L}$)
on the validation data set. Empirically, and perhaps surprisingly, $\mathbb{L}$
has a well-defined maximum as the variance grows from zero (which corresponds
to the baseline model). Conveniently, calibration level of predictions also
tends to grow favorably until the peak of $\mathbb{L}$ is reached. In most
experiments, PEP provides a small improvement in performance, and, in some
cases, a substantial improvement in empirical calibration. We show that this
"PEP effect" (the gain in log-likelihood) is related to the mean curvature of
the likelihood function and the empirical Fisher information. Experiments on
ImageNet pre-trained networks including ResNet, DenseNet, and Inception showed
improved calibration and likelihood. We further observed a mild improvement in
classification accuracy on these networks. Experiments on classification
benchmarks such as MNIST and CIFAR-10 showed improved calibration and
likelihood, as well as the relationship between the PEP effect and overfitting;
this demonstrates that PEP can be used to probe the level of overfitting that
occurred during training. In general, no special training procedure or network
architecture is needed, and in the case of pre-trained networks, no additional
training is needed.
