  We propose a new metaheuristic training scheme that combines Stochastic
Gradient Descent (SGD) and Discrete Optimization in an unconventional way. Our
idea is to define a discrete neighborhood of the current SGD point containing a
number of "potentially good moves" that exploit gradient information, and to
search this neighborhood by using a classical metaheuristic scheme borrowed
from Discrete Optimization.
  In the present paper we investigate the use of a simple Simulated Annealing
(SA) metaheuristic that accepts/rejects a candidate new solution in the
neighborhood with a probability that depends both on the new solution quality
and on a parameter (the temperature) which is modified over time to lower the
probability of accepting worsening moves. We use this scheme as an automatic
way to perform hyper-parameter tuning, hence the title of the paper. A
distinctive feature of our scheme is that hyper-parameters are modified within
a single SGD execution (and not in an external loop, as customary) and
evaluated on the fly on the current minibatch, i.e., their tuning is fully
embedded within the SGD algorithm.
  The use of SA for training is not new, but previous proposals were mainly
intended for non-differentiable objective functions for which SGD is not
applied due to the lack of gradients. On the contrary, our SA method requires
differentiability of (a proxy of) the loss function, and leverages on the
availability of a gradient direction to define local moves that have a large
probability to improve the current solution.
  Computational results on image classification (CIFAR-10) are reported,
showing that the proposed approach leads to an improvement of the final
validation accuracy for modern Deep Neural Networks such as ResNet34 and VGG16.
