  We present a novel definition of the reinforcement learning state, actions
and reward function that allows a deep Q-network (DQN) to learn to control an
optimization hyperparameter. Using Q-learning with experience replay, we train
two DQNs to accept a state representation of an objective function as input and
output the expected discounted return of rewards, or q-values, connected to the
actions of either adjusting the learning rate or leaving it unchanged. The two
DQNs learn a policy similar to a line search, but differ in the number of
allowed actions. The trained DQNs in combination with a gradient-based update
routine form the basis of the Q-gradient descent algorithms. To demonstrate the
viability of this framework, we show that the DQN's q-values associated with
optimal action converge and that the Q-gradient descent algorithms outperform
gradient descent with an Armijo or nonmonotone line search. Unlike traditional
optimization methods, Q-gradient descent can incorporate any objective
statistic and by varying the actions we gain insight into the type of learning
rate adjustment strategies that are successful for neural network optimization.
