Before revision:  The point our algorithm converges to is stable when the maximizing player can respond using any sequence of steps which increase the loss at each step, and the minimizing player is empowered to simulate the maximizing player 's response for arbitrarily many steps but is restricted to move according to updates sampled from a stochastic gradient oracle.
After revision:  To achieve convergence, we 1) give a novel approximation to the minimizing player is empowered to simulate the maximizing player 's response for arbitrarily many steps but is restricted to move according to updates sampled from a stochastic gradient oracle.