In gradient-based optimization of photonic devices, within the overall design parameter space, one iteratively performs a line search in a one-dimensional subspace as spanned by the search direction. While the search direction can be efficiently determined with the adjoint variable method, there has not been an efficient algorithm that determines the optimal learning rate that controls the distance one moves along the search direction. Here we introduce an efficient algorithm of determining the optimal learning rate, using the Shanks transformation in the Lippmann-Schwinger formalism. Our approach can determine very accurately the optimal learning rates at each epoch, with only a modest increase of computational cost. We show that this approach can significantly improve the figure of merits of the final structure, as compared to conventional methods for estimating the learning rate.