  We study the statistical and computational complexities of the Polyak step
size gradient descent algorithm under generalized smoothness and Lojasiewicz
conditions of the population loss function, namely, the limit of the empirical
loss function when the sample size goes to infinity, and the stability between
the gradients of the empirical and population loss functions, namely, the
polynomial growth on the concentration bound between the gradients of sample
and population loss functions. We demonstrate that the Polyak step size
gradient descent iterates reach a final statistical radius of convergence
around the true parameter after logarithmic number of iterations in terms of
the sample size. It is computationally cheaper than the polynomial number of
iterations on the sample size of the fixed-step size gradient descent algorithm
to reach the same final statistical radius when the population loss function is
not locally strongly convex. Finally, we illustrate our general theory under
three statistical examples: generalized linear model, mixture model, and mixed
linear regression model.
