Before revision:  It is notable that our analysis span two popular theoretical viewpoints: stability and uniform convergence. To be specific  , in stability regime, we present high probability  rates of order O (1/n) w.r.t. the sample size n for ERM and SGD with milder assumptions in convex learning and similar high probability rates of order O (1/n) in nonconvex learning, rather than in expectation.
After revision:  It is notable that our analysis span two popular theoretical viewpoints: stability , in stability regime, we present high probability rates of order O (1/n) w.r.t. the sample size n for ERM and SGD with milder assumptions in convex learning and similar high probability rates of order O (1/n) in nonconvex learning, rather than in expectation.