  This paper focuses on the online gradient and proximal-gradient methods with
stochastic gradient errors. In particular, we examine the performance of the
online gradient descent method when the cost satisfies the Polyak-\L ojasiewicz
(PL) inequality. We provide bounds in expectation and in high probability (that
hold iteration-wise), with the latter derived by leveraging a sub-Weibull model
for the errors affecting the gradient. The convergence results show that the
instantaneous regret converges linearly up to an error that depends on the
variability of the problem and the statistics of the sub-Weibull gradient
error. Similar convergence results are then provided for the online
proximal-gradient method, under the assumption that the composite cost
satisfies the proximal-PL condition. In the case of static costs, we provide
new bounds for the regret incurred by these methods when the gradient errors
are modeled as sub-Weibull random variables. Illustrative simulations are
provided to corroborate the technical findings.
