  We study the learning performance of gradient descent when the empirical risk
is weakly convex, namely, the smallest negative eigenvalue of the empirical
risk's Hessian is bounded in magnitude. By showing that this eigenvalue can
control the stability of gradient descent, generalisation error bounds are
proven that hold under a wider range of step sizes compared to previous work.
Out of sample guarantees are then achieved by decomposing the test error into
generalisation, optimisation and approximation errors, each of which can be
bounded and traded off with respect to algorithmic parameters, sample size and
magnitude of this eigenvalue. In the case of a two layer neural network, we
demonstrate that the empirical risk can satisfy a notion of local weak
convexity, specifically, the Hessian's smallest eigenvalue during training can
be controlled by the normalisation of the layers, i.e., network scaling. This
allows test error guarantees to then be achieved when the population risk
minimiser satisfies a complexity assumption. By trading off the network
complexity and scaling, insights are gained into the implicit bias of neural
network scaling, which are further supported by experimental findings.
