  Proving algorithm-dependent generalization error bounds for gradient-type
optimization methods has attracted significant attention recently in learning
theory. However, most existing trajectory-based analyses require either
restrictive assumptions on the learning rate (e.g., fast decreasing learning
rate), or continuous injected noise (such as the Gaussian noise in Langevin
dynamics). In this paper, we introduce a new discrete data-dependent prior to
the PAC-Bayesian framework, and prove a high probability generalization bound
of order $O(\frac{1}{n}\cdot
\sum_{t=1}^T(\gamma_t/\varepsilon_t)^2\left\|{\mathbf{g}_t}\right\|^2)$ for
Floored GD (i.e. a version of gradient descent with precision level
$\varepsilon_t$), where $n$ is the number of training samples, $\gamma_t$ is
the learning rate at step $t$, $\mathbf{g}_t$ is roughly the difference of the
gradient computed using all samples and that using only prior samples.
$\left\|{\mathbf{g}_t}\right\|$ is upper bounded by and and typical much
smaller than the gradient norm $\left\|{\nabla f(W_t)}\right\|$. We remark that
our bound holds for nonconvex and nonsmooth scenarios. Moreover, our
theoretical results provide numerically favorable upper bounds of testing
errors (e.g., $0.037$ on MNIST). Using a similar technique, we can also obtain
new generalization bounds for certain variants of SGD. Furthermore, we study
the generalization bounds for gradient Langevin Dynamics (GLD). Using the same
framework with a carefully constructed continuous prior, we show a new high
probability generalization bound of order $O(\frac{1}{n} +
\frac{L^2}{n^2}\sum_{t=1}^T(\gamma_t/\sigma_t)^2)$ for GLD. The new $1/n^2$
rate is due to the concentration of the difference between the gradient of
training samples and that of the prior.
