  Gaussian processes (GP) provide a prior over functions and allow finding
complex regularities in data. Gaussian processes are successfully used for
classification/regression problems and dimensionality reduction. In this work
we consider the classification problem only. The complexity of standard methods
for GP-classification scales cubically with the size of the training dataset.
This complexity makes them inapplicable to big data problems. Therefore, a
variety of methods were introduced to overcome this limitation. In the paper we
focus on methods based on so called inducing inputs. This approach is based on
variational inference and proposes a particular lower bound for marginal
likelihood (evidence). This bound is then maximized w.r.t. parameters of kernel
function of the Gaussian process, thus fitting the model to data. The
computational complexity of this method is $O(nm^2)$, where $m$ is the number
of inducing inputs used by the model and is assumed to be substantially smaller
than the size of the dataset $n$. Recently, a new evidence lower bound for
GP-classification problem was introduced. It allows using stochastic
optimization, which makes it suitable for big data problems. However, the new
lower bound depends on $O(m^2)$ variational parameter, which makes optimization
challenging in case of big m. In this work we develop a new approach for
training inducing input GP models for classification problems. Here we use
quadratic approximation of several terms in the aforementioned evidence lower
bound, obtaining analytical expressions for optimal values of most of the
parameters in the optimization, thus sufficiently reducing the dimension of
optimization space. In our experiments we achieve as well or better results,
compared to the existing method. Moreover, our method doesn't require the user
to manually set the learning rate, making it more practical, than the existing
method.
