Optimization based on k -step contrastive divergence (CD) has become a common way to train restricted Boltzmann machines (RBMs). The k -step CD is a biased estimator of the log-likelihood gradient relying on Gibbs sampling. We derive a new upper bound for this bias. Its magnitude depends on k , the number of variables in the RBM, and the maximum change in energy that can be produced by changing a single variable. The last reflects the dependence on the absolute values of the RBM parameters. The magnitude of the bias is also affected by the distance in variation between the modeled distribution and the starting distribution of the Gibbs chain.

Iterative learning algorithms that approximate the solution of support vector machines (SVMs) have two potential advantages. First, they allow online and active learning. Second, for large data sets, computing the exact SVM solution may be too time-consuming, and an efficient approximation can be preferable. The powerful LASVM iteratively approaches the exact SVM solution using sequential minimal optimization (SMO). It allows efficient online and active learning. Here, this algorithm is considerably improved in speed and accuracy by replacing the working set selection in the SMO steps. A second-order working set selection strategy, which greedily aims at maximizing the progress in each single step, is incorporated.

Gradient-based optimizing of gaussian kernel functions is considered. The gradient for the adaptation of scaling and rotation of the input space is computed to achieve invariance against linear transformations. This is done by using the exponential map as a parameterization of the kernel parameter manifold. By restricting the optimization to a constant trace subspace, the kernel size can be controlled. This is, for example, useful to prevent overfitting when minimizing radius-margin generalization performance measures. The concepts are demonstrated by training hard margin support vector machines on toy data.