Notes on Regularization

In some cases, e.g., if the data is sparse, the iterative algorithms underlying the parameter inference functions might not converge. A pragmatic solution to this problem is to add a little bit of regularization.

Inference functions in choix provide a generic regularization argument: alpha. When \(\alpha = 0\), regularization is turned off; setting \(\alpha > 0\) turns it on. In practice, if regularization is needed, we recommend starting with small values (e.g., \(10^{-4}\)) and increasing the value if necessary.

Below, we briefly how the regularization parameter is used inside the various parameter inference functions.

Markov-chain based algorithms

For Markov-chain based algorithms such Luce Spectral Ranking and Rank Centrality, \(\alpha\) is used to initialize the transition rates of the Markov chain.

In the special case of pairwise-comparison data, this can be loosely understood as placing an independent Beta prior for each pair of items on the respective comparison outcome probability.

Minorization-maximization algorithms

In the case of Minorization-maximization algorithms, the exponentiated model parameters \(e^{\theta_1}, \ldots, e^{\theta_n}\) are endowed each with an independent Gamma prior distribution, with scale \(\alpha + 1\). See Caron & Doucet (2012) for details.

Other algorithms

The scipy-based optimization functions use an \(\ell_2\)-regularizer on the parameters \(\theta_1, \ldots, \theta_n\). In other words, the parameters are endowed each with an independent Gaussian prior with variance \(1 / \alpha\).