There is a large literature on this problem.

Consequently, a line of work has focused on characterizing the implicit biases of global optimum reached by various optimization algorithms. For example, Gunasekar et al. [ 2017 ] consider the problem of matrix factorization and show that gradient descent (GD) on un-regularized objective converges to the minimum nuclear norm solution.

Although optimization is the longstanding algorithmic backbone of machine learning, new models still require the time-consuming implementation of new solvers. As a result, there are thousands of implementations of optimization algorithms for machine learning problems.

Consequently, a line of work has focused on characterizing the implicit biases of global optimum reached by various optimization algorithms. For example, Gunasekar et al. [ 2017 ] consider the problem of matrix factorization and show that gradient descent (GD) on un-regularized objective converges to the minimum nuclear norm solution.

Due to the inherent uncertainty of data, the problem of predicting partial ranking from pairwise comparison data with ties has attracted increasing interest in recent years. However, in real-world scenarios, different individuals often hold distinct preferences. It might be misleading to merely look at a global partial ranking while ignoring personal diversity. In this paper, instead of learning a global ranking which is agreed with the consensus, we pursue the tie-aware partial ranking from an individualized perspective. Particularly, we formulate a unified framework which not only can be used for individualized partial ranking prediction, but also be helpful for abnormal user selection.

The next lemma was used in the proof of Theorem 1.

This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan.

Current numerical procedures for SLOPE, however, lack the efficiency that respective tools for the lasso enjoy, particularly in the context of estimating a complete regularization path.