Intent learning, which aims to learn users' intents for user understanding and

Learners sharing similar implicit cognitive states often display comparable observable problem-solving performances.

The matrix completion problem seeks to recover a d d ground truth matrix of low rank r d from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with d so large that even the simplest full-dimension vector operations with O ( d) time complexity become prohibitively expensive. Stochastic gradient descent (SGD) is one of the few algorithms capable of solving matrix completion on a huge scale, and can also naturally handle streaming data over an evolving ground truth. Unfortunately, SGD experiences a dramatic slow-down when the underlying ground truth is ill-conditioned; it requires at least O ( κ log(1 /ϵ)) iterations to get ϵ -close to ground truth matrix with condition number κ. In this paper, we propose a preconditioned version of SGD that preserves all the favorable practical qualities of SGD for huge-scale online optimization while also making it agnostic to κ. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to ϵ -accuracy in O (log(1 /ϵ)) iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with κ = 1 . In our experiments, we observe a similar acceleration for item-item collaborative filtering on the MovieLens25M dataset via a pair-wise ranking loss, with 100 million training pairs and 10 million testing pairs.

The task of reconstructing a matrix given a sample of observed entries is known as the matrix completion problem . It arises in a wide range of problems, including recommender systems, collaborative filtering, dimensionality reduction, image processing, quantum physics or multi-class classification to name a few. Most works have focused on recovering an unknown real-valued low-rank matrix from randomly sub-sampling its entries. Here, we investigate the case where the observations take a finite number of values, corresponding for examples to ratings in recommender systems or labels in multi-class classification. We also consider a general sampling scheme (not necessarily uniform) over the matrix entries. The performance of a nuclear-norm penalized estimator is analyzed theoretically. More precisely, we derive bounds for the Kullback-Leibler divergence between the true and estimated distributions. In practice, we have also proposed an efficient algorithm based on lifted coordinate gradient descent in order to tackle potentially high dimensional settings.

A case in point occurs in personalized recommendation systems where users' preferences are judged

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