We study an online joint assortment-inventory optimization problem, in which we assume that the choice behavior of each customer follows the Multinomial Logit (MNL) choice model, and the attraction parameters are unknown a priori. The retailer makes periodic assortment and inventory decisions to dynamically learn from the realized demands about the attraction parameters while maximizing the expected total profit over time. In this paper, we propose a novel algorithm that can effectively balance the exploration and exploitation in the online decision-making of assortment and inventory. Our algorithm builds on a new estimator for the MNL attraction parameters, a novel approach to incentivize exploration by adaptively tuning certain known and unknown parameters, and an optimization oracle to static single-cycle assortment-inventory planning problems with given parameters. We establish a regret upper bound for our algorithm and a lower bound for the online joint assortment-inventory optimization problem, suggesting that our algorithm achieves nearly optimal regret rate, provided that the static optimization oracle is exact. Then we incorporate more practical approximate static optimization oracles into our algorithm, and bound from above the impact of static optimization errors on the regret of our algorithm. At last, we perform numerical studies to demonstrate the effectiveness of our proposed algorithm.
Tensor Robust Principal Component Analysis (TRPCA), which aims to recover a low-rank tensor corrupted by sparse noise, has attracted much attention in many real applications. This paper develops a new Global Weighted TRPCA method (GWTRPCA), which is the first approach simultaneously considers the significance of intra-frontal slice and inter-frontal slice singular values in the Fourier domain. Exploiting this global information, GWTRPCA penalizes the larger singular values less and assigns smaller weights to them. Hence, our method can recover the low-tubal-rank components more exactly. Moreover, we propose an effective adaptive weight learning strategy by a Modified Cauchy Estimator (MCE) since the weight setting plays a crucial role in the success of GWTRPCA. To implement the GWTRPCA method, we devise an optimization algorithm using an Alternating Direction Method of Multipliers (ADMM) method. Experiments on real-world datasets validate the effectiveness of our proposed method.
As an effective and efficient discriminative learning method, Broad Learning System (BLS) has received increasing attention due to its outstanding performance in various regression and classification problems. However, the standard BLS is derived under the minimum mean square error (MMSE) criterion, which is, of course, not always a good choice due to its sensitivity to outliers. To enhance the robustness of BLS, we propose in this work to adopt the maximum correntropy criterion (MCC) to train the output weights, obtaining a correntropy based broad learning system (C-BLS). Thanks to the inherent superiorities of MCC, the proposed C-BLS is expected to achieve excellent robustness to outliers while maintaining the original performance of the standard BLS in Gaussian or noise-free environment. In addition, three alternative incremental learning algorithms, derived from a weighted regularized least-squares solution rather than pseudoinverse formula, for C-BLS are developed.With the incremental learning algorithms, the system can be updated quickly without the entire retraining process from the beginning, when some new samples arrive or the network deems to be expanded. Experiments on various regression and classification datasets are reported to demonstrate the desirable performance of the new methods.