[Paper Review] Assessing the Performance of Deep Learning Algorithms for Newsvendor Problem

Yanfei Zhang, Junbin Gao|arXiv (Cornell University)|Jun 9, 2017

Forecasting Techniques and Applications10 references4 citations

TL;DR

This paper proposes a deep learning approach for the multi-feature Newsvendor problem using the original non-differentiable Newsvendor loss function as the training objective, demonstrating superior performance over the commonly used quadratic loss function. The method achieves better generalization and robustness to outliers by avoiding overfitting while maintaining strong in-sample fit.

ABSTRACT

In retailer management, the Newsvendor problem has widely attracted attention as one of basic inventory models. In the traditional approach to solving this problem, it relies on the probability distribution of the demand. In theory, if the probability distribution is known, the problem can be considered as fully solved. However, in any real world scenario, it is almost impossible to even approximate or estimate a better probability distribution for the demand. In recent years, researchers start adopting machine learning approach to learn a demand prediction model by using other feature information. In this paper, we propose a supervised learning that optimizes the demand quantities for products based on feature information. We demonstrate that the original Newsvendor loss function as the training objective outperforms the recently suggested quadratic loss function. The new algorithm has been assessed on both the synthetic data and real-world data, demonstrating better performance.

Motivation & Objective

• To address the limitations of traditional Newsvendor models that rely on strong distributional assumptions for demand.
• To develop a deep learning-based approach that directly optimizes order quantities using feature information without assuming known demand distributions.
• To evaluate whether the original Newsvendor loss function outperforms the quadratic loss function in deep learning training for this problem.
• To investigate the impact of shortage and holding costs on model training and performance.

Proposed method

• Integrates the original Newsvendor loss function, defined as $ C(y) = \mathbb{E}_d[c_p(d-y)_+ + c_h(y-d)_+] $, into a deep neural network for end-to-end training.
• Adapts the backpropagation algorithm to handle the non-differentiable nature of the original loss function through a modified gradient computation.
• Uses both synthetic data and real-world retail demand datasets to train and validate the model under varying cost ratios $ c_p/c_h $.
• Employs a feedforward neural network architecture with multiple hidden layers to learn complex, non-linear relationships between input features and optimal order quantities.
• Compares model performance using the original Newsvendor loss against the quadratic loss function proposed by Oroojlooyjadid et al. (2016).
• Applies cross-validation and evaluates both training and testing errors to assess in-sample fit and out-of-sample generalization.

Experimental results

Research questions

• RQ1 Can the original Newsvendor loss function be effectively integrated into a deep learning framework despite its non-differentiability?
• RQ2 Does training with the original Newsvendor loss function lead to better performance than using the quadratic loss function in terms of generalization and robustness?
• RQ3 How do the relative values of shortage cost $ c_p $ and holding cost $ c_h $ affect the training dynamics and prediction accuracy of deep learning models in the Newsvendor setting?
• RQ4 How does the model perform in the presence of outliers in demand data, particularly in comparison to models trained with quadratic loss?

Key findings

• The original Newsvendor loss function consistently outperforms the quadratic loss function in terms of generalization, especially on test data, indicating better out-of-sample performance.
• The model trained with the original loss function shows superior robustness to outliers, avoiding overfitting to extreme demand values, whereas the quadratic loss model attempts to fit these outliers, leading to poor generalization.
• In the presence of large outliers, the training error for the original loss function is higher than for the quadratic loss in early stages, confirming that it avoids overfitting to noisy data.
• The original loss function maintains strong in-sample fit while achieving better out-of-sample performance, demonstrating a favorable trade-off between bias and variance.
• For small-range demand data, the advantage of the original loss function in preventing overfitting is less pronounced, suggesting context-dependent benefits.
• The results confirm that the original loss function is more appropriate for deep learning-based Newsvendor models, particularly when demand distributions are unknown or non-normal.