Quadratic Direct Forecast for Training Multi-Step Time-Series Forecast Models

2025-10-28

AI Summary

Existing time-series multi-step forecasting models employ pointwise losses (e.g., MSE), treating each prediction step as an independent and equally weighted task—ignoring inherent label autocorrelation and introducing theoretical bias in the training objective. To address this, we propose Quadratic Weighted Training (QWT), a novel training objective that employs a learnable, dynamic weight matrix to jointly model inter-step dependencies and heterogeneous task importance—marking the first approach to unify label autocorrelation and task weighting within a single framework. Furthermore, we design Quadratic Direct Forecasting (QDF), a learning algorithm enabling end-to-end optimization and seamless integration with diverse backbone architectures. Extensive experiments on multiple benchmark datasets demonstrate significant improvements over state-of-the-art methods. Our implementation is publicly available.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The design of training objective is central to training time-series forecasting models. Existing training objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which we found leading to the following two issues: (1) overlook the label autocorrelation effect among future steps, leading to biased training objective; (2) fail to set heterogeneous task weights for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted training objective, addressing both of the issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the most preferable weights of the forecasting tasks with varying future steps. To achieve this, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves performance of various forecast models, achieving state-of-the-art results. Code is available at https://anonymous.4open.science/r/QDF-8937.

Problem

• Addresses biased training from ignoring future step autocorrelations
• Solves heterogeneous task weighting for different forecasting horizons
• Proposes quadratic training objective to improve forecast model performance

Innovation

• Quadratic-form weighted training objective for time-series
• Weighting matrix addresses label autocorrelation and task heterogeneity
• Adaptively updated quadratic weighting improves forecast model performance