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

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Time-seriestime-series forecast
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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper proposes a quadratic-form weighted training objective that addresses label autocorrelation and heterogeneous task weighting across forecasting horizons. It resides in the 'Multi-Step and Horizon-Aware Losses' leaf, which contains only three papers total, indicating a relatively sparse research direction within the broader taxonomy of fifty papers. This leaf focuses specifically on loss functions that explicitly weight or penalize errors across different future steps, distinguishing it from other loss design approaches that emphasize shape matching, quantile regression, or domain-specific penalties.

The taxonomy reveals that the paper's immediate neighbors include 'Shape and Temporal Similarity-Based Losses' (two papers on DTW-style criteria) and 'Hybrid and Composite Loss Functions' (five papers combining multiple metrics). Nearby branches address training strategies like reinforcement learning and error propagation mitigation, as well as architectural innovations in transformers and recurrent networks. The quadratic weighting approach diverges from these by focusing on the correlation structure among forecast steps rather than architectural modifications or multi-objective optimization, positioning it at the intersection of loss design and temporal dependency modeling.

Among nineteen candidates examined, the quadratic-form objective (Contribution A) shows one refutable match out of six candidates reviewed, suggesting some prior exploration of weighted horizon losses. The QDF algorithm (Contribution B) and the identification of autocorrelation/weighting challenges (Contribution C) encountered no refutations across five and eight candidates respectively. Given the limited search scope—top-K semantic matches rather than exhaustive review—these statistics indicate that while horizon-aware weighting has precedent, the specific quadratic formulation and adaptive update mechanism appear less directly anticipated in the examined literature.

Based on the thirty-paper semantic search and the sparse three-paper leaf, the work appears to occupy a moderately explored niche. The taxonomy structure shows that while multi-step forecasting is a mature area, explicit horizon-aware loss design remains less crowded than architectural or domain-specific innovations. The analysis covers top semantic matches and does not claim exhaustive coverage of all possible prior work in weighted loss functions or temporal correlation modeling.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
19
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: Multi-step time-series forecasting with improved training objectives. The field has evolved around several complementary directions that address how models learn to predict multiple future time steps. Novel Loss Function Design explores custom objectives that go beyond standard mean squared error, including shape-aware criteria and horizon-specific penalties. Training Strategy and Optimization Enhancements focuses on curriculum learning, regularization schemes, and iterative refinement methods that improve convergence and generalization. Architecture and Model Design investigates neural network structures—ranging from transformers to decomposition-based modules—that inherently support multi-horizon prediction. Large Language Model Adaptations examines how pretrained language models can be repurposed for time-series tasks, while Application-Specific Forecasting Systems tailors solutions to domains like energy, finance, or healthcare. Probabilistic and Uncertainty Quantification Methods emphasizes distributional forecasts and confidence intervals, ensuring that predictions capture variability across horizons. Representative works such as Autotimes[2] and Galformer[3] illustrate how architectural innovations intertwine with training enhancements, while Shape Temporal Criteria[1] and Temporal Loss Consistency[38] exemplify loss-centric approaches. A particularly active line of research centers on designing losses that explicitly account for the multi-step nature of forecasting, balancing point accuracy with temporal coherence and horizon-aware weighting. Quadratic Direct Forecast[0] sits within this Multi-Step and Horizon-Aware Losses cluster, proposing a quadratic penalty that directly targets forecast horizons rather than treating each step uniformly. This contrasts with nearby efforts like Multistep Loss Dynamics[10], which analyzes how loss landscapes evolve across prediction steps, and Temporal Loss Consistency[38], which enforces smoothness constraints to maintain coherent trajectories. The trade-offs revolve around computational overhead, interpretability of the loss surface, and the degree to which horizon-specific weighting improves long-range accuracy without sacrificing near-term precision. Open questions include how to adaptively tune horizon penalties during training and whether such losses generalize across diverse application domains, from energy load forecasting to financial volatility prediction.

Claimed Contributions

Quadratic-form weighted learning objective for time-series forecasting

The authors introduce a quadratic-form weighted learning objective that simultaneously addresses label autocorrelation effect (via off-diagonal elements of the weighting matrix) and heterogeneous task weights (via non-uniform diagonal elements) for training multi-step time-series forecast models.

6 retrieved papers
Can Refute
Quadratic Direct Forecast (QDF) learning algorithm

The authors develop the QDF algorithm that trains forecast models by adaptively learning and updating the quadratic-form weighting matrix through a bilevel optimization procedure targeting model generalization performance.

5 retrieved papers
Identification of two fundamental challenges in learning objective design

The authors formally identify and characterize two key challenges that existing learning objectives fail to address: the autocorrelation effect among future steps in label sequences and the need for heterogeneous weights across different forecasting tasks.

8 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Quadratic-form weighted learning objective for time-series forecasting

The authors introduce a quadratic-form weighted learning objective that simultaneously addresses label autocorrelation effect (via off-diagonal elements of the weighting matrix) and heterogeneous task weights (via non-uniform diagonal elements) for training multi-step time-series forecast models.

Contribution

Quadratic Direct Forecast (QDF) learning algorithm

The authors develop the QDF algorithm that trains forecast models by adaptively learning and updating the quadratic-form weighting matrix through a bilevel optimization procedure targeting model generalization performance.

Contribution

Identification of two fundamental challenges in learning objective design

The authors formally identify and characterize two key challenges that existing learning objectives fail to address: the autocorrelation effect among future steps in label sequences and the need for heterogeneous weights across different forecasting tasks.

Table of Contents