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Dirichlet-Guided Group Forecasting for Alleviating Over-smoothing in Time Series Forecasting

This paper proposes Dirichlet-Guided Group Forecasting (DGF), a novel framework that mitigates over-smoothing in time series forecasting by explicitly modeling multiple plausible future modes and their selection probabilities through a Dirichlet-guided hierarchical sampling mechanism and reward-based optimization.

Original authors: Xingyu Zhang, Jingyao Wang, Xin Yu, Zeen Song, Jianqi Zhang, Changwen Zheng, Wenwen Qiang

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xingyu Zhang, Jingyao Wang, Xin Yu, Zeen Song, Jianqi Zhang, Changwen Zheng, Wenwen Qiang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Problem: The "Blurry Photo" Effect

Imagine you are trying to predict the weather for next week. You look at the history: it's been sunny, then rainy, then sunny again.

Most current forecasting models act like a blurry camera. They look at all the possible futures and try to find the "average" one.

If the future could be a sharp spike in temperature (a heatwave) OR a sharp drop (a cold snap), the model doesn't know which one will happen.
So, it averages them out. Instead of predicting a heatwave or a cold snap, it predicts a "mildly warm day."
The Result: The prediction looks smooth and safe, but it misses the exciting, dangerous, or important details. In the paper, this is called "over-smoothing." It's like taking a high-definition photo and applying a heavy blur filter; you see the general shape, but you lose the sharp edges, the sudden turns, and the specific details that actually matter.

The Root Cause: The "One-Story" Trap

Why does this happen? The paper suggests it's because of how the models are trained.

Imagine a teacher showing a student a history of events and then showing them only one possible ending (the "ground truth").
The student thinks, "Okay, the only way this story ends is this way."
But in reality, that same history could have led to many different endings (a "multi-modal" future).
Because the student only saw one ending, they try to squeeze all possibilities into that single path. When they try to predict a future that could go Up or Down, they get stuck in the middle, creating a flat, boring line.

The Solution: DGF (The "Scenario Planning" Team)

The authors propose a new method called Dirichlet-Guided Group Forecasting (DGF). Instead of trying to guess the one right answer, DGF acts like a team of scenario planners.

Here is how it works, step-by-step:

1. The "Mode" Team (Different Scenarios)

Instead of one brain trying to predict everything, DGF creates a team of K different forecasters (called "modes").

Forecaster A specializes in "Upward Trends."
Forecaster B specializes in "Downward Trends."
Forecaster C specializes in "Oscillations" (wiggles).
Forecaster D specializes in "Sudden Spikes."
Each forecaster is trained to be an expert in their specific type of future, rather than trying to be a mediocre expert at everything.

2. The "Dirichlet" Manager (The Uncertainty Boss)

This is the clever part. The model doesn't just pick one forecaster; it uses a special tool called a Dirichlet distribution to act as a manager.

Imagine the manager has a bag of marbles representing the probability of each scenario happening.
If the history is very clear (e.g., a steady trend), the manager is confident and grabs mostly "Upward" marbles.
If the history is confusing or chaotic, the manager is uncertain. The Dirichlet tool allows the manager to say, "I'm not sure which one it is, so let's keep the options open and explore a mix of possibilities."
This prevents the model from forcing a single answer when the data is ambiguous.

3. The "Group" Game (Avoiding the Blur)

The model generates predictions in groups (mini-teams).

It picks a "probability mix" (the manager's decision) for the whole group.
Then, every member of that group generates a future based on that mix.
The Reward System: The model gets points for:
- Accuracy: Did the prediction match the real data?
- Dynamics: Did it keep the sharp edges and sudden turns? (No blurring!)
- Diversity: Did the different forecasters stay different from each other? (If everyone starts predicting the same boring line, they lose points).

The Analogy: The Orchestra vs. The Soloist

Old Models (Over-smoothing): Like a soloist trying to play a complex symphony. They play the "average" note of every instrument. It sounds okay, but it's flat and lacks the punch of the drums or the melody of the violins.
DGF (The New Method): Like a full orchestra.
- The Violins (Mode 1) play the high notes.
- The Drums (Mode 2) play the sudden beats.
- The Conductor (Dirichlet) decides how loud each section should be based on the music sheet (the history).
- If the sheet says "Crescendo," the conductor boosts the volume. If the sheet is ambiguous, the conductor lets the sections improvise slightly, ensuring the music remains dynamic and full of life, rather than a single, flat tone.

The Results

The paper tested this on real-world data (like electricity usage, weather, and traffic).

Less Blurring: The predictions kept the sharp peaks and sudden drops that other models smoothed over.
Better Accuracy: Because the model wasn't forced to average out conflicting possibilities, it actually got closer to the real future.
More Useful: The model provided a set of distinct, plausible futures (e.g., "It could be a heatwave, OR it could be a cold snap") rather than one vague, middle-of-the-road guess.

Summary

The paper argues that time series forecasting shouldn't be about finding the "average" future. It should be about recognizing that the future has different distinct possibilities (modes). By using a team of specialized forecasters and a smart manager that handles uncertainty, DGF stops the "blurry photo" effect and produces sharp, dynamic, and accurate predictions.

Technical Summary: Dirichlet-Guided Group Forecasting (DGF)

1. Problem Definition: Over-Smoothing as Latent Mode Compression

The paper identifies over-smoothing as a critical limitation in time series forecasting (TSF), particularly in multi-step predictions where future dynamics are multi-modal. Traditional models often capture coarse trends but fail to preserve sharp changes, oscillations, turning points, and regime transitions.

The authors reframe over-smoothing not merely as a loss-induced artifact or frequency-domain issue, but as latent dynamical mode compression. Under the standard setting of single-realization supervision (where each training input is paired with only one realized future trajectory), the following issues arise:

Mode Averaging: Multiple plausible future modes (e.g., rising, declining, oscillatory, or regime-switching trajectories) that are incompatible under linear averaging are merged or weakened.
Non-Convexity: The set of dynamically valid trajectories for different modes is generally non-convex. Averaging an increasing and a decreasing trajectory produces a flat, dynamically invalid trajectory.
Collapse: Naive multi-head supervision, where all heads are optimized against the same ground truth, leads to a collapsed optimum where all heads converge to the same conditional mean, failing to represent distinct latent modes.

Consequently, a forecast may achieve low value-level error (e.g., MSE) while losing the structural properties required for dynamic plausibility.

2. Methodology: Dirichlet-Guided Group Forecasting (DGF)

To mitigate over-smoothing, the authors propose Dirichlet-Guided Group Forecasting (DGF), a mode-preserving probabilistic framework. DGF explicitly separates mode-conditioned prediction from mode selection and models uncertainty over the selection probabilities.

2.1 Core Components

Mode-Conditioned Predictive Distributions:
Given a historical input $X$ , the model learns $K$ distinct predictive distributions $q_{\theta,k}(Y|X)$ , each intended to capture a specific latent dynamical mode (e.g., specific trend directions or volatility states).
Dirichlet-Guided Hierarchical Sampling:
Instead of using a deterministic softmax gate to select modes, DGF models the uncertainty over the mode-selection probability vector $\pi$ .
- A Dirichlet distribution $Dir(\pi | \alpha_\theta(X))$ is parameterized by concentration parameters $\alpha_\theta(X)$ . This allows the model to represent not just the expected mode probabilities, but also the confidence (second-order uncertainty) in those probabilities.
- Group Sampling: For a mini-group of $G$ samples, a single probability vector $\pi_b$ is sampled from the Dirichlet distribution. This shared $\pi_b$ represents a group-level hypothesis about the mode composition.
- Trajectories are then generated by sampling a mode index from $\pi_b$ and drawing a trajectory from the corresponding mode-conditioned distribution. This avoids the deterministic aggregation of incompatible modes.
Hierarchical GRPO Optimization:
The model is trained using a Group Relative Policy Optimization (GRPO) objective that combines three reward signals to prevent mode collapse and ensure dynamic consistency:
- Trajectory Reward ( $r_{traj}$ ): Combines accuracy (closeness to ground truth, e.g., MSE) and dynamical consistency (matching local slopes, curvature, frequency spectra, and transition smoothness).
- Dirichlet-Level Credit Assignment: The Dirichlet concentration parameters are updated based on the average reward of the entire mini-group. This encourages the model to learn mode-composition hypotheses that generate high-quality forecast groups, rather than optimizing individual samples in isolation.
- Distribution-Level Diversity Regularization: A penalty term ensures that the $K$ mode-conditioned predictive distributions remain distinct, preventing them from collapsing into a single solution.

3. Key Contributions

The paper outlines four primary contributions:

Problem Formulation: It formally defines over-smoothing in TSF as the compression of non-convex latent dynamical modes under single-realization supervision, demonstrating that point-risk minimization can induce dynamically invalid forecasts.
Framework Proposal: It introduces DGF, a framework that jointly learns mode-conditioned predictive distributions and models second-order uncertainty over their selection probabilities using a Dirichlet distribution.
Optimization Strategy: It develops a GRPO-based training objective that utilizes accuracy, dynamical consistency, and diversity rewards to optimize the hierarchical sampling process.
Empirical Validation: It demonstrates through extensive experiments that DGF reduces over-smoothing while improving forecasting accuracy, diversity, and dynamical consistency on real-world benchmarks.

4. Experimental Results

The authors evaluated DGF on standard long-sequence forecasting (LSF) benchmarks (ETT, Electricity, Weather) and probabilistic forecasting datasets (Solar, Istanbul Traffic, Turkey Power).

Deterministic Forecasting: DGF consistently improved performance over strong baselines (including Moirai-small and Moirai-base with full fine-tuning, LoRA, and prompt tuning) across multiple datasets. It achieved the best average performance on most benchmarks, indicating that mode-preserving generation improves point-forecasting accuracy without relying on averaging incompatible modes.
Probabilistic Forecasting: DGF achieved state-of-the-art or competitive results in Continuous Ranked Probability Score (CRPS) and Mean Scaled Interval Score (MSIS), suggesting improved calibration and distributional quality.
Over-Smoothing Mitigation: Using the STRIPE protocol and DILATE metrics (sensitive to shape and temporal distortion), DGF showed significant improvements in both mean-sample and best-sample scores. The gains in DILATE specifically confirmed a reduction in dynamic-structure degradation.
Qualitative Analysis: Visualizations of generated trajectories showed that DGF produces distinct yet coherent futures (e.g., separate upward, stable, and downward trends) rather than collapsing them into a single smoothed trajectory.
Efficiency: Despite the hierarchical sampling, DGF maintained comparable or lower training and inference times compared to supervised full fine-tuning, suggesting the performance gains do not come at a prohibitive computational cost.

5. Significance and Claims

The paper claims that DGF offers a paradigm shift from forcing models to output a single averaged future to explicitly representing and reasoning about multiple plausible future modes.

Addressing the Root Cause: By treating over-smoothing as a mode-compression problem, DGF addresses the structural limitation of standard supervised learning where incompatible modes are averaged out.
Dynamic Consistency: The framework prioritizes the preservation of dynamic structures (peaks, oscillations, regime shifts) alongside value-level accuracy, which is critical for downstream decision-making in domains like energy management, weather forecasting, and traffic control.
Uncertainty Quantification: By modeling the Dirichlet concentration parameters, DGF provides a richer representation of uncertainty regarding which future mode is likely to occur, rather than just a point estimate of mode probabilities.

The authors note that while DGF is model-agnostic, it currently relies on pretrained backbones for initialization to ensure stable optimization. They also acknowledge the computational overhead of sampling but argue it is a necessary cost for preserving latent modes. The work suggests that future decision-makers can benefit from forecasts that are more dynamically consistent and mode-diverse, helping to avoid over-reliance on over-smoothed predictions.