Multi-period Learning Framework

• Multi-period Learning Framework (MLF) is a modeling approach that integrates multiple temporal periods to capture intra- and inter-period dependencies for improved forecasting and decisions.
• MLF employs techniques like self-adaptive patching, patch squeeze, and inter-period redundancy filtering to reduce bias and efficiently integrate multi-scale data.
• Applications of MLF span financial forecasting, asset-liability management, and power system optimization, emphasizing robust constraint enforcement and scalability.

A Multi-period Learning Framework (MLF) refers to a class of machine learning and optimization methodologies explicitly designed to process, model, and make predictions or decisions over temporally-structured data where multiple distinct input periods, stages, or time horizons are relevant to the downstream objective. In contemporary literature, MLFs are prominent in domains such as financial time series forecasting, asset-liability management under regime-switching, and operational planning tasks such as multi-period optimal power flow with storage and ramping constraints. The principal challenge for MLFs is to coherently integrate signal across diverse temporal windows, enforce feasible inter-temporal coupling, and deliver efficient and robust solutions that scale with problem size and real-world deployment constraints.

1. Formalization and Motivation of Multi-period Learning

MLF formalizes the learning and decision process as one where a model must (a) parse historical data over multiple, potentially non-overlapping or hierarchically nested time intervals, (b) produce integrated predictions or actions that account for both intra-period and inter-period dependencies, and (c) address structural constraints arising from temporal coupling or sequential feasibility.

A canonical mathematical template is as follows:

Given a set of $S$ historical sequences ("periods") $X_h^s \in \mathbb{R}^{n_s \times c}$ for $s=1,\ldots,S$ (with $n_1 < n_2 < \cdots < n_S$), an MLF seeks to map these to a target $X_f$ or to a trajectory of actions $\{a_t\}_{t=1}^T$. In settings such as multi-period DC-OPF, the optimization problem comprises a vector of decision variables $$\left\{g_t, p_t^{ch}, p_t^{dis}, e_t\right\}_{t=1}^T$$ subject to temporal and operational constraints. Alternatively, in reinforcement learning-based asset-liability management, the evolution of assets and liabilities $x_t$, $l_t$ is jointly modeled under regime-switching and partial observability, with the policy $\pi$ optimized over a multi-period objective function.

The motivation for MLF arises from empirical and operational observations that single-period models: (1) cannot capture multi-scale drivers of system behavior; (2) are forced to tune window sizes heuristically per task/horizon; (3) are unable to enforce end-to-end feasibility for inter-temporally linked constraints; and (4) face severe trade-offs between input length and computational tractability.

2. Representative MLF Architectures and Algorithms

Key recent contributions have instantiated the MLF paradigm in diverse ways, including:

A. MLF for Financial Time Series Forecasting

The architecture in "Multi-period Learning for Financial Time Series Forecasting" (Zhang et al., 7 Nov 2025) comprises the following major modules:

• Multi-period self-Adaptive Patching (MAP): Each historical period $X_h^s$ is converted into exactly $\widetilde N$ patches, neutralizing bias towards longer periods. For each period, stride and patch size are

$$K^s = \left\lfloor \frac{n^s}{\widetilde N} \right\rfloor, \qquad L^s = \alpha K^s \; (\alpha=2),$$

yielding $X_p^s \in \mathbb{R}^{L^s \times \widetilde N}$.

• Patch Squeeze: Each $X_d^s$ (patch-embedded period) is compressed with a learned encoder to reduce patch count by $r$ (e.g., $r=4,8$), exploiting intra-period redundancy. A reconstructor aids training via a mean-squared error loss.
• Inter-period Redundancy Filtering (IRF): For each Transformer block, the output embedding of each period $z_e^s$ is "cleaned" by subtracting the estimated redundancy $\epsilon_e^j$ of all shorter periods $j<s$, ensuring no duplicate contribution to downstream attention.
• Learnable Weighted-average Integration (LWI): Final per-period forecasts are fused via weights $a_s \in (0,1)$ computed from global features of the longest period, using MLPs and a 1D-CNN.
• End-to-End Objective: The total loss is

$$\mathcal{L} = \text{MSE}(\widehat X_f, X_f) + \frac{1}{S} \sum_{s=1}^S \mathrm{MSE}(\widehat X_p^s, X_p^s),$$

enforcing forecasting accuracy and reconstruction regularization.

B. MLF in Multi-period Asset-Liability Management

In "Multi-period Asset-liability Management with Reinforcement Learning in a Regime-Switching Market" (Gao et al., 3 Sep 2025), the MLF is realized through:

• Regime-Switching Model: Assets and liabilities are subject to hidden Markov regimes ($\epsilon_t$), with stochastic filtering ($\widehat p_t$) estimating regime probabilities.
• Exploratory Mean-Variance (EMV) RL: The policy space is regularized with an entropy term $H(\pi) = -\sum \int \pi_t(u) \ln \pi_t(u) du$, balancing exploration/exploitation.
• Actor-Critic Training: Policy $\pi_t^\phi$ is parameterized as a Gaussian, value function is quadratic in state $(x_t, l_t)$, and temporal difference-based martingale loss drives parameter updates.
• Handling Time-Inconsistency: A pre-committed strategy (decouple policy selection at $t=0$ from later stages) is adopted to avoid classical mean-variance inconsistency.

C. MLF for Multi-period Constrained Optimization (DC-OPF)

The "MPA-DNN: Projection-Aware Unsupervised Learning for Multi-period DC-OPF" (Kim et al., 10 Oct 2025) realizes MLF through:

• Feedforward DNN: Consumes the entire load trajectory $\operatorname{vec}(p^d) \in \mathbb{R}^{n_dT}$; outputs raw generation/storage trajectories $\operatorname{vec}(z) \in \mathbb{R}^{pT}$.
• Differentiable Projection Layer: Projects $z$ onto the feasible set $\mathcal{C}$ of the multi-period DC-OPF via a quadratic program:

$$\Pi_{\mathcal{C}}(z) = \hat x = \operatorname*{arg\,min}_{x \in \mathcal{C}} \frac{1}{2} \lVert x - z \rVert^2_2,$$

ensuring all operational/ramping/network constraints hold at every forward pass.

• End-to-End Unsupervised Training: Loss is total generation cost over $T$:

$L(\theta) = \sum_{t=1}^T c_t^\top \hat g_t,$

with feasibility guaranteed by the projection. Back-propagation through KKT conditions enables gradient-based learning.

3. Core Principles and Structural Modules

Recent MLFs share structural design principles:

• Explicit Multi-period Input Handling: Simultaneous access to multi-scale historical data or decision variables indexed by temporal window.
• Redundancy Reduction: Elimination of overlapping or duplicated temporal information, both across periods (e.g., IRF) and within each period (e.g., Patch Squeeze).
• Adaptive Integration: Dynamic weighting of each period's prediction or decision trace, informed by learned global or contextual features.
• Strict Feasibility Enforcement: Embedding physical, operational, or economic constraints directly into the learning pipeline (e.g., via differentiable projection layers or policy regularization).
• Jointly Optimized Objectives: End-to-end objectives that penalize both forecasting/decision error and auxiliary regularization, often supported by multi-task or multi-module architectures.

4. Empirical Performance and Benchmarking

Recent MLF implementations demonstrate substantial empirical advances over single-period and naïvely multi-period methods:

Setting	Metric	MLF Result	Comparison
Financial TSF	MSE, WMAPE	7–15% MSE reduction vs. PatchTST, 10–20% vs. Scaleformer/Pathformer (Zhang et al., 7 Nov 2025)	Single-period models
Multi-period DC-OPF	MAE (p.u.), optimality gap, violations	MAE $\leq$0.016 p.u., gap $\leq$0.024%, zero violations, 10–50$\times$ speedup (Kim et al., 10 Oct 2025)	Gurobi, SPA-DNN
Asset-Liability Management	Mean, Variance, Sharpe ratio	PoEMV-1 mean=7.9985, variance=0.0094, Sharpe=72.02 (Gao et al., 3 Sep 2025)	Closed-form, CoEMV

These results collectively highlight MLF's capability to deliver accuracy, enforce constraints, and maintain computational efficiency.

5. Applications and Practical Recommendations

MLFs are deployed across several domains:

• Financial forecasting (fund sales, multi-variate benchmarks): Use a geometric progression of period lengths (e.g., $\{5, 10, 30, 60, 120, 150\}$ days) to cover diverse time scales; patch count per period ($\widetilde N=64$ or $32$), Patch Squeeze ($r=4$ or $8$) for efficiency; daily retraining on sliding windows is recommended (Zhang et al., 7 Nov 2025).
• Power system scheduling (DC-OPF): MLF enables feasible, rapid dispatch for grids with high renewable/storage penetration; strict constraint satisfaction is maintained via projection; suitable for hour-scale to day-ahead scheduling. For large-scale instances, decomposition or warm-starts may be necessary (Kim et al., 10 Oct 2025).
• Asset-liability management: RL-based MLFs allow exploration/exploitation trade-offs, handle regime-switching and partial observability; practical for multi-year strategic planning (Gao et al., 3 Sep 2025).

Operational deployment commonly involves: periodic retraining, monitoring adaptive integration weights for drift or bias, tuning period lengths per asset class/problem domain, and deploying with mixed-precision and gradient clipping for numerical stability.

6. Limitations, Open Challenges, and Directions

While MLFs represent the state of the art in multi-temporal modeling, several challenges remain:

• Scalability: Projection-based MLFs face quadratic program scaling with the product of time horizon and system size; specialized solvers or approximate methods are required for nationwide grids or high-frequency forecasting.
• Extension to Nonlinear/Nonconvex Constraints: For full AC-OPF in power systems or nonlinear financial constraints, differentiable nonconvex projection layers or approximations (e.g., convex relaxations, deep FBSDE methods) are research frontiers.
• Uncertainty and Robustness: Current MLFs operate primarily under point forecasts or empirical regime models. Incorporating chance constraints, multi-stage stochastic programming, or adversarial approaches remains largely open.
• High-dimensionality: RL-based MLFs for large asset universes demand richer function approximators, scalable policy/value architectures, and efficient entropy regularization strategies.
• Objective Generalization: Beyond mean-square error or mean-variance, new MLF formulations are required to target alternative risk measures (e.g., CVaR, drawdown, utility-based criteria) in finance and robust/secure objectives in operations.

A plausible implication is that future MLF research will increasingly unify advances in multi-scale representation learning, differentiable optimization, and robust RL to address the above challenges and expand applicability.

7. Conclusion

Multi-period Learning Frameworks embody a systematic methodology for integrating, processing, and acting upon temporally-structured data that spans multiple periods or horizons. By incorporating modules for adaptive patching, redundancy filtering, weighted integration, and feasibility projection, MLFs have achieved leading empirical results and robust production deployment across power systems, finance, and complex dynamic scheduling domains. Ongoing research continues to address scaling, nonconvexity, and uncertainty, positioning MLF as a foundational tool for multi-horizon analytics and decision-making in data-intensive scientific and engineering applications.