f-Plan Boosting: Federated & Functional Methods

Updated 7 July 2025

f-Plan Boosting is a framework that integrates functional, federated, and meta-planning principles with ensemble boosting to iteratively minimize residual errors.
It applies stagewise correction by adding weak learners that target residual errors, ensuring effective variable selection and convergence guarantees across diverse learning tasks.
By incorporating methods like federated gradient descent and PFN boosting, f-Plan Boosting provides scalable, privacy-focused solutions for high-dimensional, tabular, and agent-based scenarios.

f-Plan Boosting refers to a class of methods that integrate functional or federated principles with ensemble-based, stagewise boosting strategies for fitting statistical or machine learning models. These approaches utilize functional, federated, or meta-planning elements to overcome challenges in high-dimensional, distributed, tabular, or agent-based learning scenarios, leveraging theoretical advancements in functional gradient descent, weak learners, and meta-guided task optimization.

1. Foundations and Key Concepts

f-Plan Boosting encompasses methodologies where boosting—a sequential technique for constructing an ensemble of weak learners—is either applied to functional data, implemented in federated/distributed environments, or enhanced via high-level plan-oriented structures for agent control.

A central mechanism is stagewise correction: a sequence of weak learners, each trained to predict residual errors or functional gradients of the preceding ensemble, is iteratively added. This can be captured, generically, by the recurrence:

$F_m(x) = F_{m-1}(x) + \gamma_m h_m(x)$

where $h_m(x)$ is the $m$ th weak learner, $\gamma_m$ is its step size, and $F_{m-1}(x)$ represents the prior ensemble.

Distinct strands have emerged, each tailored to a particular setting:

Functional regression and boosting
Federated functional boosting
Boosting of pretrained neural predictors
Planning and meta-guidance for LLM agents

2. Functional Boosting in Regression

Functional boosting methods address scenarios where predictors, responses, or both are functions—common in longitudinal biomedical data, signal processing, or spectroscopic analysis. FDboost (Brockhaus et al., 2017) exemplifies this family:

Framework: Built on the mboost infrastructure, FDboost extends component-wise gradient boosting to handle scalar-on-function, function-on-scalar, and function-on-function regression models.
Base-learners: Specialized for functional effects, including P-spline (bsignal), functional principal component (bfpc), and historical/concurrent effect learners (bhist, bconcurrent).
Model flexibility: Enables mean regression, quantile regression, and GAMLSS models by allowing arbitrary loss functions.
High-dimensional data: Only one base-learner is updated per iteration, yielding automatic variable selection and shrinkage even when covariates outnumber samples.
Applications: Shown effective for spectrometric fossil fuel data (scalar-on-function regression) and multimodal neurophysiological signals (function-on-function and function-on-scalar regression).

FDboost leverages blockwise or coordinate descent in function space, relying on early stopping for regularization and incorporating extensive visualization/tuning support for practical analysis.

3. Federated Functional Gradient Boosting

Federated Functional Gradient Boosting (FFGB) generalizes functional minimization to decentralized learning, where data is partitioned across clients with potentially heterogeneous distributions (Shen et al., 2021).

Algorithmic schema: Each client performs local restricted functional gradient descent (RFGD), approximating the true functional gradient using a weak oracle, and communicates updates to a central server.
Residual correction: Introduction of a residual variable on each client tracks and corrects the approximation error from weak learners—a mechanism critical for convergence under heterogeneity.
Extensions:
- FFGB.C incorporates L-infinity norm clipping and ties convergence neighborhood radius to the average total variation distance between client and global distributions.
- FFGB.L (for squared loss) further leverages function smoothness, shrinking the convergence radius based on the average Wasserstein-1 distance.
Convergence Results: Guarantee convergence to the global optimum (if distributions align) or to a provable neighborhood as a function of distributional divergence.
Empirical evidence: On CIFAR10 and MNIST, FFGB demonstrates superior accuracy vs. communication cost, robustness to data heterogeneity, and advantages over FedAvg in ensemble-based federated minimization.

This branch of f-Plan Boosting addresses privacy, communication, and heterogeneity in distributed settings, using functional updating and error-correcting strategies.

4. Boosting with Prior-Fitted Networks for Tabular Data

BoostPFN extends prior-fitted networks (PFNs), pretrained transformer models for tabular data, to large-scale datasets by treating each PFN invocation as a weak learner in a boosting framework (Wang et al., 3 Mar 2025).

Approach:
- Each PFN is trained on a small, weighted sample of the data; sampling weights are iteratively updated using boosting error signals (e.g., exponential Hadamard rules).
- The ensemble prediction iteratively aggregates predictions from these subsampled, in-context evaluations.
Theoretical guarantee: BoostPFN is formalized as a randomized gradient boosting machine with convergence rates of $O(\sigma/M)$ under standard smoothness assumptions, where $M$ is the ensemble size.
Empirical performance:
- Outperforms or matches standard GBDT models (LightGBM, CatBoost, XGBoost), deep learning methods, and AutoML frameworks on both small and large tabular datasets.
- Demonstrably scales PFNs to up to $50 \times$ their pretraining size, resolving prior memory/computation limitations.
Implications: BoostPFN enables the fast application of PFN priors in big-data scenarios, bridges model-based and ensemble-based tabular learning, and maintains competitive accuracy with rapid inference.

5. Tree-Based Functional Boosting

Recent advances explore boosting algorithms for regression where the explanatory variables are infinite-dimensional functions, employing decision trees adapted for functional input (Ju et al., 2021).

Functional multi-index trees: Instead of reducing the data with basis expansions or feature extraction, these methods project each functional input onto multiple directions, then fit a tree on the projected K-tuple.
Identifiability: Theorems establish that, under normalization (unit norm of projection directions) and an activity requirement (each index must affect splits), the projection set is unique up to sign.
Training strategies:
- Type A: Outer-loop optimization over projection directions and inner-loop tree fitting.
- Type B: On-the-fly, tree-construction with randomized candidate projections.
Performance: Demonstrated via simulation to achieve the lowest or near-lowest mean squared errors among linear and nonparametric competitors, especially where the regression function is nonlinear.
Real-world validation: In electricity demand forecasting, adjusting for seasonality, the estimator delivered superior predictive accuracy over both linear and additive competitors.

This line of work shows f-Plan Boosting is feasible for complex, nonlinear, high-dimensional functional regression, with rigorous identifiability and empirical robustness.

6. Boosting Planning in LLM Agents

Meta Plan Optimization (MPO) exemplifies f-Plan Boosting in LLM agent planning, integrating explicit, abstract meta-planning stages that condition and improve agent reasoning (Xiong et al., 4 Mar 2025).

Meta planning: A meta planner produces high-level, environment-agnostic plans that serve as guides for agents. These plans are inserted as prompt elements, framing the agent’s execution trajectory.
Continuous optimization: Meta plans are optimized using a Monte Carlo evaluation of execution trajectories followed by Direct Preference Optimization (DPO) on contrastive plan pairs to prefer high-success strategies.
Plug-and-play integration: Because meta plans are external and agent-agnostic, this guidance is modular, enabling compatibility with a wide range of LLM agent frameworks without retraining.
Experimental results: On benchmarks including ScienceWorld and ALFWorld, MPO delivers average reward and success rate improvements (e.g., up to 51.8% improvement in reward on Llama-3.1-8B-Instruct agents), effective on both seen and unseen tasks.
Generalization: By explicitly abstracting over low-level environmental details (e.g., “go to where the first pillow may be located”), agents generalize more robustly to new task variants.

MPO demonstrates f-Plan Boosting as meta-guided, preference-optimized explicit planning, addressing agent hallucination, retraining costs, and transferability in LLM planning tasks.

7. Theoretical and Practical Implications

f-Plan Boosting frameworks share critical theoretical and operational characteristics:

Regularization and variable selection: Early stopping, component-wise updates, and residual correction universally act as regularizing, feature-selection mechanisms in both functional and federated settings.
Convergence guarantees: Multiple strands, including functional gradient boosting, federated optimizers, and randomized ensemble strategies, furnish explicit convergence rates and error bounds, often as a function of functional space geometry or distribution match.
Scalability: By partitioning computation, either across functions (functional boosting), agents (federated), or sampled subsets (BoostPFN), f-Plan approaches are positioned for large-scale, heterogeneous, or real-time inference.
Model compression and refinement: In federated settings, ensemble size may become a practical bottleneck, prompting the use of distillation or post-hoc compression to facilitate deployment.

A plausible implication is that f-Plan Boosting will continue to bridge statistical learning, distributed optimization, deep learning, and agent-based planning via ensemble-oriented, functionally grounded, and meta-guided techniques, particularly as requirements for scalability, privacy, and explainability intensify.