PhD Candidature Confirmation

Federated Hierarchical Forecasting

Distributed Learning Methods for Hierarchical Forecasting

Yingjie Zhao

yingjie.zhao@hdr.qut.edu.au

School of Mathematical Science, Faculty of Science

Gardens Point Campus, Queensland University of Technology

7 October 2026

INTRODUCTION · 2

Presentation Outline

Presentation for PhD candidature.

Section	Title
01	INTRODUCTION
02	RELATED WORKS
03	RESEARCH QUESTIONS
04	PROGRESS TO DATE
05	NEXT STEPS
06	QUESTIONS

PhD Candidature Confirmation

Introduction

Hierarchical forecasting, multi-stakeholder systems and forecasting paradigms

INTRODUCTION / HIERARCHICAL FORECASTING · 4

Hierarchical Forecasting Basis

A forecast for every node, linked by the same aggregation structure.

Seven-node hierarchy: Total, A, B, A1, A2, B1 and B2

7 series, including 4 terminal series

Concept	Notation	Meaning
Base forecast	$\hat{\mathbf y}_{t+h\mid t}$	Forecasts before reconciliation, may not properly add up.
Coherent forecast	$\tilde{\mathbf y}_{t+h\mid t}$	Forecasts that satisfy every aggregation constraint.
Forecast reconciliation	$\hat{\mathbf y}\mapsto\tilde{\mathbf y}$	A mapping from base forecasts to coherent forecasts.

y_{\mathrm{Total},t}=y_{A,t}+y_{B,t},\quad y_{A,t}=y_{A1,t}+y_{A2,t},\quad y_{B,t}=y_{B1,t}+y_{B2,t}

INTRODUCTION / HIERARCHICAL FORECASTING · 5

Base and Coherent Forecasts

Undersanding coherence: a numerical example on the same seven-node hierarchy.

Forecast Type	Total	A	B	A1	A2	B1	B2
Base Forecasts	100	55	60	25	30	20	40
Bottom-up (BU)	115	55	60	25	30	20	40

\tilde y_{\mathrm{Total}}=\tilde y_A+\tilde y_B=115,\qquad\tilde y_A=25+30,\quad\tilde y_B=20+40

Bottom-up reconciliation retains terminal forecasts and recomputes the aggregates. Coherence does not guarantee better accuracy.

INTRODUCTION / HIERARCHICAL FORECASTING · 6

The Summing Matrix Represents The Hierarchy

All seven observations can be written as sums of four terminal observations.

\underbrace{\begin{bmatrix}y_{\mathrm{Total},t}\\y_{A,t}\\y_{B,t}\\y_{A1,t}\\y_{A2,t}\\y_{B1,t}\\y_{B2,t}\end{bmatrix}}_{\mathbf y_t}=\underbrace{\begin{bmatrix}1&1&1&1\\1&1&0&0\\0&0&1&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}_{\mathbf S\in\mathbb R^{7\times4}}\underbrace{\begin{bmatrix}y_{A1,t}\\y_{A2,t}\\y_{B1,t}\\y_{B2,t}\end{bmatrix}}_{\mathbf b_t}

A coherent forecast also meet the same form, $\quad \tilde{\mathbf y}_{t+h\mid t}=\mathbf S\tilde{\mathbf b}_{t+h\mid t}$ .

INTRODUCTION / MULTI-STAKEHOLDER SYSTEM · 7

An Example for Multi-stakeholder System

Each node retains its own history and base forecasts.

Author illustration of a generator, two substations and four homes

Distributed ownership

Households, substations and the system-level stakeholder hold data at different nodes.

Raw histories stay with their owners and are not exchanged between nodes.

Potential disclosures

Demand histories can reveal household routines or operating patterns.

Model updates, residuals and forecast-derived messages can also expose information.

There is a call for solutions to cross-entity hierarchical forecasting and reconciliation.

INTRODUCTION / FORECASTING PARADIGMS · 8

Forecasting Paradigms: A Common Terminology

Use machine-learning terminology consistently throughout the presentation.

Comparison of HTS data, localised learning, centralised learning and federated learning

Forecasting terminology	Machine-Learning terminology	Training arrangement
Local	Localised	Fit a separate model to each series or owner's local data.
Global	Centralised	Fit a shared model using pooled observations.
Collaborative	Federated	Train collaboratively while observations remain with their owners.

Federated learning is our modelling choice: independent data ownership with cross-series learning.

RELATED WORKS · 9

Federated Learning

Clients learn a shared forecasting model while observations remain local.

Shared model parameters do not impose adding-up constraints on independently generated forecasts.

PhD Candidature Confirmation

Related Works

Federated learning, forecast reconciliation and the remaining gap

RELATED WORKS · 11

Federated Learning: Taxonomy

Data alignment, federation scale, coordination and learning methods.

RELATED WORKS · 12

Horizontal and Vertical Federated Learning

Data partitioning determines what participants can learn together.

Horizontal FL

Samples / entities

Features

Owner A Owner B

Same feature space; different sample sets.
Example: hospitals train on different patients using the same measurements.

Each owner can train the full model locally; model updates are aggregated.

Vertical FL

Samples / entities

Features

Owner A Owner B

Aligned entities; different feature sets.
Example: a bank and a retailer hold complementary records for shared customers.

Training combines intermediate outputs or gradients; entity alignment is required.

FHF uses a common input representation across node-series: horizontal-style FL. Hierarchy levels do not define vertical FL.

RELATED WORKS · 13

FedAvg And FedProx

A common global objective, with different local training objectives.

\min_{\theta} F(\theta)=\sum_{i=1}^{K}p_iF_i(\theta),\qquad p_i\geq0,\quad\sum_i p_i=1

FedAvg · Local SGD

\theta_i^{r,0}=\theta^r,\qquad\theta_i^{r,e+1}=\theta_i^{r,e}-\eta\nabla F_i(\theta_i^{r,e};\xi_i^{r,e})

Start from the shared model and run several local minibatch updates.

FedProx · Constrain Local Drift

\theta_i^{r+1}\approx\arg\min_{\theta}\left[F_i(\theta)+\frac{\mu}{2}\|\theta-\theta^r\|_2^2\right]

The proximal term discourages large departures from the round’s shared model.

\text{Server aggregation:}\qquad\theta^{r+1}=\sum_{i\in\mathcal C_r}\alpha_i^r\theta_i^{r+1},\qquad\sum_{i\in\mathcal C_r}\alpha_i^r=1

$F_i$ : local empirical loss; $\xi$ : minibatch; $\mathcal C_r$ : selected clients. With matched local work, $\mu=0$ recovers FedAvg.

FedProx addresses heterogeneous local optimisation; forecast coherence and privacy still need separate mechanisms.

RELATED WORKS · 14

Three Federated Learning Architectures

The coordination topology is independent of the horizontal/vertical data partition.

Flat · Server-based

One coordinator broadcasts and aggregates models.

Typical use: a federation with a shared aggregation service.

Hierarchical · Edge–Cloud

Nearby clients aggregate at edges; edges periodically coordinate globally.

Typical use: geographically distributed devices.

Decentralised · Peer-to-peer

Clients exchange and mix neighbour models, with no central aggregator.

Typical use: owners without a common server.

“Centralised FL” refers to coordination: observations remain local. None of these topologies automatically ensures hierarchical forecast coherence.

RELATED WORKS · 15

Forecast Reconciliation: Taxonomy

Point and probabilistic reconciliation, and coherence integrated into learning.

RELATED WORKS / FORECAST RECONCILIATION · 16

Linear Forecast Reconciliation

A single linear mapping produces coherent forecasts; covariance specifications distinguish the methods.

\begin{aligned}\tilde{\mathbf y}_h^{(m)}&=\mathbf P_h^{(m)}\hat{\mathbf y}_h=\mathbf S\mathbf G_h^{(m)}\hat{\mathbf y}_h\\[0.55em]\mathbf G_h^{(m)}&=\left(\mathbf S^{\mathsf T}(\mathbf W_h^{(m)})^{-1}\mathbf S\right)^{-1}\mathbf S^{\mathsf T}(\mathbf W_h^{(m)})^{-1}\end{aligned}

S: summing matrix · W: base-error covariance · G: bottom-level mapping · ŷ / ỹ: base / coherent forecasts.
P = SG: full reconciliation matrix · h: forecast horizon · m: method; forecast origin omitted.

Method	$\mathbf W_h^{(m)}$	Interpretation
OLS	$\mathbf I_n$	Equal weighting
WLSv	$\operatorname{Diag}(\hat{\mathbf W}_h)$	Residual-variance weighting
WLSs	$\operatorname{Diag}(\mathbf S\mathbf 1_{n_b})$	Structural scaling
MinTc	$\hat{\mathbf W}_h$	Full sample covariance
MinTs	$\lambda\operatorname{Diag}(\hat{\mathbf W}_h)+(1-\lambda)\hat{\mathbf W}_h$	Shrinkage towards the diagonal

$\hat{\mathbf W}_h$ : sample covariance; $0\le\lambda\le1$ : shrinkage intensity. Table MinTs: diagonal target; PPHF MinTs: scaled-identity target.

RELATED WORKS · 17

The Research Gap

Independent stakeholders need joint forecasting, coherent outputs and explicit privacy protection.

Federated Learning

Learns across owners while keeping observations local. Coherence and update privacy still need to be addressed.

Distributed Reconciliation

Produces coherent forecasts through exchanges between owners. Exchanged information still needs privacy analysis.

Possible Privacy-Preserving Methods

Differential privacy and secure aggregation offer protection. Their trust assumptions and utility costs need evaluation.

Research question: how can independent owners learn jointly and reconcile forecasts while protecting the information they release?

PhD Candidature Confirmation

Research Questions

Research objectives before the detailed Research Question 1 - PPHF

RESEARCH QUESTIONS · 19

Overall Research Problem And Three Research Questions

From distributed learning, to full decentralisation, to coordinated decisions.

How can a cross-node distributed learning method for hierarchical forecasting be developed for multi-stakeholder systems?

RQ1

Privacy-preserving Hierarchical Forecasting

How can learning and reconciliation be developed for privacy-preserving hierarchical forecasting in a distributed multi-stakeholder systems?

RQ2

Fully Decentralised Forecasting And Reconciliation

How can learning and reconciliation be developed in a fully decentralised system?

RQ3

Forecasting And Decision-Making In Multi-Agent Systems

How can decentralised hierarchical forecasting and decision-making be integrated in a multi-agent system?

RQ1 establishes the forecasting framework; RQ2 removes central coordination; RQ3 extends coherence to stakeholder decisions.

PhD Candidature Confirmation

Progress to Date

Privacy-Preserving Hierarchical Forecasting: framework, evaluations, results and future work

PPHF / 4.1 · FEDERATED HIERARCHICAL FORECASTING · 21

Proposed FHF Framework

Train collaboratively; construct the operator at a coordinator; apply reconciliation through local contributions.

Author FHF framework: local series and model training, coordinator aggregation and matrix construction, column allocation and distributed contributions

PPHF / 4.1 · FEDERATED HIERARCHICAL FORECASTING · 22

Distributed Reconciliation: A Three-Series Example

Allocate columns, calculate local contributions, then sum the entries required by each recipient.

Three-series hierarchy and reconciliation operator: the server allocates columns to clients TS 1, TS 2 and TS 3; clients weight their local base forecasts by those columns

1 · Allocate The Columns

The server gives client i column i of the reconciliation matrix.

\mathbf P[:,1]=[\,p_{11},\ p_{21},\ p_{31}\,]^{\mathsf T}

2 · Calculate Contributions Locally

Each client multiplies its column by its own base forecast and sends each entry to its recipient.

\mathbf c_i=\mathbf P[:,i]\widehat y_i

3 · Sum At Each Recipient

Client 1 sums the first entry from every client; clients 2 and 3 sum the second and third entries.

\widetilde y_1=p_{11}\widehat y_1+p_{12}\widehat y_2+p_{13}\widehat y_3

The distributed sums reproduce Pŷ exactly. For a reconciliation operator P = SG, the resulting forecasts are coherent.

PPHF / 4.1 · FEDERATED HIERARCHICAL FORECASTING · 23

Fully Hierarchical Variants: LN-FHF And CH-FHF

Training participation and aggregation follow the hierarchy; reconciliation is shared.

Original paper topologies for flat, leaf-node and complete-hierarchy federated learning

Architecture	Training participants	Aggregation and reconciliation
FHF	All participating node-series	Flat FedAvg; coordinator-assisted reconciliation.
LN-FHF	Terminal node-series only	Recursive hierarchy-aligned averaging; parents still forecast and supply residuals.
CH-FHF	Terminal, intermediate and root series	Recursive averaging includes each parent’s own model update.

With matched participants and synchronous sample-count weights, recursive averaging reduces to flat averaging. All variants retain a coordinator.

PPHF / 4.2 · DIFFERENTIAL PRIVACY · 24

Differential Privacy In The FHF Framework

Threat model, privacy guarantee and protection across the three stages.

Honest-But-Curious Threat Model

Participants follow the protocol but may infer private information from received messages. Training trusts the server with clipped updates; residuals and forecasts are perturbed before transmission.

Differential Privacy

\Pr[\mathcal M(D)\in O]\leq e^{\varepsilon}\Pr[\mathcal M(D^{\prime})\in O]+\delta\qquad\forall D\sim D^{\prime},\ \forall O

Protection At Three Framework Stages

Stage	Exchanged message	Protected input	DP category
Training	Model updates	One node-client contribution Fixed-slot zeroing	User-level central DP $(\varepsilon,\delta)$ · noise at trusted server
Covariance estimation	Residuals	One residual coordinate Replacement	Event-level local DP $\varepsilon_e$ · noise at owner
Reconciliation	Weighted forecasts	One forecast coordinate Replacement	Event-level local DP $\varepsilon_y$ · noise at owner

Central training DP protects model releases, while local DP protects residuals and forecasts before transmission.

PPHF / 4.2 · DIFFERENTIAL PRIVACY · 25

Privacy Mechanisms Across Three Stages

Protect training updates, residuals and forecasts at their respective release boundaries.

Federated Training

User-level central DP

\bar{\boldsymbol\Delta}_k=\frac{\boldsymbol\Delta_k}{\max\!\left\{1,\lVert\boldsymbol\Delta_k\rVert_2/C\right\}}

\boldsymbol\theta^{(r)}=\boldsymbol\theta^{(r-1)}+\sum_k\alpha_k\bar{\boldsymbol\Delta}_k+\mathbf z

Clip updates; add Gaussian noise at the trusted server. Uniform averaging: SD $\sigma C/K$ ; q = 1, with RDP accounting.

Residual Covariance

Event-level local DP

\widetilde e_{i,t,h}=\operatorname{clip}(e_{i,t,h};-B_{i,h},B_{i,h})+\xi_{i,t,h},\quad \xi_{i,t,h}\sim\operatorname{Laplace}(0,b_{i,h}),\quad b_{i,h}=2B_{i,h}/\varepsilon_e

\mathbb E[\mathbf A_h\mid\overline{\mathbf E}_h]=\widehat{\mathbf W}^{\mathrm{clip}}_h+2\operatorname{diag}(b_{1,h}^{2},\ldots,b_{n,h}^{2})

Local Laplace noise; replacement sensitivity $2B_{i,h}$ . Reuse noisy residuals for covariance and weights.

Forecast Reconciliation

Event-level local DP

\widehat y^{\mathrm{DP}}_{i}=\operatorname{clip}(\widehat y_i;L_{i,s},U_{i,s})+\eta_i,\quad\eta_i\sim\operatorname{Laplace}(0,(U_{i,s}-L_{i,s})/\varepsilon_y)

\widetilde{\mathbf y}^{\mathrm{DP}}=\mathbf P\widehat{\mathbf y}^{\mathrm{DP}}=\mathbf S\mathbf G\widehat{\mathbf y}^{\mathrm{DP}}

Add local Laplace noise once per forecast; reuse the same noisy value across recipients and reconciliation methods.

Training uses fixed client slots and public weights; local releases assume fixed models, bounds and reconciliation operators.

PPHF / 5 · EVALUATIONS · 26

Data And Experimental Setup

Two unbalanced electricity-demand hierarchies; terminal nodes occur at different depths.

Property	GEF2017	AusGrid2013
Distinct series	10	360
Sampling / observations per series	Hourly / 124,171	Half-hourly / 17,520
Forecast-window length h	1, 3 or 6; average leads 1…h	1 half-hour
Root / internal / terminal roles	1 / 1 / 8	1 / 60 / 299

Common protocol: GEF2017: TOTAL, states and Massachusetts zones. AusGrid2013: general consumption, 100 postcodes, 299 retained customers; terminal roles include single-customer postcodes.

48 observations span two days in GEF2017 and one day in AusGrid2013. Restore physical units before evaluation.

PPHF / 5 · EVALUATIONS · 27

Training And Evaluation Protocol

A common LSTM specification keeps the forecasting architecture fixed.

Data pipeline

48 previous observations per input.

Chronological 80% / 20% split with overlapping leakage excluded.

Per-series scaling uses training statistics.

Model settings

Two LSTM layers, 64 hidden units each.

Dropout 0.1, Adam learning rate 0.001.

MSE loss, batch size 256, random seed 42; RTX 4070 Super.

Complete dataset-preparation figure from the paper, including hierarchy construction, rolling windows and the chronological training/test split.

Compare FHF, LN-FHF and CH-FHF with fBU, fWLSv and fMinTs; DP experiments also include fOLS. Evaluate after restoring physical units.

PPHF / 5 · EVALUATIONS · 28

Accuracy, Coherence And Statistical Evidence

Forecast accuracy, scaled aggregation mismatch and statistical comparisons answer different questions.

MASE · Mean Absolute Scaled Error

\operatorname{MASE}_i=\frac{Q_i^{-1}\sum_{q=1}^{Q_i}|y_{i,q}-\hat y_{i,q}|}{(T_i-m_i)^{-1}\sum_{t=m_i+1}^{T_i}|y_{i,t}-y_{i,t-m_i}|}

RMSSE · Root Mean Squared Scaled Error

\operatorname{RMSSE}_i=\sqrt{\frac{Q_i^{-1}\sum_{q=1}^{Q_i}e_{i,q}^{2}}{(T_i-m_i)^{-1}\sum_{t=m_i+1}^{T_i}(y_{i,t}-y_{i,t-m_i})^{2}}}

MASI · Mean Absolute Scaled Incoherence

\operatorname{MASI}_{\ell}=\frac{1}{|\mathcal P_{\ell}|}\sum_{j\in\mathcal P_{\ell}}\frac{Q_j^{-1}\sum_{q=1}^{Q_j}|\hat y_{j,q}-\sum_{i\in\operatorname{ch}(j)}\hat y_{i,q}|}{(T_j-1)^{-1}\sum_{t=2}^{T_j}|y_{j,t}-y_{j,t-1}|}

Accuracy And Coherence

MASE and RMSSE measure scaled forecast errors. MASI uses the same lag-one training scale as MASE.

Each parent contributes equally; zero MASI indicates coherent forecasts when the scale is positive.

Statistical Evidence

MCB compares method ranks across distinct series, separately for each forecast window.

Average leads 1…h within a window. A lowest mean rank does not imply statistical superiority.

MASI is dimensionless, not a percentage, and has no finite upper bound. The training denominator is shared across methods.

PPHF / 5 · EVALUATIONS · 29

Evaluations: Three Separate Privacy Experiments

One-step forecasts; mean MASE and RMSSE across distinct series; each stage is evaluated separately.

Stage	Fixed comparison	Privacy / calibration settings
Private aggregation	FedAvg vs DP-FedAvg, before and after fMinTs / fWLSv / fOLS.	100 rounds, q = 1, C = 1, δ = 10⁻⁵; σ = 0.3 (GEF) or 1.2 (AusGrid).
Private residuals	Non-private FHF forecasts fixed; construct operators from perturbed training residuals.	εe = 1; fixed B = 0.0005 or per-client quantile bounds with 99.5%, 75%, 50%, 25%, 0.5% retention.
Private forecasts	Non-private models and operators fixed; reuse each noisy forecast across recipients.	εy = 1, 5, 10, 20; slot-specific 1st/99th percentile training bounds.

Residuals use original units: 99,288 training rows/client (GEF2017) and 13,968 (AusGrid2013). Retention is the target percentage left unclipped; all rows remain in the sample.

Calibration is fixed auxiliary information. These stage-specific tests and alternative budgets do not establish one end-to-end privacy guarantee.

PPHF / 6.1 · RESULTS FOR FHF · 30

GEF2017: Forecast Accuracy And Coherence

10 hourly series · h is the forecast-window length · each configuration averages leads 1…h.

Bold: best · Underline: second-best within each architecture and column.

Architecture	Method	MASE ↓			RMSSE ↓			MASI
Architecture	Method	L1	L2	L3	L1	L2	L3	L1	L2
FHF	FedAvg	0.120	0.147	0.189	0.124	0.152	0.192	0.0212	0.0165
	fBU	0.123	0.148	0.189	0.127	0.153	0.192	0	0
	fMinTs	0.121	0.146	0.187	0.124	0.152	0.190	0	0
	fWLSv	0.122	0.147	0.188	0.125	0.152	0.192	0	0
LN-FHF	HierAvg	0.131	0.160	0.194	0.131	0.160	0.196	0.0229	0.0170
	fBU	0.134	0.160	0.194	0.134	0.160	0.196	0	0
	fMinTs	0.131	0.158	0.192	0.131	0.158	0.193	0	0
	fWLSv	0.133	0.159	0.194	0.132	0.159	0.195	0	0
CH-FHF	HierAvgPS	0.120	0.145	0.189	0.124	0.151	0.193	0.0220	0.0177
	fBU	0.123	0.146	0.189	0.127	0.151	0.193	0	0
	fMinTs	0.120	0.145	0.186	0.124	0.150	0.190	0	0
	fWLSv	0.121	0.145	0.188	0.125	0.150	0.192	0	0

Architecture	Method	MASE ↓			RMSSE ↓			MASI
Architecture	Method	L1	L2	L3	L1	L2	L3	L1	L2
FHF	FedAvg	0.253	0.314	0.377	0.278	0.348	0.408	0.0557	0.0441
	fBU	0.270	0.322	0.377	0.297	0.356	0.408	0	0
	fMinTs	0.254	0.306	0.368	0.278	0.338	0.396	0	0
	fWLSv	0.263	0.315	0.373	0.289	0.348	0.404	0	0
LN-FHF	HierAvg	0.258	0.320	0.377	0.280	0.350	0.405	0.0554	0.0441
	fBU	0.276	0.328	0.377	0.298	0.357	0.405	0	0
	fMinTs	0.258	0.311	0.365	0.280	0.340	0.391	0	0
	fWLSv	0.269	0.321	0.373	0.291	0.350	0.400	0	0
CH-FHF	HierAvgPS	0.250	0.309	0.374	0.275	0.341	0.403	0.0563	0.0493
	fBU	0.264	0.316	0.374	0.288	0.346	0.403	0	0
	fMinTs	0.250	0.303	0.367	0.275	0.334	0.394	0	0
	fWLSv	0.258	0.309	0.370	0.282	0.340	0.399	0	0

Architecture	Method	MASE ↓			RMSSE ↓			MASI
Architecture	Method	L1	L2	L3	L1	L2	L3	L1	L2
FHF	FedAvg	0.400	0.492	0.575	0.441	0.546	0.626	0.0792	0.0711
	fBU	0.424	0.505	0.575	0.465	0.557	0.626	0	0
	fMinTs	0.400	0.479	0.562	0.441	0.532	0.610	0	0
	fWLSv	0.415	0.494	0.570	0.456	0.546	0.620	0	0
LN-FHF	HierAvg	0.415	0.513	0.586	0.455	0.565	0.635	0.0842	0.0707
	fBU	0.439	0.526	0.586	0.478	0.576	0.635	0	0
	fMinTs	0.414	0.499	0.572	0.454	0.551	0.619	0	0
	fWLSv	0.430	0.514	0.580	0.469	0.565	0.629	0	0
CH-FHF	HierAvgPS	0.392	0.486	0.574	0.434	0.539	0.627	0.0840	0.0811
	fBU	0.421	0.505	0.574	0.460	0.555	0.627	0	0
	fMinTs	0.392	0.471	0.559	0.433	0.524	0.610	0	0
	fWLSv	0.410	0.490	0.568	0.449	0.541	0.620	0	0

Reconciled MASI is zero. fBU, fWLSv and fMinTs denote distributed applications of the corresponding operators.

PPHF / 6.1 · RESULTS FOR FHF · 31

GEF2017: Statistical Comparisons

MASE and RMSSE ranks across ten distinct series · select forecast window h; scores average leads 1…h.

Complete GEF2017 MASE multiple-comparisons-with-the-best chart at forecast window 6 — **MASE:** CH-FHF–fMinTs · lowest mean rank **2.90**.

Complete GEF2017 RMSSE multiple-comparisons-with-the-best chart at forecast window 6 — **RMSSE:** CH-FHF–fMinTs · lowest mean rank **2.80**.

Lower rank is better; several methods remain statistically indistinguishable from the best. The lowest mean rank does not establish a unique winner.

PPHF / 6.1 · RESULTS FOR FHF · 32

AusGrid2013: Forecast Accuracy And Coherence

360 half-hourly series · h = 1 · bold: best; underline: second-best within each architecture.

Architecture	Method	MASE ↓			RMSSE ↓			MASI
Architecture	Method	L1	L2	L3	L1	L2	L3	L1	L2
Baseline	Naive	1.267	1.164	1.171	1.199	1.140	1.134	0	0
FHF	FedAvg	1.067	1.094	1.133	0.956	1.019	1.009	0.3991	0.1754
	fBU	1.171	1.082	1.133	1.165	1.022	1.009	0	0
	fMinTs	1.025	1.100	1.176	0.914	1.005	1.006	0	0
	fWLSv	1.119	1.085	1.142	1.082	1.016	1.008	0	0
LN-FHF	HierAvg	1.250	1.114	1.149	1.126	1.026	1.010	0.3731	0.1888
	fBU	1.227	1.099	1.149	1.153	1.021	1.010	0	0
	fMinTs	1.117	1.096	1.164	1.046	1.013	1.009	0	0
	fWLSv	1.216	1.102	1.155	1.128	1.019	1.010	0	0
CH-FHF	HierAvgPS	1.069	1.093	1.132	0.960	1.019	1.010	0.3998	0.1751
	fBU	1.176	1.082	1.132	1.174	1.023	1.010	0	0
	fMinTs	1.025	1.100	1.176	0.916	1.005	1.006	0	0
	fWLSv	1.123	1.084	1.141	1.091	1.016	1.009	0	0

MinTs improves top-level FHF accuracy, but terminal MASE can increase. Hierarchical training has no uniform advantage.

PPHF / 6.1 · RESULTS FOR FHF · 33

AusGrid2013: Statistical Comparisons

MCB across all 360 series · lower mean rank is better · f denotes distributed reconciliation.

AusGrid MASE multiple comparisons with the best — MASE: CH-FHF–fBU has the lowest mean rank (2.79).

AusGrid RMSSE multiple comparisons with the best — RMSSE: FHF–fMinTs has the lowest mean rank (4.88).

The best group depends on the error measure; several methods are statistically indistinguishable.

PPHF / 6.2 · RESULTS FOR DIFFERENTIAL PRIVACY · 34

Private Aggregation: Accuracy Before And After Reconciliation

100 rounds · q = 1 · C = 1 · δ = 10⁻⁵ · mean scores at h = 1.

Dataset	Forecast	FedAvg MASE	FedAvg RMSSE	DP MASE	DP RMSSE
GEF2017	Base	0.178	0.182	0.479	0.472
GEF2017	fMinTs	0.176	0.180	0.476	0.473
GEF2017	fWLSv	0.177	0.181	0.478	0.471
GEF2017	fOLS	0.177	0.180	0.474	0.469
AusGrid2013	Base	1.126	1.011	1.153	1.014
AusGrid2013	fMinTs	1.163	1.006	1.156	1.010
AusGrid2013	fWLSv	1.132	1.010	1.154	1.013
AusGrid2013	fOLS	1.166	1.010	1.165	1.014

GEF2017: ε = 740.496, aggregate noise SD = 0.03000. AusGrid2013: ε = 75.109, SD = 0.00333. Smaller private RMSSE does not imply smaller MASE.

The reported budgets give weak formal bounds. Reconciliation has limited and metric-dependent gains after private training.

PPHF / 6.2 · RESULTS FOR DIFFERENTIAL PRIVACY · 35

Private Aggregation: Budget Alone Does Not Explain Utility

20 rounds · q = 1 · h = 1 · δ = 10⁻⁵ · mean scores across nodes · $s_{\mathrm{agg}}=\sigma C/K$ .

12 configurations per dataset · complete base and reconciled results

GEF2017 · K = 10 clients

C	σ	ε	$s_{\mathrm{agg}}$	Mean MASE ↓				Mean RMSSE ↓
C	σ	ε	$s_{\mathrm{agg}}$	Base	fMinTs	fWLSv	fOLS	Base	fMinTs	fWLSv	fOLS
0.2	0.3	184.941	0.006	3.201	3.202	3.201	3.201	2.998	2.999	2.998	2.999
	0.6	63.962	0.012	3.225	3.224	3.225	3.224	3.030	3.028	3.030	3.029
	0.9	36.204	0.018	3.271	3.268	3.271	3.270	3.078	3.074	3.078	3.077
	1.2	25.402	0.024	3.318	3.313	3.318	3.316	3.122	3.115	3.122	3.120
0.5	0.3	184.941	0.015	1.673	1.572	1.673	1.667	1.575	1.449	1.575	1.568
	0.6	63.962	0.030	2.041	1.916	2.041	2.031	1.907	1.764	1.907	1.895
	0.9	36.204	0.045	2.059	2.018	2.059	2.052	1.900	1.833	1.900	1.889
	1.2	25.402	0.060	2.720	2.689	2.719	2.760	2.514	2.487	2.512	2.554
1.0	0.3	184.941	0.030	0.450	0.448	0.450	0.448	0.436	0.436	0.435	0.436
	0.6	63.962	0.060	1.279	1.266	1.279	1.290	1.178	1.167	1.177	1.187
	0.9	36.204	0.090	3.032	3.012	3.025	3.131	3.111	3.071	3.099	3.277
	1.2	25.402	0.120	5.555	5.473	5.474	6.364	5.086	4.997	5.002	5.893

AusGrid2013 · K = 360 clients

C	σ	ε	$s_{\mathrm{agg}}$	Mean MASE ↓				Mean RMSSE ↓
C	σ	ε	$s_{\mathrm{agg}}$	Base	fMinTs	fWLSv	fOLS	Base	fMinTs	fWLSv	fOLS
0.2	0.3	184.941	0.00017	1.803	1.904	1.807	1.830	1.434	1.441	1.435	1.431
	0.6	63.962	0.00033	1.802	1.904	1.806	1.830	1.434	1.442	1.435	1.431
	0.9	36.204	0.00050	1.802	1.904	1.805	1.830	1.435	1.442	1.435	1.431
	1.2	25.402	0.00067	1.801	1.905	1.805	1.830	1.435	1.443	1.435	1.431
0.5	0.3	184.941	0.00042	1.251	1.283	1.253	1.257	1.089	1.077	1.089	1.086
	0.6	63.962	0.00083	1.251	1.282	1.252	1.256	1.089	1.077	1.089	1.086
	0.9	36.204	0.00125	1.250	1.281	1.252	1.255	1.089	1.077	1.089	1.086
	1.2	25.402	0.00167	1.250	1.279	1.251	1.255	1.089	1.077	1.088	1.085
1.0	0.3	184.941	0.00083	1.199	1.187	1.198	1.195	1.066	1.058	1.066	1.065
	0.6	63.962	0.00167	1.201	1.184	1.200	1.195	1.066	1.058	1.066	1.065
	0.9	36.204	0.00250	1.204	1.182	1.203	1.196	1.066	1.058	1.066	1.065
	1.2	25.402	0.00333	1.208	1.180	1.206	1.197	1.066	1.058	1.066	1.065

These are separate 20-round runs. Compare clipping, multiplier, population and duration alongside the per-run budget.

PPHF / 6.2 · RESULTS FOR DIFFERENTIAL PRIVACY · 36

Private Residuals: Fixed And Quantile Clipping

Non-private base forecasts fixed · h = 1 · $\varepsilon_e=1$ · $\rho=10^{-6}$ · retention = target percentage left unclipped.

8 rows per dataset · complete fMinTs and fWLSv results

GEF2017 · means across 10 series

Clipping / retention	fMinTs		fWLSv
Clipping / retention	MASE ↓	RMSSE ↓	MASE ↓	RMSSE ↓
Non-DP residuals	0.176	0.180	0.177	0.181
OLS reference	0.177	0.180	0.177	0.180
Fixed bound (B = 0.0005)	0.177	0.180	0.177	0.180
99.5%	0.177	0.181	0.177	0.181
75%	0.177	0.181	0.177	0.181
50%	0.177	0.181	0.177	0.181
25%	0.177	0.181	0.177	0.181
0.5%	0.177	0.181	0.177	0.181

AusGrid2013 · means across 360 series

Clipping / retention	fMinTs		fWLSv
Clipping / retention	MASE ↓	RMSSE ↓	MASE ↓	RMSSE ↓
Non-DP residuals	1.163	1.006	1.132	1.010
OLS reference	1.166	1.010	1.166	1.010
Fixed bound (B = 0.0005)	1.166	1.009	1.166	1.010
99.5%	1.135	1.010	1.135	1.010
75%	1.128	1.010	1.128	1.010
50%	1.127	1.011	1.127	1.011
25%	1.126	1.011	1.127	1.011
0.5%	1.127	1.011	1.128	1.010

Equal-noise limit

\mathbf W_k=\mathbf W_0+k\mathbf I\quad\Longrightarrow\quad\lim_{k\to\infty}\mathbf P(\mathbf W_k)=\mathbf P_{\mathrm{OLS}}

Quantile clipping improves AusGrid MASE relative to the fixed bound; RMSSE has no consistent improvement. A single seed limits conclusions.

PPHF / 6.2 · RESULTS FOR DIFFERENTIAL PRIVACY · 37

Private Forecasts: Coherence With A Substantial Utility Cost

Non-private FHF models and fixed operators · all four alternative privacy budgets · h = 1.

All four alternative privacy budgets · both accuracy metrics

GEF2017 · εy = 1, 5, 10, 20

εy	MinTs		WLSv
εy	MASE ↓	RMSSE ↓	MASE ↓	RMSSE ↓
1	23.114	23.535	13.763	14.356
5	4.633	4.715	2.769	2.884
10	2.330	2.369	1.407	1.461
20	1.189	1.208	0.740	0.768

εy	MinTs	WLSv
1	15.777	11.001	9.929	7.574
5	3.463	2.427	2.421	1.829
10	2.076	1.507	1.626	1.277
20	1.488	1.161	1.304	1.090

WLSv outperforms MinTs under every shown budget, but both remain less accurate than their non-private counterparts. Reusing each noisy forecast keeps the reconciled vector coherent.

Coherence survives local forecast perturbation; useful accuracy may not. Alternative budgets are separate comparisons and calibration is conditional.

PPHF / 7 · CONCLUSION AND FUTURE WORK · 38

PPHF: Conclusion And Future Work

A coordinator-assisted baseline establishes distributed coherence and separates three privacy–utility trade-offs.

Established Results

Exactness: complete owner-side contributions reproduce the selected central linear operator.

Forecasting: reconciled MASI < 1.5 × 10⁻¹⁴; accuracy effects depend on level, horizon and metric.

Privacy: training changes fitted forecasts; private residuals alter weights; direct forecast noise has substantial utility costs.

Limits And Next Research Questions

Current evidence uses a fixed LSTM, one seed, synchronous participation and one computing system.

RQ2: remove the learning server and reconciliation coordinator; unify communication and aggregation constraints.

RQ3: integrate coherent forecasts and coordinated decisions among agents with local objectives and shared constraints.

Further extensions include probabilistic outputs, asynchronous or missing clients, and new application hierarchies.

PhD Candidature Confirmation

Next Steps

Research directions beyond the current Research Question 1 - PPHF

NEXT STEPS / RQ2 · 40

RQ2: Existing Methods And Layer Decomposition

Local forecasts can be reconciled iteratively; a learned projection can be split into owner-side contributions.

Localised Forecasting + Consensus ADMM

Boyd et al. (2011) · Bai & Pinson (2019): OLS/WLS

Local forecasts: each owner fits a model on its own series and produces its base forecast.

\min_{\widetilde{\mathbf y}}\;\frac12\sum_i a_i(\widetilde y_i-\widehat y_i)^2\\\text{s.t. }\;\mathbf A\widetilde{\mathbf y}=\mathbf0

ADMM iterations: solve local correction subproblems; exchange shared quantities; update consensus and dual variables.

Coherent solution: repeat until primal and dual tolerances hold. For RQ2, shared averages must use neighbour communication.

Decompose A Learnable Layer

Tsiourvas et al. · ICML 2024

Existing model: jointly train a forecaster and structured projection using the loss on coherent outputs.

\widehat{\mathbf y}=f_\theta(\mathbf x),\qquad \widetilde{\mathbf y}=\mathbf P_\phi\widehat{\mathbf y}\\\mathbf P_\phi=\mathbf S\mathbf G_\phi,\qquad\mathbf G_\phi\mathbf S=\mathbf I

Decomposition for RQ2: allocate columns to forecast owners. Each owner computes its contribution; recipients sum them.

\widetilde y_r=\sum_i\mathbf P_\phi[r,i]\widehat y_i

The forward decomposition is exact. Distributed learning also requires gradient exchange and preservation of the projection constraints.

cADMM coordinates forecast corrections; column decomposition distributes the layer’s forward computation. These provide two starting points for RQ2.

NEXT STEPS / RQ2 · 41

RQ2: Two Routes To Fully Decentralised Forecasting

Collaborate through neighbour communication, then enforce coherence by optimisation or a trainable layer.

Shared DFL Stage · Local Training And Neighbour Model Exchange

\boldsymbol\theta_i^{k+1}=\sum_{j\in\mathcal N_i\cup\{i\}}w_{ij}\boldsymbol\theta_j^k-\eta\nabla\ell_i(\boldsymbol\theta_i^k;\mathcal D_i)

DFL + cADMM

Forecast collaboratively: learn through DFL while each owner retains its observations.

Reconcile iteratively: use peer-to-peer correction, consensus and dual updates to enforce parent–child sums.

\mathbf A\widetilde{\mathbf y}=\mathbf0,\qquad\mathbf A\mathbf S=\mathbf0

Research focus: align the two communication graphs, couple intermediate nodes across levels, and reduce nested ADMM rounds.

DFL + Decentralised Learnable Layer

Partition the layer: distribute projection parameters and forecast contributions across owners.

Learn jointly: exchange gradient messages and minimise local losses on coherent outputs.

\min_{\theta,\phi}\;\sum_i\ell_i(\widetilde y_i,y_i)\\\widetilde{\mathbf y}=\mathbf S\mathbf G_\phi\widehat{\mathbf y},\qquad\mathbf G_\phi\mathbf S=\mathbf I

Research focus: preserve the projection constraints and design gradient communication for dense or factorised layers.

Compare both routes using matched data and forecasters: accuracy, coherence residuals, communication, latency and convergence.

NEXT STEPS / RQ3 · 42

RQ3: Integrate Forecasting And Multi-Agent Decisions

Households, retailers and network operators act on forecasts with different objectives.

Controlled setting	Forecast information	Decision coordination
1 · Incoherent forecasts	Unreconciled local forecasts.	Local policies; hierarchical forecast/decision consistency is not enforced.
2 · Coherent forecasts	Decentralised reconciled forecasts from RQ2.	Forecasts are coherent, but learned actions can still be inconsistent across the hierarchy.
3 · Coherent forecasts and decisions	The same coherent forecasting basis.	Coordinate actions for the same physical quantity across agents and hierarchy levels.

Matched Experiment

Use one environment, such as CityLearn, with the same stakeholder objectives and physical operating limits in all three settings.

Evaluation

Assess stakeholder utilities, system performance, forecast accuracy, forecast/decision consistency, constraint violations and communication costs.

Vary stakeholder preferences and forecast uncertainty to test whether forecast coherence helps decisions and decision coherence adds value.

PhD Candidature Confirmation

Questions

Thanks for your attendance and coming questions

RESEARCH QUESTION 1 - PPHF BACKUP · 44

Communication Cost And Coordination

Dense reconciliation shifts work to owners but retains quadratic message complexity.

z=\operatorname{nnz}(\mathbf P),\qquad\text{computation}=O(z)

Stage	Logical scalar volume
Column allocation	z nonzero coefficients per method and horizon
Forecast application	z multiplications and associated sums
Cross-owner exchange	z_net ≤ z contributions per forecast origin
Dense, one series per owner	At most n(n − 1) cross-owner contributions

The study runs on one computing system. It does not establish real network latency or deployment speedups.

RESEARCH QUESTION 1 - PPHF BACKUP · 45

Privacy Accounting Under Full Participation

Full participation; fixed-slot zeroing adjacency with unchanged public weights and denominator.

\varepsilon(\delta)=\min_{\lambda>1}\left\{\frac{R\lambda}{2\sigma^2}+\frac{\log(1/\delta)}{\lambda-1}\right\}

R: number of Gaussian releases. σ: noise multiplier.

Each round contributes $\lambda/(2\sigma^2)$ in RDP under the stated sensitivity calibration.

Replacing one non-zero contribution by another doubles sensitivity; recalibrate noise. Each bound covers one configured run, not the parameter sweep.

Training, residual and forecast budgets require a common adjacency before an end-to-end claim.

RESEARCH QUESTION 1 - PPHF BACKUP · 46

Full Forecast-privacy Utility Results

Each row uses slot-specific sensitivity and non-private FHF base forecasts.

Dataset	εᵧ	MinTs MASE	MinTs RMSSE	WLSv MASE	WLSv RMSSE
GEF2017	1	23.114	23.535	13.763	14.356
GEF2017	5	4.633	4.715	2.769	2.884
GEF2017	10	2.330	2.369	1.407	1.461
GEF2017	20	1.189	1.208	0.740	0.768
AusGrid2013	1	15.777	11.001	9.929	7.574
AusGrid2013	5	3.463	2.427	2.421	1.829
AusGrid2013	10	2.076	1.507	1.626	1.277
AusGrid2013	20	1.488	1.161	1.304	1.090

RESEARCH QUESTION 1 - PPHF BACKUP · 47

Hierarchy Counts And Evaluation Roles

Node type and evaluation role are different classifications.

Dataset	Physical node types	Evaluation roles
GEF2017	1 total + 6 states + 3 zones = 10	1 root + 1 internal + 8 terminal
AusGrid2013	1 total + 100 postcodes + 259 customers = 360	1 root + 60 internal + 299 terminal

AusGrid terminal series comprise 40 terminal postcodes + 259 customer nodes.

Only non-terminal postcodes have explicit customer children in the deduplicated representation.

RESEARCH QUESTION 1 - PPHF BACKUP · 48

Selected Methodological References

References cited in the supplied manuscript.

McMahan et al. (2016). Communication-Efficient Learning of Deep Networks from Decentralized Data. FedAvg.

Hyndman et al. (2011). Optimal combination forecasts for hierarchical time series.

Wickramasuriya, Athanasopoulos & Hyndman (2019). Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization.

Bai & Pinson (2019). Distributed Reconciliation in Day-Ahead Wind Power Forecasting.

Mironov (2017). Rényi Differential Privacy.

Federated Hierarchical Forecasting

Presentation Outline

Introduction

Hierarchical Forecasting Basis

Base and Coherent Forecasts

The Summing Matrix Represents The Hierarchy

An Example for Multi-stakeholder System

Distributed ownership

Potential disclosures

Forecasting Paradigms: A Common Terminology

Federated Learning

Related Works

Federated Learning: Taxonomy

Horizontal and Vertical Federated Learning

Horizontal FL

Vertical FL

FedAvg And FedProx

FedAvg · Local SGD

FedProx · Constrain Local Drift

Three Federated Learning Architectures

Flat · Server-based

Hierarchical · Edge–Cloud

Decentralised · Peer-to-peer

Forecast Reconciliation: Taxonomy

Linear Forecast Reconciliation

The Research Gap

Federated Learning

Distributed Reconciliation

Possible Privacy-Preserving Methods

Research Questions

Overall Research Problem And Three Research Questions

Privacy-preserving Hierarchical Forecasting

Fully Decentralised Forecasting And Reconciliation

Forecasting And Decision-Making In Multi-Agent Systems

Progress to Date

Proposed FHF Framework

Distributed Reconciliation: A Three-Series Example

1 · Allocate The Columns

2 · Calculate Contributions Locally

3 · Sum At Each Recipient

Fully Hierarchical Variants: LN-FHF And CH-FHF

Differential Privacy In The FHF Framework

Honest-But-Curious Threat Model

Differential Privacy

Protection At Three Framework Stages

Privacy Mechanisms Across Three Stages

Federated Training

Residual Covariance

Forecast Reconciliation

Data And Experimental Setup

Training And Evaluation Protocol

Data pipeline

Model settings

Accuracy, Coherence And Statistical Evidence

MASE · Mean Absolute Scaled Error

RMSSE · Root Mean Squared Scaled Error

MASI · Mean Absolute Scaled Incoherence

Accuracy And Coherence

Statistical Evidence

Evaluations: Three Separate Privacy Experiments

GEF2017: Forecast Accuracy And Coherence

GEF2017: Statistical Comparisons

AusGrid2013: Forecast Accuracy And Coherence

AusGrid2013: Statistical Comparisons

Private Aggregation: Accuracy Before And After Reconciliation

Private Aggregation: Budget Alone Does Not Explain Utility

GEF2017 · K = 10 clients

AusGrid2013 · K = 360 clients

Private Residuals: Fixed And Quantile Clipping

GEF2017 · means across 10 series

AusGrid2013 · means across 360 series

Equal-noise limit

Private Forecasts: Coherence With A Substantial Utility Cost

GEF2017 · εy = 1, 5, 10, 20

AusGrid2013 · εy = 1, 5, 10, 20

PPHF: Conclusion And Future Work

Established Results

Limits And Next Research Questions

Next Steps

RQ2: Existing Methods And Layer Decomposition