Federated Learning with Efficient Local Adaptation for Realized Volatility Prediction
Authors: Lei Zhao
Lin Cai
Wu-Sheng Lu
Venue: Accepted by TMLR ()

Abstract: Financial markets present unique challenges for Federated Learning (FL) due to fragmented datasets, dynamic participation, and the critical need for precise and reliable predictions. Isolated local datasets often fail to capture the full spectrum of market dynamics, blocking accurate realized volatility predictions. Unlike traditional FL methods that focus on improving convergence during the training process, we propose Federated Learning with Adaptive Robustness and Efficiency for Local Adaptation (FLARE-LA), a novel framework designed to optimize predictive performance after the global training phase. FLARE-LA leverages Taylor-based local linearization and probabilistic optimization to efficiently adapt global models to local data distributions, enabling fast responsiveness to new market conditions. This adaptability ensures trained local models align with real-world scenarios, making FLARE-LA particularly suited to dynamic financial applications. Extensive experimental evaluations demonstrate FLARE-LA's superior performance, showcasing its ability to significantly enhance post-FL outcomes compared to state-of-the-art FL algorithms. The results underscore FLARE-LA's unique capability to drive advancements in financial forecasting and other high-stakes, rapidly evolving domains.

Published in Transactions on Machine Learning Research (01/2025)
Federated Learning with Efficient Local Adaptation for Real- ized Volatility Prediction
Lei Zhao
leishao@rric.cak
Deportment of Electrical and Computer Engineering University of Victoria
Lin Cai
cai@ere unic.cn
Department of Electrical and Computer Engincering University of Victoria
Wu-Sheng Lu Department of Electrical and Computer Engineering Universily of Victoria
wstu@ece uvic.cn
Reviewed on OpenReview: https: // openreview. net/ forum? id= bHAEtW5E70
Abstract
Financial markets present unique challenges for Federated Learning (FL) due to fragmented datasets, dynamic participation, and the critical need for precise and reliable predictions. Isolated local datascts often fail to capture the full spectrum of market dynamics, block- ing accurate realized volatility predictions. Unlike traditional FL methods that focus improving convergence during the training process, propose Federated Learning with Adaptive Robustness and Efficiency for Local Adaptation (FLARE-LA). novel framework designed to optimize predictive performance after the global training phase. FLARE-LA leverages Taylor-based local linearization and probabilistic optimization to efficiently adapt global models to local data distributions, enabling fast responsiveness new market condi- tions. This adaptability ensures trained local models align with real-world scenarios, making FLARE-LA particularly suited to dynamic financial applications. Extensive experimental evaluations demonstrate FLARF-LA's superior performance, showcasing its ability to signifi- cantly enhance post-FL outcomes compared to state-of-the-art FL algorithms. The results underscore FLARE-LA's 's unique capability to drive advancements in financial forecasting and other high-stakes, rapidly evolving domains.
1 Introduction
Predicting realized volatility is cornerstone of financial forecasting, essential for effective risk management and informed investment strategies within the framework of deep hedging (Buehler et al. 2019 Vuletić & Cont 2023 Mueller et al. 2024). However, financial markets naturally generate fragmented and asynchronous data across multiple trading venues. These platforms are unable to share data with third parties due stringent privacy concerns, regulatory constraints, and technical challenges (Kairouz al. 2021). The data fragmentation poses significant obstacles to the accuracy and reliability of realized volatility predictions. When data is distributed across multiple platforms, the collective market understanding becomes incomplete, limiting the ability to accurately capture price movements and liquidity dynamics. Volatility prediction, which depends on comprehensive market data, is particularly vulnerable to inaccuracies and biases under these conditions. Furthermore, discrepancies in liquidity levels and pricing for the same asset across exchanges exacerbate these challenges, potentially leading to incorrect volatility estimates when one platform's data fails reflect broader market trends (Otero. 2002 Madhavan (2000. Federated Learning (FL) offers a promising solution by enabling collaborative model training across distributed data sources while preserving

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data privacy, which addresses key privacy and regulatory concerns by ensuring data remains local to each
trading platform (Yang et al. 2019 Yu et al. 2020 Taan et al. 2022 Chen et al. 2023a Meng et al. 2024
Financial markets impose uniquely demanding requirements on FL due to the critical need for high precision, reliability, and robustness. Unlike other domains, where minor inaccuracies may be tolerable, small errors in financial forecasting can result in significant financial losses or missed opportunities (Ning et al. 2023 Volatility prediction, in particular, presents highly challenging task as it requires models that not only identify complex and rapidly evolving trends in market behavior but also provide robust and interpretable risk estimates (Bergeron et al. 2021]. The fragmented nature of financial data intensifies these challenges. Trading platforms operate independently, each generating datasets that reflect its unique market conditions, liquidity levels, and trading behaviors. The data decentralizatton introduces substantial obstacles to achieving consistent, high-quality predictions across platforms.
While data hetcrogencity and dynamic participation are common challenges in FL, their impact is amplified in financial markets. The inherent variability in datasets across trading platforms leads to significant discrepancies in local and global data distributions. Additionally, trading platforms frequently enter and exit the training process due to operational constraints, creating dynamnc and unpredictable training environment. Maintaining robustness under such conditions is essential to ensuring consistent model performance across all participants. Beyond these technical challenges, the necessity for interpretable and trustworthy predictions is particularly acute in financial applications. Models must go beyond delivering accurate forecasts, which must also quantify uncertainties effectively, enabling informed decision-making in high-stakes environments. Traditional FL methods often lack the precision, adaptability, and interpretability required for such applications, limiting their practical utility in financial forecasting.
To address these challenges, we introduce Federated Learning with Adaptive Robustness and Efficiency for Local Adaptation (FLARE-LA), cutting-edge framework designed to address the distinct requirements of financial markets. FLARE-LA utilizes Taylor- linearization to achieve computationally efficient and accurate local adaptations, effectively aligning the global model with platform-specific datasets. Moreover, it incorporates a probabilistic mechanism that, leverages the Jacobian matrix of the global model, facilitating localized optimization and delivering interpretable uncertainty quantification to enhance reliability and support informed decision-making. This integrated approach ensures robust performance in dynamic and fragmented environments while maintaining compulational efficiency. By blending global insights with fine-tuned local adjustments, F FLARE-LA substantially improves the accuracy of realized volatility predictions, satisfying the stringent precision and reliability demands of financial forecasting.
Although FLARE-LA is rigorously evaluated within the context of financial markets, its underlying principles, such as efficient local adaptation and uncertainty-aware predictions, are broadly applicable to a wide range of domains. The financial market serves as a challenging and representative scenario that underscores the framework's capabilities, providing valuable insights into its potential for other complex and dynamic environments. Experimental evaluations demonstrate that FLARE-LA consistently outperforms state-of-the- art basclines, achieving lower mean loss, Value at Risk (VaR95%), and Conditional Value at Risk (CVaR95%) across diverse platform distributions and participation rates. These results highlight the framework's 's scalability, adaptability, and robustness, making it well-suited for general federated learning tasks where data heterogeneity, dynamic participation, and computational efficiency are critical.
In the following sections, we provide an overview of related work in Section 2 positioning our contributions within the broader landscape of FL and financial forecasting. Section 3 introduces the fragmented nature of financial markets and formulates the problem addressed by our framework. In Section 4 we detail our proposed approach, highlighting the integration of probabilistic frameworks and efficient local adaptation techniques. Empirical evaluations of our method, showcasing its effectiveness and robustness across various scenarios, are detailed in Section Finally, we summarize our findings, and outline potential directions for future research in Section
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2
Related Work
In the context of financial markets, predicting realized volatilitt using order book data is challenging task due to the decentralized nature of data acquisition (Banabilah et al. 2022). Order books, which capture buy and sell orders for securilies, form a dynamic and fragmented data environment, often referred to as "dala islands." These characteristies make FL an appealing approach for such environments (Hasbrouck) 2007). FL enables collaborative model training across distributed data sources while preserving data privacy, a critical requirement in financial markets.
Existing FL methods face specific challenges when applied to predicting realized volatility in financial markets, such as high data hetcrogencity, rapid changes in data patterns, and the need for timely model updates. Methods like FedProx (Li et al., 2020), which introduces a proximal term to the local objective function to stabilize optimization, mitigate the effects of heterogeneity by reducing the impact of local updates that deviate significantly from the global model. However, while FedProx offers stability, its proximal term may not fully capture the dynamic nature of financial data, and it can slow convergence, a critical drawback in fast-paced trading environments (Arthur et al., 2018 Cantillon & Yin, 2011). To address local-global mis-alignments, SCAFFOLD (Karimireddy et al.j 2020 incorporates control variates to correct the drift in local updates, improving aligument with the global model. However, the rapidly evolving financial landscape can still lead to misalignments that adversely affect prediction accuracy (Boukherouaa et al. 2021). While effective in certain scenarios, the introduction of control variates increases computational complexity and communication overhead, posing challenges for deployment in high-frequency trading environments.
Personalized FL methods, such as FedPer (Arivazhagan et al. 2019), decouple shared global parameters from client-specific local parameters to provide personalization. Despite this, the high variability and unpredictability in financial markets require frequent adjustments to personalized models, making the process Resource-intensin and limiting scalability in large-scale financial networks. Similarly, L.G (Liang) et al. 2020), APFL (Deng et al. 2020), and pFedMe (T Dinh et al. 2020) adopt approaches to balance global and local knowledge but face challenges in handling the feature and distributional heterogeneity prevalent in financial markets.
Recent innovations, including Ditto Li et al. 2021), ; FedRep (Collins et al. 2021 and SuPerFed (Hahn ct al. )022), have explored various personalization tochniques, such as interpolating global and local models or applying proximity regularization. These methods highlight progress in adapting global models to client- specific data. However, they often require fine-tuning or additional computational resources for new clients, reducing scalability in highly dynamic environments like financial trading. [eng et al. (2024) explores techniques to enhance global generalization and local personalization through adaptive aggregation and dual optimization, which aligns with our goal of striking a balance between global insights and local adaptations in heterogencous FL settings. While their work focuses on represcntation learning and aggregation strategies, our method introduces a Taylor-based linearization approach combined with a probabilistic framework to achieve more precise and interpretable local adaptation. (Tan et al. (2022) provides insights into handling client-specific model updates using personalized layers and meta-learning, offering a solution for improving local performance in non-IID settings. This is relevant to our method in FLARE-LA, which also aims to achieve strong local adaptation but does so through efficient linearized updates and probabilistic adjustments, eliminating the need for additional network layers or meta-learning components. Chen et al. (2023a) focuses on sparse model adaptation to enhance scalability and computational efficiency in personalized FL, a strategy particularly useful in resource-const rained environments. Similarly, Yu et al. (2020) highlights the importance of localized training adjustments to address the limitations of fodcrated aggregation in hetcrogencous datascts. Both approaches emphasize the need for effective local training, which resonates with our method's focus on dynamic participation and efficient adaptation. However, our approach extends these ideas by leveraging Jacobian-based lincarization and uncertainty quantification, enabling robust local updates tailored to the fragmented and rapidly changing nature of financial data.
Advances in neural network behavior further inspire solutions for FL. Research has revealed that infinitely wide deep neural networks (DNNs) exhibit behaviors similar to their Taylor expansions around initialization (Chizat) ct al. 2019). Extensions of this analysis to finite-witth DNNs demonstrate that their training dynamics resemble linear models (Seleznova & Kutyniok 2022). while the inductive biases of linearized neural networks
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effectively summarize full network functions (Maddox et al. 2021 These findings motivate the development of FL, approaches that better address the unique characteristies of local trading platforms, where global models often fail to capture localized data patterns.
To address these challenges, the proposed FLARE-LA framework introduces adaptive local training mechanisms that go beyond traditional FI, approaches by focusing on post-training performance optimization. FLARF--A leverages insights from neural network linearization to enable precise and computationally efficient local adaptations, ensuring that global models are effectively refined to meet the specific needs of individual trading platforms. Unlike existing methods, FLARE-LA is designed to address the dynamic and heterogeneous nature of financial markets by rapidly adapting to new data and evolving conditions, ensuring that local models remain robust and aligned with real-world scenarios. The proposed innovative approach is a transformative solution for achieving aighiprecision and reliable predictions in decentralized and fragmented financial environments.
3 Fragmented Financial Markets
3.1 Background
In financial markets, trading occurs across a wide range of exchanges and platforms, resulting in highly fragmented datasets. Each platform independently maintains transaction and order book data, capturing buy and sell orders as well as their exccution details. This fragmentation provides an incomplete view of market activity for any given asset, with notable variations in pricing, liquidity, and order depth across platforms [Hasbrouck] 2007).
The order book plays a critical role in market analysis, offering traders insights into short-term trading dynamics. By displaying order imbalances and identifying potential support and resistance levels for stock, the order book supports informed trading decisions. Heightened market activity and uncertainty are often reflected in increased realized volatility, which arises from frequent directional price movements. Trading data, which records executed transactions, complements the order book by offering valuable insights into market dynamics, such as price trends, trading volumes, and liquidity conditions.
Predicting short-term realized volatility is essential for effective risk management and the development of trading strategies (Chen et al. 2023b). By analyzing order book and trade data over fixed time intervals, traders and institutions can forocast future volatility levels, enabling improved decision-making and enhanced risk mitigation. Realized volatility predictions help market participants manage exposure, optimize portfolio allocations, and design robust trading strategies.
Extracting meaningful insights from order book data is vital for understanding market dynamics and assessing stock values. Key metrics, such as the bid-ask spread, weighted average price, and volume-related indicators, provide a wealth of information about market liquidity and potential volatility. However, the fragmented nature of financial markets poses significant challenges for comprehensive analysis, as data silos limit access to the full scope of market activity.
3.2 Problem Formulation
By leveraging diverse data sources, FL facilitates the development of a robust global model that enhances local predictions, preserving both data privacy and confidentiality, which allows trading platforms to benefit from comprehensive understanding of market dynamics while maintaining compliance with regulatory requirements and addressing privacy concerns.
Consider a distributed dataset consisting of n data sample pairs (x;, y across |E| trading fortforms. Each data sample pair represents features extracted from order book and trading data, with x, denoting the feature vector and y; representing the corresponding label, which is the volatility. There are 363 features for each sample generated from order book and trading data, capturing essential market dynamics such bid-ask spreads, price movements, and trading volumes.
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We denote the local dataset of the e-th trading platform as P.. which contains ne training samples. The union of all local datasets from each trading platform, P1 U P3 U...U P E]: covers the entire dataset, guaranteeing that each sample is assigned to exactly one trading platform's 's dataset. For trading platform c, the labels (Whee, represent the volatility levels observed in the corresponding platform's trading data. These volatility labels are used as the ground truth for training the predictive model.
We aim develop a predictive model, represented by deep neural network function f, which maps an input feature vector x to an output volatility prediction y. The model is trained using the distributed dataset across multiple trading platforms, leveraging the features extracted from order book and trading data to prodict future volatility levels accurately. The local objective function for trading platform is defined as
1 minimize Le(w) = Σ (f(2;, w) yi)² for c 1,,, ,18,, iEP
(1)
where w represents the trainable parameters of the model. Meanwhile, the global objective function, aggregating the local objectives across all trading platforms, is given by
E Σ La(w).
minimize u
L(w)
(2)
F.
Realized volatility prediction poses unique challenges due to the fragmented nature of financial data, and the rapidly evolving market conditions. Each platform's dataset captures only localized perspective of the broader market, leading to non-IID data distributions that complicate the development of unified model. Furthermore, the predictive lask demands a model capable of capturing complex, non-linear interactions between features to provide reliable and interpretable outputs. By leveraging FL, trading platforms can train global model that integrates diverse data sources, improving prodictive accuracy while maintaining data privacy. This collaborative framework enables financial institutions to optimize their trading strategics, enhance risk management, and make informed decisions in volatile market conditions.
4 Federated Learning with Adaptive Robustness and Efficiency for Local Adaptation
In this section, we introduce our proposed approach FLARE-LA, a framework designed to address the challenges posed by heterogencous local datasets and dynamic participation in financial markets. While the global objective function in (2) captures general patterns across all trading platformss, it may fail to fully represent the unique characteristics of each platform's local dataset, potentially resulting in suboptimal performance for individual platforms as outlined in []. To overcome this limitation, FLARE-LA provides an innovative mechanism for trading platforms to adapt the globally trained model to their specific local data.
The FLARE-LA franework operates through a two-stage process. In the initial stage, trading platforms collaboratively train a global model while ensuring strict data privacy. This is accomplished by aggregating model updates from each platform without transmitting raw data, thereby preserving confidentialityy and compliance with privacy regulations. The global model, built from the collective knowledge of all participating platforms, effectively captures shared patterns and structures across the distributed datasets. This stage provides a robust baseline for subsequent local adaptation. Moreover, FLARE-LA is designed to seamlessly integrate advanced federated learning techniques into this collaborative training stage, enhancing flexibility and adaptability to various use cases.
In the second stage, FLARE-LA introduces an innovative local adaptation mechanism that fine-tunes the globally trained model to align with the specific characteristics of each trading platform's 's dataset. This process leverages based lincarization and probabilistic frameworks to achieve computational efficiency and precision. By utilizing the Jacobian matrix of the global model, FLARE-LA integrates localized optimization with interpretable uncertainty quantification, enabling platforms to adapt the global model dynamically while maintaining robustness in predictive accuracy. This approach ensures that FLARE-LA excels in addressing the challenges of non-IID data distributions, enhancing model performance in diverse and fragmented environments.
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By combining the strengths of FL with tailored local adaptations, FLARE-LA effectively addresses the inherent heterogeneity of financial market datasets and accommodates the dynamic participation of trading platforms. This dual-stage approach ensures that each platform benefits from the collaborative insights of FL while achieving optimal performance for its specific market conditions.
4.1 Federated Training equipped with Efficient Local Adaptation for Financial Market Dynamics
The initialization of model weights plays a pivotal role in determining the efficiency and stability of the training process. Arbitrary or poorly chosen initialization methods can block progress, leading to issues such as slow convergence or training stagnation (Xie et al. 2017). To address these challenges, it is essential to ensure that the activation distributions maintain consistent variance the network deepens, preventing common pit,alls like vanishing or exploding gradients. To achieve this, the initial weights are drawn from Gaussian distribution with mean of zero and standard deviation inversely proportional to the square root of the number of input units foeding into the layer as
w ~ N N((0.
(3)
Rin
where w denotes the initial weight vector, and Bin represents the number of input units the layer. This tailored initialization ensures balanced variance in the activation distributions across layers, fostering a smoother gradient flow and more stable training dynamics.
Federated training operates in a dynamically evolving environment where the participation of trading platforms fluctuates umpredictably. At each training round, subset of trading platforms, denoted as S' S [E), is selected to participate. This mirrors real-world scenarios where platform availability is influenced by operational constraints, market activity, or other factors. To capture these dynamics, the set S is sampled from predefined distributions, including Exponential, Geometric, Gamma, and Chi-square distributions, reflecting variety of participation patterns.
The dynamic nature of platform participation introduces additional complexity to the federated training process, as the global model must adapt to fluctuating contributions without compromising performance. By incorporating realistic participation patterns into our simulation, we ensure that the training procedure reflects the challenges of real-world financial environments, enhancing the robustness and applicability of our approach.
Once the active participants for round are determined, the current global model w is distributed to the selected trading platforms in St Each platform initializes its local model for the training round
{we = w'
(4)
where wn. represents the initial local model weights for trading platform c at the onset of round t. This initialization ensures that all participating platforms begin the round with identical copics of the global model, fostering a collaborative and unified starting point in the dynamic participation environment.
During local training trading platform e, the model undergoes iterative updates using financial market data. The k-th step of this update process is defined as:
11 = w' - aj (w.
(5)
where ag the local learning rate, tailored to the unique dynamics and characteristics of cach platform's dataset. This localized learning process allows each platform to refine the model in alignment, with its specific market conditions. The local training procedure continues for K iterations, resulting in final local model as
K
th
w
Oi V (e
(6)
which integrates the cumulative effects of gradient- cradeent-based updates, highlighting how each platform adapts the global model to its specific data through weighted gradient descents.
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To quantify the divergence between the locally adapted model and the initial global model, we define the model discrepancy for trading platform e after K iterations as
Aw = w'.k - w':
(7)
measuring the extent to which each platform's local updates diverge from the global model parameters, and reflecting the influence of its unique market data on the learning process.
The aggregated local updates are used to compute the global model for the next ileration as
w w
9
Σ Aw
(8)
S
where o' is the global learning rate for round i, and the contribution of each local model is normalized by the number of participating platforms |.S%). This normalization ensures equitable integration of local updates into the global model, promoting fairness and robustness across platforms. The updated global model w++1 marks the conclusion of the 1-th round of training and serves as the starting point for the next round of FL.
The above iterative process allows FLARE-LA to adaptively refine the global model by incorporating diverse contributions from participating platforms within the dynamic and heterogencous environment of financial markets. Essentially, the federated iterations in FLARE-LA are designed to be modular, enabling the seamless integration of any advanced FL solutions. This flcxibility enhances the scalability and generalization of the framework, allowing to adapt to evolving methods and leverage state-of-the-at advancements in FL. By combining tailored local training with equitable aggregation, FLARE-LA effectively addresses the challenges of data heterogeneity and fluctuating participation rates, ensuring robust performance and broad applicability.
The global model w", obtained after FL training, may not be fully optimized or may exhibit poor local performance due to the diverse nature of local datasets and the dynamic participation. Nonetheless, it serves as the baseline for adaptive local training. To derive the local adaptive training strategy, we consider a given neural network model function f. We can approximate f around the trained model parameters w" using a Taylor expansion
f(;; w) f(;; wn) + Jw. (x=) -w",
where (x) denotes the Jacobian matrix of partial derivatives of f with respect to the model parameters at w", with dimensions x PPa| and we use to denote the size of the model parameters. This Jacobian represents the sensitivity of the output with respect to changes in the model parameters near w".
We formulate the probabilistic model governing the output y, given input features x extracted from order book and trading data, and model parameters w as
[omw)) 20
plyla x. w) A (f))) V2TO²
(10)
where of represents the variance associated with the Gaussian noise, capturing the inherent uncertainty and noise in the model predictions of volatility. This distribution's mean is specified by the linear approximation obtained from the Taylor expansion of f, with a variance o?-
For volatility prediction in financial markets using FL, deviations from the baseline global model w² influence the mean prediction through the Jacobian adjustment, while the Gaussian term WO o2) accounts for the stochastic nature of the predictions. This framework establishes a robust basis for trading platforms to adapt and retrain the global model locally, ensuring performance optimization tailored to the unique characteristics of individual datasets.
For each trading platform e with its local dataset {(0, 1 the likelihood function quantifies the probability of observing the given data. It incorporates both the individual variances from the Gaussian noise and the deviations of the model predictions from actual data points. This integration is captured by the model's output and its linear approximation around wn which is formulated as
IPdl
P
1
1
Po(w)
(2
PP
exp
20
(u - (f(x;;)) + Jw (zzi) w w²)))²
(11)
1=1
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which enables trading platforms to effectively assess the fit between their local data and the global model, guiding them in refining the model parameters to better capture the underlying patterns in volatility dynamics. For rapid local adaptation within our financial market volatility prediction, we transform the likelihood function into its logarithmic form as
Pd Σ i=1
[Pal 2
log 1og(2 1
log(P.(w))
y1 - (f(x, + Juo (x:)"(w w' w²)))²
(12)
20
which simplifies the expression by converting the product of probabilities into a sum of logarithms, linearizing the effects of the paramcters and enhancing the tractability of the optimization problem. Importantly, 20% Σ Eul f(æ;; u2) represents the sum of squared residuals, adjusted by the inverse of the noise
variance o². Therefore, the local adaptation process can be formulated as minimizing the following loss function
1 P.l
E y3 (f(x;; Jw. vw (xx;)((w w w²)))² Pa Le(w) 20
log (2g()))
(13)
which comprises a term that evaluates the sum of squared deviations between the predicted volatility and the actual volatility, scaled by the noise variance of, and a constant term that standardizes the loss based on the dataset size and noise level in the context of local financial market data.
We define - = (Jw- K as the collection of Jacobian matrices of the model's predictions with respect to the features generated from order book and trading data, evaluated at w". The sum of the outer products of these Jacobian matrices across all data points forms a symmetric matrix as
Σ Jw (w:)) (x:)) = Jw m
(14)
which reflects the covariance structure of the gradients, capturing the sensitivity of the model's predictions to the features derived from the trading platforms' data. To simplify computations in practice, the loss function can be reformulated as
20 Ju . JT- w- (w - -w") - (w - w²y 1 Jue - fc) 0²
Luww)= = (w--ww -
w²²T ww-=
(15)
+ 20 (ye fc) (ye fc) + IPal
log log(2
2
where fc = (f(æ;; un)) - and ye = (u It quantifies the balance between the model's internal predictions and the observed deviations from the actual volatility outcomes, scaled by the noise variance, o?: This local loss function is critical for adapting the global model to better fit the specific characteristies of the local trading platform's data. The local model adaptation is achieved by setting the gradient of the designed local loss function, v (w), to zero as
1 _ Jw J (w ww)
1 Ju (ye - fc) = 0. o²
v viv)
(16)
??
By solving the stationary condition of the linearized local loss, we identify the local adaptation for platform as
w = JT- Jnn (Be- - fc) + w".
(17)
which typically serves as minimum for well-defined convex function. This formulation indicates that the local model adaptation is proportional to the inverse of the aggregated Jacobian product, adjusted by the residuals between the observed volatility Ye and the model's baseline predictions fc- Importantly, the matrix-vector product (J=-J -' only needs to be computed once per platform, providing significant computational efficiency during repeated local adaptations. And
0 = f(x;; w²) + Jw- (x0x)" (Jw- Ju (ye -c) + NN(0
(18)
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governs the local model's predictions for new data sample 2;, where the term (N aه) captures the inherent noise variance in volatility forecasting.
w- is invertible, rank-deficient or ill-conditioned Jacobians can arise under real- world data constraints, e.g., when local data are limited or collinear. To ensure reliable adaptation in such scenarios, we introduce small regularization parameter À on the diagonal of Jw Ju and we can obtain
Although 17 and 18) assume Jw
w w" + (Jw-J J + + Jw (ye fc).
(19)
In this modification, every eigenvalue of Jw -J is shifted by at least X, thereby ensuring positive definiteness and invertibility, Therefore, the predictive step becomes
Di [( w²) + J (a:;) (Jw- + AT) +
" Jw- fo) + OO
(20)
This diagonal shift has minimal impact the model's geometry while resolving degenerate eigenvalues and stabilizing local adaptation. Therefore, FLARE-LA accommodates wide range of data conditions and ensures the final adaptation step remains both computationally feasible and theoretically sound, even when the Jacobian product matrix is poorly conditioned.
Algorithm 1 Federated Learning with Adaptive Robustness and Efficiency for Local Adaptation (FLARE-LA)
Input: Global model initialization w", learning rales as, {a/h, participation distribution, local datasets (Deleec, regularization parameter À 2 0, ill-condition tolerance E >
Output: Updated global model w",, optionally adapted local models {w]}
1: Initialize: w ~ 10,
2: for t= 1 to T do
3: Sample active platforms St ~ Participation Distribution for each platform e E St in parallel do Initialize local model w w² for k 1 to K do w', w " ag VLe(w',n-1)
8: end for
9: wi,k + final local model after K steps
10: Lw +- --w,, w
11: end for
12: Aggregate global model: w +- w' + 1 Ecest Aw
13: end for
14: Set w" + w- (Final global model after T rounds)
15: Efficient Local Adaptation:
16: for each platform e (if local adaptation is needed) do
17: Compute Jacobian J on De. and residuals Te < y fc
18: Compute the eigendecomposition: Ju JTh w. = U A U
19:
Obtain the smallest eigenvalue: Amin = min; X;
20:
if Amiin then
21:
Jw- JT Ju JT AI
22:
end if
23: Update local model: w + wt + (Ju- (Jw-J''') J --
Jw
24:
Prediction for new data æ; ds + f(x;; w²) + Jw (2x)² (T. Jww - Te + NN,,
25:
end for
By combining the baseline prediction using the global model parameters f(æ;; w²) with a local refinement. term Ju- (xx)" (J JT ) - (ye - fc) and explicitly modeling the inherent variability through a Gaussian noise term (O, o2), FLARE-LA offers an adaptive stratogy tailored cach trading platform's data heterogencity. This hybrid prediction process leverages the global model as baseline while swiftly capturing local requirements, ensuring robust and accurate volatility forecasts under dynamic market conditions. When

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Jw. is rank-deficient ill-conditioned, small diagonal shift AI is introduced to preserve positive definiteness and maintain a well-defined local update. The full procedure of FLARE-LA, incorporating both federated training and efficient local adaptation, is summarized in Algorithm !
4.2 Analysis of the Local Adaptation Mechanism
FLARE-LA integrates the federated training phase with Taylor-based local adaptation mechanism, effectively addressing the challenges of non-IID data and dynamic participation in financial markets. By uniting collective insights from distributed platforms with precise local refinements, FLARE-LA produces volatility prediction model that balances global consistency with platform-specific accuracy. To complement the federated training phase, FLARE-LAS LA introduces an advanced Taylor- based linearization strategy for computationally efficient and precise local adaptations. By leveraging the Jacobian matrix the global model, FLARE-LA approximates complex local adjustments, enabling trading platforms to quickly tailor the global model their unique data distributions without extensive computational overhead. Additionally, the framework integrates probabilistic modeling to capture prediction uncertainties, enhancing interpretability and reliability, which is key for high-stakes financial decision-making. This modular and extensible approach ensures that FLARE-LA remains scalable, adaptive, and generalizable framework for FL in financial markets and beyond.
FLARE-LA incorporates adaptive regularization and probabilistic modeling to address the volatility and heterogeneity typical in financial data. Each platform characterized by its own of encapsulating local noise levels. The Hessian of the local objective, (2(x, is assumed to be bounded within (14, B:), and we have
1 3/ Jw J S Br.
14 S
(21)
07
To ensure numerical stability and predictable convergence during local adaptation, we first provide the analysis that the local gradient w.r.t w* can be bounded. According to the quadratic upper bound and the linear lower bound of the local objective function, we can obtain the inequality
L,(w)) - L,(w) I,(=*) - L((2) + L,(2) L,(w)
=) + VL,(w) T (= w) + 3r - +|= wll
< (L.(w*)"(("") (wo" =)
(22)
(i: (w" -w) + (VL;(W") ( =) + Bi
- || = w|l
2
We define
== 1 VL ([([w)-V[((w)))
(23)
3
and then, we have
(VL VL:(w)) (w - 2) = 1
VV VL.
(24)
B: || w|12 1
28, VLi(w*) VL,(w)113
hence,
L((w)) - L.(w)= S VL,(w*) (w= (w-- -w) 23,
(25)
which leads to
[,(w)= - Ii(w0* V (ww w²) 2 VL:(W*) - (L))1
(26)
23,
Since
(w, L,(w=)) L(w) L",
(27)
E
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then, we have
28,(1w))- (,) 2 F
VLi(w)) - VLi(2))11
(28)
iEE
The bound on the local gradient can be found as
1 E
1
[L;(w)
VVL(()))V(((()) + VL,(W*)I
E
E
E
2
[L;(w) VL;
[w
(29)
E
E
EEE
iEE
- (3)
+
2
Σ \Vi(2*)11 iEE
Therefore, we define the local gradient w.r.t w² is bounded by 6;, ie., (w:(w*)) S 6i, and wi is the optimal model for trading platform i. The improvement of local adaptation in the model space can be bounded by
w|w w || ||00- - =I(V''a(u"))~<(<<ccc)) + (w w=) wIll
- -
(30)
Since
Vl((w)) = (L:(")+++++ V','"'') - w²))
(31)
we obtain that
|w ww-will (w"" - vL:(w']] " Li
(32)
We assume the local gradient w.r.t w" is bounded by c, Then, we have
S (1
_
(33)
1ii
Furthermore, we have
[<'' - <
(L, vi:(w")
3:
(34)
| w"
Therefore, we have
ur - will 5
31 || [4g
Ei w² +
(35)
14
to illustrate how closely each platform's adapted weights, w, align with its own local optimum, w This result shows that the deviation of the adapted solution from the local optimum depends on the curvature factors (M, 8.) and the magnitude of the global model's misalignment with the true local optimum. Therefore, even if the global model w" is not fully optimized for platform i, FLARE-LA's 's closed-form local adaptation draws the parameters significantly closer to w.
Another key benefit of local adaptation appears in the reduction of the local objective function. By expanding (,(w) in sccond-order Taylor scries around w", it can be shown that
L,(w) = -,(w") 1 Vi.("") (V= -I "L((w*).
(36)
Hence, the local adaptation step decreases the objective by
i,(w") - [.(w) = vi 'V((w"))
(37)
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Since vLe(w") is proportional to o/ Jw- (yt ft), the extent of this loss reduction is bounded by curvature terms and the squared residual y/ fill², giving
B. 1 Sall?.
L,(w=) (i(
(38)
of pii
This result confirms that local adaptation can substantially cut down on the local error to provide the promising local adaptation.
By combining global insights with localized precision, FLARE-LA addresses the fragmentation of market data without sacrificing data privacy model stability. Its Taylor-based lincarization efficiently refines model parameters, while the adaptive regularization and variance-aware modeling capture uncertainly in volatile markets. The bounding of Hessian cigenvalues and gradients ensures that each trading platform's updates remain well-controleed and that performance gains are both siguificant and predictable. Through the seamless integration of federated collaboration and an efficient yet robust local adaptation mechanism, FLARE-LA ensures enhanced performance across diverse and dynamic environments.
5 Experimental Evaluation
This experimental evaluation validates the efficacy and adaptability of the proposed FLARE-LA framework in addressing the challenges of FL across both domain-specifie and general scenarios. Our primary focus lies in the financial domain, where the demands for high precision, robustness, and scalability are particularly pronounced. We first utilize dataset for realized volatility prediction, consisting of order book and trade data from multiple trading platforms. These experiments aim to demonstrate FLARE-LA's 's ability to handle extreme data heterogeneity, dynamic participation, and the fragmented nature of financial datasets while maintaining robust predictive performance.
To further evaluate the generalizability of FLARE-LA, we extend our experiments to CIFAR10 and MNIST, two well-established datasets in FL. research. These datasets allow to test FLARE-LA's performance under non-llD data distributions, varying client participation rates, and label noise scenarios, mimicking real-world challenges. By incorporating these datasets, we provide complementary evidence of FLARE-LA's versatility and scalability, demonst rating its utility across diverse applications beyond financial forecasting.
In our experiments, client participation is dynamically regulated using a participation ratio, simulating high variability in client engagement during federated training. Non-IID data distributions are modeled using a Dirichlet distribution, with the concentration parameter controlling the heterogeneity of client data. For a + 0, clients primarily have data from a single class, while oo results in a uniform distribution of classes across clients. We evaluate the performance of FLARE-LA against several state-of-the-at FL methods, including FedProx (Li et al. 2020). SCAFFOLD (Karimiredyy et al. 2020). FedPer (Arivazhagan et al. 2019). LG-FedAvg (Liang et al. 2020)) Mee (T Dinh et al. (2020), Ditto Li et al. 2021), FedRep (Collins et al., 2021), and SuPerFed (Hahn et al. 2022). For local training, we utilize the ResNet model, which provides a robust architecture for handling diverse data distributions. The evaluation metrics include mean loss, VaR95%, and CVaR95%, offering a comprehensive assessment of model performance. These metrics underscore FLARE-LA's ability to deliver superior predictive accuracy, adapt effectively to dynamic environments, and maintain computational efficiency, even in challenging FL scenarios.
5.1 Experiments on Realized Volatility Prediction
We aim to forecast short-term volatility for stocks spanning multiple sectors (Andrew Meyer 2021]. The dataset comprises both order book and trade data for these stocks, aggregated into multiple time buckets. The values in the order book represent the latest snapshots of market activity, taken at one-sccond intervals. Each time bucket comprises order book data spanning the 600 seconds. Our experiments involve predicting the volatility for each time bucket of the stocks. There are 428, 932 samples in the entire dataset, where 107 of the stocks have data for 3830 time buckets, while 3 stocks have data for 3829 time buckets, stock has data for 3820 time buckets, and another stock has data for 3815 time buckets. The entire dataset is divided into 10, 000 trading platforms based on Dirichlet distribution--ased non-IID setting (Hsu al.
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2019h. The Dirichlet distribution's concentration parameter, a, determines the stock distribution for each trading platform which is set to 0.5 in our experiments. Each trading platform randomly splits its data into a training set and a test set, with 20% allocated for testing. This setup allows us to estimate the performance of cach FL algorithm on each trading platform's 's test set using its personalized model.
5.1.1
Performance Comparison
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Figure 1: Comparison of FLARE-LA with Individual Train and other FL baselines (FedProx, SCAFFOLD, and FedPer) across different epochs.
In Fig. 1 we compare the performance of FLARE-LA against Individual Train and baseline FL methods FedProx, SCAFFOLD, and FedPer over 50 epochs as shown in Fig. [a] and 200 epochs shown in Fig. [b). The Individual Train baseline performs training independently for each trading platform without leveraging federated collaboration. Despite identical numbers of paramcter updates, FLARE-LA demonstrates significantly superior performance, emphasizing the value of FL in leveraging global insights while lailoring models to local data.
As shown in Fig. aa), after 50 epochs, FLARE-LA achieves a remarkably lower mean loss of 7.726 10-5 compared to Individual Train (0.0132), FedProx (0.0031), SCAFFOLD (0.0015), and FedPer (0.0017). In Fig. [b), after 200 epochs, FLARE-LA continues to outperform all basclines, maintaining its lead in terms of mean loss, VaR95%, and CVaR95%. The results highlight FLARE-LA'S ability to balance global knowledge with precise local adaptation, resulting in superior performance in federated settings. This demonstrates that FLARE-LA not only accelerates convergence but also ensures higher accuracy and robustness compared to individual and bascline fodcrated training methods.
Across all experimental settings as shown Fig- 2 FLARE-LA consistently outperforms baseline methods, including FedProx, SCAFFOLD, FedPer, and SuPerFed, terms of Mean Loss, VaR95%, and CVaR955 for realized volatility prediction tasks.
With low client participation rates 1s10% FLARE-LA demonstrates exceptional robustness and precision. As shown in Fig. aaa, after 5 training rounds, FLARE-LA achieves a Mean Loss of approximately 0.0001, compared to significantly higher values for SuPerFed (0.0008), FedPer (0.0017), SCAFFOLD (0.0031), and FedProx (0.004). The advantage becomes even more pronounced risk-sensitive metrics such as VaR95% and CVaR95%, where FLARE-LA achioves much lower values, highlighting ability to effectively manage tail risks even with limited trading platforms participation. These trends persist in Fig. bb, with 20 training rounds further consolidating FLARE-LA'S dominance in all metrics.
When the participation rate incrcases to 30% as shown in Figs. [(c and dd) the overall model performance improves across all methods. However, FLARE-LA retains clear advantage, achieving substantially lower Mean Loss values. For instance, in Fig. dd, FLARE-LA reaches Mean Loss of approximately 0.00005, outperforming SuPerFed (0.0003), FedPer (0.0005), SCAFFOLD (0.0010), and FedProx (0.002). The improvement in FLARE-LA's performance with higher participation rates underscores ability to fully leverage the increased availability of local data while maintaining its computational efficiency and accuracy.
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Figure 2: Performance comparison with varying participation rates and training rounds realized volatility prediction.
Furthermore, the impact of increasing the number of federated training rounds is evident. FLARE-LA demonstrates rapid convergence low loss values within few rounds, significantly reducing the computational burden compared to other methods. Even after just 5 training rounds, FLARE-LA achieves results comparable to better than the bascline methods after 20 rounds, as shown in Figs. 2 and ((... This efficiency highlights FLARE-LA's 's capability to deliver robust performance even in scenarios with limited training rounds or participation rates.
As shown in Fig. 8 we provide the comparative analysis across various participation distributions to evaluate addressing the inherent challenges of FL with dynamic participation. The experiments were conducted with a 20% participation rate, 10 federated rounds, and 10 local epochs in each round. As shown in Fig. 3 ()) where trading platforms are sampled from exponential distribution, with scale parameter of 1.0, FLARE-LA demonstrates remarkable ability to achieve mean loss of 7.358 x 10 VaR95% of 2.284 10-1, and R95% of 3.978 10-1, out perforriin FedProx, SCAFFOLD, and FedPer by an order. As shown in Fig. 3(b), where trading platforms are sampled from a geometric distribution with a probability of success of an individual trial set at 0.35, FLARE-LA once again emerges as the top- rop-performing algorithm. Fig. ()c) explores the performance of algorithms when trading platforms are sampled from a Gamma distribution, with a shape parameter of 2.0 and scale parameter of 1.0. In Fig. 3)), where trading flatforms are sampled from chi-square distribution with the number of degrees of freedom set 2.0, FLARE-LA LA continues to outshine the baseline algorithms.
the efficacy of FLARE-LA
By consistently delivering superior performance metrics, FLARE-LA showcases its adaptability in scenarios characterized by varying levels of data availability and participation. By consistently achieving lower mean loss, VaR95%, and CVaR95% values, FLARE-LA underscores its resilience and adaptability optimizing fodcrated model training across a spectrum of trading platform distributions. These results demonstrate FLARE-LA's capability in managing volatile market conditions and optimizing federated model training despite unpredictable trading platform participation patterns.
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Figure 3: Illustrations of distributions: (a) Exponential, (b) Geometric, (c) Camma, and (d) Chi-square.
Table 1: The Computation Cost Comparison for One Round
Participation Rate
Fedprox (s)
SCAFFOLD (s)
FedPer (s)
SuPerFed (s)
FLARE-LA (s)
30%
74.39
105.9
46.18
99.63
48.16
60%
144.83
208.51
90.42
186.67
94.43
5.1.2
Computation Cost Comparison
The computation cost comparison for one round of FL is presented in Table ! showcasing FLARE-LA's computational efficiency across varying participation rates. All experiments were conducted experimental form featuring 8-core CPU, 14-core GPU, and 16GB of RAM. This setup ensures consistent. benchmarking across all evaluated FL methods.
participation rate of 30%. FLARE-LA achieves a computation time 48.16 seconds, significantly outperforming SCAFFOLD (105.9 seconds) and SuPerFed (99.63 seconds). While FedPer demonstrates a slightly faster computation time of 46.18 scconds, its slower convergence rate necessitates more training rounds to achieve comparable results, thereby increasing the overall computational burden. FLARE-LA'S superior balance between computational demands and model accuracy ensures efficient and timely model updates, even under challenging participation scenarios.
When the participation rate increases to 60%, FLARE-LA continues to excel with a computation time of 94.43 seconds, outperforming Fedprox (144.83 seconds) and SCAFFOLD (208.51 seconds) by substantial margins. Although FedPer achieves a comparable time of 90.42 seconds, FLARE-LA's faster convergence significantly reduces the total training cost, making it a more efficient and scalable solution for large-scale FL applications in financial markets.
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The experimental results demonstrate FLARE-LA's robustness and adaptability in addressing challenges such as fragmented datasets and irregular client participation. By achieving competitive computation times, FLARE-LA ensures privacy-preservin collaboration and timely model updates, addressing critical requirements for FL in docentralized financial environments. Its ability to provide reliable volatility prodictions is particularly valuable for ivfective risk management and investment decision-making dynamic financial markets.
5.2 Experiments with CIFARI0 and MNIST
To extend our evaluation beyond the financial domain, we conducted experiments using the CIFARIO and MNIST datasets, partitioned into 1000 clients. These datasets serve as benchmarks to demonstrate the generalizability and robustness of FLARE-LA in broader FL scenarios.
To simulate rcal-world challenges, we introduced artificial label noise into the training sets, employing two commonly used noise schemes: pairwise flipping (Han et al.] 2018) and symmetric flipping (Van Rooyen et al. 2015). The pairwise flipping scheme models scenarios where labels transition to semantically similar neighboring labels with noise ratio c, while retaining the correct label with a probability of - The symmetric flipping scheme assumes uniform mislabeling across all incorrect labels, distributing the noise ratio evenly among them, while preserving the correct label with a probability of 6
For both schemes, the test sets remain clean to ensure a fair and accurate evaluation of model performance. This setup allows us to rigorously assess MARRE-L''s ability handle noisy labels, which is a critical capabiliiy for FL. applications in dynamic and unpredictable environments. By addressing label noise effectively, FLARE-LA demonstrates its robustness and adaptability in diverse scenarios, further validating its utility across domains.
5.2.1 Convergence Analysis
We assessed the convergence performance of FLARE-LA compared to state-of-the-art FL methods using the CIFAR10 and MNIST datasets. These experiments evaluated FLARE-LA's ability handle FL challenges, such non-IID) data distributions, dynamic participation, and prolonged training, while maintaining robust convergence across various training rounds. Model performance was measured in terms of Mcan Loss, VaR95% and CVaR95%.
On the CIFAR1O dataset, as shown in Fig. ! FLARE-LA consistently outperformed the baselines, including FedRep, Ditto, and SuPerFed, across all training stages. At 300 training rounds as illustrated in Fig. 4(),, FLARE-LA achieved a Mean Loss of approximately 1.2, significantly lower than SuPerFed with 2.1, Ditto with 2.7, and FedRep with 4.3, while demonstrating superior tail-risk management with reduced VaR95% and CVaR95%. As training progressed to 400, 500, and 600 rounds shown in Figs. (d), FLARE-LA maintained its dominance, achieving final Mean Loss of 0.8, solidifying its position as the most robust and reliable method. Compared to the baselines, which struggled with slower convergence and less effective local adaptation, FLARE-LA demonstrated faster convergence and superior overall performance.
We also compare their performance on the MNIST dataset as shown in Fig. 5. FLARE-LA outperformed LG-FedAvg, PFedMe, and FedRop across all training rounds. As shown Fig. 5(a) with 300 rounds, FLARE-LA achieved a Mean Loss of 0.3, compared to 1.0 for LG-FedAvg and 0.8 for FedRep, while also demonstrating lower VaR95% and CVaR95% reflecting its superior risk management. As training progressed to 400 and 500 rounds as shown in Figs. (b) and 5(cc), FLARE-LA's 's advantage became more pronounced, achieving a Mcan Loss of 0.2 by 500 rounds, compared to 0.7 for LG-FedAvg and 0.6 for FedRep. Even as PFedMe slightly narrowed the gap, it failed to match FLARE-LA' 's efficiency in managing tail risks. By 600 training rounds as shown in Fig. d)d),, FLARE-LA achieved a Mean Loss of 0.1, significantly outperforming FedRep with 0.5 and LG-FedAvg with 0.6, while maintaining lower VaR95% and CVaR95%. These results shown in both Fig. A and Fig. 5 highlight FLARE-LA's scalability and adaptability, demonstrating faster convergence and superior performance across diverse FL settings.
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Figure 4: Convergence analysis of model performance across different federated training rounds using the CIFAR10 dataset.
5.2.2 Analysis of Label Noise
We evaluated the robustness of FLARE-LA under symmetric and pairwise label noise conditions using the CIFAR10 and MNIST datasets. These experiments tested FLARE-LA's A's ability to maintain strong performance under varying levels of label noise, comparing it to several baseline methods in terms of Mean Loss, VaR95%, and CVaR95%.
As shown Fig. 6 FLARE-LA consistently outperformed FedRep, Ditto, and SuPerFed across noise ratios ranging from 0.2 to 0.8 on the CIFAR10 dataset. a noise ratio of 0.2 as shown Fig. G(a), FLARF-LA achieved Mean Loss of approximately 1.5, significantly lower than SuPerFed with 2.3, Ditto with 2.0, and FedRep with 3.2. As noise levels increased to 0.4 and 0.6 illustrated in Figs. ((b and Gc). FLARE-LA maintained its competitive edge, with lower losses and risk metrics than the baselines. showcasing its resilience to intermediate noise levels. Even under extreme noise at 0.8 as shown in Fig. d). FLARE-LA demonstrated remarkable robustness, maintaining strong performance, while the baselines exhibited significant degradation, emphasizing FLARE-LA's superior adaptability.
As shown in Fig. 7 FLARE-LA demonstrated similar robustness against symmetric label noise, outperforming LG-FedAvg, Me, and FedRep across all noise levels on the MNIST dataset. At a noise ratio of 0.2, FLARE-LA effectively mitigated the adverse effects of noise, achieving superior predictive accuracy and risk management. As noise levels increased to 0.4, 0.6, and 0.8, FLARE-LA maintained substantial performance advantage, with significantly lower losses and risk metrics, highlighting its scalability and effectiveness in handling challenging scenarios with noisy labels.
Under pairwise label noise conditions on the CIFARIO dataset, as shown Fig. FLARE-LA continued demonstrate resilience. At noise ratio of 0.2 illustrated in Fig. Bla), FLARE-LA achieved a Mean Loss 2.2, outperforming Su SuPerFed with 2.5, Ditto with 2.5, and FedRep with 3.0, along with significantly lower VaR95%
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Figure 5: Convergence analysis of model performance across different federated training rounds using the MNIST dataset.
and CVaR95%. At a higher noise ratio of 0.6 as shown in Fig. B(). FLARE-LA maintained its superiority, achieving CVaR95% of 3.5, compared to 3.8 for SuPerFed, 4.1 for Ditto, and 6.6 for FedRepp These results emphasize the efficacy of FLARE-LA's probabilistic local adaptation mechanism in mitigating noise effects and leveraging the structure of local data distributions. FLARF-LA consistently demonstrated superior resilience to label noise across both symmetric and pairwise sccnarios, outperforming baseline methods in all metrics. Its robust local adapiation mechanism ensures stable and accurale performance under challenging noisy label conditions, making it reliable solution in woaldworld settings.
5.2.3 Analysis of Local Data Heterogeneity
We analyzed the impact of local data heterogeneity on the performance of FLARE-LA compared to baseline methods using the CIFAR10 and MAIST datasets. Data heterogeneity was controlled through the Dirichlet distribution's concentration parameter a, where smaller a values indicate higher heterogeneity, i.e., non-lID data, and larger a values correspond to more homogencous distributions, i.e., near- data.
The performance compared on the CIFAR10 dataset is shown in Fig. where FLARE-LA consistently outperformed FedRep, Ditto, and SuPerFed across both heterogeneity settings. Under the moderately heterogeneous scenario with 10.0 shown in Fig. a), FLARE-LA achieved Mean Loss of approximately 1.2, significantly lower than SuPerFed, Ditto, and FedRep, all of which reported values above 2.5. FLARE- LA also achieved lower VaR95% and CVaR95%, demonstrating its superior ability to handle diverse data distributions and miligate tail risks. In the near-IID scenario with = 1000.0 as shown in Fig. 9(b), all methods improved due to reduced heterogeneity, but FLARE-LA maintained its performance advantage, achieving a Mean Loss of 1.5, compared 2.6 for SuPerFed and Ditto, and 3.0 for FedRop. These results highlight FLARE-LA's adaptability across varying levels of heterogeneity, consistently delivering robust performance.
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When compare their performance on the MNIST dataset as shown in Fig. 10 FLARE-LA demonstrated similar dominance over LG-FedAvg, PFodMe, and FedRep. In the highly heterogencous scenario with a 10.0 illustrated in Fig. 10fa), FLARE-LA achieved the lowest Mean Loss, significantly outperforming the baselines while also maintaining lower VaR95% and CVaR95% reflecting its superior risk management capabilities. The baselines struggled to generalize effectively under these conditions, with LG-FedAvg and FedRep showing higher losses and slower convergence. Although PFedMe performed competitively, it lagged behind FLARE-LA in accuracy and convergence speed. In the near-IID) scenario with a = 1000.0 as shown Fig. 10.b), the performance gap narrowed due to the more uniform data distribution. However, FLARE-LA maintained advantage, achieving the best results in terms of Mean Loss and risk metrics, while also demonstrating computational efficiency by requiring fewer training rounds to achieve convergence. These results shown in both Fig. 9 and Fig. 10 underscore FLARE-LA's ability to adapt to varying levels of data heterogencity, consistently out purperooriing baseline methods across diverse FL scenarios. Its probabilistic local adaptation mechanism and Taylor- based linearization effectively balance global and local contributions, ensuring robust and efficient performance in both non-IID and IID-like settings.
5.2.4 Analysis of Client Participation Rates
Then, analyzed the impact of varying client participation rates the performance of FLARE-LA compared to baseline methods using the CIFAR10 and MNIST datasets. These experiments simulate dynamic participation scenarios where only a small subset of clients is involved in each training round, evaluating FLARE-LA's robustness under extreme constraints.
On the CIFAR10 dataset, as shown in Fig. 11 two participation rates were tested, i.c., 1% and 2%. At the 1% participation rate as illustrated in Fig. 11 a). FLARE-LA demonstrated exceptional robustness and stability, significantly outperforming FedRep, Ditto, and SuPerFed in both convergence speed and predictive
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accuracy. The probabilistic local adaptation mechanism in FLARE-LA ensures effective personalization while preserving generalization, even with limited client engagement. At the 2% participation rate as shown in Fig. bb), all methods improved due to increased client involvement, but FLARE-LA maintained its advantage with faster convergence and superior performance metrics.
As shown in Fig. 12 four participation rates were tested, i.e., 0.5%, 1%, 1.5%, and 2.0% on the MNIST datasct. At the lowest participation rate of 0.5% as illustrated Fig. 12 a), FLARE-LA achieved a mean loss of approximately 0.2, significantly outperforming LG-FodAvg with 1.4, PFedMe with 0.6, and FedRep with 0.4. Its risk metrics, VaR95% and CVaR95%, consistently lower than those of the baselines, reflecting superior
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robustness. As participation rates increased to 1%, 1.5%, and 2.0% shown Figs. 12 (c), FLARE-LA maintained its performance lead, achieving the lowest mean loss and risk metrics across all settings. At the highest participation rate of 2.0%, FLARE-LA effectively leveraged increased client engagement achieve the most favorable results, further underscoring its scalability and adaptability. The above results highlight FLARF-LA's 's ability to maintain robustness and generalization across dynamic participation scenarios, consistently outperforming baseline methods. Its efficient local adaptation mechanism ensures stable and accurate performance, even under extreme constraints or varying client participation rates.
5.3 Performance Comparison with Additional Local Updates
further highlight the superiority of FLARE-LA adaptive local updating, beyond the innovative two-stage framework, we conducted comparative analysis by equipping existing bascline methods with additional local updates following the FL training phase. Specifically, we modified LG-FedAvg (Liang al.) 2020), APFL (Deng al. 2020), FedRep (Collins et al. 2,021), FedPer (Arivazhagan et al.) 2019 Ditto (Li et al. 2021), and SuPerFed (Hahn et al. 2022) to include this adjustment, resulting in the variants L.G-FedAvg-LEE APFL-LE, FedRep-LE, FedPer- LE,
LE, Ditto-LE, and SuPerFed-LE.
The purpose this comparison to demonstrate that FLARE-LA's design offers significant advantages over these locally enhanced basclines, even when they incorporate additional efforts improve local performance. Unlike these approaches, FLARE-LA is explicitly designed address the trade-off between global knowledge preservation and localized adaptation, ensuring both robustness and generalization across heterogeneous
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Figure 12: Impact of varying client participation rates on model performance during federated training with the MNIST dataset.
datasels. The experiments, conducted multiple datasets, show that FLARE-LA achieves superior performance without sacrificing the benefits of the collaborative FL training.
5.3.1 Performance Comparison on the Volatility Prediction Dataset
We evaluated the performance of FLARE-LA against locally enhanced basclines LG-FodAvg-LE, Ditto-LE, APFT.-LE, and FedPer-T.F using the volatility prediction dataset. The experiments involved 20% of 10,000
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Figure 13: Performance comparison of FLARE-LA against FL basclines enhanced with increasing numbers of local updates after the FL training procedure the volatility prediction data set..
trading platforms, with 10 rounds of FL training and 10 local epochs round. We varied the number of additional local updates applicd to the basclines, increasing from 5 in Fig. 13 aa to 40 in Fig. 13 sdd..
The results demonstrate clear and consistent advantage of FLARE-LA over the locally enhanced baselines, which achieves a significantly lower mean loss and risk metrics compared all baselines. As shown in Fig. 13 a), FLARE-LA's mcan loss is an order of magnitude lower than the best-performing bascline, FedPer- LE, and its risk values, i.e., VaR95% and CVaR95% FLARF-LA's 's ability to effectively balance global knowledge and localized adaptation, avoiding the overfitting and drift that additional local updates can introduce.
reduced by more than 95%. These results highlight
As the number of local updates increases from 5 to 40, the baselines show an expected improvement personalization, reflected in slightly lower mean losses. However, this comes at a significant cost generalization and robustness. For instance, while baselines like APFL-LE and FedPer-LE demonstrate marginal improvements in their mean loss with more local updates, their VaR95% and CVaR95% values degrade, indicating higher risk of poor performance extreme cases. These results confirm the effectiveness of FLARE-LA' two-stage framework. While the locally enhanced basclines attempt to improve personalization through additional updates, they cannol balance between personalization, robustness, and generalization. The proposed saylor-based linearization and probabilistic local adaptation mechanisms in FLARE-LA ensure that local updates enhance model performance without deviating from the collaborative objectives of federated training efforts. Therefore, FLARE-LA achieves superior performance both mean loss and risk metrics, making it reliable and efficient solution for volatility prediction tasks.
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Figure 14: Performance comparison of FLARE-LA against FL basclines enhanced with increasing numbers of local updates after the FL training procedure the CIFAR10 dataset.
5.3.2 Performance Comparison on the CIFAR10 and MNIST Datasets
Furthermore, also evaluated the performance of FLARE-LA against the extra local updating euhanced baselines using the CIFAR10 and MNIST datasets. Both experiments involved 600 federated training rounds, each with 5 local epochs, followed by additional local updates for the baselines, ranging from 5 to 50, Across all settings, FLARE-LA consistently outperformed the basclines in terms of both mean test loss and risk metrics, i.., VaR95% and CVaR95%, demonstrating its robustness and generalization capabilities.
As shown in Fig. 14 FLARE-LA achieved mean test loss of 1.2928 and CVaR95% of 2.6734, remaining stable across all local update settings the CIFAR10 dataset. In contrast, the baselines showed slight improvements in mean test loss with additional local updates but exhibited significant inconsistencies in risk metrics. Even with 50 updates, FedRep-LE achieved a mean loss of 1.9511 but had CVaR95% of 4.1711, highlighting its lack of robustness compared to FLARE-LA.
On the MNIST dataset, as shown Fig. 15 FLARE-LA demonstrated even greater stability, achieving mcan test loss of 0.0427 and a CVaR95% of 0.4679, irrespective of the number of local updates applied to the baselines. By contrast, the baselines showed considerable variability, with FedRep-LE achieving its best mean test loss of 0.1745 50 updates, but its CVaR95% remained high at 1.2898. Similar trends were observed with L.G-FeAAvg-LEF and Ditto-LE, which failed to consistently balance local personalization and global generalization, leading to unstable risk metrics.
The results as shown in both Fig. 14 and Fig. [5 demonstrate that FLARE-LA effectively integrates global and local knowledge, maintaining superior performance without requiring additional local updates. The stability and robustness of FLARE-LA, evidenced by consistently low risk metrics across both datasets, highlight. the effectiveness of its Taylor-based linearization and probabilistic local adaptation mechanisms. Tn
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Figure 16: Performance comparison of FLARE-LA against FL basclines enhanced with increasing numbers of local updates after the FL training procedure the MNIST dataset.
contrast, the baselines struggled with significant variability and reduced generalization, even when enhanced with increasing local updates.
6 Conclusions and Discussions
This work proposed FLARE-LA, novel framework addressing the challenges of FL in diverse and dynamic environments, particularly financial markets. By integrating Taylor-based linearization for efficient local adaptation with a probabilistic mechanism leveraging the Jacobian matrix, FLARE-LA achieves precise optimization, robust performance, and interprotable uncertainty quantification. Extensive experiments demonstrated its superiority over state-of-the-art baselines, achieving higher accuracy, faster convergence, and resilience to label noise and dynamic participation. Tn financial applications, FLARR--LA excelled metrics like mean loss, VaR95%, and CVaR95%, underscoring its suitability for high-stakes, heterogeneous environments. With its ability to adapt global models to local distributions, handle fragmented datasets, and ensure compulational efficiency, FLARR--A offers scalable and versatile solution for FT, challenges. Future directions include extending the framework to domains like healthcare and loT,, integrating advanced optimization techniques, positioning FLARE-LA as a foundation for advancing FL innovations.
Acknowledgments
This work was supported in part by the Natural Sciences and Engincering Research Council of Canada and Compute Canada.
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Reviews:
[{"review_id": "Fy3O7ZX9jg", "reviewer": "Reviewer_fzmC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**[Strengths]**   \n1. The paper studies the critical challenge of dataset heterogeneity in the prediction of financial volatility, a significant barrier in financial analytics.   \n2. The technique of linearizing neural networks for local adaptation within a federated learning framework represents a reasonable innovation. This approach allows for more precise and efficient prediction of local datasets, improving overall prediction performance.  \n3. The experimental results demonstrate that FLELA achieves efficient prediction compared to other baselines.   \n  \n**[Weakness]**   \n1. The paper does not explore or compare against simpler baseline such as direct fine-tuning of parameter w on local datasets using standard techniques like SGD (by minimizing the prediction error). This omission raises questions about whether the proposed complex method is necessary or optimal.  \n2. The paper did not discuss several related literature that focus on adaptation strategies in federated learning. Recent literature [1,2,3,4] offers insights into personalized model training within federated frameworks, which are directly relevant to this work.  \n3. It lacks a comparative analysis with methods specifically designed for enhanced local adaptation within federated learning contexts, such as [3,4]. This comparison is essential to fully evaluate the effectiveness of FLELA in scenarios that demand strong local performance.  \n\n[1]  Meng, Lei, et al. \"Improving Global Generalization and Local Personalization for Federated Learning.\" IEEE Transactions on Neural Networks and Learning Systems (2024).  \n[2] Tan, Alysa Ziying, et al. \"Towards personalized federated learning.\" IEEE transactions on neural networks and learning systems 34.12 (2022): 9587-9603.  \n[3] Chen, Daoyuan, et al. \"Efficient personalized federated learning via sparse model-adaptation.\" International Conference on Machine Learning. PMLR, 2023.  \n[4] Yu, Tao, Eugene Bagdasaryan, and Vitaly Shmatikov. \"Salvaging federated learning by local adaptation.\" arXiv preprint arXiv:2002.04758 (2020).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies the problem of predicting realized volatility in financial markets using federated learning.  This paper introduces an approach called Federated Learning with Efficient Local Adaptation (FLELA).  By incorporating localized linearization techniques and probabilistic frameworks, FLELA is designed to enhance the fit of a global model to heterogeneous data across different trading platforms. The authors demonstrate that FLELA achieves accurate and efficient predictions.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The authors should discuss why direct fine-tuning on local datasets was not considered as a baseline approach. Including an experimental comparison with this strategy would help to validate the necessity and effectiveness of the proposed method.  \n2. It is crucial to discuss relevant literature on adaptation and personalization within federated learning, such as references [1, 2, 3, 4].  \n3. The paper should compare FLELA against other federated learning methods that focus on local adaptation, such as those cited in references [3, 4].", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "hVTBysLpyN", "reviewer": "Reviewer_4V13", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n1. The authors realize the property of financial markets that is heterogeneous and dynamic in FL.\n2. The proposed algorithm, FLELA, uses a local adaptation method to fit such a dynamic system well.\n3. The empirical results demonstrate that FLELA outperforms existing FL algorithms in performance (mean loss, VaR, and CVaR) and computation time.\n\n**Weaknesses:**\n1. The paper only considers the financial field. The properties of heterogeneity and dynamic participation are common in FL, not just in the financial case. Why not consider a general case and test this algorithm on more datasets/benchmarks?\n2. The experiment is on only one dataset and is a bit limited.\n3. The chosen baselines don't include recent methods. FedProx is proposed in 2020, and SCAFFOLD and FedPer in 2019.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes Federated Learning with Efficient Local Adaptation (FLELA) to improve realized volatility prediction in decentralized financial markets. Given the data heterogeneity across trading platforms and dynamic participation in FL, the authors developed a novel FL approach combining global model training with locally adaptive updates. FLELA leverages probabilistic frameworks and linearized local adaptations to optimize model predictions for each trading platform. The experimental results show that FLELA outperforms some existing FL methods in both performance and computation time.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Explain why FLELA only focuses on the financial market field.\n2. Please test FLELA and other baselines on other datasets.\n3. Add more recent personalized and local adaptation FL methods as baselines.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "8i43VOSwy3", "reviewer": "Reviewer_ciru", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper focus on improving local performance for Federated learning. To be honest, I am not an expertise in this area, but I will try my best to help you. It use linearization on the optimal solution in FL part and adapt it into local model, so can easily speed up on time. There are several benefit to do it: Linearization around the optimal solution in FL part is quite easy; we only need to do it once for each local model; it can help improve performance much if the local distribution is different from global. But the way seems a little bit naive to me -- can we just fine tune the model on different dataset to get a better performance for each dataset, or explore the distribution of different local dataset during the global gradient decent step? I am also concerning about whether linear-model is sufficient for a large model -- now the experiment only takes about 40 neurons and that seems to be a small network.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper focus on a specific situation of financial market, which may have isolated datasets. Federated learning (FL) can optimize over all dataset, but may lack performance in some of the local dataset. This paper present Federated Learning with Efficient Local Adaptation (FLELA), which can make a local adaptation step after global optimization. The method first do global optimization based on square loss, then linearize the model at every local dataset and chose the optimal local minimum for every local dataset, using maximum likelihood loss. The algorithm performs good on various dataset and method.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would like to request experiment on comparison for fine-tuning the model on each dataset. And also some case that linear function cannot capture the local model efficiently. Also for the paper writing I would suggest to make the figure smaller and dense(not to put it in one page).", "broader_impact_concerns": "This paper’s focus on improving local adaptation in federated learning has potential benefits for privacy and accuracy in financial markets, where data heterogeneity is prevalent. It can significantly improve the local performance of FL, and help speed up the time comparing to alternative methods. However, relying on linear approximations may limit its effectiveness in capturing complex local patterns, particularly in diverse datasets.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents an empirical study on federated learning, with an emphasis on financial applications. A new algorithm is proposed which consists of two stages: a global training stage utilizing the data across all nodes to obtain a shared model parameter, and a local adaptation stage that fine-tunes this globally-trained model using local data. In particular, instead of using SGD for fine-tuning, the local adaptation stage linearizes the globally-trained model before applying the quadratic loss, which converts the conventional SGD-based fine-tuning to least square. Experiments are performed to demonstrate the good performance of this approach in a variety of settings. 

On the positive side, all reviewers found the paper well-motivated. A notable strength is that the proposed algorithm is tested in a fairly large amount of settings, against plenty of baselines. Reviewers also found the related works to be sufficiently discussed. 

However, currently there are two major issues obstructing an unconditioned acceptance decision. 
- Despite the variety of tested baselines, an important one is missing: using SGD in the local adaptation stage (initialized at the model parameter obtained from global training). Comparing to this baseline is crucial for justifying the main algorithmic components proposed by this paper, namely linearization and least square. This is different from the Individual Train baseline considered in Section 5.1.1. Admittedly, it's possible that such a SGD-fine-tuning baseline is already considered in this paper but all reviewers and I missed it. In that case, the authors might add a note, or certain pointers in the paper to make it clearer. 
- Some theoretical analysis is presented in Appendix A, but after a skimming I found it unclear and sloppy. A requested change is to reorganize this section into a theorem-proof format, with all assumptions clearly stated in or before the theorem. Then, certain components of the proof could be separately presented as lemmas, which will improve its readability. I also encourage the authors to reuse existing theoretical results, instead of proving everything from scratch. Is the analysis following any existing analytical template? If so then this needs to be made clear. 
- Rather than revising Appendix A, an alternative option is to remove it altogether. After all this is primarily an empirical paper, and it is fine to present an algorithm with very good empirical performance but no proofs. 

Summarizing the above, I would recommend acceptance conditioned on the above two issues being properly addressed. 

Other suggestions and minor issues

- The authors are also suggested to polish the writing in the revision. Just to name a few issues:

1. The inline citations should be (author, 2024) rather than author (2024), unless the name of the author is a part of the sentence. 

2. Showing the loss values in the abstract has no tangible meaning without introducing the experiment setup, therefore it is suggested to remove those. 

3. Another suggestion is to add a pseudocode of the proposed algorithm, and clearly introduce the required hyperparameters. 

4. Overall, the authors may also consider being more concise and crisp in the writing style. Currently the paper seems unnecessarily lengthy. Certain parts of the paper, e.g., Section 4.2, are removable. 

- On the technical side, I encourage the authors to consider different settings of conditioning, i.e., the eigenvalues of the matrix $JJ^T$. The degenerate case ($JJ^T$ is not invertible) needs to be addressed in the presentation of the algorithm. 

To finally summarize, the reviewers and I find more strengths than weaknesses in this paper, therefore we would be glad to support its acceptance to TMLR, as long as the above two major issues are properly addressed in the final revision. Here is a gentle reminder of TMLR's acceptance standard: acceptance is conditioned on the claims being supported by sufficient evidences. To meet this standard, one could either add evidences to support existing claims in a paper, or adjust the claims and clearly state the limitations. 

I look forward to checking the final version of this paper.
Decision: Accept (poster)