Online-Within-Online Fair Multi-Task Learning

• The paper introduces a two-timescale algorithm where an outer loop learns optimal model initialization while an inner loop dynamically adapts fairness through user-priority updates.
• It reformulates fairness via a convex-conjugate approach, employing a primal-dual framework to balance utility across heterogeneous users over sequential tasks.
• Empirical results demonstrate 20–40% fairness gains and 10–30% utility improvements, highlighting its potential for efficient resource allocation in AI-enabled Radio Access Networks.

Online-Within-Online Fair Multi-Task Learning (OWO-FMTL) denotes a fairness-aware, adaptive multi-task learning framework with two nested online processes: an outer online loop that learns a good model initialization across rounds, and an inner online loop that adapts a shared model within each round while dynamically rebalancing user priorities to enforce fairness. In the AI-RAN formulation, OWO-FMTL serves heterogeneous users with time-varying learning tasks over shared edge resources, quantifies equity through generalized $\alpha$-fairness, and evaluates performance through round-average fairness regret (RAF regret) against the hindsight fairest model for each round (Raptis et al., 9 Feb 2026).

1. Problem setting and target criterion

The canonical OWO-FMTL setting introduced for AI-enabled Radio Access Networks operates on two time scales: rounds $t=1,\dots,T$ and slots $i=1,\dots,m$. At each round $t$, every user $k\in\mathcal K$ has a task $(k,t)$, and each task consists of $m$ sequential jobs, one per slot. The system uses a shared model hosted at the edge or RAN rather than one dedicated model per user, which makes fairness across heterogeneous users a first-class optimization target rather than a byproduct of independent per-user training (Raptis et al., 9 Feb 2026).

The shared model is represented by an initialization $x_t \in \mathcal X \subset \mathbb R^d$ at round $t$, and by slot-wise parameters $\theta_{ti} \in \Theta \subset \mathbb R^d$ within the round. Each user-slot-job $t=1,\dots,T$0 has a concave utility

$t=1,\dots,T$1

where $t=1,\dots,T$2 is the task loss and $t=1,\dots,T$3 is a high-loss reference level. Higher utility therefore means lower loss or better service.

Fairness is not imposed slotwise. Instead, the framework maximizes generalized $t=1,\dots,T$4-fairness over the round-averaged utility vector: $t=1,\dots,T$5 with the round benchmark

$t=1,\dots,T$6

The performance criterion is RAF regret: $t=1,\dots,T$7 The design target is $t=1,\dots,T$8, meaning asymptotic equivalence to the hindsight-optimal fair model at every round (Raptis et al., 9 Feb 2026).

2. Nested online architecture and primal-dual mechanics

The defining structural feature of OWO-FMTL is the nesting of an inner online fair adaptation loop inside an outer online initialization-learning loop. The inner loop updates both the shared model $t=1,\dots,T$9 and a vector of user-priority variables $i=1,\dots,m$0. The outer loop updates the next round’s initialization $i=1,\dots,m$1 from the fairness-adaptation performance observed in the current round (Raptis et al., 9 Feb 2026).

A key technical step is a convex-conjugate reformulation of fairness. The framework introduces

$i=1,\dots,m$2

with

$i=1,\dots,m$3

This yields the saddle representation

$i=1,\dots,m$4

Within a round, the shared model is updated by Online Gradient Ascent from $i=1,\dots,m$5: $i=1,\dots,m$6 where

$i=1,\dots,m$7

Thus the server aggregates user gradients through dynamic fairness weights $i=1,\dots,m$8, not through static averaging.

The dual variables are updated by strongly convex OGD: $i=1,\dots,m$9 with

$t$0

These weights act as learned user priorities: if a user’s utility lags, the dual dynamics raise its influence in subsequent shared-model updates.

Across rounds, the outer loop optimizes the initialization-sensitive bound

$t$1

via OGD: $t$2 This makes OWO-FMTL a two-timescale primal-dual meta-learner: inner online fairness balancing within rounds, outer online transfer across rounds (Raptis et al., 9 Feb 2026).

3. Fairness semantics and theoretical guarantees

The fairness parameter $t$3 controls the efficiency-equity trade-off. The intended interpretation is standard: $t$4 gives utilitarian or sum-utility behavior, $t$5 gives proportional fairness with $t$6, and larger $t$7 increasingly emphasizes low-performing users and approaches max-min-like behavior. In this framework, the fairness object is the round-averaged utility vector $t$8, not raw per-slot loss (Raptis et al., 9 Feb 2026).

The main theoretical result is Theorem 1: $t$9 The bound decomposes into inner-loop primal adaptation cost, outer-loop initialization error, and dual fairness-balancing cost. The leading term is $k\in\mathcal K$0, the $k\in\mathcal K$1 term is lower order, and the paper interprets the result as vanishing fairness regret and diminishing performance disparity over time (Raptis et al., 9 Feb 2026).

The outer loop achieves

$k\in\mathcal K$2

under $k\in\mathcal K$3, while the dual inner loop admits

$k\in\mathcal K$4

Relative to single-round learning (SRL), the asymptotic scaling improves from $k\in\mathcal K$5 to $k\in\mathcal K$6, giving a multiplicative gain of $k\in\mathcal K$7 when the fair solutions across rounds are clustered (Raptis et al., 9 Feb 2026).

Several scope conditions are explicit. The formal guarantee is convex; deep learning results are empirical. The analysis assumes bounded utilities, bounded gradients, compact feasible sets, and a “standard perturbation-restrictiveness assumption” so that the appendix residual term

$k\in\mathcal K$8

is $k\in\mathcal K$9. The theory also assumes knowledge of $(k,t)$0, although the authors note that guessing mechanisms from prior work can be adapted (Raptis et al., 9 Feb 2026).

4. Algorithmic pipeline, systems realization, and empirical behavior

Algorithm 1 takes as inputs the feasible initialization set $(k,t)$1, the model parameter set $(k,t)$2, the fairness parameter $(k,t)$3, and the utility range $(k,t)$4. It derives the dual domain

$(k,t)$5

initializes $(k,t)$6, and then for each round sets $(k,t)$7, chooses $(k,t)$8, runs the inner slotwise primal-dual updates, computes $(k,t)$9, and performs the outer update $m$0 (Raptis et al., 9 Feb 2026).

The method is explicitly lightweight. Per slot it performs one first-order primal update and one first-order dual update; per round it performs one outer OGD step. The paper emphasizes that it does not need to store a separate gradient history for each user and that, for deep learning, the weighted gradient can be obtained from a single backpropagation on a weighted aggregated loss. This is one reason the framework is positioned as suitable for edge deployment (Raptis et al., 9 Feb 2026).

Empirical evaluation covers both convex and deep settings, under stochastic and adversarial environments. In the convex experiment, a polynomial-kernel linear regression model is used with $m$1, $m$2, and $m$3; the reported result is that fairness regret decreases sublinearly with $m$4, consistent with the theory. In the deep experiment, a two-user Rainbow MNIST setup uses a LeNet CNN, $m$5 rounds, $m$6, utility bounds $m$7, $m$8, batch size $m$9 per user, and $x_t \in \mathcal X \subset \mathbb R^d$0 slots per task. Because exact $x_t \in \mathcal X \subset \mathbb R^d$1 is impractical in the nonconvex case, the paper evaluates approximations named LAST and AVG (Raptis et al., 9 Feb 2026).

Against constant weighting schemes and SRL, OWO-FMTL, especially LAST, achieves the best balance between fairness and user utilities across rounds. The reported gains are approximately 20–40% higher fairness and approximately 10–30% higher utilities across users, especially under adversarial conditions. SRL stays near random-guess performance because the in-round horizon is too short, while OWO-FMTL steadily lowers test loss over rounds, which experimentally isolates the benefit of the outer loop (Raptis et al., 9 Feb 2026).

5. Intellectual antecedents and neighboring formulations

The most direct non-fair antecedent is Coordinated Online Learning (CoOL), which studies asynchronous online multi-task learning with task-specific online learners coupled through a convex structural constraint set and periodically coordinated by a weighted projection

$x_t \in \mathcal X \subset \mathbb R^d$2

The paper does not formulate fairness, but it states that if fairness can be encoded as a convex constraint on the joint parameter vector, then one can define

$x_t \in \mathcal X \subset \mathbb R^d$3

and preserve the projection-based architecture. This makes CoOL a projection-based precursor for OWO-FMTL, especially when fairness surrogates are convex in parameter space (Hirnschall et al., 2017).

A second precursor is FFML, a fairness-aware online meta-learning method that learns a shared primal-dual meta-prior $x_t \in \mathcal X \subset \mathbb R^d$4 across sequential tasks and uses fairness constraints based on Decision Boundary Covariance. FFML gives $x_t \in \mathcal X \subset \mathbb R^d$5 loss regret and $x_t \in \mathcal X \subset \mathbb R^d$6 cumulative fairness violation, but its inner loop is support-set adaptation rather than a true within-task online sequence. It therefore overlaps with OWO-FMTL as an outer-online, inner-adaptive fairness-constrained framework, while remaining only a partial instance of the stronger online-within-online formulation (Zhao et al., 2021).

A third neighboring line is L2T-FMT, which recasts fairness-aware multi-task learning as dynamic per-task objective selection. A teacher DQN chooses, at each epoch and for each task, whether the student should optimize the accuracy loss $x_t \in \mathcal X \subset \mathbb R^d$7 or the fairness loss $x_t \in \mathcal X \subset \mathbb R^d$8. This is a nested adaptive system and a strong training-time analogue of OWO structure, but it is explicitly described as an offline-training method with an internal online adaptive decision process rather than a formal online learning algorithm (Roy et al., 2022).

Other fair MTL frameworks supply reusable fairness layers without providing OWO structure. FairGrad formulates MTL optimization as $x_t \in \mathcal X \subset \mathbb R^d$9-fair utility maximization over task-wise loss decrease rates $t$0, thereby importing generalized $t$1-fairness into gradient allocation (Ban et al., 2024). EMTL defines equitable MTL through the task-level ratio

$t$2

regularizes the variance of this relative contribution, and reports a 5% watch-time improvement in an online A/B test for multi-task recommendation (Yuan et al., 2023). FairMT addresses heterogeneous task types and incomplete supervision through asymmetric task-specific fairness violations, AHFDA aggregation, and a primal-dual constrained optimizer, but it is an offline framework rather than an online one (Hu et al., 29 Nov 2025).

6. Scope, misconceptions, and limitations

A common misconception is to treat every nested or adaptive multi-task learner as OWO-FMTL. The literature is more structured. CoOL is online multi-task coordination without fairness (Hirnschall et al., 2017). FFML is fair online meta-learning with cumulative fairness control across tasks, but not truly online within task (Zhao et al., 2021). L2T-FMT is internally online-adaptive but trained offline and without regret analysis (Roy et al., 2022). OWO-FMTL, in the explicit AI-RAN sense, is distinguished by round-wise outer online adaptation, slot-wise inner online primal-dual fairness balancing, and RAF regret against the hindsight fairest model (Raptis et al., 9 Feb 2026).

Another misconception is to assume that “fairness” has a fixed meaning across fair MTL. In OWO-FMTL for AI-RANs, fairness is generalized $t$3-fairness over round-averaged user utilities (Raptis et al., 9 Feb 2026). FFML uses DBC-based long-term cumulative fairness constraints related to demographic parity (Zhao et al., 2021). L2T-FMT optimizes a robust log-loss surrogate aligned with equalized odds (Roy et al., 2022). EMTL addresses fairness across tasks, not protected groups (Yuan et al., 2023). FairMT uses asymmetric per-task fairness violations for classification, detection, and regression under incomplete supervision (Hu et al., 29 Nov 2025). These are not interchangeable objectives.

The explicit limitations of the AI-RAN OWO-FMTL formulation are also clear. The formal theory is convex, while deep-learning evidence is empirical. The outer-loop update depends on the hindsight fairest model $t$4, which must be approximated in nonconvex settings. The analysis requires bounded utilities and gradients, and vanishing fairness regret depends on the residual perturbation assumption noted in the appendix (Raptis et al., 9 Feb 2026). A plausible implication is that broader OWO-FMTL deployments will need alternative outer-loop surrogates, richer fairness notions, and stronger theory under nonconvexity, distribution shift, or heterogeneous output spaces—directions that adjacent work has begun to articulate but not yet unify (Hu et al., 29 Nov 2025).