FAIR-k: Freshness & Magnitude Aware Top-k

• FAIR-k is a selection algorithm that combines magnitude-based and age-of-update principles to ensure timely and important updates in high-dimensional data scenarios.
• It employs a dual budget approach with explicit Markov chain staleness analysis, effectively balancing convergence speed with fairness and resource constraints.
• Empirical results demonstrate that FAIR-k accelerates model convergence and reduces energy usage by integrating top-k and round-robin strategies under communication limits.

Freshness-mAgnItude awaRe top-k (FAIR-k) refers to a class of selection algorithms designed to optimize both the “freshness” (timeliness of update) and “importance” (activity or magnitude) when disseminating or collecting the k most critical entries from a high-dimensional data source under resource and/or communication constraints. Two canonical domains for FAIR-k are: (i) over-the-air federated learning (OAC-FL) under channel constraints, and (ii) time-critical, energy-efficient top-k sensing in networked systems. FAIR-k balances the repeated selection of high-magnitude entities (as in Top-k) with periodic forced updates of stale or underupdated entries (as in Age-of-Update or Round-Robin), combining their strengths for robust performance across heterogeneous, resource-limited settings.

1. Mathematical Definition and Selection Rule

In an archetypal OAC-FL setting, the FAIR-k selection for a $d$-dimensional parameter vector $g_t \in \mathbb{R}^d$ at round $t$ proceeds as follows. Let $A_t \in \mathbb{N}^d$ be the Age-of-Update (AoU) vector tracking the number of rounds since each coordinate was last updated.

Selection parameters $k_M$ (magnitude-budget) and $k_A = k - k_M$ (age-budget), with $k \ll d$, define the partition:

• $v_t^M = \mathrm{Top}(g_t, k_M)$, the binary mask of the $k_M$ coordinates with largest $|g_{t,i}|$.
• $v_t^A = \mathrm{Top}(A_t \circ (1 - v_t^M), k_A)$, the $k_A$ stalest coordinates outside the magnitude-selected set.
• The next-round mask: $S_{t+1} = v_t^M + v_t^A$, $\|S_{t+1}\|_1 = k$.

This yields, in limit regimes, pure Top-k ($k_M = k$) or pure AoU/Round-Robin ($k_A = k$) as special cases (Zhu et al., 22 Dec 2025).

In content-based wake-up sensor networks for top-k retrieval under deadline constraints, FAIR-k refers to threshold-type wake-up rules that activate nodes likely to possess fresh, extremal readings while minimizing energy, tunable via a transmission threshold and lead time before deadline (Shiraishi et al., 2024).

2. Detailed Algorithmic Flow

The OAC-FL FAIR-k update cycle operates as follows (Zhu et al., 22 Dec 2025):

1. Initialization:
   • $w_0 \in \mathbb{R}^d$ (global model), $A_0 = 0$ (all fresh), $S_0 = 1$ (update all).
   • Partition $k = k_M + k_A$.
2. At round $t$:
   • Server broadcasts $(w_t, S_t)$ to $N$ clients.
   • Each client $n$ performs $H$ steps of local SGD, yielding local gradient $\Delta_{n,t}$.
   • Clients sparsify: $g_{n,t} = S_t \circ \Delta_{n,t}$, transmit $k$ coordinates over $k$ orthogonal waveforms.
   • Server receives noisy sum: $$\check g_t = \frac{1}{N} \sum h_{n,t} \check g_{n,t} + \xi_t$$.
   • The $k$-dimensional aggregation is expanded to full $d$ by zero insertion:

   $$g_t = \frac{1}{N} \sum h_{n,t} (S_t \circ \Delta_{n,t}) + (1 - S_t)\circ g_{t-1} + \tilde \xi_t$$

• Model update: $w_{t+1} = w_t - \eta g_t$.
• AoU update: $A_{t+1} = (A_t + 1) \circ (1 - S_t)$.
• Next selection mask: $$S_{t+1} = \mathrm{Top}(g_t, k_M) + \mathrm{Top}(A_t \circ (1 - \mathrm{Top}(g_t, k_M)), k_A)$$.

For networked sensor FAIR-k, threshold and timing are selected to minimize expected energy subject to a target top-k Query Age of Information (k-QAoI), via grid-search over threshold $V_{th}$ and lead time $\zeta$ to find Pareto optimal energy-freshness trade-offs (Shiraishi et al., 2024).

Step	OAC-FL FAIR-k (Zhu et al., 22 Dec 2025)	IoT Sensing FAIR-k (Shiraishi et al., 2024)
Key variable	$(w_t, A_t, S_t, g_t)$	$(V_{th}, \zeta, \text{k-QAoI})$
Resource constraint	$k \ll d$ waveforms (spectrum)	Sensor energy, channel usage

3. Markov Chain Staleness Analysis

The evolution of parameter staleness under FAIR-k is captured by a Markov chain on the AoU of coordinates (Zhu et al., 22 Dec 2025). For each coordinate $i$, its state is the time since last update $\tau_i = A_{t,i}$ at its next selection. The transition matrix $P$ has three regimes:

• AoU-prioritized: $1\leq i\leq k_A$.
• Magnitude-prioritized: $k_A+1\leq i\leq k$.
• Unselected: $i\geq k+1$.

Transitions depend on the random fraction $k_0$ of entries swapped between the magnitude set and its complement, leading to $p_1 = k_0 / k_M$, $p_2 = k_0 / (d-k_M)$. The steady-state distribution $\pi$ over $\{1,\dots,d\}$ is unique and enables calculation of the distribution of staleness $\tau$:

$$P(\tau = \ell) = \sum_{i=1}^d \pi_i \left[ (P^\ell_{(1,k_A+1)} P)_{i,1} + (P^\ell_{(1,k_A+1)} P)_{i,k_A+1} \right]$$

The average staleness $E[\tau]$ directly impacts convergence and fairness of parameter updates.

4. Convergence Rate and Assumptions

The convergence rate of OAC-FL with FAIR-k is analyzed under the following assumptions (Zhu et al., 22 Dec 2025):

• Global $L_g$-smoothness.
• Pseudo-Lipschitz heterogeneity with $L_h$.
• Unbiased SGD with variance $\leq \sigma_s^2$.
• Bounded gradient norms $\mathbb{E} \|\nabla f_n(w;\theta)\|^2 \leq G^2$.
• Bounded gradient divergence $\|\nabla f_n(w) - \nabla f(w)\|^2 \leq \sigma_g^2$.

The main theorem asserts:

$$\min_{0\leq t<T} E[\|\nabla f(w_t)\|^2] = O\left( \frac{f(w_0) - f(w^*)}{\eta \mu_c H T} + \frac{\eta d L_g \sigma_z^2}{\mu_c H N^2} + \frac{\eta L_g \sigma_s^2(\mu_c^2 + \sigma_c^2)}{\mu_c N} + (H-1)^2 \eta_l^2 L_h^2 \sigma_g^2 + (H-1) \eta_l^2 \sigma_s^2 (L_h^2 + L_g^2/N) + \frac{\eta L_g E[\tau]}{H} \left( \frac{d \sigma_z^2}{N^2} + G^2 H^2(1+\mu_c^2 + \sigma_c^2 ) \right) \right)$$

Key terms: the first quantifies optimization progress ($O(1/T)$), next two are noise-induced residuals (OTA and SGD noise), the fourth/fifth express heterogeneity-induced drift, the final term grows with $E[\tau]$ (average staleness), indicating accelerated convergence with fresher updates. Tighter data heterogeneity constants ($L_g$, $L_h$) are empirically much smaller than conventional single-Lipschitz bounds, enabling stable long-epoch local training (Zhu et al., 22 Dec 2025).

5. Practical Implications, Trade-offs, and Performance

FAIR-k offers a tunable interpolation between Top-k and Round-Robin selection, mitigating two key issues: unbounded staleness in pure Top-k, and suboptimal resource allocation in AoU-only schemes. Empirically, on CIFAR-10/100 with ResNet-18, FAIR-k consistently converges faster—70% test accuracy achieved nearly 30% sooner at ρ=10% (sparsity) over Top-k (Zhu et al., 22 Dec 2025). AoU metrics confirm a halving of average AoU and more uniform coordinate selection, reducing selection bias and improving fairness.

In networked sensor applications, a FAIR-k design—implemented via Content-based Wake-up (CoWu)—enables strict reductions in both energy consumption and k-QAoI versus Round-Robin. For example, with $N=100$, $k=5$, energy and k-QAoI can be reduced by 30–50% compared to RR (Shiraishi et al., 2024). The achievable region strictly outperforms RR for $k \ll N$; the scalability is governed by physical process statistics and communication reliability.

Algorithm	OAC-FL: Fairness, Convergence	IoT Sensing: Energy, Freshness
Top-k	Fastest high-magnitude, poor fairness	Freshest extreme values, but energy-inefficient
Round-Robin/AoU	Uniform staleness, ignores magnitude	Good fairness, high energy/transmission cost
FAIR-k	Balanced: convergence, staleness, fairness	Balanced: lower k-QAoI, lower energy

6. Distinctive Features and Theoretical Innovations

FAIR-k's core distinction lies in unifying importance (magnitude-based) and fairness (AoU-driven) principles within a resource-constrained selection policy, supported by exact Markov-chain analysis for staleness and a convergence bound that exposes the interplay between communication constraints, system noise, and heterogeneity. This multipronged approach accounts for practical channel effects, gradient staleness, and real data/local model drift. The scheme supports extended local computation, robust under non-i.i.d. data and realistic over-the-air noise. In IoT, FAIR-k codifies an explicit Pareto frontier for (energy, freshness), and can be dynamically re-optimized under changing process statistics or network conditions (Zhu et al., 22 Dec 2025, Shiraishi et al., 2024).

7. Outlook and Applicability

The FAIR-k paradigm is broadly applicable across distributed learning and resource-efficient sensing. By quantifying and managing parameter staleness and providing explicit controls for importance/freshness trade-off, FAIR-k supports efficient, fair, and timely parameter aggregation and sensing across high-dimensional, heterogeneous, and dynamic systems. Its empirical and analytical superiority over existing approaches is documented for both simulation and hardware prototypes, with applicability to federated learning, timely IoT sensing, and other systems where both selection relevance and temporal freshness are critical (Zhu et al., 22 Dec 2025, Shiraishi et al., 2024).