Client-Confident Convergence Overview

• Client-confident convergence is a federated learning framework that treats client participation as a first-class object, ensuring robust convergence without needing explicit estimates of client availability.
• Techniques such as availability weighting, AoI-based scheduling, and cyclic participation regularization are employed to balance client contributions and improve convergence rates.
• The framework extends to personalized guarantees and privacy-aware aggregation, fostering per-client risk reduction and defense against participation inference.

Searching arXiv for papers on federated learning convergence under client participation, selection, and robustness. “Client-confident convergence” (Editor’s term) denotes a family of federated learning convergence guarantees in which training remains well characterized under intermittent, biased, unknown, or otherwise heterogeneous client participation, and, in stronger formulations, also provides balanced participation, representative client cohorts, per-client risk decrease, personalization, robustness to malicious behavior, or privacy against participation inference. Recent work places this notion across several technical lines: agnostic FedAvg under unknown non-uniform availability, balanced scheduling via Age of Information (AoI), regularized and cyclic participation, cohort filtering, consensus-based per-client descent, personalized clustering, privacy-aware aggregation, and client-centric adaptive optimization (Herlock et al., 14 Jul 2025, Javani et al., 8 May 2025, Malinovsky et al., 2023, Cho et al., 2023, Fourati et al., 2023, Zheng et al., 2023, Werner et al., 2023, Díaz et al., 3 Feb 2025, Mulay, 2022, Sun et al., 17 Jan 2025).

1. Conceptual scope

At its narrowest, client-confident convergence refers to guarantees that do not rely on knowing, controlling, or estimating client availability, yet still identify a well-defined optimization problem and prove convergence of the actual training dynamics. At its broader end, it includes guarantees that clients are selected in a balanced way, that the chosen cohort is representative of the population, that each client’s risk decreases, or that participation cannot be reliably inferred from the released model (Herlock et al., 14 Jul 2025, Javani et al., 8 May 2025, Zheng et al., 2023, Díaz et al., 3 Feb 2025).

Paradigm	Central mechanism	Representative papers
Availability-aware convergence	Optimize the objective induced by participation	(Herlock et al., 14 Jul 2025, Wang et al., 2022)
Participation regularization	Control inter-selection intervals or meta-epoch coverage	(Javani et al., 8 May 2025, Malinovsky et al., 2023, Cho et al., 2023)
Cohort quality control	Filter or sample representative clients	(Fourati et al., 2023, Zheng et al., 2023)
Per-client or personalized guarantees	Consensus descent or cluster-specific models	(Zheng et al., 2023, Werner et al., 2023)
Privacy-aware convergence	Add calibrated privacy noise while preserving utility	(Díaz et al., 3 Feb 2025, Mulay, 2022)
Realistic systems support	Asynchrony, arbitrary participation, heterogeneous local work	(Sun et al., 17 Jan 2025)

This suggests that the term names not a single algorithmic primitive but a convergence viewpoint: the participation process is treated as a first-class mathematical object rather than an inconvenience abstracted away.

2. Availability-induced objectives

A central result is that convergence under partial participation depends on the objective actually induced by participation. In agnostic FedAvg, with random subset $S^t \subseteq [N]$ drawn from an arbitrary distribution $q(\mathcal{A})$ with $q(\emptyset)=0$, the relevant client weights are

$$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$

and the induced objective is

$f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$

The key claim is that simple averaging over available clients is not an approximation to the uniform client objective under unknown biased availability; it is a stochastic approximation to this availability-weighted objective. Under convex, possibly nonsmooth losses, compact convex constraints, bounded local gradient variance, and bounded gradient norms, agnostic FedAvg converges to the minimizer of $f(\theta)$ at rate $\mathcal{O}(1/\sqrt{T})$, without knowledge of $q(\mathcal{A})$ or $\{p_i\}$ (Herlock et al., 14 Jul 2025).

The same line of work identifies a sample-to-model inequality,

$$\mathbb{E}\big[\|\hat{\theta}^{t} - \theta^*\|^2 \mid \mathcal{F}_{t-1}\big] \le \sum_{i \in [N]} p_i \, \|\theta_i^{t-1} - \theta^*\|^2,$$

which formalizes that random aggregation is “no worse” than the availability-weighted average over client parameters. This is the precise sense in which convergence remains client-confident under unknown participation: the server need not estimate the participation law in order for the stochastic dynamics to be convergent (Herlock et al., 14 Jul 2025).

A broader unified analysis of arbitrary participation encodes the effect of participation into a single time-averaged bias term $q(\mathcal{A})$0, defined through the discrepancy between the true global gradient and the $q(\mathcal{A})$1-round average of participation-weighted gradients. When participation is regularized so that each client has equal average weight over each interval, $q(\mathcal{A})$2, and generalized FedAvg with amplified updates every $q(\mathcal{A})$3 rounds matches either the lower bound of stochastic gradient descent or the state of the art in specific stochastic settings (Wang et al., 2022). This suggests that the practical question is often not whether participation is random, but whether time-averaged participation is sufficiently balanced.

A common misconception follows from older FedAvg analyses: convergence under partial participation does not automatically imply convergence to the uniform client objective. Under biased availability, the learned model is availability-weighted, not client-uniform or data-size-weighted, unless the sampling law enforces those weights (Herlock et al., 14 Jul 2025).

3. Participation regularization and scheduling

A second strand of client-confident convergence seeks not merely to tolerate irregular participation, but to shape it. AoI-based scheduling formalizes the inter-selection interval $q(\mathcal{A})$4 as a load metric and identifies it with the peak Age of Information. The policy objective is to minimize $q(\mathcal{A})$5 subject to equal marginal selection probability $q(\mathcal{A})$6. The resulting decentralized Markov scheduling policy uses age-dependent selection probabilities $q(\mathcal{A})$7, produces nearly deterministic inter-selection intervals around $q(\mathcal{A})$8, and yields a convergence bound of order $q(\mathcal{A})$9 whose constants depend explicitly on the total variance of aggregation weights $q(\emptyset)=0$0, selection skew $q(\emptyset)=0$1, and heterogeneity $q(\emptyset)=0$2 (Javani et al., 8 May 2025).

The empirical interpretation is direct. In experiments with 100 clients and 15 selected per round, AoI-Markov scheduling reduced $q(\emptyset)=0$3 from approximately $q(\emptyset)=0$4 under random selection to approximately $q(\emptyset)=0$5 under the optimal Markov policy, and achieved 7.5–20% fewer communication rounds to reach the same accuracy across MNIST, CIFAR-10, and CIFAR-100 under IID and non-IID settings (Javani et al., 8 May 2025). The mechanism is fully decentralized: each client tracks only its own age and uses globally known $q(\emptyset)=0$6.

Regularized client participation pushes the same idea further. In RR-CLI, each client joins the learning process every $q(\emptyset)=0$7 communication rounds, defining a meta epoch in which every client participates exactly once. Combined with client reshuffling and local data reshuffling, this reduces client-sampling variance from a term linear in step size to a term quadratic in step size. Under strong convexity, this yields a convergence rate $q(\emptyset)=0$8, compared with $q(\emptyset)=0$9 for with-replacement client sampling, and remains valid under arbitrary client availability so long as each client is available once per meta epoch (Malinovsky et al., 2023).

Cyclic client participation provides a related deterministic structure. Clients are partitioned into disjoint groups $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$0, traversed in a fixed cyclic order, with uniform sampling within each active group. Under the PL condition, the analysis decomposes heterogeneity into intra-group $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$1 and inter-group $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$2. For local GD, if $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$3 and $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$4, the $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$5 term vanishes and the remaining bound is $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$6, faster than vanilla FedAvg under uniform client participation (Cho et al., 2023). The effect is weaker under local SGD because stochastic variance remains.

These results clarify that “balanced participation” is not a heuristic fairness add-on; it directly alters the convergence constants and, under strong structural assumptions, the asymptotic rate.

4. Cohort quality, representativeness, and filtered participation

A third line of work argues that client-confident convergence depends on the quality of the selected cohort, not merely on the marginal participation rate. FilFL introduces a public proxy dataset $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$7 and assigns a reward to a subset $$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$8 of available clients by

$$p_i = \sum_{j=1}^{2^N} \frac{q(\mathcal{A}_j)}{|\mathcal{A}_j|}\,\mathbb{I}[i\in \mathcal{A}_j],$$9

The server then applies deterministic or randomized greedy filtering to obtain a filtered-in set $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$0, on top of which any downstream selector can operate. Under smoothness, strong convexity, bounded stochastic gradient variance, and bounded heterogeneity, the resulting convergence guarantee has the form

$f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$1

where the asymptotic neighborhood term $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$2 decreases as the expected filtering gap $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$3 increases (Fourati et al., 2023).

The filtering objective is explicitly combinatorial rather than client-wise: a client is assessed as part of a combination. On a small CIFAR-10 setup where exhaustive search is possible, both DGF and RGF achieve $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$4, and across CIFAR-10, FEMNIST, and Shakespeare the method yields improved learning efficiency, faster convergence, and up to 10% higher test accuracy than training without filtering (Fourati et al., 2023).

Representative sampling also appears in FedCOME. In partial participation, the server maintains a similarity table $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$5 based on gradient cosine similarity and selects a subset $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$6 of size $f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$7 by minimizing the pairwise similarity sum

$f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$8

approximately solved by simulated annealing. The purpose is not simply diversity for its own sake, but representativeness of the global data distribution; empirically, training on these selected clients with the consensus mechanism leads to risk decrease for clients that are not selected (Zheng et al., 2023).

A plausible implication is that cohort design introduces a second convergence object alongside the objective function: the training dynamics depend on whether the participating clients form a coherent and representative basis for the update.

5. Per-client descent, personalization, robustness, and privacy

The strongest interpretation of client-confident convergence is per-client monotonic improvement. FedCOME formulates the round-wise constrained problem

$f(\theta)=\sum_{i=1}^{N} p_i f_i(\theta).$9

and enforces it by projecting each client gradient $f(\theta)$0 to a corrected gradient $f(\theta)$1 that solves

$f(\theta)$2

Under full participation, full batch, one local epoch, smoothness, and sufficiently small learning rate, this consensus mechanism ensures $f(\theta)$3 for every client, while the global objective retains a standard nonconvex convergence rate in terms of averaged gradient norm (Zheng et al., 2023).

Personalized clustering extends the notion from shared convergence to client-specific convergence. In model-per-cluster federated optimization, Threshold-Clustering identifies clients with similar objectives using gradients or momenta. For client $f(\theta)$4, under smoothness, bounded stochastic gradient variance, intra-cluster similarity, inter-cluster separation, and a malicious fraction $f(\theta)$5, the personalized rate is

$f(\theta)$6

which asymptotically matches the oracle rate obtained when the true clustering is known, up to the clustering and Byzantine terms (Werner et al., 2023). Here client confidence is literal: each honest client has a convergence statement for its personalized model rather than for a single global average.

Privacy-aware convergence adds a different confidence criterion: participation should not be inferable. In medical imaging FL, metric-privacy calibrates server-side Gaussian noise to a round-wise distance

$f(\theta)$7

and empirically improves utility over standard global DP across six aggregation strategies while offering similar protection against a client inference attack (Díaz et al., 3 Feb 2025). LOCKS gives a complementary user-level DP analysis: clipped and noised client outputs yield convergence bounds whose DP term scales with model dimensionality, and the paper shows that the expected gradient variance is minimized by approximately $f(\theta)$8 rounds (Mulay, 2022).

Finally, client-centric federated adaptive optimization places arbitrary client participation, asynchronous server aggregation, and heterogeneous local computing inside a single nonconvex theory. Clients decide independently when to participate and how many local steps $f(\theta)$9 to take; the server aggregates buffered normalized model differences and applies FedAdagrad-, FedAdam-, or AMSGrad-style updates. Under bounded staleness $\mathcal{O}(1/\sqrt{T})$0, bounded local and global variance, and appropriate step sizes, the framework achieves

$\mathcal{O}(1/\sqrt{T})$1

and recovers the best-known rate $\mathcal{O}(1/\sqrt{T})$2 when $\mathcal{O}(1/\sqrt{T})$3 is sufficiently large and staleness is moderate (Sun et al., 17 Jan 2025).

6. Misconceptions, limitations, and open directions

Several misconceptions recur across this literature. First, convergence under partial participation is not synonymous with convergence to the intended uniform objective: agnostic FedAvg converges to an availability-weighted objective, and fairness concerns remain because frequently available clients receive larger effective weight (Herlock et al., 14 Jul 2025). Second, balanced participation is not guaranteed by equal marginal selection probability alone; AoI-based work shows that low variance of inter-selection intervals and low variance of aggregation weights matter for stability (Javani et al., 8 May 2025). Third, per-client monotonic descent is not a generic property of FL; in the cited results it is proved in the idealized full-participation FedSGD setting with a consensus correction, not for arbitrary stochastic FedAvg deployments (Zheng et al., 2023).

The strongest rates also rely on strong assumptions. The $\mathcal{O}(1/\sqrt{T})$4 improvements from regularized participation and cyclic participation are established under strong convexity or PL-type conditions and structured schedules, not for general deep nonconvex models (Malinovsky et al., 2023, Cho et al., 2023). Agnostic FedAvg’s availability-robust guarantee is convex, possibly nonsmooth, and assumes i.i.d. participation across rounds (Herlock et al., 14 Jul 2025). FilFL’s theory assumes strong convexity and a public proxy dataset, while its guarantees are in expectation rather than high probability (Fourati et al., 2023). Personalized clustering requires sufficiently large cluster separation $\mathcal{O}(1/\sqrt{T})$5 and sufficiently small malicious fraction $\mathcal{O}(1/\sqrt{T})$6 (Werner et al., 2023). Metric-privacy is evaluated under a trusted-server model and does not provide a full multi-round composition analysis in the cited work (Díaz et al., 3 Feb 2025). LOCKS exposes an explicit dimensional burden for privacy noise (Mulay, 2022). Client-centric adaptive optimization still admits an unavoidable $\mathcal{O}(1/\sqrt{T})$7 floor under worst-case arbitrary participation (Sun et al., 17 Jan 2025).

The current frontier therefore lies less in proving that FL can converge under client uncertainty than in specifying what kind of convergence is sought. One line aims to identify the correct objective under unknown participation; another seeks to regularize participation so the objective is better aligned with the desired one; a third improves cohort quality through filtering or representative sampling; a fourth strengthens the guarantee to per-client descent or personalization; and a fifth couples convergence with privacy against participation inference. Taken together, these works suggest that client-confident convergence is becoming a unifying criterion for realistic federated optimization: convergence should remain mathematically valid when clients are intermittent, heterogeneous, resource-constrained, privacy-sensitive, and only partially represented in any given round.