Learning OTA: A Unified Framework for Edge Sensing, Computation, and Communication

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Learning OTA: A Unified Framework for Edge Sensing, Computation, and Communication

arXiv:2610.04718v1 [cs.IT] 3 Oct 2026

Mehdi Karbalayghareh, Member, IEEE, David J. Love, Fellow, IEEE, and Christopher G. Brinton, Senior Member, IEEE

Abstract—Most federated learning (FL) studies implicitly assume that training data is readily available at participating edge devices. In practice, this data must first be acquired through heterogeneous sensors with potentially different measurement modalities, and the quality and cost of this acquisition process can directly affect the learning performance. This dependence creates a coupled relationship among sensing, computation, and communication, necessitating a system model that captures their joint effect on resource efficiency and learning accuracy. This paper develops such a unified treatment for an over-the-air (OTA) FL system in which each device configures its sensing modality, power, resolution, and sample size, with aggregations conducted OTA at the server. We derive models for sensing noise, energy consumption, latency, and OTA aggregation distortion, and establish a non-convex convergence bound quantifying how these factors influence the global model trajectory. Guided by this analysis, we formulate a per-round optimization that balances energy, latency, and learning quality under physical device constraints. The resulting mixed optimization is addressed through an alternating procedure with closed-form and one-dimensional updates for continuous variables and a lightweight discrete search. Numerical experiments corroborate the advantage of jointly optimizing sensing, computation, and communication decisions. Index Terms—Federated learning, over-the-air (OTA) computation, joint sensing–computation–communication.

I. I NTRODUCTION AI-native sixth generation (6G) wireless networks are expected to rely heavily on distributed edge learning, where heterogeneous devices collaboratively train models under latency, reliability, bandwidth, and energy constraints [2]–[7]. Federated learning (FL) provides a scalable framework for this setting by enabling devices to compute local updates and exchange model information rather than raw data [8]–[15]. Prior work on wireless FL has largely focused on communication and computation bottlenecks. Communicationefficient FL schemes reduce downlink/uplink overhead through compression, quantization, sparsification, and partial device M. Karbalayghareh, D. J. Love, and C. G. Brinton are with the Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA (e-mails: mkarbala@purdue.edu; djlove@purdue.edu; cgb@purdue.edu). This work was supported in part by the Office of Naval Research (ONR) under grant N00014-21-1-2472 and by the National Science Foundation (NSF) under grants CNS2212565, CNS2225578, and CPS2313109. A preliminary version of this work has been accepted to the 2026 IEEE Information Theory Workshop (ITW) [1], focusing only on sensing-parameter optimization. This journal version extends to a unified joint sensing– computation–communication design with OTA aggregation and convergenceaware resource optimization under link impairments.

participation [16]–[19]. In wireless settings, over-the-air federated learning (OTA-FL) leverages the superposition property of multiple access channels (MAC) to aggregate updates in the uplink (at the server), significantly reducing latency and spectral usage [20]–[23]. Complementary studies incorporate receiver beamforming, power control, and convergence-driven resource allocation to mitigate aggregation distortion [24]. In parallel, computation-aware FL designs adapt local processing frequency and workload allocation to balance energy consumption and training latency [25], [26]. Collectively, these works establish a mature understanding of how communication and computation resources shape learning performance. Despite this progress, most FL formulations rely on an implicit but critical assumption: training data at each device is already available and of fixed statistical quality. Under this abstraction, the learning pipeline begins with local model updates and ends with global aggregation, while the process by which data is acquired is entirely omitted. In practice, however, training data is generated through physical sensing processes that consume energy, incur delay, and introduce measurement distortion. As a result, learning performance is fundamentally constrained not only by how efficiently model updates are communicated and computed, but also by how data is sensed and generated at the devices. In many edge learning applications, devices are connected to heterogeneous sensors such as cameras, microphones, radar units, or LiDAR that acquire observations from the physical environment. The statistical quality of the sensed data depends on several factors, including sensing modality, sensing resolution, and sensing power. Different sensing modalities capture distinct physical phenomena and produce data with inherently different noise characteristics and feature distributions. Resolution governs the granularity of the acquired measurements and measurement precision, while sensing power determines the effective signal-to-noise ratio (SNR) of the acquired samples. These sensing decisions directly affect the fidelity of the training data and, in turn, influence learning performance and final model accuracy. Consequently, sensing, computation, and communication form a tightly coupled pipeline in practical FL systems; isolating any single stage fails to capture the true system-level tradeoffs. A. Related Works Recent works have started to incorporate data acquisition and sensing into federated learning. Liu et al. [27] studied task-oriented sensing, computation, and communication integration for federated edge learning, where sensing time,

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energy consumption, and communication resources are jointly considered. Liang et al. [28] introduced an integrated sensing– communication–computation (ISCC) formulation for OTA-FL, showing that the number of sensed samples, computation resources, and uplink transmission design should be jointly optimized when data are acquired during training. Du et al. [29] further considered ISCC-enabled OTA-FL in wireless networks and analyzed the impact of sensing and aggregation errors on learning performance. Fu et al. [30] studied a joint sensing–communication–computation (JSCC) design for aerial-network-enabled OTA-FL, where sensing, communication, and computation resources are optimized under aerialnetwork constraints. More recently, Wen et al. [31] developed an ISCC-enabled OTA-FL framework that characterizes the effects of sensing noise and aggregation distortion. Sensing-aware FL has also been studied in vertical or feature-partitioned learning architectures. Liu et al. [32] proposed a vertical federated edge learning (V-FEEL) framework with distributed integrated sensing and communication, where devices observe the same target from different sensing views. Cao et al. [33] further developed an ISCC-enabled V-FEEL framework for edge perception networks, where sensing noise, multi-view feature diversity, and OTA embedding aggregation jointly affect learning performance. These V-FEEL works are closely related in motivation, but they focus on featurepartitioned data and embedding aggregation, whereas this paper considers horizontal OTA-FL with sample-partitioned local datasets and model-update aggregation. Despite these advances, existing sensing-aware FL formulations still represent sensing quality through a limited set of controls, such as sample count or sensing power. These variables are important, but they do not fully capture the sensing-quality tradeoffs that arise in diverse edge systems. In particular, two devices collecting the same number of samples may produce data of different fidelity due to heterogeneous sensors. Conversely, increasing sensing resolution or sensing power may reduce measurement distortion without increasing the sample count, at the cost of higher energy and latency.

Global model broadcast

Local computation

Σ

Device 1 Device 2

Uplink MAC and OTA aggregation

Device 3

local SGD steps, CPU frequency

Device 𝑁

Uplink, OTA aggregation power, OTA aggregation, combining vector

image/video voice radar

lidar IoT sensors

Heterogeneous sensing

Fig. 1: System overview of the joint sensing, computation, and communication design for OTA-FL under heterogeneous sensing and wireless link impairments. aggregation error. Guided by this analysis, we formulate a perround mixed discrete–continuous optimization problem that balances learning quality, sensing energy, computation latency, and communication distortion. The problem is solved using a low-complexity alternating method with closed-form and onedimensional updates for continuous variables and finite search over discrete sensing configurations. The main contributions of this work are summarized as follows: • We introduce a sensing-quality-aware OTA-FL model that captures the impact of sensing modality, resolution, power, and sample size on data fidelity and learning performance. • We derive a convergence characterization that jointly accounts for sensing-induced distortion, stochastic gradient variability, and OTA aggregation error. • We formulate and solve a unified sensing–computation– communication optimization problem using a lowcomplexity alternating approach. Notation: matrices and vectors are denoted by bold capital and bold small letters, respectively. For a matrix A, the transpose is denoted A⊤ , the Hermitian AH , and the conjugate A∗ . Im is the m × m identity matrix, and the dimension is omitted if clear from the context. Expectation is denoted E(·).

B. Contributions This paper addresses the above gap by adopting a sensingcentric view of OTA-FL, where sensing quality is treated as a multidimensional controllable component rather than an implicit property of the local dataset. Specifically, each device jointly selects its sensing modality, sensing resolution, sensing power, and per-round sample size, while local computation and OTA communication resources are optimized within the same learning-aware framework. This distinction is important because sensing distortion corrupts the training data before local optimization and therefore propagates through subsequent gradient updates, unlike communication noise that affects only the aggregation stage. To capture this effect, we employ an additive sensing noise model that relates sensing configuration to data fidelity in a tractable manner. We then derive a convergence characterization that captures the combined effects of sensinginduced distortion, stochastic gradient variability, and OTA

Global broadcast

Local multimodal sensing sensing mode, power, resolution, sample size

Parameter server (PS)

Base station (BS)

under downlink impairments

II. S YSTEM M ODEL In this section, we describe the system model for a general distributed learning paradigm over wireless channels, depicted in Fig. 1. We present the models for global model broadcast, local data sensing, local computation, uplink channel estimation, and uplink model aggregation. A. Preliminaries We consider a federated learning system with a parameter server (PS) and N participating devices (see Fig. 1). The server is located at the base station (BS), which has M antennas. All devices have a single antenna. The model training process is carried out over T communication rounds. Throughout the paper, we use the terms “device”, “receiver”, and “user” interchangeably to refer to the receiving terminals in the downlink. The system operates under physical, network, data, and computation resource constraints and heterogeneity.

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Each device n ∈ [N ] begins with an initial local dataset Dn,0 of size Dn,0 = |Dn,0 | ≥ 0, which may be zero. In each communication round t ∈ [T ], device n senses a fresh batch of Dn,t = |Dn,t | new samples through the local sensing process described in Section II-C. These newly acquired samples are appended to the device’s accumulated dataset: ( Dn,t = Dn,t−1 ∪ Dn,t . (1) Dn,t = Dn,t−1 + Dn,t PN Define Dt ≜ n=1 Dn,t , and let f denote the per-sample loss function. The local empirical loss at device n in round t for a d-dimensional model θ ∈ Rd is X 1 f (θ, u), (2) Fen,t (θ) ≜ Dn,t u∈Dn,t

and the corresponding global loss is Fet (θ) =

N X Dn,t

Dt n=1

Fen,t (θ).

(3)

Note that both Fen,t and Fet evolve deterministically with the accumulation history {Dn,ℓ }ℓ≤t . FL aims to minimize Fet (θ) through iterative collaboration between the PS and the N edge devices. During each round, the BS broadcasts the global model θt to the devices over the downlink channel, which is subject to link impairments. Each device n receives a noisy version of the model, denoted by θn,t , and performs τ steps of mini-batch stochastic gradient descent (SGD) using its accumulated local data. At local step (i) i ∈ [τ ], device n draws a mini-batch ξn,t of fixed size B uniformly from Dn,t and updates its local model as   (i+1) (i) (i) (i) θn,t = θn,t − η ∇Fen,t θn,t ; ξn,t , i ∈ [τ ], (4) (1)

where θn,t = θn,t and η > 0 is the global stepsize. After τ local iterations, device n forms its model update (τ +1)

∆θn,t ≜ θn,t

(1)

− θn,t .

(5)

Each device n simultaneously transmits a scaled version of its ∆θn,t to the BS over the uplink, as described in Sections II-E and II-F. The PS employs the received superposition to update the global model, which in effect becomes θt+1 = θt +

N X Dn,t

Dt n=1

c n,t , ∆θ

(6)

The received signal at device n in round t is dl yn,t = hH n,t xt + wn,t ,

(7)

M where xdl is the downlink transmit signal across M t ∈ C H dl dl antennas, satisfying the power constraint E[(xdl t ) xt ] = p 2 at the BS, and wn,t ∼ CN (0, σw ) is the AWGN at device n. The corresponding per-device downlink SNR is

SNRdl n,t =

∥hn,t ∥2 pdl . 2 σw

(8)

Since the BS uses a common broadcast transmission, the achievable downlink rate in round t is limited by the weakest link (device). Therefore,   (9) Rtdl = W dl log2 1 + minn∈[N ] SNRdl n,t . The time required to broadcast the global model to all devices in round t is Sm Sm  . Ttdl = dl = (10) Rt W dl log 1 + minn SNRdl 2

n,t

The corresponding downlink energy at the BS is Etdl = pdl Ttdl .

(11)

Remark 1. The system operates under time-division duplexing (TDD) mode with perfect reciprocity for simplicity. Therefore, we do not consider downlink channel training, as the BS can estimate the channels via uplink pilots. Our framework is also applicable to frequency-division duplexing (FDD) systems, with additional overhead for downlink training and feedback. C. Local Data Sensing In the downlink round t, after receiving θt , each device n performs local sensing to collect new data for its local training. We consider a finite set of sensing modalities M = {1, · · · , Mmod }, which may represent, e.g., camera, microphone, LiDAR, and RF sensing. Device n selects exactly one modality in round t, (m) from Mn ⊆ M, using binary selection variables zn,t ∈ {0, 1} such that X (m) zn,t = 1, ∀n, t. m∈Mn

c n,t is a noisy version of each device’s update. This where ∆θ process continues over T training rounds.

Each device n also chooses a sensing resolution qn,t (the number of measurements captured per sample, e.g., pixels per frame or points per scan) from a device-specific finite set

B. Global Model Broadcast In each communication round t ∈ [T ], the BS sends the current global model θt to all devices over the downlink. Let hn,t ∈ CM denote the channel between the BS and device n in round t, with entries CN (0, 1). Further, let Sm and W dl denote the number of bits1 and the downlink bandwidth, respectively.

Qn = {qn(1) , . . . , qn(Qn ) }.

1 The value of S m depends on the transmission mode (analog or digital), coding rate, quantization level, and other practical factors that determine how it relates to the model dimension d. A detailed discussion of these aspects is outside the scope of this work; see [34] for more information.

Recall that Dn,t denotes the size of newly sensed data in round t at device n. Let sn,t denote the underlying physical signal. The sensed sample (feature vector) is modeled as xn,t = sn,t + vn,t ,

(12)

where vn,t is
the additive sensing noise distributed vn,t ∼ 2 N 0, σv,n,t I . The noise variance depends on the selected modality, resolution, and sensing power. Let psn,t ∈ (0, Pns,max ]

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denote the allocated power for sensing at device n in round t, where Pns,max is the maximum sensing power supported by the sensor front-end. We model the sensing noise variance as (m)

2 σv,n,t =

(m)

X

zn,t

m∈Mn

c1,n , s pn,t qn,t

(13)

(m)

where c1,n > 0 is a modality- and device-specific constant characterizing the sensing hardware (e.g., sensor sensitivity and front-end noise figure). This captures the natural trade-off that higher sensing power and higher resolution yield lower sensing noise. The time to sense Dn,t samples under modality m depends (m) on a modality-specific sensing rate rs,n > 0 (samples per second) and on the acquisition resolution qn,t : finer resolution captures more measurements per sample, lowering the effec(m) tive sampling rate to rs,n /qn,t . The sensing latency is s = Tn,t

X m∈Mn

(m) D n,t qn,t , (m) rs,n

zn,t

(14)

and the corresponding sensing energy is s s En,t = psn,t Tn,t .

(15)

After sensing, the newly acquired samples Dn,t are added to the existing dataset Dn,t−1 to form Dn,t . Remark 2. The additive Gaussian sensing noise model in (12) and (13) is adopted as a tractable abstraction for capturing the dependence of sensing fidelity on modality, power, and resolution. While certain modalities exhibit structured or signal-dependent noise (e.g., Poisson statistics in optical sensing or multiplicative noise in radar), the Gaussian model is the entropy-maximizing choice among additive noises with the same variance [35] and is adopted here as a tractable information-theoretic worst case. Extending the analysis to signal-dependent noise models is left to future work. D. Local Computation Model Following the local SGD updates in (4) and the model update definition in (5), we now model the latency and energy c of the local computation phase at each device. Let fn,t denote the CPU frequency (cycles per second) at device n in round t, and let κ denote the number of CPU cycles required to compute one sample gradient (a system constant determined by the model architecture and SGD configuration). Since device n performs τ local SGD steps in round t, each on a mini-batch of size B, the total number of sample-gradient evaluations per round is τ B. The local computation time is therefore κτ B c Tn,t = c . (16) fn,t We adopt the standard dynamic voltage and frequency scaling (DVFS) model for processor energy [36], [37], under which the energy consumed for a workload of κτ B cycles at frec quency fn,t is
2 c c En,t = κc κ τ B fn,t , (17)

where κc > 0 is the effective switched capacitance coefficient of the processor. The CPU frequency at each device is constrained as c fnmin ≤ fn,t ≤ fnmax ,

∀n, t.

(18)

c The computation cost depends on B, τ , and fn,t , but not on the accumulated dataset size Dn,t . The newly sensed sample count Dn,t remains a sensing-side decision variable that affects the sensing latency and energy in (14)–(15) and the accumulated size Dn,t entering the convergence analysis in Section III.

E. Uplink Channel Training Before transmitting the local model updates, the BS acquires channel state information (CSI) via uplink pilot transmission. At the beginning of round t, each device transmits an uplink pilot sequence of length πp symbols. We assume orthogonal pilots (no pilot contamination). Let ppn,t denote the pilot transmit power of device n in round t. The received pilot signal corresponding to device n at the BS is q p (19) yn,t = πp ppn,t hn,t + zt , where zt ∼ CN (0, σz2 IM ) is the AWGN in the received signal at the BS. Observing (19), the MMSE estimate of hn,t is [38] q πp ppn,t b yp . (20) hn,t = πp ppn,t + σz2 n,t The corresponding estimation error is b n,t , en,t ≜ hn,t − h

(21)

which is Gaussian and uncorrelated with the channel estimate b n,t [38]. Therefore, h   b n,t h b H = (1 − σ 2 )IM , E h (22) n,t e,n,t   H 2 E en,t en,t = σe,n,t IM , (23) where the estimation error variance is πp ppn,t σz2 2 = . σe,n,t =1− p p πp pn,t + σz2 πp pn,t + σz2

(24)

F. Uplink OTA Aggregation of Model Updates Each device n transmits its model update ∆θn,t to the BS in the uplink. We focus on a per-coordinate description of the transmission, which is standard in OTA-FL: each element of ∆θn,t is transmitted in a symbol interval, and the OTA aggregation is performed coordinate-wise. For brevity, we consider one generic coordinate, denoted by ∆θn,t , and omit the coordinate index. We assume that the update coordinates are powernormalized, with the normalization/de-normalization factors absorbed into the transmit/receive scaling and omitted from the notation. Let pun,t ∈ [0, Pnu,max ] denote the transmit power used by device n in round t, where Pnu,max is the maximum transmit power of the device’s power amplifier. The received signal at the BS across its M antennas is N q X (25) yt = hn,t pun,t ∆θn,t + zt , n=1

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where zt ∼ CN (0, σz2 IM ). Using (21), (25) becomes yt =

N X

b n,t h

q

pun,t ∆θn,t +

n=1

N X

en,t

q pun,t ∆θn,t + zt .

are omitted from the per-round accounting. Hence, the perround latency is
s c Tt = Ttdl + max Tn,t + Tn,t + Ttu . (34) n

n=1

(26) The BS applies a linear combining vector vt ∈ CM ×1 to obtain a scalar aggregated estimate ỹt = vtH yt .

(27)

The desired aggregated update for the considered coordinate is the weighted sum

The total energy consumption in round t is the sum of the devices’ sensing, computation, and uplink communication energies. Since the downlink energy is typically dominated by the BS and does not impact device-side optimization, we focus on the device energies and define Et =

N  X

 s c u . En,t + En,t + En,t

(35)

n=1

gt ≜

N X

αn,t ∆θn,t ,

(28)

n=1

where αn,t ≜ Dn,t /Dt . Due to channel distortion, estimation errors, and noise, ỹt will deviate from gt . We define the meansquare OTA aggregation error (per coordinate) as   b n,t }N , (29) MSEut ≜ E |ỹt − gt |2 {h n=1 where the expectation is taken over the channel estimation error, receiver noise, and normalized update coordinates. Specifically, we assume that the normalized update coordinates have unit second moment and are second-order orthogonal across devices: E[|∆θn,t |2 ] = 1,

∗ E[∆θn,t ∆θj,t ] = 0, n ̸= j, ∀n, t. (30)

The normalized update coordinates are also independent of the channel estimation error and receiver noise. Under these assumptions and the MMSE channel estimation model in (20)(24), we have MSEut =

N X

b n,t vtH h

q

pun,t − αn,t

2

III. C ONVERGENCE A NALYSIS In this section, we study the learning performance of the proposed framework when the global loss is non-convex. For each sample vector u ∈ Dn,t , let s(u) denote its underlying clean signal in the sensing model (12). We define the clean reference local and global losses as X 1 f (θ, s(u)), (36) Fn,t (θ) ≜ Dn,t u∈Dn,t

Ft (θ) ≜

N X Dn,t n=1

Dt

Fn,t (θ).

(37)

These losses are used only for analysis, while the actual local updates use Fen,t in (2). Since Ft evolves with the accumulated PT −1  data, we track T1 t=0 E ∥∇Ft (θt )∥2 . A. Preliminaries and Assumptions

n=1

+ ∥vt ∥2

We will later leverage the expressions in (34) and (35), together with a convergence bound on the FL process, to formulate and solve a joint sensing–computation–communication optimization problem.

N X



2 pun,t σe,n,t + σz2 .

(31)

n=1

In the sequel, we will use (31) as the OTA aggregation distortion metric that enters the convergence bound and optimization problem. In analog OTA aggregation, each of the d model coordinates occupies one uplink symbol interval, so the uplink transmission time is common across devices and given by d Ttu = u , W u En,t = pun,t Ttu ,

(32) (33)

where W u is the uplink symbol rate. G. Per-Round Latency and Energy In round t, the total latency includes the downlink broadcast time, the maximum sensing and computation time across devices, and the common uplink aggregation time. Since the pilot length and pilot powers are fixed and are not optimization variables, the corresponding channel estimation latency and energy are constant with respect to the design variables and

We work under standard OTA-FL assumptions [22], [24], [39], together with sensing-specific conditions which apply uniformly across rounds. A1. L-smoothness: The per-sample loss f (θ, u) is L-smooth in θ for every u, ∥∇f (θ, u) − ∇f (θ ′ , u)∥ ≤ L∥θ − θ ′ ∥,

∀θ, θ ′ , u. (38)

Hence Fn,t and Ft are L-smooth in θ for every n and t. A2. Bounded loss: There exist Finf , Fsup ∈ R such that Finf ≤ f (θ, s) ≤ Fsup for every clean sample s and all θ; consequently Finf ≤ Ft (θ) ≤ Fsup ,

∀θ, t.

(39)

Let ∆F ≜ Fsup − Finf . A3. Bounded gradient norm: There exists G > 0 such that ∥∇Fn,t (θ; ξ)∥ ≤ G,

∀θ, n, t, ξ,

(40)

where ξ is any admissible clean mini-batch. A4. Mini-batch variance decomposition: Let Fn∞ (θ) ≜ Es∼Pn [f (θ, s)] denote the clean population loss at device (i) n. For a mini-batch ξn,t , i ∈ [τ ], with size B drawn

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uniformly from Dn,t , the stochastic gradient error around the population gradient satisfies 2 h i σ2 σpop (i) E ∥∇Fn,t (θ; ξn,t ) − ∇Fn∞ (θ)∥2 ≤ B + , B Dn,t (41) 2 where σB bounds the per-sample variance around the 2 empirical mean and σpop bounds the per-sample variance around the population mean. A5. Lipschitz-in-input gradient: For each device n, there exists csn > 0 such that, for any clean signal s and any sensing-noise realization v,   Ev ∥∇f (θ; s + v) − ∇f (θ; s)∥2 ≤ csn E∥v∥2 . (42)

This holds when f (θ, ·) has a uniformly bounded crossJacobian ∇s ∇θ f , consistent with Jacobian-based sensitivity analyses under input perturbations [40]. A6. First-order unbiased sensing perturbation: Under the additive sensing model in (12), the zero-mean sensing noise induces a sample-gradient perturbation that is unbiased to first order, Ev [∇f (θ; s + v) − ∇f (θ; s)] = 0,

(43)

consistent with first-order analyses of learning under additive input noise [40], [41]. Remark 3. Throughout the convergence analysis, we assume θn,t = θt for every n and t, i.e., the BS broadcast is decoded reliably at every device. This isolates the effects of sensing, local computation, and uplink OTA aggregation on the learning trajectory. Extending the analysis to the imperfect downlink case in Section II-B follows the same template: an additional perturbation term θn,t − θt propagates through the τ local SGD steps and contributes a separate term to the perround bound. We omit this for brevity. Let ζt denote the aggregate perturbation, which in the per-round update arises from three physical sources: sensing noise in the local data acquisition phase, stochasticity and finite-data effects in the local computation phase, and uplink OTA aggregation distortion due to fading, imperfect CSI, and receiver noise at the BS. In the following, we derive the decomposition explicitly and separate each contribution into its bias and variance components. B. Decomposition of the Per-Round Update Error We begin by writing the BS-side update in a form that exposes the deviation from an ideal full-gradient descent step on Ft . Under Remark 3, each device initializes its local SGD (1) at θn,t = θt . From (5) and (6), we have θt+1 − θt =

N X Dn,t

Dt n=1

c n,t . ∆θ

(44)

We add and subtract the ideal τ -step full-gradient reference on Ft , namely −ητ ∇Ft (θt ), to obtain θt+1 − θt = −ητ ∇Ft (θt ) + ζt ,

(45)

where the aggregate perturbation ζt is defined as ζt ≜

N X Dn,t n=1

Dt

c n,t + ητ ∇Ft (θt ). ∆θ

(46)

The expression in (46) captures every deviation between the actual BS update and the ideal full-gradient step. We decompose ζt along the three physical sources previewed in Section III-A by inserting the noise-free local update as an intermediate quantity. clean Let ∆θn,t denote the hypothetical local update that device n would have produced in the absence of sensing noise, using the same initialization and the same sequence of minibatch indices as the actual local SGD, but replacing each noisy observation xn,t by its corresponding clean signal sn,t . Adding P D P D clean ∆θn,t and n Dn,t ∆θn,t in (46) and subtracting n Dn,t t t results in (47), shown at the top of the next page. The three terms respectively capture the uplink OTA aggregation distortion (ϵt ), the propagation of sensing noise through the local SGD updates (γts ), and the deviation of the noisefree τ -step local SGD from the ideal full-gradient reference (γtc ). Each contribution is expressed directly in terms of system quantities. We separate each contribution into a deterministic given θt bias and a zero-mean variance component. Define   ζ̄t ≜ E ζt θt , ζt0 ≜ ζt − ζ̄t , (48) so that E[ζt0 | θt ] = 0 and     E ∥ζt ∥2 θt = ∥ζ̄t ∥2 + E ∥ζt0 ∥2 θt .

(49)

Applying this split inside (47), the three contributions split as follows. By Assumption A6, the per-sample sensing perturbation is zero-mean in a first-order sense, so γts contributes only to the variance. The computation contribution γtc has a nonzero conditional mean from client drift over the τ local steps on heterogeneous Fn,t , so it contributes to both the bias and the variance. The communication contribution ϵt has a conditional mean whose power is the channel misalignment term in (31) and a zero-mean residual from the MMSE channel estimation error and the receiver noise. C. Bounding the Bias and Variance We now bound each contribution in (47) using Assumptions A1–A6, separating bias and variance where appropriate. 1) Sensing contribution: For each accumulated sample in Dn,t , the noise variance is the one that prevailed in the round ℓ ≤ t when the sample was acquired: (m

2 σv,n,ℓ =

)

c1,nn,ℓ . psn,ℓ qn,ℓ

(50)

s clean Let γn,t ≜ ∆θn,t − ∆θn,t . Since the actual and clean local updates use the same mini-batch indices, comparing their SGD recursions and applying A1 and A5 with ητ L ≤ 1/2 gives t

s E γn,t

2

≤ 4η 2 τ 2

csn X 2 Dn,ℓ σv,n,ℓ . Dn,t ℓ=1

(51)

7

ζt =

N X Dn,t n=1

Dt

|

N N
X
X Dn,t Dn,t clean clean c n,t − ∆θn,t + ∆θn,t + ητ ∇Ft (θt ) ∆θ ∆θn,t − ∆θn,t + Dt Dt n=1 n=1 {z } | {z } | {z } ≜ γts

≜ ϵt

The factor 4 follows Pτ −1by unrolling the noisy–clean SGD recursion and using i=0 (1 + ηL)i ≤ τ eηLτ < 2τ under ητ L ≤ 1/2. Aggregating across devices with weights αn,t = Dn,t /Dt and applying Jensen’s inequality, E∥γts ∥2 ≤ 4η 2 τ 2

N X Dn,t n=1

Dt

t

csn ·

1 X 2 Dn,ℓ σv,n,ℓ ≜ Ets . Dn,t ℓ=1 (52)

The sensing floor depends on the entire history of sensing parameters used to acquire samples in Dn,t , not just on the round-t choices. 2) Computation contribution: The computation term γtc measures the deviation of the noise-free τ -step local SGD from the ideal full-gradient reference −ητ ∇Ft (θt ). We split it into a bias from client drift over heterogeneous Fn,t and a zero-mean variance from mini-batch sampling. Lemma 1 (Local-step drift bound). Under Assumptions A1 and A3 with the stepsize ητ L ≤ 1/2, the conditional mean of γtc satisfies 2 E[γtc | θt ] ≤ L2 η 4 τ 4 G2 .

(53)

Proof: See Appendix A.

Proof: See Appendix B. Combining Lemmas 1 and 2 via E∥γtc ∥2 = ∥E[γtc |θt ]∥2 + E∥γtc − E[γtc |θt ]∥2 , we have E∥γtc ∥2 ≤ L2 η 4 τ 4 G2 + 3η τ

N X Dn,t

Dt n=1

≜ γtc

power. The first term in MSEut is the channel misalignment error controlled by (vt , pun,t ) jointly, and the second term is the variance from the MMSE channel estimation residual and receiver noise. 4) Total perturbation power: Lemma 3 (Tightened total perturbation bound). Under Assumptions A1–A6 and the stepsize ητ L ≤ 1/2, with the sensing perturbation interpreted to first order as in A6, the second moment of the aggregate perturbation ζt satisfies E∥ζt ∥2 ≤ 2Ets + 2Etc + 2Etu .

(57)

Proof: Let Ξt denote the complete mini-batch index sequence in round t. Under the first-order sensing model of A6, conditioning on θt , the clean data, and Ξt and linearizing the noisy–clean local recursion gives E[γts | θt , Ξt ] = 0. Since γtc is fixed under this conditioning, the tower property yields E[γts⊤ γtc | θt ] = 0. Applying Young’s inequality to ζt = (γts + γtc ) + ϵt gives ∥ζt ∥2 ≤ 2∥γts + γtc ∥2 + 2∥ϵt ∥2 .

(58)

Taking expectations and using the above orthogonality, E∥ζt ∥2 ≤ 2E∥γts ∥2 + 2E∥γtc ∥2 + 2E∥ϵt ∥2

Lemma 2 (Mini-batch variance propagation). Under Assumptions A1, A3, and A4 with the stepsize ητ L ≤ 1/2, the variance of γtc around its conditional mean satisfies ! N 2 2 X σpop Dn,t σB 2 2 c c E γt − E[γt | θt ] ≤ 3η τ + . Dt B Dn,t n=1 (54)

2

(47)

2 2 σpop σB + B Dn,t

! ≜ Etc .

(55)

The bound exhibits three regimes: a local-step drift term controlled by ητ , a mini-batch variance controlled by B, and an empirical-to-population variance that vanishes as Dn,t grows. 3) Uplink OTA contribution: The OTA aggregation produces a noisy estimate of the weighted sum gt = P n αn,t ∆θn,t through the per-coordinate model in Section II-F. Under the normalized update model in (30), the per-coordinate MSE in (31) aggregates over d coordinates to E∥ϵt ∥2 = d · MSEut ≜ Etu .

(56)

The factor d converts the per-coordinate aggregation error in (31) into a model-dimension scaled gradient perturbation

≤ 2Ets + 2Etc + 2Etu .

(59)

This completes the proof. Define the per-round learning penalty as Ψt ≜ 2Ets + 2Etc + 2Etu ,

(60)

which gathers all the sensing, computation, and communication effects in round t. The penalty is fully expressed in terms of system design variables: sensing history {(mn,ℓ , psn,ℓ , qn,ℓ , Dn,ℓ )}ℓ≤t , local SGD configuration b n,t (τ, B, η), uplink powers pun,t , MMSE channel estimates h 2 with error variances σe,n,t , and the BS combining vector vt . With the per-round penalty Ψt in place, we now state the main convergence result. Theorem 1. Under Assumptions A1–A6 and the stepsize condition ητ L ≤ 1/2, the iterates generated by the proposed scheme satisfy
T −1  4 E[F0 (θ0 )] − Finf 1 X  2 E ∥∇Ft (θt )∥ ≤ T t=0 ητ T T −1 T −1  4 2L  1 X 4 X + Ψ + δt , (61) t η2 τ 2 ητ T t=0 ητ T t=0   where δt ≜ E Ft+1 (θt+1 )−Ft (θt+1 ) is the per-round objective drift induced by the dataset accumulation P from round t to round t+1, bounded under A2 by δt ≤ ∆F n Dn,t+1 /Dt+1 .

+

Proof: See Appendix C.

8

Inequality (61) decomposes the average squared gradient norm into three parts. The first term is the optimization term, decaying as O(1/T ) from the initialization gap. The second is the per-round learning penalty Ψt , which collects the sensing, computation, and communication distortions and does not vanish with T ; it sets the accuracy floor that the system design controls. The third is the objective-drift term δt , which arises because the global objective Ft changes as devices accumulate freshly sensed data. Each term in the penalty maps to a distinct design lever: Ets decreases with improved sensing quality through higher sensing power and resolution; Etc shrinks with larger mini-batches, smaller local horizons, and larger datasets that reduce the empirical-topopulation gap; and Etu shrinks with better channel estimation, uplink power control, and combining. The drift δt reflects the cost of changing the learning objective by sensing new data, and is bounded under A2 by the per-round change in the dataset composition. P Remark 4. The drift δt ≤ ∆F n Dn,t+1 /Dt+1 is set by the fraction of newly sensed data appended in round t + 1. As the accumulated datasets grow, this fraction vanishes and the bound recovers the standard fixed-objective non-convex rate. IV. P ROBLEM F ORMULATION AND S OLUTION A PPROACH In this section, we first formalize the optimization problem arising from the interaction of the sensing, computation, and communication modules described earlier. We then develop a structured solution method that exploits the problem’s separability and convexity properties. Each communication round contributes to the global model update and is affected by three distortion sources: sensing noise, local computation limitations, and uplink OTA aggregation. These effects are captured through the per-round penalty Ψt , given in (60), in which Ets , Etc , and Etu follow from the convergence bounds in Section III. Improvements in sensing quality decrease Ets , tighter SGD variance control decreases Etc , and better OTA aggregation at the server reduces Etu . A. Joint Optimization Problem Recall the total latency and energy requirements calculated in (34) and (35), respectively. To balance energy consumption, latency, and learning quality, we consider a weighted per-round objective Jt = ωE Et + ωT Tt + ωL Ψt ,

(62)

Let Θ ≜ {Θt }Tt=1 . The joint optimization problem over T rounds is T X min Jt (64a) Θ

s.t.

t=1

0 < psn,t ≤ Pns,max ,

(64b)

0 ≤ pun,t ≤ Pnu,max ,

(64c)

c fnmin ≤ fn,t ≤ fnmax , Dn,t ∈ {1, . . . , Dnmax },

(64d) (64e)

qn,t ∈ Qn , (m) zn,t ∈ {0, 1},

(64f) X

(m) zn,t = 1,

(64g)

m∈Mn

vt ∈ CM .

(64h)

The problem is non-convex due to the OTA aggregation term, the coupling between sensing parameters and learning performance, and the discrete nature of sample count, modality, and resolution choices. Since the rounds couple only through the accumulated dataset sizes determined by past decisions, we solve (64) in a per-round manner, minimizing Jt over Θt given the history from rounds ℓ < t. To develop a tractable solution, we group variables with similar functional roles as  Θ = Θsens , Θcomp , Θcomm , Θdisc , (65) where  s  Θsens = {pn,t }n,t  Θ c comp = {fn,t }n,t Θcomm = {pun,t }n,t ∪ {vt }t    (m) Θdisc = {Dn,t , qn,t , zn,t }n,t

.

(66)

Each block has a distinct functional role, and when the other blocks are fixed, the subproblem for each block becomes either convex or low-dimensional. B. Alternating Optimization Framework We adopt an alternating optimization (AO) method that updates one block at a time while holding the others fixed. This approach is effective because each block leads to a simplified subproblem that is either solvable in closed form or via a onedimensional search. (m) 1) Sensing power update: With (Dn,t , qn,t , zn,t ) fixed, the sensing contribution to Jt reduces to a strictly convex function of psn,t : (m

where the positive weights (ωE , ωT , ωL ) are set based on system priorities. The overall optimization problem then minimizes the sum of these per-round objectives over all communication rounds. For each communication round t ∈ {1, . . . , T }, we collect all device- and BS-side optimization variables into  N (m) c Θt ≜ Dn,t , qn,t , zn,t , psn,t , fn,t , pun,t n=1 ∪ {vt }. (63) All of these variables are defined in Section II. Uplink pilot power is not optimized and is assumed fixed during all channel estimation slots.

)

c1,nn,t s 2 s ωE En,t + ωL λsn,t σv,n,t = ωE psn,t Tn,t + ωL λsn,t s , pn,t qn,t (67) where λsn,t ≜ 8η 2 τ 2 csn Dn,t /Dt is the per-device sensing coefficient in round t from (52). Setting the derivative of (67) to zero yields the closed-form update v u u ωL λs c(mn,t ) n,t 1,n s⋆ pn,t = t , (68) s ωE qn,t Tn,t followed by projection onto (0, Pns,max ].

9

2) CPU frequency update: The CPU frequencies are coupled through the maximum latency in (34). Introducing an auxiliary bottleneck latency χt , the CPU frequency subproblem is min

c }N {fn,t n=1 , χt

s.t.

ωE κc κτ B

N X

c fn,t

2

+ ωT χt

n=1 s Tn,t +

(69)

κτ B ≤ χt , c fn,t

c fnmin ≤ fn,t ≤ fnmax ,

∀n.

For a fixed feasible χt , the energy-minimizing frequency is   κτ B c⋆ fn,t (χt ) = max fnmin , . (70) s χt − Tn,t Hence, χt is obtained from the one-dimensional convex problem N X
2 c⋆ χ⋆t = arg min ωE κc κτ B fn,t (χt ) + ωT χt , χmin ≤χt ≤χmax t t

n=1

(71)

s where χmin ≜ maxn
Tn,t + κτ B/fnmax and χmax ≜ t t min s maxn Tn,t + κτ B/fn . The scalar problem can be solved c⋆ = efficiently by one-dimensional search, after which fn,t ⋆ c⋆ fn,t (χt ).

3) Uplink power and combining vector update: For fixed uplink powers, the OTA distortion is quadratic in vt , yielding the linear MMSE-type update
−1 H vt = AH At αt , (72) t At + γ t I p u b p u b H where At = [ p1,t h1,t , . . . , pN,t hN,t ] , αt = P 2 + σz2 . [α1,t , . . . , αN,t ]H , and γt = n pun,t σe,n,t Given vt , each device solves a one-dimensional optimization over 0 ≤ pun,t ≤ Pnu,max , which is a bounded subproblem that can be handled efficiently by line search or projected Newton updates. 4) Discrete variable updates: We now describe how to (m) update the discrete variables (Dn,t , qn,t , zn,t ) for each device. The sensed sample count is bounded as 1 ≤ Dn,t ≤ Dnmax , and we write the candidate set as Dn ≜ {1, 2, . . . , Dnmax },

(73)

so that the discrete design variables at device n are m ∈ Mn , q ∈ Qn , and D ∈ Dn . When updating device n in round t, all other variables are held fixed. For a candidate count D, the accumulated dataset size is Dn,t =PDn,t−1 + D from (1), and the network total is Dt = Dn,t + j̸=n Dj,t . For a candidate triple (m, q, D), the sensing latency and energy are s Tn,t (m, q, D) =

qD

, (m) rs,n s s En,t (m, q, D) = psn,t Tn,t (m, q, D),

(74)

Algorithm 1: Alternating Optimization for Joint Sensing–Computation–Communication Design Input: System parameters, feasible sets {Qn , Dn }N n=1 , modality set Mn , weights (ωE , ωT , ωL ), and constraints. Output: Per-round decisions {Θt }Tt=1 . (0) T 1 Initialize {Θt }t=1 2 for t = 1, . . . , T do 3 k←0 4 repeat s,(k+1) N 5 Update {pn,t }n=1 by (68) and project onto s,max (0, Pn ] (k+1) c,(k+1) N 6 Update χt by (71) and {fn,t }n=1 by min (70) and project onto [fn , fnmax ] (k+1) 7 Update vt by (72) u,(k+1) N 8 Update {pn,t }n=1 by solving the 1-D subproblems and projecting onto [0, Pnu,max ] 9 for n = 1, . . . , N do ⋆ disc 10 (m⋆ , q ⋆ , D ) ← arg min Jn,t (m, q, D) ⋆ (m⋆ ) (k+1) 11 Set (zn,t , qn,t , Dn,t ) ← (1, q ⋆ , D ), (m) with other zn,t = 0 12 k ←k+1 13 until Jt converges or k = kmax (k) 14 Θt ← Θt

Define the bottleneck latency among all
devices except des c vice n as χ−n,t ≜ maxj̸=n Tj,t + Tj,t . The local surrogate discrete cost at device n in round t is
disc s c Jn,t (m, q, D) ≜ ωE En,t (m, q, D) + En,t  c s + ωT max χ−n,t , Tn,t (m, q, D) + Tn,t  λc  2 (77) + ωL λsn,t σv,n,t (m, q) + t , Dt P 2 where Dt = Dn,t−1 + D + j̸=n Dj,t , and λct ≜ 6η 2 τ N σpop from (55). The first two terms account for the sensing and computation energy and the round bottleneck latency, while the last term reflects the impact of sensing noise and the accumulated sample count on the round-t learning penalty Ψt . The discrete update at device n is the finite optimization ⋆

⋆ (m⋆n,t , qn,t , Dn,t ) =

arg min

disc Jn,t (m, q, D). (78)

m∈Mn , q∈Qn , D∈Dn

This rule specifies how the modality, resolution, and number of newly sensed samples are chosen. Since Mn , Qn , and Dn are finite and typically small, the optimization is carried out by direct search over all candidate triples (m, q, D). The overall AO procedure for the sensing, computation, and communication co-design is summarized in Algorithm 1.

(75)

2 and the sensing noise variance σv,n,t (m, q) is given in (13). The computation latency and energy follow the mini-batch model in (16)–(17) and are independent of D: κτ B c c c 2 Tn,t = c , En,t = κc κτ B(fn,t ) . (76) fn,t

C. Computational Complexity We analyze the per-round cost of Algorithm 1. Let K denote the number of AO iterations per round, and let Ic and Iu denote the numbers of iterations used by the one-dimensional searches for χt and pun,t , respectively.

10

(a) Modality training loss (MNIST).

(b) Test accuracy (CIFAR-10), i.i.d.

(c) Test accuracy (CIFAR-10), non-i.i.d.

Fig. 2: Simulation results with the MNIST and CIFAR-10 datasets. (a) Training loss for the three sensing modalities in Table I. (b) Test accuracy for the proposed scheme and baselines with i.i.d. data partitioning. (c) Same as (b) for non-i.i.d. partitioning. Within one AO iteration, the five blocks scale as follows. The sensing power update is closed-form and costs O(N ), while the CPU frequency block requires a one-dimensional search over χt with O(N ) cost per search iteration, yielding O(N Ic ). The combining vector vt requires constructing an M × M Gram matrix from the N estimated channels and solving one linear system, costing O(N M 2 +M 3 ). The uplink power update solves a bounded p one-dimensional problem per device (convex in the amplitude pun,t ), costing O(N Iu ). The discrete update enumerates the triple (m, q, D) for each device over Mn × Qn × Dn , costing O(N |M||Q||D|), where |M| = maxn |Mn |, |Q| = maxn |Qn | and |D| = maxn |Dn |. Over K AO iterations, the per-round optimization cost is 
 O K N Ic + N M 2 + M 3 + N Iu + N |M||Q||D| . (79) On top of this, each device runs τ local SGD steps on mini-batches of size B, costing O(N τ BCmodel ) across the network in a round, where Cmodel is the cost of a single forward-backward pass of the model. At the BS, the OTA aggregation produces a length-d estimate by applying vt to the received signal at cost O(M d) per round, and MMSE channel estimation from the pilot phase contributes O(N M πp ). V. N UMERICAL E XPERIMENTS A. Simulation Setup We conduct experiments using the MNIST [42], CIFAR10, and CIFAR-100 [43] datasets. For training on MNIST, we use the convolutional neural network (CNN) architecture with fully connected layers. For CIFAR-10 and CIFAR-100, we respectively employ the ResNet-18 and ResNet-20 architectures. Unless stated otherwise, M = 8, N = 20, τ = 5, 2 B = 16, σw = σz2 = 1, Dnmax = 100, Qn = {1, 2, 4}, min fn = 0.5 GHz, fnmax = 2 GHz, and Pns,max = 1 W for all devices. The average SNR is set to 10 dB for the downlink and uplink channels (both pilot and data phases). The downlink bandwidth is set to 10 MHz, and the uplink symbol rate is set to 10 Msymbol/s. Both i.i.d. and non-i.i.d. data distributions are considered across the devices. At each communication round t ∈ [T ], device n selects a sensing configuration consisting of a modality, resolution, power, and the number of newly sensed samples. The sensing process follows the model in Section II-C: clean samples

TABLE I: Sensing modality parameters used in simulations. Modality m 1 2 3

(m)

c1 1.0 0.5 0.2

(m)

rs 20 10 1

Description Low sensing quality, fast acquisition Moderate quality and acquisition speed High sensing quality, slow acquisition

drawn from the underlying dataset are corrupted according to the selected sensing configuration and then used for local training. The sensing modalities are modeled as abstract configurations with different quality–latency tradeoffs, as summa(m) rized in Table I, where the rate rs is measured in samples/s. This captures heterogeneous sensing capabilities without tying the simulations to specific sensor physics. Device sensing heterogeneity is incorporated by allowing each device to access only a subset of the available sensing modalities. Among the N = 20 devices, 6 have access to modality m = 1 only, 7 can choose between m = 1 and m = 2, and the remaining 7 have access to all three modalities. B. Results and Discussion Fig. 2(a) shows the training loss for the three sensing modalities in Table I on the MNIST dataset under i.i.d. data distribution with T = 200. Modality m = 3 achieves the fastest convergence and lowest steady-state loss, followed by m = 2 and m = 1. This confirms that higher sensing quality reduces data distortion, which improves local gradient estimation and speeds up convergence. Fig. 2(b) and Fig. 2(c) show the test accuracy versus communication rounds (T = 500) for the proposed scheme and benchmark schemes on the CIFAR-10 dataset under i.i.d. and non-i.i.d. data partitions. Several benchmark schemes are included to isolate the contribution of each resourcecontrol block in the proposed joint design. Specifically, fixed communication disables the communication-resource adaptation, fixed computation disables the CPU frequency adaptation, fixed sensing keeps the sensing configuration fixed (we assume m = 2 for all devices) throughout training, and ISCC optimizes the number of newly sensed samples along with the computation/communication parameters (the scheme given in [28]). The proposed joint sensing–computation– communication design achieves the highest accuracy among all practical schemes. For instance, in the non-i.i.d. case, the proposed scheme achieves approximately 18% higher accuracy

11

Test Accuracy

80

i.i.d. non-i.i.d.

60 40 20 0

] g 1) ing ation ation [28 nsin lg. ns t (A CC t se se nic pu u S d d I m c e e rfe os mm d co Fix co pe op e Pr .1/ ed Fix x g i l F A

(a) Training loss, i.i.d.

(c) Final test accuracy, T = 1000

(b) Test accuracy, non-i.i.d.

Fig. 3: Simulation results with the CIFAR-100 dataset. (a) Training loss for the proposed scheme and baselines with i.i.d. partitioning. (b) Test accuracy for the proposed scheme and baselines with non-i.i.d. partitioning. (c) Final test accuracy after T = 1000 communication rounds. The shaded regions and error bars indicate one standard deviation over three seeds. than the ISCC scheme in [28]. The proposed scheme with perfect sensing and perfect CSI serves as an oracle upper bound, corresponding to the ideal case in which the sensed data are free of measurement noise and the channel estimates are error-free. Fig. 3(a) shows the training loss versus communication rounds for the proposed scheme and benchmark schemes on the CIFAR-100 dataset under i.i.d. data partitioning, with T = 1000 rounds. The proposed joint design achieves the fastest loss reduction and the lowest steady-state training loss among all practical schemes, confirming its effectiveness on the more challenging CIFAR-100 task. The baselines exhibit higher loss because each optimizes only part of the sensing– computation–communication pipeline. Fig. 3(b) shows the test accuracy versus communication rounds on CIFAR-100 under non-i.i.d. data partitioning, where the proposed scheme consistently outperforms all practical baselines by jointly adapting sensing, computation, and communication resources. Fig. 3(c) compares the final test accuracy after T = 1000 rounds under both i.i.d. and non-i.i.d. data distributions. The proposed scheme achieves the highest final accuracy among all practical schemes in both cases, while the moderate gap between i.i.d. and non-i.i.d. results reflects the impact of data heterogeneity. For instance, the proposed scheme achieves approximately 12% higher accuracy than the closest practical benchmark under non-i.i.d. partitioning. VI. C ONCLUSION In this paper, we investigate a unified wireless edge-learning framework that jointly optimizes sensing, computation, and communication in federated learning. Unlike formulations that assume data is already available at the devices, the framework models the sensing process explicitly and captures the coupled effects of sensing quality, local computation, and OTA aggregation under imperfect CSI on the learning trajectory. We derive a non-convex convergence bound that exposes these effects and use it to formulate a joint resource allocation problem balancing learning quality, energy, and latency, which we solve through a low-complexity alternating optimization method. The results highlight sensing as a fundamental and controllable component of wireless edge learning, with a significant impact on learning performance, and provide a principled basis for

the joint design of sensing, computation, and communication under noisy multimodal sensing and imperfect CSI in future 6G edge-intelligence systems. A PPENDIX A P ROOF OF L EMMA 1 PN clean +ητ ∇Ft (θt ), where By definition, γtc = n=1 αn,t ∆θn,t clean αn,t = Dn,t /Dt and ∆θn,t is the result of τ noise-free minibatch SGD steps on Fn,t starting at θt . Taking conditional expectation over the mini-batch sampling at device n replaces the stochastic gradient by the empirical gradient: clean E[∆θn,t | θt ] = −η

τ X   (i) E ∇Fn,t (θn,t ) θt ,

(80)

i=1 (i)

where θn,t denotes the noise-free local iterate at step i (1) with θn,t = θt . Substituting into the definition of γtc and rearranging, E[γtc | θt ] = −η =η

N X

αn,t

n=1 τ X

τ X   (i) E ∇Fn,t (θn,t ) θt + ητ ∇Ft (θt ) i=1

∇Ft (θt ) −

N X

  (i) αn,t E ∇Fn,t (θn,t ) θt .

n=1

i=1

(81) Adding and subtracting each summand of (81),

P

n αn,t ∇Fn,t (θt ) = ∇Ft (θt ) inside

E[γtc | θt ] =η

τ X N X

   (i) αn,t ∇Fn,t (θt ) − E ∇Fn,t (θn,t ) θt . (82)

i=1 n=1

Taking norms, applying the triangle inequality, and using the L-smoothness of Fn,t (A1), E[γtc | θt ] ≤ η

τ X N X

 (i) αn,t L E θt − θn,t

 θt .

(83)

i=1 n=1

The local-iterate drift from the noise-free recursion is bounded (j+1) (j) (j) (j) by induction. At step j, θn,t = θn,t − η∇Fn,t (θn,t ; ξn,t ), so (j+1)

θt − θn,t

(j)

(j)

(j)

≤ θt − θn,t + η ∇Fn,t (θn,t ; ξn,t )

12

(j)

≤ θt − θn,t + ηG,

(84)

where the last step uses the bounded gradient norm in A3. (1) (i) Unrolling the recursion from θn,t = θt gives ∥θt − θn,t ∥ ≤ (i − 1)ηG ≤ ητ G for i ≤ τ . Substituting into (83) and squaring,
2 2 E[γtc | θt ] ≤ η · τ · L · ητ G = L2 η 4 τ 4 G2 , (85) ■

which is (53). A PPENDIX B P ROOF OF L EMMA 2 (i)

(i)

(i)

(i)

(i)

(i)

Let rn,t ≜ ∇Fn,t (θn,t ; ξn,t ) − E[∇Fn,t (θn,t ; ξn,t ) | θn,t ] denote the mini-batch noise at local step i on device n, (i) (i) conditional on the iterate θn,t . The sequence {rn,t }τi=1 forms a martingale-difference sequence conditioned on the local (i) iterates. By A4 and Jensen’s inequality, conditioning on θn,t , 2  (i) σpop σ2 (i)  E ∥rn,t ∥2 θn,t ≤ B + ≜ Vn,t . B Dn,t

(86)

2 (i+1) (i+1) θn,t − E[θn,t | θt ] 2 (i) (i) ≤ (1 + ηL)2 E θn,t − E[θn,t | θt ] + η 2 Vn,t .

(87)

(1)

Since θn,t = θt , unrolling the recursion gives clean clean E ∆θn,t − E[∆θn,t | θt ]

2

≤ η 2 Vn,t

− ητ ∇Ft (θt ) + ζt

τ −1 X

(1 + ηL)2i

i=0

≤ 3η 2 τ Vn,t ,

(88)

where the last inequality follows from (1 + ηL)2τ ≤ e2ηLτ ≤ e < 3 under ητ L ≤ 1/2. Aggregating across devices using the convex-combination structure and Jensen’s inequality,

clean clean αn,t E ∆θn,t − E[∆θn,t | θt ]

and collecting terms in (91),  Lητ  Ft (θt+1 ) ≤ Ft (θt ) − ητ 1 − ∥∇Ft (θt )∥2 2 L + (1 − Lητ )∇Ft (θt )⊤ ζt + ∥ζt ∥2 . (93) 2 We take the conditional expectation given θt . The perturbation splits as ζt = ζ̄t +ζt0 with E[ζt0 | θt ] = 0, where ζ̄t = E[ζt | θt ] collects the client-drift and OTA-misalignment biases. Hence   E ∇Ft (θt )⊤ ζt θt = ∇Ft (θt )⊤ ζ̄t . (94) For ητ L ≤ 1/2 we have 0 ≤ 1 − Lητ ≤ 1. Applying Cauchy– Schwarz and Young’s inequality with weight ητ /2,

≤ 3η τ

X

2

αn,t Vn,t .

Since ητ (1 − Lητ /2) ≥ ητ /2 for ητ L ≤ 1/2, the net coefficient of ∥∇Ft (θt )∥2 after subtracting the ητ /4 term in (95) is at most −ητ /4. Taking total expectation, ητ E[Ft (θt+1 )] ≤ E[Ft (θt )] − E∥∇Ft (θt )∥2 4 1 L + E∥ζ̄t ∥2 + E∥ζt ∥2 . (96) ητ 2 The bias norm satisfies ∥ζ̄t ∥2 ≤ E∥ζt ∥2 ≤ Ψt by Jensen’s inequality and Lemma 3, and E∥ζt ∥2 ≤ Ψt directly. Substituting both, ητ E[Ft (θt+1 )] ≤ E[Ft (θt )] − E∥∇Ft (θt )∥2 4  1 L + Ψt . (97) + ητ 2

(89) ■

A PPENDIX C P ROOF OF T HEOREM 1 By the L-smoothness of Ft (A1), Ft (θt+1 ) ≤ Ft (θt ) + ∇Ft (θt )⊤ (θt+1 − θt ) L + ∥θt+1 − θt ∥2 . (90) 2 Substituting the perturbed update θt+1 − θt = −ητ ∇Ft (θt ) + ζt from (45), Ft (θt+1 ) ≤ Ft (θt ) − ητ ∥∇Ft (θt )∥2 + ∇Ft (θt )⊤ ζt

(98)

Substituting into (97),

n

Substituting Vn,t yields (54).

1 ητ ∥∇Ft (θt )∥2 + ∥ζ̄t ∥2 . (95) 4 ητ

E[Ft+1 (θt+1 )] = E[Ft (θt+1 )] + δt .

n 2

= η 2 τ 2 ∥∇Ft (θt )∥2 − 2ητ ∇Ft (θt )⊤ ζt + ∥ζt ∥2 , (92)

n

≤

(91)

To telescope across rounds with the time-varying objective, we bridge Ft and Ft+1 . By the definition of the drift,

2

E γtc − E[γtc | θt ] X
2 clean clean ≤E αn,t ∆θn,t − E[∆θn,t | θt ] X

2

(1 − Lητ )∇Ft (θt )⊤ ζ̄t ≤

Since the local SGD mapping is (1 + ηL)-Lipschitz under A1, the conditional variance propagates as E

L 2 − ητ ∇Ft (θt ) + ζt . 2 Expanding the squared norm, +

ητ E[Ft+1 (θt+1 )] ≤ E[Ft (θt )] − E∥∇Ft (θt )∥2 4  1 L + Ψt + δ t . + ητ 2

(99)

Rearranging and summing over t = 0, . . . , T −1, the E[Ft (θt )] terms telescope: T −1

ητ X E∥∇Ft (θt )∥2 ≤ E[F0 (θ0 )] − E[FT (θT )] 4 t=0 +

 1 ητ

+

T −1 T −1 X L X Ψt + δt . (100) 2 t=0 t=0

Using E[FT (θT )] ≥ Finf from A2 and dividing by ητ T /4,
T −1 4 E[F0 (θ0 )] − Finf 1 X 2 E∥∇Ft (θt )∥ ≤ T t=0 ητ T

13

+

T −1 T −1  4 2L  1 X 4 X + Ψ + δt . (101) t η2 τ 2 ητ T t=0 ητ T t=0

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