On Provisioning Cellular Networks for Distributed Inference

TL;DR

This paper develops an analytical framework to understand how cellular network design impacts distributed ML inference accuracy at the network edge, highlighting the importance of access point density and bandwidth.

Contribution

It introduces a novel analytical model linking cellular network parameters to distributed inference accuracy, guiding network provisioning for edge ML applications.

Findings

Minimum AP density is needed for target inference accuracy.

Asymptotic accuracy is limited by air-interface bandwidth.

Edge inference accuracy depends inversely on AP density and bandwidth.

Abstract

Wireless traffic attributable to machine learning (ML) inference workloads is increasing with the proliferation of applications and smart wireless devices leveraging ML inference. Owing to limited compute capabilities at these "edge" devices, achieving high inference accuracy often requires coordination with a remote compute node or "cloud" over the wireless cellular network. The accuracy of this distributed inference is, thus, impacted by the communication rate and reliability offered by the cellular network. In this paper, an analytical framework is proposed to characterize inference accuracy as a function of cellular network design. Using the developed framework, it is shown that cellular network should be provisioned with a minimum density of access points (APs) to guarantee a target inference accuracy, and the inference accuracy achievable at asymptotically high AP density is limited by the air-interface bandwidth. Furthermore, the minimum accuracy required of edge inference to deliver a target inference accuracy is shown to be inversely proportional to the density of APs and the bandwidth.

Figures

Tables

Table 1. TABLE I: Notation and simulation parameters

Notation	Parameter	Value (if not specified
${\mathrm{𝚂𝙸𝙽𝚁}}_x$, ${\mathrm{𝚁𝚊𝚝𝚎}}_x$	Uplink (x:u), downlink (x:d) $\mathrm{𝚂𝙸𝙽𝚁}$ and $\mathrm{𝚁𝚊𝚝𝚎}$
$m_d$, $m_c$, $m_t$, $m$	Edge, cloud, target, and average inference MSE	$m_d=1.5m_c$
$\lambda$, ${\lambda }_u$	density of APs and devices
$b$	uplink transmission bandwidth
$N$	number of uplink devices in AP serving the typical device
$d_t$, $d_c$, $D$	target delay budget, cloud compute delay, and cloud inference delay
$q$	cumulative (uplink and downlink) size of cloud inference input and output
$r_{\min }$	inference rate	$\frac{q}{b\left(d_t-d_c\right)}$

Equations

$${\mathtt{SINR}}_u=\frac{H_{0,{\mathcal{B}}_0}}{{\mathtt{SNR}}^{-1}+{\sum }_{X\in {\Phi }_u^b}{L}_X{H}_{X,{\mathcal{B}}_0}L(X,{\mathcal{B}}_0{)}^{-1}},$$

$${\mathtt{Rate}}_u=\frac{b}{N}\log \left(1+{\mathtt{SINR}}_u\right),$$

$$y_o=\{\begin{array}{ll}{y}_c,& \text{ if }D\le d_t,\\ y_d,& \text{ otherwise, }\end{array}$$

$$D=\frac{q}{2{\mathtt{Rate}}_u}+\frac{q}{2{\mathtt{Rate}}_d}+d_c,$$

$$m_d=\mathbb{E}\left[(y-y_d{)}^2\right]\text{ and }{m}_c=\mathbb{E}\left[(y-y_c{)}^2\right],$$

$$\bar{m}\triangleq \mathbb{E}\left[\text{mathbbm}1(D\le d_t)m_c\right]+\mathbb{E}\left[\text{mathbbm}1(D>d_t)m_d\right],$$

$$D=\frac{q}{{\mathtt{Rate}}_u}+d_c.$$

$$N\approx \bar{n}=1+\frac{1.28}{\hat{\lambda }},$$

$$\begin{gathered} \mathcal{D}(d)\triangleq \mathbb{P}(D\le d)=\exp \left(-\mathrm{C}\circ \mathrm{T}\left(\bar{n}{r}_{\min }\frac{d_t-d_c}{d-d_c}\right)\right) \\ \forall d>d_c, \end{gathered}$$

$$\bar{m}=m_d-(m_d-m_c)\exp \left(-\mathrm{C}\circ \mathrm{T}\left(\bar{n}{r}_{\min }\right)\right).$$

$$m_{\mathrm{asy}}$$

$$\text{Also }\underset{r_{\min }\to \infty }{\lim }{m}_{\mathrm{asy}}=m_d\text{ and }\underset{r_{\min }\to 0}{\lim }{m}_{\mathrm{asy}}=m_c.$$

$$\begin{gathered} {\lambda }_c\triangleq 1.28{\lambda }_u{\left\{\frac{1}{r_{\min }}{\log }_2\left(1+{\mathrm{C}}^{-1}\left[\log \left(\frac{m_d-m_c}{m_d-m_t}\right)\right]\right)-1\right\}}^{-1}, \\ \forall m_t>m_{\mathrm{asy}}. \end{gathered}$$

$$m_{d,\max }\triangleq m_c\frac{m_t/m_c-\exp \left(-\mathrm{C}\circ \mathrm{T}\left(\bar{n}{r}_{\min }\right)\right)}{1-\exp \left(-\mathrm{C}\circ \mathrm{T}\left(\bar{n}{r}_{\min }\right)\right)}$$

$$\mathbb{P}(D\le d)=\mathbb{P}\left({\mathtt{Rate}}_u\ge \frac{q}{d-d_c}\right)$$

$$\stackrel{(a)}{=}\mathbb{P}\left({\mathtt{SINR}}_u\ge \mathrm{T}\left(\frac{\bar{n}q}{b(d-d_c)}\right)\right)$$

$$\stackrel{(b)}{=}\exp \left(-\mathrm{C}\circ \mathrm{T}\left(\frac{\bar{n}q}{b(d-d_c)}\right)\right),$$

$$\begin{gathered} \bar{m}\le m_t \\ \stackrel{(a)}{⟹}{m}_d-(m_d-m_c)\exp \left(-\mathrm{C}\circ \mathrm{T}\left(\bar{n}{r}_{\min }\right)\right)\le m_t, \\ \text{ or }\mathrm{T}\left(\bar{n}{r}_{\min }\right)\le {\mathrm{C}}^{-1}\left(\log \left(\frac{m_d-m_c}{m_d-m_t}\right)\right), \end{gathered}$$

Full text

I Introduction

With the proliferation of machine learning (ML) enabled applications (e.g. Siri, GoogleHome, Alexa, etc.) and wireless devices (e.g. smart speakers, phones, wireless cameras), the traffic attributable to such applications is bound to increase [1]. These applications gather data, like voice commands or security videos, through smart devices and run inference leveraging ML model(s)111The term “model” refers to a parametric mapping function (e.g. neural network, decision tree, random forest, etc.) fitted using a data driven training procedure., which are becoming increasingly computationally expensive. Owing to the limited compute, power, and storage capabilities at these devices or “edge”, achieving high inference accuracy with low delay is challenging [2]. As a result, offloading of inference is attractive, wherein a partial (or full) inference may run on a remote compute resource (with more computational power) or “cloud”. Wireless cellular network, being the communication medium, plays a key role in enabling such a coordination between edge and cloud.

Providing seamless end-user experience for these applications creates unprecedented challenges for wireless network operators. Cellular network’s rate and reliability, being inherently bounded, has to be appropriately provisioned to meet the delay and accuracy demands of this distributed inference paradigm. Moreover, the application developers also need to be cognizant of the uncertainty added by the communication delay to the inference accuracy and, in turn, application performance, and design the edge model appropriately.

Most of the prior work in cellular network design for computation offload has focussed on the design of offloading strategies aiming to optimize for a myriad of objectives (see [3, 4] for a survery). The work in [5] proposed a stochastic geometry based model for a wireless mobile edge computing network and characterized the average computation and communication latency as function of network parameters. The analysis was further extended by [6], which considered the impact of user association in a heterogeneous cellular network on the communication/compute delay distribution. The work in [7] characterized the impact of these delays on energy consumption. None of the prior works, however, characterized the impact of cellular network design on inference accuracy achievable in a distributed inference framework, and the consequent inter-play in provisioning of cellular network and the edge inference accuracy. This paper is aimed to bridge this gap.

This paper proposes a tractable approach to characterize the impact of cellular network design on application inference performance for distributed inference. In particular, application inference accuracy, as measured by mean squared error (MSE)222Inference MSE implies the inverse of accuracy throughout this paper., is derived as a function of the key system parameters: density of the wireless access points (APs), transmission bandwidth, edge and cloud model’s accuracy. Using the developed framework,

• •

it is shown that average inference accuracy improves with increasing AP density, but saturates at a level inversely proportional to the bandwidth.

• •

the minimum AP density required to achieve a target inference accuracy for a given edge and cloud inference accuracy is analytically characterized, and shown to decrease with bandwidth.

• •

the minimum edge model accuracy required to guarantee an overall target inference accuracy is derived and it is shown that higher AP density and/or bandwidth allows application developer to use a less accurate edge model, while guaranteeing the overall target inference accuracy.

II System Model

The APs are assumed to be distributed uniformly in $2$ as a homogeneous Poisson Point process (PPP) $\Phi$ of density ${\lambda}$. The devices in the network are assumed to be distributed according to an independent homogeneous PPP $\Phi_{u}$ with density $\lambda_{u}$. The power received from an AP at $X\in 2$ transmitting with power $\mathrm{P}_{X}$ at a device at $Y\in 2$ is $\mathrm{P}_{X}H_{X,Y}L(X,Y)^{-1}$, where $H\in+$ is the fast fading power gain and assumed to be Rayleigh distributed with unit average power, i.e., $H\sim\exp(1)$, and $L(X,Y)\triangleq S_{X,Y}\|X-Y\|^{4}$, where $S\in+$ denotes the large scale fading (or shadowing). Both small and large scale fading are assumed i.i.d across all device-AP pair. The analysis in this paper is done for a typical device located at the origin.

II-A Uplink $\mathtt{SINR}$ and $\mathtt{Rate}$

Let $L_{X}$ be the path loss between the device at $X\in 2$ and its serving AP. A full pathloss-inversion based power control is assumed for uplink transmission, where a device at $X$ transmits with a power spectral density (dBm/Hz) $\mathrm{P}_{X}=\mathrm{P}_{u}L_{X}$, and $\mathrm{P}_{u}$ is the open loop power spectral density. Orthogonal access is assumed in the uplink and hence at any given resource block, there is at most one device transmitting in each cell. Let $\Phi_{u}^{b}$ be the point process denoting the location of devices transmitting on the same resource as the typical device. The uplink $\mathtt{SINR}$ of the typical device (at origin) on a given resource block is

[TABLE]

where $\mathcal{B}_{0}$ denotes the AP serving the typical device, $\mathtt{SNR}\triangleq\frac{\mathrm{P}_{u}\mathrm{L_{0}}}{{\mathrm{N}_{0}}}$ with $\mathrm{N}_{0}$ being the thermal noise spectral density, and $\mathrm{L_{0}}$ is the free space path loss at a reference distance. Every device is assumed to be using minimum path loss for association and is assumed that each AP has at least one device with data to transmit in uplink. Assuming an equal partitioning of the total uplink resources at an AP among the associated uplink users, the uplink rate of the typical device is

[TABLE]

where $b$ is the uplink bandwidth, ${N}$ denotes the total number of devices served by the AP. Along similar lines (as in [8]), downlink rate $\mathtt{Rate}_{d}$ can be defined.

II-B Inference Framework

An inference framework is assumed wherein, for each inference input (e.g. a chunk of speech or an image), denoted by $x$, the device transmits the inference input to the cloud while, concurrently, computing a local inference output (say $y_{d}$) using the edge model. If the device receives the inference result from the cloud (say $y_{c}$) within target delay budget (denoted by $d_{t}$), it is used as the final output $y_{o}$; otherwise the device uses the edge model’s output $y_{d}$. Therefore,

[TABLE]

where $D$ denotes the cumulative delay incurred in receiving $y_{c}$ at the device. Assuming inference input and output has a fixed (over the air) payload size, i.e. $|x|=|y|=q/2$, the cloud inference delay is

[TABLE]

where the first two terms correspond to communication delays (in uplink and downlink respectively) and $d_{c}$ is the compute delay (assumed to be fixed333the communication delay associated with the cloud transport network is assumed to be incorporated in $d_{c}$.) incurred by the cloud inference model.

Denoting the actual inference output by $y$, device and cloud model inference accuracy are defined by their mean square errors (MSE’s),

[TABLE]

respectively, where the expectation is over the data distribution, and cloud model’s accuracy is assumed to be more accurate than that of edge model, i.e., $m_{c}\leq m_{d}$.

As a result of the inference mechanism in (3), the average MSE (denoted by $\bar{m}$) for a typical device is

[TABLE]

where $\mathbbm{1}(A)$ denotes the indicator of the event $A$.

The notation used in this paper is summarized in Table I.

III Inference Accuracy

The accuracy of the distributed inference model is characterized as a function of the network parameters in this section. The following two assumptions are taken to simplify the analysis.

Assumption 1**.**

The downlink rate of a typical device is assumed to be equal to that of the uplink.

In [8], it was shown that downlink rate stochastically dominates uplink rate, hence the above assumption leads to over-estimation of the cloud inference delay in (4), which simplifies to

[TABLE]

Using (3) and (6), the minimum uplink rate required for a device to use cloud inference, i.e. $y_{o}=y_{c}$, is $\frac{q}{d_{t}-d_{c}}$. Henceforth, this minimum rate normalized by the transmission bandwidth, $r_{\mathrm{min}}\triangleq\frac{q}{b(d_{t}-d_{c})}$, is referred to as the inference rate.

Assumption 2**.**

The load on the AP serving the typical device is assumed to be constant and equal to its average value (denoted by $\bar{n}$), i.e.,

[TABLE]

where ${\hat{\lambda}}\triangleq{\lambda}/\lambda_{u}$.

This assumption was taken in past works (see [8] and references therein) without loss of generality of design insights.

Lemma 1**.**

Delay distribution. The cloud inference delay distribution experienced by a typical device is

[TABLE]

where $\mathrm{C}(x)=\sqrt{x}\arctan(\sqrt{x})$ and $\mathrm{T}(x)=2^{x}-1$.

Proof.

See Appendix A. ∎

Since both $\mathrm{C}$ and $\mathrm{T}$ are monotonically increasing function, cloud inference delay is proportional to the inference rate and average load.

Lemma 2**.**

Average MSE. The average output inference MSE of a typical device is

[TABLE]

Proof:

Follows by using Lemma 1 with (5). ∎

As can be observed from Lemma 2, $m_{c}\leq\bar{m}\leq m_{d}$. The term on the right captures the average MSE improvement provided by the cloud inference. This improvement diminishes with increasing inference rate ($r_{\mathrm{min}}$) and higher device load ($\bar{n}$). The following section further formalizes these insights.

IV Performance analysis and insights

Figure 1 shows the variation of average inference MSE normalized by the cloud MSE (i.e. $\bar{m}/m_{c}$) with the normalized AP density (${\hat{\lambda}}$) and inference rate444the values of system parameters not specified explicitly are as per Table I.. As observed, for a given AP density average inference MSE increases with $r_{\mathrm{min}}$ (or decreases with bandwidth) and approaches that of the edge, as with the increase in communication delay device has to rely on the edge model output. Moreover, average MSE decreases with increasing AP density, but saturates eventually. This is formalized with the following.

Corollary 1**.**

Asymptotic MSE. The average output MSE at asymptotically high AP density is

[TABLE]

Proof.

Follows by replacing $\lim_{{\lambda}\to\infty}\bar{n}=1$ in Lemma 2. ∎

This shows that at high AP density, the gain in MSE from cloud inference is limited by the inference rate, as even if the entire bandwidth is allocated to a single device, the cloud inference delay may not be lower than the delay budget. However, increasing bandwidth or decreasing inference rate reduces this MSE.

[TABLE]

At low inference rate, the asymptotic MSE approaches that of the cloud model; and that of the edge at high inference rate. Figure 2 shows the variation of normalized asymptotic MSE ($m_{\mathrm{asy}}/m_{c}$) between these extremes.

Corollary 2**.**

Critical density. The minimum AP density to guarantee an inference accuracy ($m_{t}$) is

[TABLE]

Proof.

See Appendix B. ∎

Figure 3 shows the variation of normalized critical density (${\lambda}_{c}/\lambda_{u}$) with the normalized target MSE $m_{t}/m_{c}\,\forall\,\,m_{t}\text{ s.t. }m_{\mathrm{asy}}<m_{t}\leq m_{d}$. As can be seen, the network needs to be provisioned with higher AP density with increasing accuracy demands. Moreover, critical density also increases with increase in inference rate (or decrease in bandwidth) for a target MSE.

As highlighted by Corollary 1, even at high infrastructure density the gain from cloud inference may be limited – guaranteeing an overall accuracy, thus, requires a minimum accuracy from the edge inference model.

Corollary 3**.**

Critical edge MSE. The maximum allowed device MSE required to guarantee an overall MSE lower than $m_{t}$ is

[TABLE]

Proof.

See Appendix B. ∎

From above, it is evident that as $\bar{n}r_{\mathrm{min}}\to\infty$, $m_{d,\mathrm{max}}\to m_{t}$, or as the cloud inference delay increases beyond the target delay, the edge inference accuracy needs to be at least at par with the target accuracy. Figure 4 shows the variation of normalized critical edge MSE ($m_{d,\mathrm{max}}/m_{c}$) for different system parameters. It can be seen that, for any given inference rate, higher AP density allows edge inference model to be less accurate, while meeting the overall target inference accuracy. Corollary 3 and the aforementioned insights emphasize the constraints imposed by the network parameters on the design of edge inference model.

V Conclusion

This paper proposes an analytical framework for characterizing the performance of distributed inference in wireless cellular networks. To the author’s best knowledge, this is the first work to present the trade-offs involved in the co-design of cellular networks and distributed inference. This work can be extended to analyze and compare different policies for offloading inference and their impact on network design. The network model can be extended to analyze the impact of congestion in the cloud transport network (as in [5]) and user association in heterogeneous cellular networks (as in [6]) on distributed inference.

Appendix A

Proof:

[TABLE]

where (a) follows using (2) and (7) and $\mathrm{T}(x)=2^{x}-1$; and (b) follows by using uplink $\mathtt{SINR}$ distribution from [8]. ∎

Appendix B

Proof:

For the average inference MSE to be less than a target, i.e.,

[TABLE]

where (a) follows using Lemma 2. The critical density is arrived at by replacing (7) in above. ∎

Proof:

Follows by algebraic manipulation on (a) above.∎