FL-MAC-RDP: Federated Learning over Multiple Access Channels with Rényi Differential Privacy

Abstract

Federated Learning (FL) is a promising paradigm, where the local users collaboratively learn models by repeatedly sharing information while the data is kept distributing on these users. FL has been considered in multiple access channels (FL-MAC), which is a hot issue. Even though FL-MAC has many advantages, it is still possible to leak privacy to a third party during the whole training process. To avoid privacy leakage, we propose to add Rényi differential privacy (RDP) into FL-MAC. At the same time, to maximize the convergent rate of users under the constraints of transmission rate and privacy, the quantization stochastic gradient descent (QSGD) is performed by users. We also illustrate our results on MNIST, and the illustration demonstrate that our scheme can improve the model accuracy with a little loss of communication efficiency.

Appendix A: Proof of Theorem 1

Proof

Since for all $\omega , \omega ^{\prime } \in {R^{d}}$, $\bar l$ is λ-strongly convex, then for any subgradient g of $\bar l$ at ω we have

$$ \bar l(\omega^{\prime}) - \bar l(\omega) \geq \langle g, \omega^{\prime} - \omega \rangle + \frac{\lambda}{2} \left\| \omega^{\prime} - \omega \right\|^{2} . $$

(18)

such that

$$ \begin{array}{@{}rcl@{}} \langle {g_{b}^{k}}, \omega_{b} - \omega \rangle & = &-\langle {g_{b}^{k}}, \omega - \omega_{b} \rangle \\ &\geq &\bar l(\omega_{b}) - \bar l(\omega^{*}) +\frac{\lambda}{2} \left\| \omega_{b} - \omega^{*} \right\|^{2} \\ &\geq& 0 . \end{array} $$

(19)

Denote ω^∗ is the optimum value of ω_b, and

$$ \begin{array}{@{}rcl@{}} \bar l(\omega_{b}) - \bar l(\omega^{*}) & \geq& \langle g^{*}, \omega_{b}-\omega^{*} \rangle +\frac{\lambda}{2} \left\| \omega_{b} - \omega^{*} \right\|^{2} \\ & \geq& \left\| \omega_{b} - \omega^{*} \right\|^{2} . \end{array} $$

(20)

Meantime, $\bar l$ is a μ-smooth function about the optimum point ω^∗, i.e., for any ω_b, we have

$$ \bar l(\omega_{b}) - \bar l(\omega^{*}) \leq \frac{\mu}{2} \left\| \omega_{b} - \omega^{*} \right\|^{2} , $$

(21)

Then

$$ E\left[ \bar l(\omega_{b}) - \bar l(\omega^{*}) \right] \leq E\left[ \frac{\mu}{2} \left\| \omega_{b} - \omega^{*} \right\|^{2}\right] , $$

(22)

where the expected value of bounding mean square error $E\left [\left \| \omega _{b} - \omega ^{*} \right \|^{2}\right ]$ is aiming to observe the extent of ω_b converges to its optimum ω^∗, we use cyclic formula $\omega _{b+1} = \omega _{b} -\eta _{b} \hat {g}_{b}$ to obtain the correlation formula of ω_b is

$$ \begin{array}{@{}rcl@{}} \left\| \omega_{(b+1)} - \omega^{*} \right\|^{2} &=& \left\| \omega_{(b+1)} - \omega^{*} - \eta_{b}\cdot \sum\limits_{k=1}^{K} \hat {g_{b}^{k}} \right\|^{2} \\ & = &\left\| \omega_{b} - \omega^{*} \right\|^{2} + {\eta_{b}^{2}} \left\| \sum\limits_{k=1}^{K} \hat {g_{b}^{k}} \right\|^{2} - 2\eta_{b} \langle \omega_{b} - \omega^{*}, \sum\limits_{k=1}^{K} \hat {g_{b}^{k}} \rangle . \end{array} $$

(23)

For the second term, we have

$$ \begin{array}{@{}rcl@{}} {\eta_{b}^{2}} \left\| \sum\limits_{k=1}^{K} \hat {g_{b}^{k}} \right\|^{2} & = &{\eta_{b}^{2}} \left\| \sum\limits_{k=1}^{K} (\bar {g_{b}^{k}} + z_{k})\right\|^{2} \\ & =& {\eta_{b}^{2}} \left\| \sum\limits_{k=1}^{K} \bar {g_{b}^{k}} + z\right\|^{2}, \end{array} $$

(24)

where $z=\sum \limits _{k=1}^{K} z_{k}$, and $z \thicksim N(0, \sigma ^{2} I_{d})$. Thus,

$$ \begin{array}{@{}rcl@{}} &&E\left[ {\eta_{b}^{2}} \left\| \sum\limits_{k=1}^{K} (\bar {g_{b}^{k}} + z_{k})\right\|^{2} \right] +E\left[2{\eta_{b}^{2}} \sum\limits_{k=1} K \langle \bar g, z \rangle \right] + E\left[ {\eta_{b}^{2}} \left\|z\right\|^{2} \right]\\ & \leq& {\eta_{b}^{2}} +d{\eta_{b}^{2}}\sigma^{2} = {\eta_{b}^{2}}(1+d\sigma^{2}). \end{array} $$

(25)

The third term of (23) can be obtained by

$$ -2\eta_{b} \langle \omega_{b} - \omega^{*}, \sum\limits_{k=1}^{K} \hat {g_{b}^{k}} \rangle = -2\eta_{b} \sum\limits_{k=1}^{K} \langle \omega_{b} - \omega^{*}, \hat {g_{b}^{k}} \rangle . $$

(26)

The expectation of the above formula is

$$ \begin{array}{@{}rcl@{}} && E\left[\langle \omega_{b} - \omega^{*}, \hat {g_{b}^{k}} \rangle \right]\\ & =&E\left[\langle \omega_{b} - \omega^{*}, \bar {g_{b}^{k}} \rangle \right] + E\left[ \langle \omega_{b} - \omega^{*}, z^{k} \rangle\right] \\ & =&E \left[ \frac{1}{K} \langle \omega_{b} - \omega^{*}, \bar {g_{b}^{k}} \rangle \right] . \end{array} $$

(27)

Then we have

$$ \begin{array}{@{}rcl@{}} && -2\eta_{b} E \left[\langle \omega_{b} - \omega^{*}, \sum\limits_{k=1}^{K} \hat {g_{b}^{k}} \rangle \right] \\ & =& -2\eta_{b} \sum\limits_{k=1} K E\left[\frac{1}{K} \langle \omega_{b} - \omega^{*}, \bar {g_{b}^{k}} \rangle \right] \\ & \leq &-2\eta_{b} \frac{1}{K} E\left[ \bar l(\omega_{b}) - l(\omega^{*}) + \frac{\lambda}{2} \left\| \omega_{b} - \omega^{*}\right\|^{2}\right] \\ & \leq &-2\eta_{b} \frac{1}{K} E\left[\lambda \left\| \omega_{b} - \omega^{*} \right\|^{2} \right] . \end{array} $$

(28)

Hence,

$$ \begin{array}{@{}rcl@{}} && E\left[\left\| \omega_{(b+1)} - \omega^{*} \right\|^{2} \right] \\ & \leq& E\left[\left\| \omega_{b} - \omega^{*} \right\|^{2} \right] + {\eta_{b}^{2}}(1+d\sigma^{2}) - 2 \eta_{b} \frac{1}{K} \lambda E\left[ \langle \omega_{b} - \omega^{*} \rangle^{2}\right] \\ & \leq& \left( 1-2\eta_{b} \frac{1}{K}\right) E\left[ \langle \omega_{b} - \omega^{*} \rangle \right] + {\eta_{b}^{2}}(1 + d \sigma^{2}) . \end{array} $$

(29)

By properties of random unbiased gradient and bounded second moment, i.e., $E\left [ \left \| \hat g_{b}\right \|_{2}^{2} \right ] \leq K^{2}$, and the learning rate of η_b = 1/λ, we have

$$ \begin{array}{@{}rcl@{}} && E\left[\left\| \omega_{(b+1)} - \omega^{*} \right\|^{2} \right] \\ & \leq& \left( 1-\frac{2}{b}\right)E\left[ \left\| \omega_{b} - \omega^{*} \right\|^{2} \right] + \frac{K^{2}(1+ d\sigma^{2})}{\lambda^{2} b^{2}}. \end{array} $$

(30)

By (23), we have

$$ E\left[\left\| \omega_{b} - \omega^{*} \right\| \right] \leq \max\{ 2, 1 + d \sigma^{2}\} \frac{2K^{2}}{\lambda^{2} b}. $$

(31)

Using this bound,we can obtained a lower bound of the convergence rate with μ-smoothness as follows:

$$ \begin{array}{@{}rcl@{}} E\left[ \bar l(\omega_{b} )- \bar l(\omega^{*})\right] & \leq& \frac{\mu}{2} E\left[ \left\| \omega_{b} - \omega^{*} \right\|^{2}\right] \\ & \leq& \max\{2, 1+d\sigma^{2}\} \frac{\mu K^{2}}{\lambda^{2} b} . \end{array} $$

(32)

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