count: true ## We want A *lot* of IoT devices want to access to a gateway of base station. - Insert them in a **crowded wireless network**. - With a protocol **slotted in time and frequency**. - Each device has a **low duty cycle** (a few message per day). ### Goal - Maintain a **good Quality of Service**. - **Without** centralized supervision! ### How? - Use **learning algorithms**: devices will learn on which frequency they should talk! ---- ## Outline 1. Introduction and motivation 2. Model and hypotheses 3. Baseline algorithms : to compare against naive and efficient centralized approaches 4. Two Multi-Armed Bandit algorithms : UCB, Thompson sampling 5. Experimental results 6. Perspectives and future works 7. Conclusion ---- ## Model - Discrete time $t\geq1$ and $N_c$ radio channels (*e.g.*, 10) (*known*)  - $D$ **dynamic** devices try to access the network *independently* - $S=S_1+\dots+S_{N_c}$ **static** devices occupy the network : $S_1,\dots,S_{N_c}$ in each channel (*unknown*). ---- ## Hypotheses ### Emission model - Each device has the same *low* emission probability: each step, each device sends a packet with probability $p$. (this gives a duty cycle proportional to $1/p$) ### Background traffic - Each static device uses only one channel. - Their repartition is fixed in time. > $\implies$ Background traffic, bothering the dynamic devices! ### Dynamic radio reconfiguration - Each **dynamic device decides the channel it uses to send every packet**. - It has memory and computational capacity to implement basic decision algorithm. ### Problem - *Goal* : *maximize packet loss ratio ($=$ number of received `Ack`) in a *finite-space discrete-time Decision Making Problem*. - *Solution ?* **Multi-Armed Bandit algorithms**, **decentralized** and used **independently** by each device. ---- ## A naive strategy : uniformly random access - **Uniformly random access**: dynamic devices choose uniformly their channel in the pull of $N_c$ channels. - Natural strategy, dead simple to implement. - Simple analysis, in term of **successful transmission probability** (for every message from dynamic devices) : $$\mathbb{P}(\text{success}|\text{sent}) = \sum_{i=1}^{N_c} \underbrace{(1 - p / N_c)^{D-1}}_{\text{No other dynamic device}} \times \underbrace{(1-p)^{S_i}}_{\text{No static device}} \times\; \frac{1}{N_c}.$$ - Works fine only if all channels are similarly occupied,\newline but **it cannot learn** to exploit the best (more free) channels. ---- ## Optimal centralized strategy - If an oracle can decide to affect $D_i$ dynamic devices to channel $i$, the **successful transmission probability** is: $$\mathbb{P}(\text{success}|\text{sent}) = \sum_{i=1}^{N_c} \underbrace{(1 - p)^{D_i - 1}}_{\;\;D_i - 1 \;\text{others}\;\;} \times \underbrace{(1 - p)^{S_i}}_{\;\;\text{No static device}\;\;} \times \underbrace{ D_i / D }_{\;\;\text{Sent in channel}\; i}.$$ - The oracle has to solve this **optimization problem**: $$\underset{D_1,\dots,D_{N_c}}{\arg\max}\;\;\; & \sum_{i=1}^{N_c} D_i (1 - p)^{S_i + D_i -1}\\ \text{such that}\;\;\; & \sum_{i=1}^{N_c} D_i = D \; \text{and} \; D_i \geq 0, \; \; \forall 1 \leq i \leq N_c .$$ - We solved this quasi-convex optimization problem with *Lagrange multipliers*, only numerically. - $\implies$ Very good performance, maximizing the transmission rate of all the $D$ dynamic devices ### But unrealistic But **not achievable in practice**: no centralized oracle! ### Let see *realistic decentralized approaches* - $\hookrightarrow$ Machine Learning ? - $\hookrightarrow$ Reinforcement Learning ? - $\hookrightarrow$ *Multi-Armed Bandit* ! ---- ## Multi-Armed Bandit formulation A dynamic device tries to collect *rewards* when transmitting : - at each time step $\tau$ when its Bernoulli emission process tells it to communicate, - chooses a channel $A(\tau) \in \{1,\dots,N_c\}$, - if `Ack` (no collision) $\implies$ reward $r_{A(\tau)} = 1$, - if collision (no `Ack`) $\implies$ reward $r_{A(\tau)} = 0$. ### Reinforcement Learning interpretation Maximize transmission rate $\equiv$ **maximize cumulated rewards** $$\max_{\text{algorithm}\;A} \;\; \sum_{\tau=1}^{\text{horizon}} r_{A(\tau)}.$$ ## Upper Confidence Bound algorithm ($\mathrm{UCB}_1$) A dynamic device keeps $\tau$ number of sent packets, $T_k(t)$ selections of channel $k$, $X_k(t)$ successful transmission in channel $k$. 1. For the first $N_c$ steps ($\tau=1,\dots,N_c$), try each channel *once*. 2. Then for the next steps $t \geq N_c$ : - Compute the index $g_k(\tau) := \underbrace{\frac{X_k(\tau)}{N_k(\tau)}}_{\text{Mean}\; \widehat{\mu_k}(\tau)} + \underbrace{\sqrt{\frac{\log(\tau)}{2 N_k(\tau)}}.}_{\text{Upper Confidence Bound}}$ - Choose channel $A(\tau) = \mathop{\arg\max}\limits_{k} \; g_k(\tau)$, - Update $T_k(\tau+1)$ and $X_k(\tau+1)$. > *References*: [Lai \& Robbins, 1985], [Auer et al, 2002], [Bubeck \& Cesa-Bianchi, 2012]. ---- ## Thompson Sampling : Bayesian approach A dynamic device assumes a stochastic hypothesis on the background traffic, modeled as Bernoulli distributions. - Rewards $r_k(\tau)$ are assumed to be *i.i.d.* samples from a Bernoulli distribution $\mathrm{Bern}(\mu_k)$. - A **binomial Bayesian posterior** is kept on the mean availability $\mu_k$ : $\mathrm{Bin}(1 + X_k(\tau), 1 + N_k(\tau) - X_k(\tau))$. - Starts with a *uniform prior* : $\mathrm{Bin}(1, 1) \sim \mathcal{U}([0,1])$. 1. Each step $\tau \geq 1$, a sample is drawn from each posterior $i_k(t) \sim \mathrm{Bin}(a_k(\tau), b_k(\tau))$, 2. Choose channel $A(\tau) = \mathop{\arg\max}\limits_k \; i_k(\tau)$, 3. Update the posterior after receiving `Ack` or if collision. > *References*: [Thompson, 1935], [Kaufmann et al, 2012]. ---- ## Experimental setting ### Simulation parameters - $N_c = 10$ channels, - $S + D = 10000$ devices in total, - $p = 10^{-3}$ probability of emission, - $\text{horizon} = 10^5$ time slots ($\simeq 100$ messages $/$ device), - The proportion of dynamic devices $D/(S+D)$ varies, - Various settings for $(S_1,\dots,S_{N_c})$ static devices repartition. ### What do we show - After a short learning time, MAB algorithm are almost as efficient as the oracle solution. - Never worse than the naive solution. - Thompson sampling is even more efficient than UCB. ---- ## $10\%$ of dynamic devices  ---- ## $30\%$ of dynamic devices  ---- ## $50\%$ of dynamic devices  ---- ## $100\%$ of dynamic devices (extreme case)  ---- ## Dependence on $D/(S+D)$  ---- ## Perspectives ### Theoretical results - MAB algorithms have performance guarantees for *stochastic settings*, - But here the collisions cancel the *i.i.d.* hypothesis, - Not easy to obtain guarantees in this mixed setting (*i.i.d.* emission process, game theoretic collisions). ### Real-world experimental validation ? - Real-world radio experiments will help to validate this. *In progress*... ---- ## Other direction of future work - *More realistic emission model*: maybe driven by number of packets in a whole day, instead of emission probability. - Validate this on a *larger experimental scale*. ---- ### We showed numerically... - After a learning period, MAB algorithms are almost as efficient as the oracle solution, - Never worse than the naive solution. - Thompson sampling is even more efficient than UCB. - Simple algorithms are up-to $16\%$ more efficient than the naive approach. ### But more work is still needed... - **Theoretical guarantees** are still missing. - Maybe study **other emission models**. - And also implement this on **real-world radio devices**. > **Thanks!** *Question?*