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slides.html | 2018-03-07 19:25 | 16K | MAB Learning in IoT Networks. Decentralized Multi-Player Multi-Arm Bandits | 23 Nov 2017 | Lilian Besson | |

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--- author: __Lilian Besson__ *Advised by Christophe Moy Émilie Kaufmann title: MAB Learning in IoT Networks subtitle: Decentralized Multi-Player Multi-Arm Bandits institute: PhD Student, Team SCEE, IETR, CentraleSupélec, Rennes & Team SequeL, CRIStAL, Inria, Lille date: SCEE Seminar - 23 November 2017 lang: english --- # Motivation: *Internet of Things* problem A *lot* of IoT devices want to access to a single base station. - Insert them in a possibly **crowded wireless network**. - With a protocol **slotted in both time and frequency**. - Each device has a **low duty cycle** (a few messages 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 and references 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, TS 5. Experimental results 6. An easier model with theoretical results 7. Perspectives and future works Main references are my recent articles (on HAL): - [*Multi-Armed Bandit Learning in IoT Networks and non-stationary settings*, Bonnefoi, Besson, Moy, Kaufmann, Palicot. CrownCom 2017](https://hal.inria.fr/hal-01575419), - [*Multi-Player Bandits Models Revisited*, Besson, Kaufmann. arXiv:1711.02317](https://hal.inria.fr/hal-01629733), ---- # First model - Discrete time t >= 1 and K radio channels (*e.g.*, 10) (*known*) ![Protocol in time and frequency, with an Acknowledgement.](crowncom/protocol.png) - D **dynamic** devices try to access the network *independently* - S=S1+...+S_K **static** devices occupy the network : Si 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. > → 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 simple **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 K channels. - Natural strategy, dead simple to implement. - Simple analysis, in term of **successful transmission probability** (for every message from dynamic devices) : P(success|sent) = sum_{i=1...K} (1 - p/K)^(D-1) × (1-p)^Si × (1/K) ## No learning - Works fine only if all channels are similarly occupied, 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: P(success|sent) = sum_{i=1...K} (1 - p)^(Di - 1) × (1 - p)^Si × Di/D. - The oracle has to solve this **optimization problem**: + argmax_{D1,...,DK} sum_i^K Di (1-p)^(Si + Di - 1), + such that sum_i^K Di = D, + and Di >= 0 for all 1 <= i <= K. - We solved this quasi-convex optimization problem with *Lagrange multipliers*, only numerically. - → Very good performance, maximizing the transmission rate of all the D dynamic devices ## But unrealistic But **not achievable in practice**: no centralized control and no oracle! ## Now let see *realistic decentralized approaches* - → Machine Learning ? - → Reinforcement Learning ? - → *Multi-Armed Bandit* ! ## Greedy approximation - Still very efficient - More reasonable - But still unachievable: this is a *baseline* ---- # Multi-Armed Bandit formulation A dynamic device tries to collect *rewards* when transmitting : - it transmits following a Bernoulli process (probability p of transmitting at each time step t), - chooses a channel A(tau) in {1,...,K}, + if `Ack` (no collision) → reward r(A(tau)) = 1, + if collision (no `Ack`) → reward r(A(tau)) = 0. ## Reinforcement Learning interpretation Maximize transmission rate = **maximize cumulated rewards** max on algorithm A of sum_{tau=1}^{horizon} r(A(tau)). ---- # Upper Confidence Bound algorithm (UCB1) Dynamic device keep tau number of sent packets, Tk(tau) selections of channel k, Xk(tau) successful transmission in channel k. 1. For the first K steps (tau=1,...,K), try each channel *once*. 2. Then for the next steps t > K : - Compute the index gk(tau) := Xk(tau)/Nk(tau) + sqrt(\log(tau) / (2 Nk(tau)), - Choose channel A(tau) = argmax gk(tau)$, - Update Tk(tau+1) and Xk(tau+1). > References: [Lai & Robbins, 1985], [Auer et al, 2002], [Bubeck & Cesa-Bianchi, 2012]. ---- # Thompson Sampling : Bayesian approach A dynamic device assumes a stochastic hypothesis on the background traffic, modeled as Bernoulli distributions. - Rewards rk(tau) are assumed to be *i.i.d.* samples from a Bernoulli distribution Bern(mu_k). - A **binomial Bayesian posterior** is kept on the mean availability mu_k : Bin(1 + Xk(tau), 1 + Nk(tau) - Xk(tau)). - Starts with a *uniform prior* : Bin(1, 1) ~ U([0,1]). 1. Each step tau >= 1, draw a sample from each posterior ik(tau) ~ Bin(ak(tau), bk(tau)), 2. Choose channel A(tau) = argmax ik(tau)$, 3. Update the posterior after receiving `Ack` or if collision. > References: [Thompson, 1933], [Kaufmann et al, 2012]. ---- # Experimental setting ## Simulation parameters - K = 10 channels, - S + D = 10000 devices **in total**. Proportion of dynamic devices D/(S+D) varies, - p = 10^-3 probability of emission, for all devices, - Horizon = 10^6 time slots, (about 100 messages by device) - Various settings for (S1,...,SK) static devices repartition. ## What do we show (for static Si) - After a short learning time, MAB algorithms are almost as efficient as the oracle solution ! - Never worse than the naive solution. - Thompson sampling is more efficient than UCB. - Stationary alg. outperform adversarial ones (UCB >> Exp3). ---- # 10% of dynamic devices ![10% of dynamic devices. 7% of gain](crowncom/10intelligent.png) ---- # 30% of dynamic devices ![30% of dynamic devices. 3% of gain but not much is possible](crowncom/30intelligent.png) ---- # Dependence on D/(S+D) ![*Almost optimal*, for any proportion of dynamic devices, *after a short learning time*. Up-to 16% gain over the naive approach!](crowncom/perf_learning.png) ---- # Section 6 A brief presentation of a different approach... Theoretical results for an easier model ---- # An easier model ## Easy case - M <= K dynamic devices **always communicating** (p=1). - Still interesting: many mathematical and experimental results! ## Two variants - *With sensing*: Device first senses for presence of Primary Users (background traffic), then use `Ack` to detect collisions. Model the "classical" Opportunistic Spectrum Access problem. Not exactly suited for IoT networks like LoRa or SigFox, can model ZigBee, and can be analyzed mathematically... (*cf* Wassim's and Navik's theses, 2012, 2017) - *Without sensing*: like our IoT model but smaller scale. Still very hard to analyze mathematically. ---- # Notations for this second model ## Notations - K channels, modeled as Bernoulli (0/1) distributions of mean mu_k, is a background traffic from *Primary Users*, - M devices use channel Aj(t) in {1,...,K} at each time step, - Reward: rj(t) := Y(Aj(t),t) 1(Cj(t)) = 1(uplink & `Ack`) + with sensing information Y(k,t) ~ Bern(mu_k), + collision for device j, Cj(t) = 1(*alone on arm Aj(t)*). ## Goal : *decentralized* reinforcement learning optimization! - Each player wants to **maximize its cumulated reward**, - With no central control, and no exchange of information, - Only possible if : each player converges to one of the M best arms, orthogonally (without collisions) ---- # Centralized regret ## New measure of success - Not the network throughput or collision probability, - Now we study the **centralized regret** R_T(mu, M, rho) := sum_{k=1}^{M} mu_k^* T - Expectancy_mu [sum_{t=1}^T sum_{j=1}^M rj(t) ]. ## Two directions of analysis - Clearly R_T = O(T), but we want a sub-linear regret - *What is the best possible performance of a decentralized algorithm in this setting?* → **Lower Bound** on regret for **any** algorithm ! - *Is this algorithm efficient in this setting?* → **Upper Bound** on regret for **one** algorithm ! ---- # Asymptotic Lower Bound on regret For any algorithm, decentralized or not, we have R_T(mu, M, rho) = sum_{k in Mworst} (mu_M^* - mu_k) E_mu[Tk(T)] + sum_{k in Mbest} (mu_k - mu_M^*) E_mu[T-Tk(T)] + sum_{k=1}^{K} mu_k E_mu[C_k(T)]. ## Small regret can be attained if... 1. Devices can quickly identify the bad arms M-worst, and not play them too much (*number of sub-optimal selections*), 2. Devices can quickly identify the best arms, and most surely play them (*number of optimal non-selections*), 3. Devices can use orthogonal channels (*number of collisions*). ## Lower-bounds - The first term E_mu[Tk(T)], for sub-optimal arms selections, is lower-bounded, using technical information theory tools (Kullback-Leibler divergence, entropy), - And we lower-bound collisions by... 0 : hard to do better! ## Theorem 1 [Besson & Kaufmann, 2017]. - For any uniformly efficient decentralized policy, and any non-degenerated problem mu, - lim inf R_T(mu, M, rho) / log(T) >= M sum_{k in Mworst} (mu_M^* - mu_k) / kl(mu_k, mu_M^*). > Where kl(x,y) := x log(x/y) + (1 - x) log((1-x)/(1-y))$ is the binary Kullback-Leibler divergence. ---- # Illustration of the Lower Bound on regret ![Any such lower-bound is very asymptotic, usually not satisfied for small horizons. We can see the importance of the collisions!](alt/figures/main_RegretCentralized____env3-4_2092905764868974160.png) ---- # Algorithms for this easier model ## Building blocks : separate the two aspects 1. **MAB policy** to learn the best arms (use sensing Y(Aj(t),t), 2. **Orthogonalization scheme** to avoid collisions (use Cj(t)). ## Many different proposals for *decentralized* learning policies - Recent: MEGA and MusicalChair, [Avner & Mannor, 2015], [Shamir et al, 2016] - State-of-the-art: **RhoRand policy** and variants, [Anandkumar et al, 2011] - **Our proposals**: [Besson & Kaufmann, 2017] + With sensing: **RandTopM** and **MCTopM** are sort of mixes between RhoRand and MusicalChair, using UCB indexes or more efficient index policy (KL-UCB), + Without sensing: Selfish use a UCB index directly on the reward rj(t) : like the first IoT model ! ---- # Illustration of different algorithms ![Regret, M=6 players, K=9 arms, horizon T=5000, against 500 problems mu uniformly sampled in [0,1]^K. RhoRand < RandTopM < Selfish < MCTopM in most cases](alt/figures/MP__K9_M6_T5000_N500__4_algos/all_RegretCentralized____env1-1_8318947830261751207.png) ---- # Regret upper-bound for \MCTopM-KL-UCB ## Theorem 2 [Besson & Kaufmann, 2017]. If all M players use MCTopM with KL-UCB, then for any non-degenerated problem mu, there exists a problem dependent constant G(M,mu), such that the regret satisfies: R_T(mu, M, rho) <= G(M,mu) \log(T) + o(log T). ## Remarks - Hard to prove, we had to carefully design the MCTopM algorithm to conclude the proof, - For the suboptimal selections, we *match our lower-bound* ! - We also *minimize the number of channel switching*: interesting as it costs energy, - Not yet possible to know what is the best possible control of collisions... ---- # In this model The Selfish decentralized approach = device don't use sensing, just learn on the receive acknowledgement, - Like our first IoT model, - It works fine in practice! - Except... when it fails drastically! - In small problems with M and K = 2 or 3, we found small probability of failures (*i.e.*, linear regret), and this prevents from having a generic upper-bound on regret for Selfish. Sadly... ---- # Illustration of failing cases for Selfish ![Regret for M=2 players, K=3 arms, horizon T=5000, 1000 repetitions and mu = [0.1, 0.5, 0.9]. Axis x is for regret (different scale for each), and Selfish have a small probability of failure (17 cases of R_T = Omega(T), out of 1000). The regret for the three other algorithms is very small for this ``easy'' problem.](alt/figures/MP__K3_M2_T5000_N1000__4_algos/all_HistogramsRegret____env1-1_5016720151160452442.png) ---- # Perspectives ## Theoretical results - MAB algorithms have guarantees for *i.i.d. settings*, - But here the collisions cancel the *i.i.d.* hypothesis, - Not easy to obtain guarantees in this mixed setting (*i.i.d.* emissions process, "game theoretic" collisions). - For OSA devices (always emitting), we obtained strong theoretical results, - But harder for IoT devices with low duty-cycle... ## Real-world experimental validation ? - Radio experiments will help to validate this. Hard ! ---- # Other directions 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*. - Extend the theoretical analysis to the large-scale IoT model, first with sensing (*e.g.*, models ZigBee networks), then without sensing (*e.g.*, LoRaWAN networks). - And also conclude the Multi-Player OSA analysis (remove hypothesis that objects know M, allow arrival/departure of objects, non-stationarity of background traffic etc) ---- # Conclusion ## We showed - Simple Multi-Armed Bandit algorithms, used in a Selfish approach by IoT devices in a crowded network, help to quickly learn the best possible repartition of dynamic devices in a fully decentralized and automatic way, - For devices with sensing, smarter algorithms can be designed, and analyze carefully. - Empirically, even if the collisions break the *i.i.d* hypothesis, stationary MAB algorithms (UCB, TS, KL-UCB) outperform more generic algorithms (adversarial, like Exp3). ## But more work is still needed... - **Theoretical guarantees** are still missing for the IoT model, and can be improved (slightly) for the OSA model. - Maybe study **other emission models**. - Implement this on **real-world radio devices** (*TestBed*). ## **Thanks!** *Any question?*