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--- author: __Lilian Besson__ *Advised by Christophe Moy Émilie Kaufmann title: Multi-Player Bandits Models Revisited subtitle: Decentralized Multi-Player Multi-Arm Bandits institute: PhD Student, Team SCEE, IETR, CentraleSupélec, Rennes & Team SequeL, CRIStAL, Inria, Lille date: SequeL Seminar - 22 December 2017 lang: english --- # Motivation We control some communicating devices, they want to access to a single base station. - Insert them in a **crowded wireless network**. - With a protocol **slotted in both time and frequency**. ## Goal - Maintain a **good Quality of Service**. - **With no centralized control** as it costs network overhead. ## How? - Devices can choose a different radio channel at each time → learn the best one with sequential algorithm! ---- # Outline and reference 1. Introduction 2. Our model: 3 different feedback levels 3. Decomposition and lower bound on regret 4. Quick reminder on single-player MAB algorithms 5. Two new multi-player decentralized algorithms 6. Upper bounds on regret for MCTopM 7. Experimental results 8. An heuristic (Selfish), and disappointing results 9. Conclusion This is based on our latest article: - "Multi-Player Bandits Models Revisited", Besson & Kaufmann. [arXiv:1711.02317](https://arXiv.org/abs/1711.02317) ---- # Our model - K radio channels (*e.g.*, 10) (*known*) - Discrete and synchronized time t >= 1. Every time frame t is: ![Protocol in time and frequency, with an Acknowledgement](figures/protocol.png) ## Dynamic device = dynamic radio reconfiguration - It decides **each time** the channel it uses to send **each packet**. - It can implement a simple **decision algorithm**. ---- # Our model ## "Easy" case - M <= K devices **always communicate** and try to access the network, *independently* without centralized supervision, - Background traffic is *i.i.d.*. ## Two variants : with or without *sensing* 1. *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 *Internet of Things*, but can model ZigBee, and can be analyzed mathematically... 2. *Without sensing*: same background traffic, but cannot sense, so only `Ack` is used. More suited for "IoT" networks like LoRa or SigFox. (Harder to analyze mathematically.) ---- # Background traffic, and rewards ## *i.i.d.* background traffic - K channels, modeled as Bernoulli (0/1) distributions of mean µ_k = background traffic from *Primary Users*, bothering the dynamic devices, - M devices, each uses channel A^j(t) in {1,...,K} at time t. ## Rewards r^j(t) := Y_{A^j(t),t} × 1(not C^j(t)) = 1(uplink & Ack). - with sensing information for all k, Y_{k,t} ~ Bern(µ_k) in {0, 1}, - collision for device j : C^j(t) = 1(*alone on arm A^j(t)*). → joint binary reward **but not** from two Bernoulli! ---- # 3 feedback levels r^j(t)} := Y_{A^j(t),t} × 1(not C^j(t)) 1. "Full feedback": observe both Y_{A^j(t),t} *and* C^j(t) separately, → Not realistic enough, we don't focus on it. 2. "Sensing": first observe Y_{A^j(t),t}, *then* C^j(t) only if Y_{A^j(t),t} != 0, → Models licensed protocols (ex. ZigBee), our main focus. 3. "No sensing": observe only the joint Y_{A^j(t),t} × 1(not C^j(t)), → Unlicensed protocols (ex. LoRaWAN), harder to analyze ! But all consider the same instantaneous reward r^j(t). ---- # Goal ## Problem - *Goal* : *minimize packet loss ratio* (= maximize nb of received `Ack`) in a *finite-space discrete-time Decision Making Problem*. - *Solution ?* **Multi-Armed Bandit algorithms**, **decentralized** and used **independently** by each dynamic device. ## *Decentralized* reinforcement learning optimization! - Max transmission rate = **max cumulated rewards** max_{text{algorithm} A} sum_{t=1}^{T} sum_{j=1}^M r^j_{A(t)}. - 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 ## A measure of success - Not the network throughput or collision probability, - We study the **centralized** (expected) **regret**: R_T(µ, M, rho) := E_µ[ sum_{t=1}^T sum_{j=1}^M µ_j^* - r^j(t)] = (sum_{k=1}^{M}µ_k^*) T - E_µ[sum_{t=1}^T sum_{j=1}^M r^j(t)] ## Two directions of analysis - Clearly R_T = O(T), but we want a sub-linear regret, as small as possible! - *How good a decentralized algorithm can be in this setting?* → **Lower Bound** on regret, for **any** algorithm ! - *How good is my decentralized algorithm in this setting?* → **Upper Bound** on regret, for **one** algorithm ! ---- # Lower bound 1. Decomposition of regret in 3 terms, 2. Asymptotic lower bound of one term, 3. And for regret, 4. Sketch of proof, 5. Illustration. ---- # Decomposition on regret ## Decomposition For any algorithm, decentralized or not, we have R_T(µ, M, rho) = sum_{k in M-worst} (µ_M^* - µ_k) E_µ[T_k(T)] + sum_{k in M-best} (µ_k - µ_M^*) (T - E_µ[T_k(T)]) + sum_{k=1}^{K} µ_k E_µ[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*). # Asymptotic Lower Bound on regret ## 3 terms to lower bound... - The first term for sub-optimal arms selections is lower bounded asymptotically, for all player j, bad arm k, lim inf (T → oo) E_µ[T_k^j(T)] / \log T >= 1 / kl(µ_k, µ_M^*), using technical information theory tools (Kullback-Leibler divergence, entropy), - And we lower bound the rest (including collisions) by... 0 T - E_µ[T_k(T)] >= 0 and E_µ[C_k(T)] >= 0, 😭 we should be able to do better! # Asymptotic Lower Bound on regret ## Theorem 1 [Besson & Kaufmann, 2017] - For any uniformly efficient decentralized policy, and any non-degenerated problem µ, lim inf (T → oo) R_T(µ, M, rho) / log(T) >= M × ( sum_{k in M-worst} (µ_M^* - µ_k) / kl(µ_k, µ_M^*) ) . Where kl(x,y) := x log(x / y) + (1 - x) log(1-x / 1-y) is the *binary* Kullback-Leibler divergence. ## Remarks - The centralized *multiple-play* lower bound is the same without the M multiplicative factor... Ref: [Anantharam et al, 1987] → "price of non-coordination" = M = nb of player? - Improved state-of-the-art lower bound, but still not perfect: collisions should also be controlled! ---- # 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!](figures/main_RegretCentralized____env3-4_2092905764868974160.png) ---- # Sketch of the proof - Like for single-player bandit, focus on E_µ[T_k^j(T)] expected number of selections of any sub-optimal arm k. - Same information-theoretic tools, using a "change of law" lemma. Ref: [Garivier et al, 2016] - It improved the state-of-the-art because of our decomposition, not because of new tools. > → See our paper for details! ---- # Single-player MAB algorithms 1. Index-based MAB deterministic policies, 2. Upper Confidence Bound algorithm : UCB, 3. Kullback-Leibler UCB algorithm : klUCB. ---- # Upper Confidence Bound algorithm UCB1 The device keep t number of sent packets, T_k(t) selections of channel k, X_k(t) successful transmissions in channel k. 1. For the first K steps (t=1,...,K), try each channel *once*. 2. Then for the next steps t > K : - Compute the index g_k(t) := X_k(t) / T_k(t) + sqrt{log(t) / 2 T_k(t)} - Choose channel A(t) = arg max_k g_k(t), - Update T_k(t+1) and X_k(t+1). References: [Lai & Robbins, 1985], [Auer et al, 2002], [Bubeck & Cesa-Bianchi, 2012] ---- # Kullback-Leibler UCB algorithm klUCB The device keep t number of sent packets, T_k(t) selections of channel k, X_k(t) successful transmissions in channel k. 1. For the first K steps (t=1,...,K), try each channel *once*. 2. Then for the next steps t > K : - Compute the index g_k(t) := sup_{q in [a, b]} { q : kl(X_k(t) / T_k(t), q) <= log(t) / T_k(t) } - Choose channel A(t) = arg max_k g_k(t), - Update T_k(t+1) and X_k(t+1). *Why bother?* klUCB is proved to be more efficient than UCB, and asymptotically optimal for single-player stochastic bandit. References: [Garivier & Cappé, 2011], [Cappé & Garivier & Maillard & Munos & Stoltz, 2013] ---- # Multi-player decentralized algorithms 1. Common building blocks of previous algorithms, 2. First proposal: RandTopM, 3. Second proposal: MCTopM, 4. Algorithm and illustration. ---- # Algorithms for this easier model ## Building blocks : separate the two aspects 1. **MAB policy** to learn the best arms (use sensing Y_{A^j(t),t}), 2. **Orthogonalization scheme** to avoid collisions (use C^j(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 policies (klUCB), - *Without sensing*: Selfish use a UCB index directly on the reward r^j(t). ---- # A first decentralized algorithm ![The MCTopM algorithm](figures/algo1.png) ---- # The RandTopM algorithm ![The MCTopM algorithm](figures/algo2.png) ---- # The MCTopM algorithm ![The MCTopM algorithm](figures/algo3.png) ---- # The MCTopM algorithm ![The MCTopM algorithm](figures/algo4.png) ---- # Regret upper bound 1. Theorem, 2. Remarks, 3. Idea of the proof. ---- # Regret upper bound for MCTopM ## Theorem 2 [Besson & Kaufmann, 2017] - If all M players use MCTopM with klUCB, then for any non-degenerated problem µ, there exists a problem dependent constant G_{M,µ} , such that the regret satisfies: R_T(µ, M, rho) \leq G_{M,µ} log(T) + \smallO{\log T}. ## How? - Decomposition of regret controlled with two terms, - Control both terms, both are logarithmic: + Suboptimal selections with the "classical analysis" on klUCB indexes + Collisions are harder to control... ---- # Regret upper bound for MCTopM ## Remarks - Hard to prove, we had to carefully design the MCTopM algorithm to conclude the proof, - The constant G_{M,µ} scales as M^3, way better than rhoRand's constant scaling as M{2M-1 \choose M}, - We also *minimize the number of channel switching*: interesting as changing arm costs energy in radio systems, - For the suboptimal selections, we *match our lower bound* ! - Not yet possible to know what is the best possible control of collisions... ---- # Sketch of the proof 1. Bound the expected number of collisions by M times the number of collisions for non-sitted players, 2. Bound the expected number of transitions of type (3) and (5), by \bigO{\log T} using the klUCB indexes and the forced choice of the algorithm: g_k^j(t-1) \leq g^j_{k'}(t-1), and} g_k^j(t) > g^j_{k'}(t) when switching from k' to k, 3. Bound the expected length of a sequence in the non-sitted state by a constant, 4. So most of the times (O(T - log T)), players are sitted, and no collision happens when they are all sitted! > → See our paper for details! ---- # Experimental results > Experiments on Bernoulli problems µ in [0,1]^K. 1. Illustration of regret for a single problem and M = K, 2. Regret for uniformly sampled problems and M < K, 3. Logarithmic number of collisions, 4. Logarithmic number of arm switches, 5. Fairness? ---- # Constant regret if M = K Regret, M=9 players, K=9 arms, horizon T=10000, 200 repetitions. Only RandTopM and MCTopM achieve constant regret in this saturated case (proved). ![](figures/MP__K9_M9_T10000_N200__4_algos/all_RegretCentralized____env1-1_2306423191427933958.png) # Illustration of regret of different algorithms Regret, M=6 players, K=9 arms, horizon T=5000, against 500 problems µ uniformly sampled in [0,1]^K. Conclusion : rhoRand < RandTopM < Selfish < MCTopM in most cases. ![](figures/MP__K9_M6_T5000_N500__4_algos/all_RegretCentralized____env1-1_8318947830261751207.png) # Logarithmic number of collisions Cumulated number of collisions. Also rhoRand < RandTopM < Selfish < MCTopM in most cases. ![](figures/MP__K9_M6_T5000_N500__4_algos/all_CumNbCollisions____env1-1_8318947830261751207.png) # Logarithmic number of arm switches Cumulated number of arm switches. Again rhoRand < RandTopM < Selfish < MCTopM, but no guarantee for rhoRand. ![](figures/MP__K9_M6_T5000_N500__4_algos/all_CumNbSwitchs____env1-1_8318947830261751207.png) # Fairness Measure of fairness among player. All 4 algorithms seem fair in average, but none is fair on a single run It's quite hard to achieve both efficiency and single-run fairness! ![](figures/MP__K9_M6_T5000_N500__4_algos/all_FairnessSTD____env1-1_8318947830261751207.png) ---- # An heuristic, Selfish For the harder feedback model, without sensing. 1. Just an heuristic, 2. Problems with Selfish, 5. Illustration of failure cases. ---- # The Selfish heuristic The Selfish decentralized approach = device don't use sensing, just learn on the reward (acknowledgement or not, r^j(t)). Reference: [Bonnefoi & Besson et al, 2017] ## Works fine... - More suited to model IoT networks, - Use less information, and don't know the value of M: we expect Selfish to not have stronger guarantees. - It works fine in practice! ## *But why would it work?* - Sensing was *i.i.d.* so using UCB to learn the µ_k makes sense, - But collisions are not *i.i.d.*, - Adversarial algorithms are more appropriate here, - But empirically, Selfish with UCB or klUCB works much better than, *e.g.*, Exp3... ## Works fine... - 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. ---- # Illustration of failing cases for Selfish ![Regret for M=2, K=3, T=5000, 1000 repetitions and µ = [0.1, 0.5, 0.9]. Axis x is for regret (different scale for each), and Selfish have a small probability of failure (17/1000 cases of R_T \gg \log T). The regret for the three other algorithms is very small for this "easy" problem.](figures/MP__K3_M2_T5000_N1000__4_algos/all_HistogramsRegret____env1-1_5016720151160452442.png) ---- # Sum-up ## *Wait, what was the problem ?* - 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). ## Theoretical results - With sensing ("OSA"), we obtained strong results: a lower bound, and an order-optimal algorithm, - But without sensing ("IoT"), it is harder... our heuristic Selfish usually works but can fail! ---- # Other directions of future work ## Conclude the Multi-Player OSA analysis - Remove hypothesis that objects know M, - Allow arrival/departure of objects, - Non-stationarity of background traffic etc - *More realistic emission model*: maybe driven by number of packets in a whole day, instead of emission probability. ## Extend to more objects M > K - 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). ---- # Conclusion - In a wireless network with an *i.i.d.* background traffic in K channels, - M devices can use both sensing and acknowledgement feedback, to learn the most free channels and to find orthogonal configurations. ## We showed 😀 - Decentralized bandit algorithms can solve this problem, - We have a lower bound for any decentralized algorithm, - And we proposed an order-optimal algorithm, based on klUCB and an improved Musical Chair scheme, MCTopM ## But more work is still needed... 😕 - **Theoretical guarantees** are still missing for the "IoT" model (without sensing), and can be improved (slightly) for the "OSA" model (with sensing). - Maybe study **other emission models**... - Implement and test this on **real-world radio devices** → demo (in progress) for the ICT 2018 conference! ## **Thanks!** 😀 *Any question or idea ?*