Algorithms for sustainable data centers

Transcription

1 Algorithms for sustainable data centers Adam Wierman (Caltech) Minghong Lin (Caltech) Zhenhua Liu (Caltech) Lachlan Andrew (Swinburne) and many others

2 IT is an energy hog The electricity use of data centers is 2-3% of the US total, and it is growing 12% a year! Total US use grows ~1% a year! x 60 co 2 co 2

3 Huge push from academia & industry Lots of engineering improvements

4 Power Usage Effectiveness (PUE) = >3 <1.1 Power to building Power to computing lower PUE lower energy consumption more sustainable

5 Power Usage Effectiveness (PUE) = >3 <1.1 Power to building Power to computing The building is efficient, it s time to work on the algorithms

6 This talk: An overview of algorithmic questions & challenges 1. Two examples 2. A general model & some results 3. A case study

7 Example 1: Dynamic resizing in a data center Workload Highly non-stationary Idling servers use 30-50% of peak power Active Servers Dynamic Controversial! resizing Goal: Adapt provisioning to minimize cost. Challenges: Switching is costly & prediction is hard.

8 Example 1: Dynamic resizing in a data center Workload (Interactive) Highly non-stationary Idling servers use 30-50% of peak power Active Servers Dynamic resizing Workload (Batch) Large, delay tolerate jobs, typically with deadlines Deadlines are often on the order of hours Load shifting Valley filling

9 Example 1: Dynamic resizing in a data center Electricity prices Active Servers Solar availability PUE Dynamic resizing Load shifting PUE Valley filling

10 Example 2: Geographical load balancing Data centers: Diverse, time varying electricity prices, renewable availabilities, cooling efficiencies, etc. Solar availability Electricity prices Proxy/Mapping Nodes: Highly non-stationary workload

11 Example 2: Geographical load balancing Goal: Adapt routing & provisioning to minimize cost. Challenges: Switching is costly & prediction is hard.

12 Dynamic resizing Geographical load balancing [Chase et al. 01] [Pinheiro et al. 01] [Chen et al. 05] [Ghandi et al. 09] [Khargharia et al. 10] [Kansal et al. 10] [LWAT10] [Pakbaznia et al. 09] [Qureshi et al. 09] [Rao et al. 10] [Stanojevic et al. 10] [Wendell et al 10] [Le et al 2010] [LLWLA 11] [LLWAL11] [LAW12], Goal: Adapt routing & provisioning to minimize cost. Challenges: Switching is costly & prediction is hard.

13 A (familiar) model

14 x t λ t # of active servers 0 T ~10min workload time ~1month

15 Note: λ t and x t can be scalar or vector x t e.g. a homogeneous data center e.g. geographical load balancing λ t # of active servers 0 T workload time

16 Data center goal: min x t F (stability constraints) t (operating c t x t + cost) x+ t (switching x t 1 cost)

17 Data center goal: min x t F t stability constraints x t N N max x t 0 x t SLA(λ t ) (operating c t x t + cost) x+ t (switching x t 1 cost)

18 Data center goal: min x t F t (operating c t x t + cost) x+ t (switching x t 1 cost)

19 Data center goal: min x t F t c t x t + x t x t 1 E.g. A homogeneous data center E.g. Cost to switch server on/off β x t x t 1 c t x t ; λ t x t = min g t (λ t λ i ) it i=1 g t l (delay) d(l) + (energy p t e lcost) r + t

20 Data center goal: min x t F t c t x t convex + x t x t 1

21 Stochastic / Adversarial x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms to minimize cost

22 c 1 c 1 (x 1 ) F x 1

23 c 2 c 1 c 2 (x 2 ) F x 1 x 2 x 2 x 1

24 SOCO comes up in many applications geographical load balancing dynamic capacity management video streaming electricity generation planning product/service selection portfolio management labor markets penalized estimation

25 x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms to minimize cost 2 Communities Online Learning Online Algorithms

26 x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms to minimize cost 2 Communities Online Learning Online Algorithms 2 Metrics Regret Competitive ratio

27 Regret(Alg) = Cost Alg Cost(Static_Opt) Competitive ratio(alg) = Cost Alg Cost Offline_Opt

28 Can an algorithm maintain sub-linear regret? [KV02] [BBCM03] [Z03] [HAK07] [LRAMW12] Competitive ratio(alg) = Cost Alg Cost Offline_Opt

29 Can an algorithm maintain sub-linear regret? [KV02] [BBCM03] [Z03] [HAK07] [LRAMW12] Can an algorithm maintain a constant competitive ratio? [BKRS92] [BLS92] [BB00] [LWAT11][LRAMW12] Can an algorithm maintain sub-linear regret and a constant competitive ratio?

30 x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms with sub-linear regret 2 Communities Online Learning Online Algorithms 2 Metrics Regret Competitive ratio

31 x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Online convex optimization Goal: Algorithms with sub-linear regret 2 Communities Online Learning Online Algorithms 2 Metrics Regret Competitive ratio

32 x 1, c 1, x 2, c 2, x 3, c 3, min x t F t c t c t x t 2 < C Online convex optimization Goal: Algorithms with sub-linear regret Many algorithms achieve sub-linear regret Online gradient descent (OGD) [Z03] x t+1 = Proj(x t η t c t x t )

33 Theorem [Z03] When η t = Θ(1/ t), online gradient descent has O T -regret for OCO. Theorem [Z03,H06] Any algorithm for OCO must incur Ω on linear cost functions. T -regret

34 x 1, c 1, x 2, c 2, x 3, c 3, min x t F t c t c t x t 2 < C Online convex optimization Goal: Algorithms with sub-linear regret Many algorithms achieve sub-linear regret - Online gradient descent (OGD) [Z03] - Multiplicative weights [AHK05] [FS99] - Newton s method based [HKKA06][HAK07]

35 x 1, c 1, x 2, c 2, x 3, c 3, min x t F t c t c t x t 2 < C Online convex optimization Goal: Algorithms with sub-linear regret Do OCO algorithms maintain sub-linear regret for SOCO?

36 x 1, c 1, x 2, c 2, x 3, c 3, min x t F t c t c t x t 2 < C + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms with sub-linear regret Do OCO algorithms maintain sub-linear regret for SOCO?

37 Corollary: When η t = Θ(1/ t), online gradient descent still has O T -regret for SOCO. Proof: t x t x t 1 M x t x t 1 2 (Equivalence of norms) M η t c t 1 (x t 1 ) 2 (Projection is non-expansive) MC η t = O T (Bounded gradient)

38 Can an algorithm maintain sub-linear regret? Yes: Online gradient descent has O T -regret. Can an algorithm maintain a constant competitive ratio? [BKRS92] [BLS92] [BB00] [LWAT11][LRAMW12] Can an algorithm maintain sub-linear regret and a constant competitive ratio?

39 x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms with constant comp. ratio 2 Communities Online Learning Online Algorithms 2 Metrics Regret Competitive ratio

40 c 1, x 1, c 2, x 2, c 3, x 3, x 1, c 1, x 2, c 2, x 3, c 3, online min x t F t c t x t convex + x t x t 1 smoothed Metrical task system (MTS) Goal: Algorithms with constant comp. ratio 2 Communities Online Learning Online Algorithms 2 Metrics Regret Competitive ratio

41 c 1, x 1, c 2, x 2, c 3, x 3, online min x t F t c t x t convex + x t x t 1 smoothed Metrical task system (MTS) Goal: Algorithms with constant comp. ratio Is it possible to achieve a constant competitive ratio?

42 n is the size of the action set Theorem [BLS92] [BKRS92]: Any deterministic algorithm is Ω(n)-competitive. Any randomized algorithm is Ω log n/log log n -competitive

43 c 1, x 1, c 2, x 2, c 3, x 3, online min x t F t convex c t x t + x t x t 1 smoothed Metrical task system (MTS) Goal: Algorithms with constant comp. ratio Is it possible to achieve a constant competitive ratio?

44 c 1, x 1, c 2, x 2, c 3, x 3, online min x t F t convex c t x t + x t x t 1 scalar smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms with constant comp. ratio Is it possible to achieve a constant competitive ratio?

45 Theorem: Lazy Capacity Provisioning (LCP) is 3-competitive. Further Cost LCP Cost OPT + 2 SwitchingCost(OPT) Is it possible to achieve a constant competitive ratio?

46 t Minimize c s (x s t s=1 ) + s=1 1 (x s 1 x s ) + Subject to x s F x t U : last entry in Lazy Capacity Provisioning (LCP) time x t L : last entry in t Minimize c s (x s t s=1 ) + s=1 1 (x s x s 1 ) + Subject to x s F

47 Theorem: Lazy Capacity Provisioning (LCP) is 3-competitive. Further Cost LCP Cost OPT + 2 SwitchingCost(OPT) Lemma: The offline optimal solution is lazy in reverse time.

48 t Minimize c s (x s t s=1 ) + s=1 1 (x s 1 x s ) + Subject to x s F x t U : last entry in Lazy Capacity Provisioning (LCP) time x t L : last entry in Optimal t Minimize c s (x s t s=1 ) + s=1 1 (x s x s 1 ) + Subject to x s F

49 c 1, x 1, c 2, x 2, c 3, x 3, online min x t F t convex c t x t + x t x t 1 scalar smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms with constant comp. ratio Is it possible to achieve a constant competitive ratio?

50 c t x t min x t F, c t+1,, c t+w t convex c t x t predictions + x t x t 1 smoothed Smoothed online convex optimization (SOCO) Goal: Algorithms with constant comp. ratio Is it possible to achieve a constant competitive ratio?

51 Classic Approach: Receding Horizon Control Is it possible to achieve a constant competitive ratio?

52 Classic Approach: Receding Horizon Control, c t 1, c t, c t+1,, c t+w 1, c t+w, c t+w+1, x t 1 x t x t+1

53 Receding Horizon Control (RHC): Choose x t to minimize cost over t, t + w, given prediction window and x t 1., c t 1, c t, c t+1,, c t+w 1, c t+w, c t+w+1, x t 1 x t x t+1

54 Receding Horizon Control (RHC): Choose x t to minimize cost over t, t + w, given prediction window and x t 1. Theorem: For one-dimensional SOCO, RHC is O(1 + 1 /w)- competitive. But, in general, RHC is Ω 1 -competitive. Does not improve as w grows!

55 , c t 1, c t, c t+1,, c t+w 1, c t+w, c t+w+1, RHC x t 1 x t FHC x t, x t+1,, x t+w AFHC x t is the average

56 Averaging Fixed Horizon Control (AFHC): Choose x t as the average of w + 1 fixed horizon control algorithms. AFHC x t is the average

57 Averaging Fixed Horizon Control (AFHC): Choose x t as the average of w + 1 fixed horizon control algorithms. Theorem: AFHC is O(1 + 1 /w)- competitive

58 Can an algorithm maintain sub-linear regret? 1 Yes: Online gradient descent has Θ T -regret. Can an algorithm maintain a constant competitive 2 ratio? Yes. (in the scalar case): LCP is 3-competitive. Yes. (in the vector case): AFHC is O( 1 /w)-comp 3 Can an algorithm maintain sub-linear regret and a constant competitive ratio? No!

59 Theorem: For arbitrary γ > 0 and any online algorithm A, there exist linear cost functions such that: Regret A CR A + T Deterministic or randomized γ

60 What now? Decide which metric you really care about? - Is εt-regret good enough? - Is log log T - competitive good enough? - What about under stochastic assumptions?

61 Decide which metric you really care about? - Is εt-regret good enough? - Is log log T - competitive good enough? - What about under stochastic assumptions? Theorem: Given a 1-dimensional SOCO, for arbitrary, increasing f T, Randomly Biased Greedy (RBG) is: (1 + f(t)/ 1 )-competitive with O(max {T/f(T), f(t)})-regret

62 Randomly Biased Greedy (f(t)) x t = argmin w t (x t ) + x t r Where: 1 RBG = f(t). w t x = min{w t 1 y + c t y + x y RBG } y r = UnifRand 1 RBG, 1 RBG Theorem: Given a 1-dimensional SOCO, for arbitrary, increasing f T, Randomly Biased Greedy (RBG) is: (1 + f(t)/ 1 )-competitive with O(max {T/f(T), f(t)})-regret

63 1 Can an algorithm maintain sub-linear regret? Yes: Online gradient descent has Θ T -regret. Can an algorithm maintain a constant competitive 2 ratio? Yes. (in the scalar case): LCP is 3-competitive. Yes. (in the vector case): AFHC is O( 1 /w)-comp 3 Can an algorithm maintain sub-linear regret and a constant competitive ratio? No! But (in the scalar case): RBG is O(f T )-competitive with O(max {T/f(T), f(t)}) regret.

64 Back to the applications

65 An implementation: Dynamic resizing Electricity prices Active Servers Solar availability PUE Dynamic resizing Load shifting PUE Valley filling

66 An implementation: Dynamic resizing WSJ ARTICLE SCREEN SCHOT HP collaborators: Yuan Chen, Cullen Bash, Martin Arlitt, Daniel Gmach, Zhikui Wang, Manish Marwah and Chris Hyser

67 An implementation: Dynamic resizing HP collaborators: Yuan Chen, Cullen Bash, Martin Arlitt, Daniel Gmach, Zhikui Wang, Manish Marwah and Chris Hyser

68 A case study: Geographical load balancing Data centers: Google data center locations Real traces for wind, solar, electricity price, etc. Solar availability Wind availability Proxy/Mapping Nodes: Scaled by population density

69 A case study: Geographical load balancing GLB: geographical load balancing & dynamic resizing vs. LOCAL: Route to the closest data center & dynamic resizing

70 # Servers ON The good Follow the renewables routing emerges. east coast west coast Day 1 Day 2

71 The good 8 Follow the renewables routing emerges. Huge reductions in grid usage become possible. Capacity needed to reduce grid usage by 85% 6 LOCAL 4 2 GLB Solar capacity

72 The good Follow the renewables routing emerges. Huge reductions in grid usage become possible. The bad GLB uses more energy if data centers don t have renewables available. GLB LOCAL Day 1 Day 2

73 The good Follow the renewables routing emerges. Huge reductions in grid usage become possible. The bad GLB uses more energy if data centers don t have renewables available. The ugly GLB uses dirtier grid electricity too!

74 Demand response can help! The ugly GLB uses dirtier grid electricity too!

75 Final thoughts

76 Dynamic resizing Geographical load balancing 𝑥1, 𝑐1, 𝑥 2, 𝑐 2, 𝑥 3, 𝑐 3, 𝑡 min 𝑥 𝑡 𝐹 𝑐 𝑥 𝑡 𝑡 convex 𝑡 + 𝑥 𝑥 online 𝑡 1 smoothed Smoothed online convex optimization Implementation

77 Lots of interesting theoretical questions remain: Stochastic models for cost functions better algorithms? Predictions are extremely important in practice Stochastic variations? The full cost function is not observed in practice Bandit versions? Can an algorithm balance regret and the competitive ratio? x 1, c 1, x 2, c 2, x 3, c 3, min x t F t c t x t convex online + x t x t 1 smoothed Smoothed online convex optimization

78 Algorithms for sustainable data centers Adam Wierman (Caltech) Minghong Lin (Caltech) Zhenhua Liu (Caltech) Lachlan Andrew (Swinburne) and many others Minghong Lin, Adam Wierman, Lachlan Andrew, and Eno Thereska. Dynamic right-sizing for power proportional data centers. Infocom, Best Paper award winner. Zhenhua Liu, Minghong Lin, Adam Wierman, Steven Low, and Lachlan Andrew. Greening geographical load balancing. Sigmetrics Zhenhua Liu, Minghong Lin, Adam Wierman, Steven Low, and Lachlan Andrew. Geographical load balancing with renewables. Greenmetrics, Best Student Paper award winner. Zhenhua Liu, Yuan Chen, Cullen Bash, Adam Wierman, et al. Renewable and cooling aware workload management for sustainable data centers. Sigmetrics Minghong Lin, Lachlan Andrew, and Adam Wierman. Online Algorithms for Geographical Load Balancing. Green Computing Conference, Best Paper award winner.