The connection between information theory and networked control

Transcription

1 The connection between information theory and networked control Anant Sahai based in part on joint work with students: Tunc Simsek, Hari Palaiyanur, and Pulkit Grover Wireless Foundations Department of Electrical Engineering and Computer Sciences University of California at Berkeley Tutorial Seminar at the Global COE Workshop on Networked Control Systems Kyoto University Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

2 Networked Control Systems Communication Links 2 Controller 6 System 6 Sensor Relay System 6 User Systems, sensors, and users connected with network links over noisy channels. Signals evolve in real time and the communication links carry ongoing and interacting streams of information. Holistic approach: overall cost function. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

3 Ho, Kastner, and Wong (1978)... sporadic and not too successful attempts have been made to relate Shannon s information theory with feedback control system design. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

4 Shannon tells us Separate source and channel coding Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

5 Shannon tells us Separate source and channel coding But delay is the price of reliability. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

6 Shannon tells us Separate source and channel coding But delay is the price of reliability. [The duality between source and channel coding] can be pursued further and is related to a duality between past and future and the notions of control and knowledge. Thus we may have knowledge of the past and cannot control it; we may control the future but have no knowledge of it. Claude Shannon 1959 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

7 Shannon tells us Separate source and channel coding But delay is the price of reliability. [The duality between source and channel coding] can be pursued further and is related to a duality between past and future and the notions of control and knowledge. Thus we may have knowledge of the past and cannot control it; we may control the future but have no knowledge of it. Claude Shannon 1959 What is this relationship since delays hurt control? Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

8 Outline 1 A bridge to nowhere? From control to information theory. A simple control problem A connection to information theory Fixing information theory and filling in the gaps. 2 Coming back to the control problem What is wrong with random coding The role of noiseless feedback 3 Taking control thinking to the forefront of information theory. The holy grail problem Control thinking to the rescue! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

9 A simple distributed control problem U t 1 1 Step Delay W t Unstable System X t Designed Observer O Possible Control Knowledge U t Control Signals Designed Controller C Possible Channel Feedback Delay Fortified Channel 1 Step X t+1 = λx t + U t + W t Unstable λ > 1, bounded initial condition and disturbance W. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

10 A simple distributed control problem U t 1 1 Step Delay W t Unstable System X t Designed Observer O Possible Control Knowledge U t Control Signals Designed Controller C Possible Channel Feedback Delay Fortified Channel 1 Step X t+1 = λx t + U t + W t Unstable λ > 1, bounded initial condition and disturbance W. Goal: Performance = sup t>0 E[ X t η ] K for some target K <. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

11 Review: Entirely noiseless channel t Window known to contain X t λ t will grow by factor of λ > Encode which control U t to apply Sending R bits, cut window by a factor of 2 R grows by Ω on each side 2 giving a new window for X t+1 t+1 As long as R > log 2 λ, we can have stay bounded forever. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

12 The separation-principle oriented program Use entropy and mutual information Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

13 The separation-principle oriented program Use entropy and mutual information Tatikonda s insight: directed mutual information captures causality Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

14 The separation-principle oriented program Use entropy and mutual information Tatikonda s insight: directed mutual information captures causality Write out entropic inequalities Key Inequality: Directed data-processing inequality Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

15 The separation-principle oriented program Use entropy and mutual information Tatikonda s insight: directed mutual information captures causality Write out entropic inequalities Key Inequality: Directed data-processing inequality Set up a mapping between bits and performance You probably don t care about the entropy of the state. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

16 The separation-principle oriented program Use entropy and mutual information Tatikonda s insight: directed mutual information captures causality Write out entropic inequalities Key Inequality: Directed data-processing inequality Set up a mapping between bits and performance You probably don t care about the entropy of the state. Lower bound performance assuming nested information Equivalent to estimation Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

17 The separation-principle oriented program Use entropy and mutual information Tatikonda s insight: directed mutual information captures causality Write out entropic inequalities Key Inequality: Directed data-processing inequality Set up a mapping between bits and performance You probably don t care about the entropy of the state. Lower bound performance assuming nested information Equivalent to estimation Rate-distortion theory can be developed Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

18 The separation-principle oriented program Use entropy and mutual information Tatikonda s insight: directed mutual information captures causality Write out entropic inequalities Key Inequality: Directed data-processing inequality Set up a mapping between bits and performance You probably don t care about the entropy of the state. Lower bound performance assuming nested information Equivalent to estimation Rate-distortion theory can be developed Get tight upper bounds and architectures? Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

19 The rate-distortion part 1 Squared error distortion "Sequential" Rate distortion (obeys causality) Rate distortion curve (non causal) Stable counterpart (non causal) Rate (in bits) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

20 How bad can entropic bounds be? Consider a system with λ = 2 for the dynamics Real packet-drop channel (C = ) Z t = { Yt with Probability with Probability 1 2 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

21 How bad can entropic bounds be? Consider a system with λ = 2 for the dynamics Real packet-drop channel (C = ) Z t = { Yt with Probability with Probability 1 2 No other constraints, so design is obvious: Y t = X t and U t = λz t Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

22 How bad can entropic bounds be? Consider a system with λ = 2 for the dynamics Real packet-drop channel (C = ) Z t = { Yt with Probability with Probability 1 2 No other constraints, so design is obvious: Y t = X t and U t = λz t X t+1 = { Wt with Probability 1 2 2X t + W t with Probability 1 2 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

23 How bad can entropic bounds be? Consider a system with λ = 2 for the dynamics Real packet-drop channel (C = ) Z t = { Yt with Probability with Probability 1 2 No other constraints, so design is obvious: Y t = X t and U t = λz t X t+1 = { Wt with Probability 1 2 2X t + W t with Probability 1 2 Under stochastic disturbances, the variance of the state is asymptotically infinite. (St. Petersburg Lottery Style) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

24 Delay-universal (anytime) communication B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 Y 11 Y 12 Y 13 Y 14 Y 15 Y 16 Y 17 Y 18 Y 19 Y 20 Y 21 Y 22 Y 23 Y 24 Y 25 Y 26 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 Z 12 Z 13 Z 14 Z 15 Z 16 Z 17 Z 18 Z 19 Z 20 Z 21 Z 22 Z 23 Z 24 Z 25 Z 26 bb 1 bb 2 bb 3 bb 4 bb 5 bb 6 bb 7 bb 8 bb 9 fixed delay d = 7 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

25 Delay-universal (anytime) communication B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 Y 11 Y 12 Y 13 Y 14 Y 15 Y 16 Y 17 Y 18 Y 19 Y 20 Y 21 Y 22 Y 23 Y 24 Y 25 Y 26 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 Z 12 Z 13 Z 14 Z 15 Z 16 Z 17 Z 18 Z 19 Z 20 Z 21 Z 22 Z 23 Z 24 Z 25 Z 26 bb 1 bb 2 bb 3 bb 4 bb 5 bb 6 bb 7 bb 8 bb 9 fixed delay d = 7 Fixed-delay reliability α is achievable if there exists a sequence of encoder/decoder pairs with increasing end-to-end delays d j such that lim j 1 d j ln P(B i ˆB j i ) = α. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

26 Delay-universal (anytime) communication B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 Y 11 Y 12 Y 13 Y 14 Y 15 Y 16 Y 17 Y 18 Y 19 Y 20 Y 21 Y 22 Y 23 Y 24 Y 25 Y 26 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 Z 12 Z 13 Z 14 Z 15 Z 16 Z 17 Z 18 Z 19 Z 20 Z 21 Z 22 Z 23 Z 24 Z 25 Z 26 bb 1 bb 2 bb 3 bb 4 bb 5 bb 6 bb 7 bb 8 bb 9 fixed delay d = 7 α is achievable delay-universally or in an anytime fashion if a single encoder works for all sufficiently large delays d. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

27 Delay-universal (anytime) communication B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 Y 11 Y 12 Y 13 Y 14 Y 15 Y 16 Y 17 Y 18 Y 19 Y 20 Y 21 Y 22 Y 23 Y 24 Y 25 Y 26 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 11 Z 12 Z 13 Z 14 Z 15 Z 16 Z 17 Z 18 Z 19 Z 20 Z 21 Z 22 Z 23 Z 24 Z 25 Z 26 bb 1 bb 2 bb 3 bb 4 bb 5 bb 6 bb 7 bb 8 bb 9 fixed delay d = 7 The anytime capacity Cany(α) is the supremal rate at which reliability α is achievable in a delay-universal way. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

28 Separation theorem for scalar control Necessity: If a scalar system with parameter λ > 1 can be stabilized with finite η-moment across a noisy channel, then the channel with noiseless feedback must have Cany(η ln λ) ln λ In general: If P( X > m) < f(m), then K : Perror(d) < f(kλ d ) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

29 Separation theorem for scalar control Necessity: If a scalar system with parameter λ > 1 can be stabilized with finite η-moment across a noisy channel, then the channel with noiseless feedback must have Cany(η ln λ) ln λ In general: If P( X > m) < f(m), then K : Perror(d) < f(kλ d ) Sufficiency: If there is an α > η ln λ for which the channel with noiseless feedback has Cany(α) > ln λ then the scalar system with parameter λ 1 with a bounded disturbance can be stabilized across the noisy channel with finite η-moment. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

30 Separation theorem for scalar control Necessity: If a scalar system with parameter λ > 1 can be stabilized with finite η-moment across a noisy channel, then the channel with noiseless feedback must have Cany(η ln λ) ln λ In general: If P( X > m) < f(m), then K : Perror(d) < f(kλ d ) Sufficiency: If there is an α > η ln λ for which the channel with noiseless feedback has Cany(α) > ln λ then the scalar system with parameter λ 1 with a bounded disturbance can be stabilized across the noisy channel with finite η-moment. Captures stabilization only. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

31 Some easy implications If we want P( X t > m) f(m) = 0 for some finite m, we require zero-error reliability across the channel. Also required (for DMCs) if we want the controller to be finite memory. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

32 Some easy implications If we want P( X t > m) f(m) = 0 for some finite m, we require zero-error reliability across the channel. Also required (for DMCs) if we want the controller to be finite memory. For generic DMCs, anytime reliability with feedback is upper-bounded: f(kλ d ) ζ d 1 ζ log 2 λ f(m) K m log 2 A controlled state can have at best a power-law tail. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

33 Some easy implications If we want P( X t > m) f(m) = 0 for some finite m, we require zero-error reliability across the channel. Also required (for DMCs) if we want the controller to be finite memory. For generic DMCs, anytime reliability with feedback is upper-bounded: f(kλ d ) ζ d 1 ζ log 2 λ f(m) K m log 2 A controlled state can have at best a power-law tail. If we just want lim m f(m) = 0, then just Shannon capacity > log 2 λ is required for DMCs. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

34 Some easy implications If we want P( X t > m) f(m) = 0 for some finite m, we require zero-error reliability across the channel. Also required (for DMCs) if we want the controller to be finite memory. For generic DMCs, anytime reliability with feedback is upper-bounded: f(kλ d ) ζ d 1 ζ log 2 λ f(m) K m log 2 A controlled state can have at best a power-law tail. If we just want lim m f(m) = 0, then just Shannon capacity > log 2 λ is required for DMCs. Almost-sure stabilization for W t = 0 follows by simple time-varying transformation. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

35 Asymptotic communication problem hierarchy The easiest: Shannon communication Asymptotically: a single figure of merit C Equivalent to most estimation problems of stationary ergodic processes with bounded distortion measures. Feedback does not matter. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

36 Asymptotic communication problem hierarchy The easiest: Shannon communication Asymptotically: a single figure of merit C Equivalent to most estimation problems of stationary ergodic processes with bounded distortion measures. Feedback does not matter. Hardest level: Zero-error communication Single figure of merit C0 Feedback matters. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

37 Asymptotic communication problem hierarchy The easiest: Shannon communication Asymptotically: a single figure of merit C Equivalent to most estimation problems of stationary ergodic processes with bounded distortion measures. Feedback does not matter. Intermediate families: Anytime communication Multiple figures of merit: ( R, α) Feedback case equivalent to stabilization problems Related nonstationary estimation problems fall here also Does feedback matter? Hardest level: Zero-error communication Single figure of merit C0 Feedback matters. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

38 My favorite example: the BEC δ 1 δ e δ 1 δ 0 Simple capacity 1 δ bits per channel use With perfect feedback, simple to achieve: retransmit until it gets through Time till success: Geometric(1 δ) Expected time to get through: 1 1 δ Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

39 My favorite example: the BEC δ 1 δ e δ 1 δ 0 Error Exponent (base 2) Simple capacity 1 δ bits per channel use With perfect feedback, simple to achieve: retransmit until it gets through Time till success: Geometric(1 δ) Expected time to get through: 1 1 δ Rate (in bits) Classical bounds Sphere-packing bound D(1 R δ) Random coding bound max ρ [0,1] E 0 (ρ) ρr Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

40 My favorite example: the BEC δ 1 δ e δ 1 δ 0 Error Exponent (base 2) Simple capacity 1 δ bits per channel use With perfect feedback, simple to achieve: retransmit until it gets through Time till success: Geometric(1 δ) Expected time to get through: 1 1 δ Rate (in bits) Classical bounds Sphere-packing bound D(1 R δ) Random coding bound max ρ [0,1] E 0 (ρ) ρr What happens with feedback? Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

41 BEC with feedback and fixed blocks At rate R < 1, have Rn bits to transmit in n channel uses. Typically (1 δ)n code bits will be received. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

42 BEC with feedback and fixed blocks At rate R < 1, have Rn bits to transmit in n channel uses. Typically (1 δ)n code bits will be received. Block errors caused by atypical channel behavior. Doomed if fewer than Rn bits arrive intact. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

43 BEC with feedback and fixed blocks At rate R < 1, have Rn bits to transmit in n channel uses. Typically (1 δ)n code bits will be received. Block errors caused by atypical channel behavior. Doomed if fewer than Rn bits arrive intact. Feedback can not save us. D(1 R δ) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

44 BEC with feedback and fixed blocks At rate R < 1, have Rn bits to transmit in n channel uses. Typically (1 δ)n code bits will be received. Block errors caused by atypical channel behavior. Doomed if fewer than Rn bits arrive intact. Feedback can not save us. D(1 R δ) Dobrushin-62 showed that this type of behavior is common: E + (R) = E sp (R) for symmetric channels. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

45 BEC with feedback and fixed delay R = 1 2 example: δ 2 δ 2 δ 2 δ 2 1 (1 δ) 2 (1 δ) 2 (1 δ) 2 (1 δ) Birth-death chain: positive recurrent if δ < 1 2 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

46 BEC with feedback and fixed delay R = 1 2 example: δ 2 δ 2 δ 2 δ 2 1 (1 δ) 2 (1 δ) 2 (1 δ) 2 (1 δ) Birth-death chain: positive recurrent if δ < 1 2 Delay exponent easy to see: P(D d) = P(L > d 2 ) = K( δ 1 δ )d Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

47 BEC with feedback and fixed delay R = 1 2 example: δ 2 δ 2 δ 2 δ 2 1 (1 δ) 2 (1 δ) 2 (1 δ) 2 (1 δ) Birth-death chain: positive recurrent if δ < 1 2 Delay exponent easy to see: P(D d) = P(L > d 2 ) = K( δ 1 δ )d vs for block-coding with δ = 0.4 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

48 BEC with feedback and fixed delay R = 1 2 example: δ 2 δ 2 δ 2 δ 2 1 (1 δ) 2 (1 δ) 2 (1 δ) 2 (1 δ) Birth-death chain: positive recurrent if δ < 1 2 Delay exponent easy to see: P(D d) = P(L > d 2 ) = K( δ 1 δ )d vs for block-coding with δ = 0.4 Block-coding is misleading! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

49 Pinsker s bounding construction explained Without feedback: E + (R) continues to be a bound. Consider a code with target delay d Use it to construct a block-code with blocksize n d Genie-aided decoder: has the truth of all bits before i B feedforward delay Fixed bb Y delay decoder Y 1 Fixed delay XOR XOR decoder... t eb (t d)r bb (t d)r 1 1 eb Causal X encoder DMC B (t d)r 1 B Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

50 Pinsker s bounding construction explained Without feedback: E + (R) continues to be a bound. Consider a code with target delay d Use it to construct a block-code with blocksize n d Genie-aided decoder: has the truth of all bits before i Error events for genie-aided system depend only on last d B feedforward delay Fixed bb Y delay decoder Y 1 Fixed delay XOR XOR decoder... t eb (t d)r bb (t d)r 1 1 eb Causal X encoder DMC B (t d)r 1 B Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

51 Pinsker s bounding construction explained Without feedback: E + (R) continues to be a bound. Consider a code with target delay d Use it to construct a block-code with blocksize n d Genie-aided decoder: has the truth of all bits before i Error events for genie-aided system depend only on last d Apply a change of measure argument B feedforward delay Fixed bb Y delay decoder Y 1 Fixed delay XOR XOR decoder... t eb (t d)r bb (t d)r 1 1 eb Causal X encoder DMC B (t d)r 1 B Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

52 Using E sp to bound α in general Past behavior λn λ 1 λ d Future (1 λ)n d λr n bits within deadline bits to ignore The block error probability is like exp( α(1 λ)n) which cannot exceed the Haroutunian bound exp( E sp (λr)n) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

53 Using E sp to bound α in general Past behavior λn λ 1 λ d Future (1 λ)n d λr n bits within deadline bits to ignore The block error probability is like exp( α(1 λ)n) which cannot exceed the Haroutunian bound exp( E sp (λr)n) α (R) E sp(λr) 1 λ Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

54 Using E sp to bound α in general Past behavior λn λ 1 λ d Future (1 λ)n d λr n bits within deadline bits to ignore The block error probability is like exp( α(1 λ)n) which cannot exceed the Haroutunian bound exp( E sp (λr)n) α (R) E sp(λr) 1 λ The error events involve both the past and the future. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

55 Uncertainty-focusing bound for symmetric DMCs Minimize over λ for symmetric DMCs to sweep out frontier by varying ρ > 0: Using the Gallager function: R(ρ) = E 0(ρ) ρ E a + (ρ) = E 0 (ρ) E 0 (ρ) = max ln q j ( q i p i ) 1+ρ 1 1+ρ ij Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

56 Uncertainty-focusing bound for symmetric DMCs Minimize over λ for symmetric DMCs to sweep out frontier by varying ρ > 0: Using the Gallager function: R(ρ) = E 0(ρ) ρ E a + (ρ) = E 0 (ρ) E 0 (ρ) = max ln q j ( q i p i ) 1+ρ 1 1+ρ ij Same form as Viterbi s convolutional coding bound for constraint-lengths, but a lot more fundamental! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

57 Upper bound tight for the BEC with feedback 7 Error Exponent (base 2) Rate (in bits) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

58 Implications for scalar moment stabilization Error Exponent (base e) Rate (in nats) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

59 Implications for scalar moment stabilization 10 8 Moments stabilized Open loop unstable gain Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

60 Outline 1 A bridge to nowhere? A simple control problem A connection to information theory Fixing information theory and filling in the gaps. 2 Coming back to control What is wrong with random coding The role of noiseless feedback 3 Taking control thinking to the forefront of information theory. The holy grail problem Control thinking to the rescue! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

61 Random coding bound is relatively easy to achieve Randomly label the uniformly quantized state! t=0 t=1 t=2 t=3 t= R = log 2 3 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

62 Random coding bound is relatively easy to achieve Randomly label the uniformly quantized state! Stable system state renews itself. t=0 t=1 t=2 t=3 t= R = log 2 3 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

63 Random coding bound is relatively easy to achieve Randomly label the uniformly quantized state! Stable system state renews itself. It diverges locally whenever the channel misbehaves. t=0 t=1 t=2 t=3 t= R = log 2 3 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

64 Random coding bound is relatively easy to achieve Randomly label the uniformly quantized state! Stable system state renews itself. It diverges locally whenever the channel misbehaves. Semi-reasonable implementation complexity. t=0 t=1 t=2 t=3 t= R = log 2 3 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

65 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

66 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

67 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Gallager s E r (R branch ) emerges as the governing exponent. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

68 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Gallager s E r (R branch ) emerges as the governing exponent. Apply the control based on current ML state. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

69 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Gallager s E r (R branch ) emerges as the governing exponent. Apply the control based on current ML state. Computational nightmare: effort grows exponentially with time. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

70 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Gallager s E r (R branch ) emerges as the governing exponent. Apply the control based on current ML state. Computational nightmare: effort grows exponentially with time. Use Stack-based greedy search algorithm instead. Log likelihoods are additive. The score of a path is a random walk with drift. Bias it so that the true path goes up and false ones down. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

71 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Gallager s E r (R branch ) emerges as the governing exponent. Apply the control based on current ML state. Computational nightmare: effort grows exponentially with time. Use Stack-based greedy search algorithm instead. Log likelihoods are additive. The score of a path is a random walk with drift. Bias it so that the true path goes up and false ones down. Classical results tell us that with appropriate bias, achieve E r (R branch ) for error probability and hence power-law in state Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

72 Controller and Computations All false disjoint paths through the trellis are pairwise independent with the true path. Bound the number of distinct paths by assuming no remerging. Gallager s E r (R branch ) emerges as the governing exponent. Apply the control based on current ML state. Computational nightmare: effort grows exponentially with time. Use Stack-based greedy search algorithm instead. Log likelihoods are additive. The score of a path is a random walk with drift. Bias it so that the true path goes up and false ones down. Classical results tell us that with appropriate bias, achieve E r (R branch ) for error probability and hence power-law in state At the cost of only finite expected computation. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

73 Catch up all-at-once phenomenon Simulation Parameters: λ = 1.1 ε = 0.05 Ω = 2.0 = Bias = 0.55 T = ,000 Blocks 17 seconds to run Decoded Bin Actual Bin Rate = Capacity = 0.71 Bin Block Number Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

74 Truth in advertising: computation revisited Although we are doing better than exponential growth, we still have power laws on both sides. What if we needed a finite speed computer in the controller? Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

75 Truth in advertising: computation revisited Although we are doing better than exponential growth, we still have power laws on both sides. What if we needed a finite speed computer in the controller? Bad news: Assume 0 control applied if we can not decode yet. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

76 Truth in advertising: computation revisited Although we are doing better than exponential growth, we still have power laws on both sides. What if we needed a finite speed computer in the controller? Bad news: Assume 0 control applied if we can not decode yet. Power law for comp. implies power low for waiting. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

77 Truth in advertising: computation revisited Although we are doing better than exponential growth, we still have power laws on both sides. What if we needed a finite speed computer in the controller? Bad news: Assume 0 control applied if we can not decode yet. Power law for comp. implies power low for waiting. Exponentially rare doubly exponentially bad states! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

78 How to hit the higher bound? Error Exponent (base e) Rate (in nats) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

79 How to hit the higher bound? 10 8 Moments stabilized Open loop unstable gain Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

80 Fortified channels Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Some mix of noisy and noiseless channels Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

81 Fortified channels Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Some mix of noisy and noiseless channels Is it all or nothing? Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

82 Noiseless channel can enable event-based sampling Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Need to allow for gradual progress during bad periods. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

83 Noiseless channel can enable event-based sampling Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Need to allow for gradual progress during bad periods. Use the noiseless channel for supervisory information: Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

84 Noiseless channel can enable event-based sampling Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Need to allow for gradual progress during bad periods. Use the noiseless channel for supervisory information: Have the observer do event-based sampling of the state. Outer net to quantize and encode the state Inner catchment area to resample the state Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

85 Noiseless channel can enable event-based sampling Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Need to allow for gradual progress during bad periods. Use the noiseless channel for supervisory information: Have the observer do event-based sampling of the state. Quantization net grows as needed, but has only e nr boxes. Outer net to quantize and encode the state Inner catchment area to resample the state Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

86 Noiseless channel can enable event-based sampling Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Need to allow for gradual progress during bad periods. Use the noiseless channel for supervisory information: Have the observer do event-based sampling of the state. Quantization net grows as needed, but has only e nr boxes. Noiseless channel tells controller when it has resampled. Outer net to quantize and encode the state Inner catchment area to resample the state Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

87 Noiseless channel can enable event-based sampling Noisy forward channel uses Fortification noiseless forward channel uses S 1 S 2 S 3 S 4 S 5 S 6 Need to allow for gradual progress during bad periods. Use the noiseless channel for supervisory information: Have the observer do event-based sampling of the state. Quantization net grows as needed, but has only e nr boxes. Noiseless channel tells controller when it has resampled. Use the noisy channel for variable-length block-coding. Outer net to quantize and encode the state Inner catchment area to resample the state Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

88 Why gradual progress is better: intuition 130 Progres (in bits) time (in BEC uses) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

89 Why this works: proof strategy Lift problem by using large nr Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

90 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

91 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Stopping time for variable-length channel is like n + T, where T is geometric exp( E 0 (ρ)). Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

92 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Stopping time for variable-length channel is like n + T, where T is geometric exp( E 0 (ρ)). Interpret with ln λ < R = E 0(ρ) ρ < E0(η+ǫ) η+ǫ Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

93 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Stopping time for variable-length channel is like n + T, where T is geometric exp( E 0 (ρ)). Interpret with ln λ < R = E 0(ρ) ρ < E0(η+ǫ) η+ǫ Behaves like a virtual packet-erasure channel. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

94 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Stopping time for variable-length channel is like n + T, where T is geometric exp( E 0 (ρ)). Interpret with ln λ < R = E 0(ρ) ρ < E0(η+ǫ) η+ǫ Behaves like a virtual packet-erasure channel. Each packet carries n(r ln λ) nats. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

95 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Stopping time for variable-length channel is like n + T, where T is geometric exp( E 0 (ρ)). Interpret with ln λ < R = E 0(ρ) ρ < E0(η+ǫ) η+ǫ Behaves like a virtual packet-erasure channel. Each packet carries n(r ln λ) nats. Disturbances grow by factor O(λ n ) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

96 Why this works: proof strategy Lift problem by using large nr Very few noiseless channel uses required Stopping time for variable-length channel is like n + T, where T is geometric exp( E 0 (ρ)). Interpret with ln λ < R = E 0(ρ) ρ < E0(η+ǫ) η+ǫ Behaves like a virtual packet-erasure channel. Each packet carries n(r ln λ) nats. Disturbances grow by factor O(λ n ) Erasure probability exp( E0 (ρ)) Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

97 Outline 1 A bridge to nowhere? A simple control problem A connection to information theory Fixing information theory and filling in the gaps. 2 Coming back to control What is wrong with random coding The role of noiseless feedback 3 Taking control thinking to the forefront of information theory. The holy grail problem Control thinking to the rescue! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

98 The holy grail: understanding complexity Classical goal: arbitrarily low probability of error. Classical assumption: not delay sensitive at all. New twist: minimize total power consumption Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

99 The holy grail: understanding complexity Classical goal: arbitrarily low probability of error. Classical assumption: not delay sensitive at all. New twist: minimize total power consumption Important technology trends Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

100 The holy grail: understanding complexity Classical goal: arbitrarily low probability of error. Classical assumption: not delay sensitive at all. New twist: minimize total power consumption Important technology trends Moore s law allows billions of transistors, and but only mildly reduces power-consumption per transistor. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

101 The holy grail: understanding complexity Classical goal: arbitrarily low probability of error. Classical assumption: not delay sensitive at all. New twist: minimize total power consumption Important technology trends Moore s law allows billions of transistors, and but only mildly reduces power-consumption per transistor. New short-range applications: swarm behavior, in-home networks, dense meshes, personal-area networks, UWB, between-chip communication, etc. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

102 Decoding power vs communication range γ (db) mm 10mm 1m 100m 10km Distance Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

103 Dense linear codes with brute-force decoding Total power Optimal transmit power Decoding power Shannon waterfall log 10 ( P e ) Power Decoding Power nr2 nr, Error Prob 2 Esp(R,P)n Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

104 Convolutional codes with Viterbi decoding Total power Optimal transmit power Decoding power Shannon waterfall log 10 ( P e ) Power Decoding Power L c R2 LcR, Error Prob 2 Econv(R,P)Lc Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

105 Convolutional with magical sequential decoding Shannon waterfall Optimal transmit power Decoding power Total power log 10 ( P e ) Power Decoding Power L c R, Error Prob 2 Econv(R,P)Lc Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

106 Dense linear codes with magical syndrome decoding Shannon waterfall Optimal transmit power Total power Decoding power log 10 ( P e ) Power Decoding Power (1 R)nR, Error Prob 2 Esp(R,P)n Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

107 A new hope: iterative decoding Make assumptions about the decoder implementation rather than the code. Rich enough to capture LDPC, RA, Turbo, etc. codes. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

108 A new hope: iterative decoding Make assumptions about the decoder implementation rather than the code. Massively parallel computational nodes. Each connected to at most α + 1 other nodes. Rich enough to capture LDPC, RA, Turbo, etc. codes. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

109 A new hope: iterative decoding Make assumptions about the decoder implementation rather than the code. Massively parallel computational nodes. Each connected to at most α + 1 other nodes. Each consumes Enode energy per iteration and can send arbitrary messages to its neighbors. Rich enough to capture LDPC, RA, Turbo, etc. codes. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

110 A new hope: iterative decoding Make assumptions about the decoder implementation rather than the code. Massively parallel computational nodes. Each connected to at most α + 1 other nodes. Each consumes Enode energy per iteration and can send arbitrary messages to its neighbors. Some nodes initialized with a single received codeword symbol. Some nodes responsible for decoding a single message bit. Rich enough to capture LDPC, RA, Turbo, etc. codes. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

111 A new hope: iterative decoding Make assumptions about the decoder implementation rather than the code. Massively parallel computational nodes. Each connected to at most α + 1 other nodes. Each consumes Enode energy per iteration and can send arbitrary messages to its neighbors. Some nodes initialized with a single received codeword symbol. Some nodes responsible for decoding a single message bit. Run for a fixed number of iterations i. Rich enough to capture LDPC, RA, Turbo, etc. codes. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

112 A new hope: iterative decoding Make assumptions about the decoder implementation rather than the code. Massively parallel computational nodes. Each connected to at most α + 1 other nodes. Each consumes Enode energy per iteration and can send arbitrary messages to its neighbors. Some nodes initialized with a single received codeword symbol. Some nodes responsible for decoding a single message bit. Run for a fixed number of iterations i. Rich enough to capture LDPC, RA, Turbo, etc. codes. Power-consumption ie node per received sample. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

113 B Y B R N Y 1 Y B Y 2 Y X Y Y 7 X Y 8 How to lower-bound the number of iterations? o o t b i t o d e B i Y i Key concept: decoding neighborhoods = information patterns B X X Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

114 B Y B R N Y 1 Y B Y 2 Y X Y Y 7 X Y 8 How to lower-bound the number of iterations? o o t b i t o d e B 1 2 B i Y i 7 8 Key concept: decoding neighborhoods = information patterns Decoding neighborhood size n 1 + (α + 1)α i 1 α i. X X Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

115 B Y B R N Y 1 Y B Y 2 Y X Y Y 7 X Y 8 How to lower-bound the number of iterations? X 4 5 o o t 5 b i t o d e B B i X Y i Key concept: decoding neighborhoods = information patterns Decoding neighborhood size n 1 + (α + 1)α i 1 α i. Need to lower-bound average probability of bit error in terms of n. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

116 B Y B R N Y 1 Y B Y 2 Y X Y Y 7 X Y 8 How to lower-bound the number of iterations? X 4 5 o o t K b i t o d e B B i X Y i Key concept: decoding neighborhoods = information patterns Decoding neighborhood size n 1 + (α + 1)α i 1 α i. Need to lower-bound average probability of bit error in terms of n. Key insight: n is playing a role analogous to delay. Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

117 A local sphere-packing bound for the AWGN Decoding neighborhood size n 1 + (α + 1)α i 1 α i. P e sup σ 2 G >σp 2 µ(n): C(G)<R h 1 b (δ(g)) exp 2 ( nd(σ 2 G σ 2 P) 1 2 C(G) = 1 2 log 2 (1 + P T ), δ(g): 1 C(G) σg 2 R ( σ 2 )) φ(n, h 1 b (δ(g))) G σp 2 1 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

118 A local sphere-packing bound for the AWGN Decoding neighborhood size n 1 + (α + 1)α i 1 α i. P e sup σ 2 G >σp 2 µ(n): C(G)<R h 1 b (δ(g)) exp 2 ( nd(σ 2 G σ 2 P) 1 2 C(G) = 1 2 log 2 (1 + P T ), δ(g): 1 C(G) σg 2 R µ(n) = 1 2 (1 + 1 T(n)+1 + 4T(n)+2 nt(n)(1+t(n)) ) T(n) = W L ( exp( 1)(1/4) 1/n ) W L (x) solves x = W L (x) exp(w L (x)) ) φ(n, y) = n(w L ( exp( 1)( y 2 ) 2 n + 1) ( σ 2 )) φ(n, h 1 b (δ(g))) G σp 2 1 Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

119 A local sphere-packing bound for the AWGN Decoding neighborhood size n 1 + (α + 1)α i 1 α i. P e sup σ 2 G >σp 2 µ(n): C(G)<R h 1 b (δ(g)) exp 2 ( nd(σ 2 G σ 2 P) 1 2 C(G) = 1 2 log 2 (1 + P T ), δ(g): 1 C(G) σg 2 R µ(n) = 1 2 (1 + 1 T(n)+1 + 4T(n)+2 nt(n)(1+t(n)) ) T(n) = W L ( exp( 1)(1/4) 1/n ) W L (x) solves x = W L (x) exp(w L (x)) ) φ(n, y) = n(w L ( exp( 1)( y 2 ) 2 n + 1) ( σ 2 )) φ(n, h 1 b (δ(g))) G σp 2 1 Double-exponential potential return on investments in decoding power! Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54

120 Waterslide curves for general AWGN case 2 4 γ = 0.4 γ = 0.3 γ = 0.2 Shannon limit log 10 ( P e ) Power Anant Sahai (UC Berkeley) IT + Control Mar 26, / 54