Communication cost of consensus for nodes with limited memory

Motivated by applications in blockchains and sensor networks, we consider a model of $n$ nodes trying to reach consensus on their majority bit. Each node $i$ is assigned a bit at time zero, and is a finite automaton with $m$ bits of memory (i.e., $2^m$ states) and a Poisson clock. When the clock of $i$ rings, $i$ can choose to communicate, and is then matched to a uniformly chosen node $j$. The nodes $j$ and $i$ may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that when $m>3 \log\log\log(n)$, consensus can be reached at linear communication cost, but this is impossible if $m<\log\log\log(n)$. We also study a synchronous variant of the model, where our upper and lower bounds on $m$ for achieving linear communication cost are $2\log\log\log(n)$ and $\log\log\log(n)$, respectively. A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low.Comment: 62 pages, 5 figure

Adaptive Information Gathering via Imitation Learning

In the adaptive information gathering problem, a policy is required to select an informative sensing location using the history of measurements acquired thus far. While there is an extensive amount of prior work investigating effective practical approximations using variants of Shannon's entropy, the efficacy of such policies heavily depends on the geometric distribution of objects in the world. On the other hand, the principled approach of employing online POMDP solvers is rendered impractical by the need to explicitly sample online from a posterior distribution of world maps. We present a novel data-driven imitation learning framework to efficiently train information gathering policies. The policy imitates a clairvoyant oracle - an oracle that at train time has full knowledge about the world map and can compute maximally informative sensing locations. We analyze the learnt policy by showing that offline imitation of a clairvoyant oracle is implicitly equivalent to online oracle execution in conjunction with posterior sampling. This observation allows us to obtain powerful near-optimality guarantees for information gathering problems possessing an adaptive sub-modularity property. As demonstrated on a spectrum of 2D and 3D exploration problems, the trained policies enjoy the best of both worlds - they adapt to different world map distributions while being computationally inexpensive to evaluate.Comment: Robotics Science and Systems, 201

Exact minimum number of bits to stabilize a linear system

We consider an unstable scalar linear stochastic system, X_(n + 1) = aX_n + Z_n – U_n.; where a ≥ 1 is the system gain, Z_n's are independent random variables with bounded α-th moments, and U_n'S are the control actions that are chosen by a controller who receives a single element of a finite set {1, …, M} as its only information about system state X_i. We show that M = [a] + 1 is necessary and sufficient for ß- moment stability, for any ß < a. Our achievable scheme is a uniform quantizer of the zoom-in / zoom-out type. We analyze its performance using probabilistic arguments. We prove a matching converse using information-theoretic techniques. Our results generalize to vector systems, to systems with dependent Gaussian noise, and to the scenario in which a small fraction of transmitted messages is lost

Stabilizing a System with an Unbounded Random Gain Using Only Finitely Many Bits

We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X_(n+1) = A_n X_n +W_n –U_n, where the A_n's are drawn independently at random at each time n from a known distribution with unbounded support, and where the controller receives at most R bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite R. While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of A_n is typical, and an emergency mode (or zoom-out), where the realization of A_n is exceptionally large

Stabilizing a system with an unbounded random gain using only a finite number of bits

We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X_(n+1) = A_nX_n+W_n−U_n, where the A_n's are drawn independently at random at each time n from a known distribution with unbounded support, and where the controller receives at most R bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite R. While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of A_n is typical, and an emergency mode (or zoom-out), where the realization of A_n is exceptionally large