20 research outputs found
Nonasymptotic Convergence Rates for Cooperative Learning Over Time-Varying Directed Graphs
We study the problem of distributed hypothesis testing with a network of
agents where some agents repeatedly gain access to information about the
correct hypothesis. The group objective is to globally agree on a joint
hypothesis that best describes the observed data at all the nodes. We assume
that the agents can interact with their neighbors in an unknown sequence of
time-varying directed graphs. Following the pioneering work of Jadbabaie,
Molavi, Sandroni, and Tahbaz-Salehi, we propose local learning dynamics which
combine Bayesian updates at each node with a local aggregation rule of private
agent signals. We show that these learning dynamics drive all agents to the set
of hypotheses which best explain the data collected at all nodes as long as the
sequence of interconnection graphs is uniformly strongly connected. Our main
result establishes a non-asymptotic, explicit, geometric convergence rate for
the learning dynamic
Learning without Recall by Random Walks on Directed Graphs
We consider a network of agents that aim to learn some unknown state of the
world using private observations and exchange of beliefs. At each time, agents
observe private signals generated based on the true unknown state. Each agent
might not be able to distinguish the true state based only on her private
observations. This occurs when some other states are observationally equivalent
to the true state from the agent's perspective. To overcome this shortcoming,
agents must communicate with each other to benefit from local observations. We
propose a model where each agent selects one of her neighbors randomly at each
time. Then, she refines her opinion using her private signal and the prior of
that particular neighbor. The proposed rule can be thought of as a Bayesian
agent who cannot recall the priors based on which other agents make inferences.
This learning without recall approach preserves some aspects of the Bayesian
inference while being computationally tractable. By establishing a
correspondence with a random walk on the network graph, we prove that under the
described protocol, agents learn the truth exponentially fast in the almost
sure sense. The asymptotic rate is expressed as the sum of the relative
entropies between the signal structures of every agent weighted by the
stationary distribution of the random walk.Comment: 6 pages, To Appear in Conference on Decision and Control 201
Distributed Learning with Infinitely Many Hypotheses
We consider a distributed learning setup where a network of agents
sequentially access realizations of a set of random variables with unknown
distributions. The network objective is to find a parametrized distribution
that best describes their joint observations in the sense of the
Kullback-Leibler divergence. Apart from recent efforts in the literature, we
analyze the case of countably many hypotheses and the case of a continuum of
hypotheses. We provide non-asymptotic bounds for the concentration rate of the
agents' beliefs around the correct hypothesis in terms of the number of agents,
the network parameters, and the learning abilities of the agents. Additionally,
we provide a novel motivation for a general set of distributed Non-Bayesian
update rules as instances of the distributed stochastic mirror descent
algorithm.Comment: Submitted to CDC201
Beliefs in Decision-Making Cascades
This work explores a social learning problem with agents having nonidentical
noise variances and mismatched beliefs. We consider an -agent binary
hypothesis test in which each agent sequentially makes a decision based not
only on a private observation, but also on preceding agents' decisions. In
addition, the agents have their own beliefs instead of the true prior, and have
nonidentical noise variances in the private signal. We focus on the Bayes risk
of the last agent, where preceding agents are selfish.
We first derive the optimal decision rule by recursive belief update and
conclude, counterintuitively, that beliefs deviating from the true prior could
be optimal in this setting. The effect of nonidentical noise levels in the
two-agent case is also considered and analytical properties of the optimal
belief curves are given. Next, we consider a predecessor selection problem
wherein the subsequent agent of a certain belief chooses a predecessor from a
set of candidates with varying beliefs. We characterize the decision region for
choosing such a predecessor and argue that a subsequent agent with beliefs
varying from the true prior often ends up selecting a suboptimal predecessor,
indicating the need for a social planner. Lastly, we discuss an augmented
intelligence design problem that uses a model of human behavior from cumulative
prospect theory and investigate its near-optimality and suboptimality.Comment: final version, to appear in IEEE Transactions on Signal Processin