8 research outputs found
EM's Convergence in Gaussian Latent Tree Models
We study the optimization landscape of the log-likelihood function and the
convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian
tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes
are observable and non-leaf nodes are unobservable. We show that the unique
non-trivial stationary point of the population log-likelihood is its global
maximum, and establish that the expectation-maximization algorithm is
guaranteed to converge to it in the single latent variable case. Our results
for the landscape of the log-likelihood function in general latent tree models
provide support for the extensive practical use of maximum likelihood
based-methods in this setting. Our results for the EM algorithm extend an
emerging line of work on obtaining global convergence guarantees for this
celebrated algorithm. We show our results for the non-trivial stationary points
of the log-likelihood by arguing that a certain system of polynomial equations
obtained from the EM updates has a unique non-trivial solution. The global
convergence of the EM algorithm follows by arguing that all trivial fixed
points are higher-order saddle points
Learning and Testing Latent-Tree Ising Models Efficiently
We provide time- and sample-efficient algorithms for learning and testing
latent-tree Ising models, i.e. Ising models that may only be observed at their
leaf nodes. On the learning side, we obtain efficient algorithms for learning a
tree-structured Ising model whose leaf node distribution is close in Total
Variation Distance, improving on the results of prior work. On the testing
side, we provide an efficient algorithm with fewer samples for testing whether
two latent-tree Ising models have leaf-node distributions that are close or far
in Total Variation distance. We obtain our algorithms by showing novel
localization results for the total variation distance between the leaf-node
distributions of tree-structured Ising models, in terms of their marginals on
pairs of leaves
Node-Max-Cut and the Complexity of Equilibrium in Linear Weighted Congestion Games
In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkable gap regarding the complexity of PNE in congestion games with weighted and unweighted players, since in case of unweighted players, a PNE can be easily computed by either a simple greedy algorithm (for series-parallel networks) or any better response dynamics (when the latency is equal to the congestion). For the latter of the results above, we need to show first that computing a local optimum of a natural restriction of Max-Cut, which we call Node-Max-Cut, is PLS-complete. In Node-Max-Cut, the input graph is vertex-weighted and the weight of each edge is equal to the product of the weights of its endpoints. Due to the very restricted nature of Node-Max-Cut, the reduction requires a careful combination of new gadgets with ideas and techniques from previous work. We also show how to compute efficiently a (1+?)-approximate equilibrium for Node-Max-Cut, if the number of different vertex weights is constant
Opinion Dynamics with Limited Information
Abstract
We study opinion formation games based on the famous model proposed by Friedkin and Johsen (FJ model). In today’s huge social networks the assumption that in each round agents update their opinions by taking into account the opinions of all their friends is unrealistic. So, we are interested in the convergence properties of simple and natural variants of the FJ model that use limited information exchange in each round and converge to the same stable point. As in the FJ model, we assume that each agent i has an intrinsic opinion
s
i
∈
[
0
,
1
]
and maintains an expressed opinion
x
i
(
t
)
∈
[
0
,
1
]
in each round t. To model limited information exchange, we consider an opinion formation process where each agent i meets with one random friend j at each round t and learns only her current opinion
x
j
(
t
)
. The amount of influence j imposes on i is reflected by the probability
p
ij
with which i meets j. Then, agent i suffers a disagreement cost that is a convex combination of
(
x
i
(
t
)
-
s
i
)
2
and
(
x
i
(
t
)
-
x
j
(
t
)
)
2
. An important class of dynamics in this setting are no regret dynamics, i.e. dynamics that ensure vanishing regret against the experienced disagreement cost to the agents. We show an exponential gap between the convergence rate of no regret dynamics and of more general dynamics that do not ensure no regret. We prove that no regret dynamics require roughly
Ω
(
1
/
ε
)
rounds to be within distance
ε
from the stable point of the FJ model. On the other hand, we provide an opinion update rule that does not ensure no regret and converges to
x
∗
in
O
~
(
log
2
(
1
/
ε
)
)
rounds. Finally, in our variant of the FJ model, we show that the agents can adopt a simple opinion update rule that ensures no regret to the experienced disagreement cost and results in an opinion vector that converges to the stable point
x
∗
of the FJ model within distance
ε
in
poly
(
1
/
ε
)
rounds. In view of our lower bound for no regret dynamics this rate of convergence is close to best possible