41 research outputs found
Estimating Kullback-Leibler Divergence Using Kernel Machines
Recently, a method called the Mutual Information Neural Estimator (MINE) that
uses neural networks has been proposed to estimate mutual information and more
generally the Kullback-Leibler (KL) divergence between two distributions. The
method uses the Donsker-Varadhan representation to arrive at the estimate of
the KL divergence and is better than the existing estimators in terms of
scalability and flexibility. The output of MINE algorithm is not guaranteed to
be a consistent estimator. We propose a new estimator that instead of searching
among functions characterized by neural networks searches the functions in a
Reproducing Kernel Hilbert Space. We prove that the proposed estimator is
consistent. We carry out simulations and show that when the datasets are small
the proposed estimator is more reliable than the MINE estimator and when the
datasets are large the performance of the two methods are close
Joint Concordance Index
Existing metrics in competing risks survival analysis such as concordance and
accuracy do not evaluate a model's ability to jointly predict the event type
and the event time. To address these limitations, we propose a new metric,
which we call the joint concordance. The joint concordance measures a model's
ability to predict the overall risk profile, i.e., risk of death from different
event types. We develop a consistent estimator for the new metric that accounts
for the censoring bias. We use the new metric to develop a variable importance
ranking approach. Using the real and synthetic data experiments, we show that
models selected using the existing metrics are worse than those selected using
joint concordance at jointly predicting the event type and event time. We show
that the existing approaches for variable importance ranking often fail to
recognize the importance of the event-specific risk factors, whereas, the
proposed approach does not, since it compares risk factors based on their
contribution to the prediction of the different event-types. To summarize,
joint concordance is helpful for model comparisons and variable importance
ranking and has the potential to impact applications such as
risk-stratification and treatment planning in multimorbid populations
Risk-Stratify: Confident Stratification Of Patients Based On Risk
A clinician desires to use a risk-stratification method that achieves
confident risk-stratification - the risk estimates of the different patients
reflect the true risks with a high probability. This allows him/her to use
these risks to make accurate predictions about prognosis and decisions about
screening, treatments for the current patient. We develop Risk-stratify - a two
phase algorithm that is designed to achieve confident risk-stratification. In
the first phase, we grow a tree to partition the covariate space. Each node in
the tree is split using statistical tests that determine if the risks of the
child nodes are different or not. The choice of the statistical tests depends
on whether the data is censored (Log-rank test) or not (U-test). The set of the
leaves of the tree form a partition. The risk distribution of patients that
belong to a leaf is different from the sibling leaf but not the rest of the
leaves. Therefore, some of the leaves that have similar underlying risks are
incorrectly specified to have different risks. In the second phase, we develop
a novel recursive graph decomposition approach to address this problem. We
merge the leaves of the tree that have similar risks to form new leaves that
form the final output. We apply Risk-stratify on a cohort of patients (with no
history of cardiovascular disease) from UK Biobank and assess their risk for
cardiovascular disease. Risk-stratify significantly improves
risk-stratification, i.e., a lower fraction of the groups have over/under
estimated risks (measured in terms of false discovery rate; 33% reduction) in
comparison to state-of-the-art methods for cardiovascular prediction (Random
forests, Cox model, etc.). We find that the Cox model significantly over
estimates the risk of 21,621 patients out of 216,211 patients. Risk-stratify
can accurately categorize 2,987 of these 21,621 patients as low-risk
individuals
Dynamic Matching and Allocation of Tasks
In many two-sided markets, the parties to be matched have incomplete
information about their characteristics. We consider the settings where the
parties engaged are extremely patient and are interested in long-term
partnerships. Hence, once the final matches are determined, they persist for a
long time. Each side has an opportunity to learn (some) relevant information
about the other before final matches are made. For instance, clients seeking
workers to perform tasks often conduct interviews that require the workers to
perform some tasks and thereby provide information to both sides. The
performance of a worker in such an interview- and hence the information
revealed - depends both on the inherent characteristics of the worker and the
task and also on the actions taken by the worker (e.g. the effort expended),
which are not observed by the client. Thus there is moral hazard. Our goal is
to derive a dynamic matching mechanism that facilitates learning on both sides
before final matches are achieved and ensures that the worker side does not
have incentive to obscure learning of their characteristics through their
actions. We derive such a mechanism that leads to final matching that achieve
optimal performance (revenue) in equilibrium. We show that the equilibrium
strategy is long-run coalitionally stable, which means there is no subset of
workers and clients that can gain by deviating from the equilibrium strategy.
We derive all the results under the modeling assumption that the utilities of
the agents are defined as limit of means of the utility obtained in each
interaction
Optimal Piecewise Local-Linear Approximations
Existing works on "black-box" model interpretation use local-linear
approximations to explain the predictions made for each data instance in terms
of the importance assigned to the different features for arriving at the
prediction. These works provide instancewise explanations and thus give a local
view of the model. To be able to trust the model it is important to understand
the global model behavior and there are relatively fewer works which do the
same. Piecewise local-linear models provide a natural way to extend
local-linear models to explain the global behavior of the model. In this work,
we provide a dynamic programming based framework to obtain piecewise
approximations of the black-box model. We also provide provable fidelity, i.e.,
how well the explanations reflect the black-box model, guarantees. We carry out
simulations on synthetic and real datasets to show the utility of the proposed
approach. At the end, we show that the ideas developed for our framework can
also be used to address the problem of clustering for one-dimensional data. We
give a polynomial time algorithm and prove that it achieves optimal clustering
The user base dynamics of websites
In this work we study for the first time the interaction between marketing
and network effects. We build a model in which the online firm starts with an
initial user base and controls the growth of the user base by choosing the
intensity of advertisements and referrals to potential users. A large user base
provides more profits to the online firm, but building a large user base
through advertisements and referrals is costly; therefore, the optimal policy
must balance the marginal benefits of adding users against the marginal costs
of sending advertisements and referrals. Our work offers three main insights:
(1) The optimal policy prescribes that a new online firm should offer many
advertisements and referrals initially, but then it should decrease
advertisements and referrals over time. (2) If the network effects decrease,
then the change in the optimal policy depends heavily on two factors i) the
level of patience of the online firm, where patient online firms are oriented
towards long term profits and impatient online firms are oriented towards short
term profits and, ii) the size of the user base. If the online firm is very
patient (impatient) and if the network effects decrease, then the optimal
policy prescribes it to be more (less) aggressive in posting advertisements and
referrals at low user base levels and less (more) aggressive in posting
advertisements and referrals at high user base levels. (3) The change in the
optimal policy when network effects decrease also depends heavily on the
heterogeneity in the user base, as measured in terms of the revenue generated
by each user. An online firm that generates most of its revenue from a core
group of users should be more aggressive and protective of its user base than a
firm that generates revenue uniformly from its users
Towards a Theory of Societal Co-Evolution: Individualism versus Collectivism
Substantial empirical research has shown that the level of individualism vs.
collectivism is one of the most critical and important determinants of societal
traits, such as economic growth, economic institutions and health conditions.
But the exact nature of this impact has thus far not been well understood in an
analytical setting. In this work, we develop one of the first theoretical
models that analytically studies the impact of individualism-collectivism on
the society. We model the growth of an individual's welfare (wealth, resources
and health) as depending not only on himself, but also on the level of
collectivism, i.e. the level of dependence on the rest of the individuals in
the society, which leads to a co-evolutionary setting. Based on our model, we
are able to predict the impact of individualism-collectivism on various
societal metrics, such as average welfare, average life-time, total population,
cumulative welfare and average inequality. We analytically show that
individualism has a positive impact on average welfare and cumulative welfare,
but comes with the drawbacks of lower average life-time, lower total population
and higher average inequality
Efficient Interference Management Policies for Femtocell Networks
Managing interference in a network of macrocells underlaid with femtocells
presents an important, yet challenging problem. A majority of spatial
(frequency/time) reuse based approaches partition the users based on coloring
the interference graph, which is shown to be suboptimal. Some spatial time
reuse based approaches schedule the maximal independent sets (MISs) in a
cyclic, (weighted) round-robin fashion, which is inefficient for
delay-sensitive applications. Our proposed policies schedule the MISs in a
non-cyclic fashion, which aim to optimize any given network performance
criterion for delay-sensitive applications while fulfilling minimum throughput
requirements of the users. Importantly, we do not take the interference graph
as given as in existing works; we propose an optimal construction of the
interference graph. We prove that under certain conditions, the proposed policy
achieves the optimal network performance. For large networks, we propose a
low-complexity algorithm for computing the proposed policy. We show that the
policy computed achieves a constant competitive ratio (with respect to the
optimal network performance), which is independent of the network size, under
wide range of deployment scenarios. The policy can be implemented in a
decentralized manner by the users. Compared to the existing policies, our
proposed policies can achieve improvement of up to 130 % in large-scale
deployments
Self-organizing Networks of Information Gathering Cognitive Agents
In many scenarios, networks emerge endogenously as cognitive agents establish
links in order to exchange information. Network formation has been widely
studied in economics, but only on the basis of simplistic models that assume
that the value of each additional piece of information is constant. In this
paper we present a first model and associated analysis for network formation
under the much more realistic assumption that the value of each additional
piece of information depends on the type of that piece of information and on
the information already possessed: information may be complementary or
redundant. We model the formation of a network as a non-cooperative game in
which the actions are the formation of links and the benefit of forming a link
is the value of the information exchanged minus the cost of forming the link.
We characterize the topologies of the networks emerging at a Nash equilibrium
(NE) of this game and compare the efficiency of equilibrium networks with the
efficiency of centrally designed networks. To quantify the impact of
information redundancy and linking cost on social information loss, we provide
estimates for the Price of Anarchy (PoA); to quantify the impact on individual
information loss we introduce and provide estimates for a measure we call
Maximum Information Loss (MIL). Finally, we consider the setting in which
agents are not endowed with information, but must produce it. We show that the
validity of the well-known "law of the few" depends on how information
aggregates; in particular, the "law of the few" fails when information displays
complementarities
Evolution of Social Networks: A Microfounded Model
Many societies are organized in networks that are formed by people who meet
and interact over time. In this paper, we present a first model to capture the
micro-foundations of social networks evolution, where boundedly rational agents
of different types join the network; meet other agents stochastically over
time; and consequently decide to form social ties. A basic premise of our model
is that in real-world networks, agents form links by reasoning about the
benefits that agents they meet over time can bestow. We study the evolution of
the emerging networks in terms of friendship and popularity acquisition given
the following exogenous parameters: structural opportunism, type distribution,
homophily, and social gregariousness. We show that the time needed for an agent
to find "friends" is influenced by the exogenous parameters: agents who are
more gregarious, more homophilic, less opportunistic, or belong to a type
"minority" spend a longer time on average searching for friendships. Moreover,
we show that preferential attachment is a consequence of an emerging doubly
preferential meeting process: a process that guides agents of a certain type to
meet more popular similar-type agents with a higher probability, thereby
creating asymmetries in the popularity evolution of different types of agents