384 research outputs found
Extremal Choice Equilibrium: Existence and Purification with Infinite-Dimensional Externalities
We prove existence and purification results for equilibria in which players choose extreme points of their feasible actions in a class of strategic environments exhibiting a product structure. We assume finite-dimensional action sets and allow for infinite-dimensional externalities. Applied to large games, we obtain existence of Nash equilibrium in pure strategies while allowing a continuum of groups and general dependence of payoffs on average actions across groups, without resorting to saturated measure spaces. Applied to games of incomplete information, we obtain a new purification result for Bayes-Nash equilibria that permits substantial correlation across types, without assuming conditional independence given the realization of a finite environmental state. We highlight our results in examples of industrial organization, auctions, and voting.
Implicit regularization of dropout
It is important to understand how dropout, a popular regularization method,
aids in achieving a good generalization solution during neural network
training. In this work, we present a theoretical derivation of an implicit
regularization of dropout, which is validated by a series of experiments.
Additionally, we numerically study two implications of the implicit
regularization, which intuitively rationalizes why dropout helps
generalization. Firstly, we find that input weights of hidden neurons tend to
condense on isolated orientations trained with dropout. Condensation is a
feature in the non-linear learning process, which makes the network less
complex. Secondly, we experimentally find that the training with dropout leads
to the neural network with a flatter minimum compared with standard gradient
descent training, and the implicit regularization is the key to finding flat
solutions. Although our theory mainly focuses on dropout used in the last
hidden layer, our experiments apply to general dropout in training neural
networks. This work points out a distinct characteristic of dropout compared
with stochastic gradient descent and serves as an important basis for fully
understanding dropout.Comment: arXiv admin note: text overlap with arXiv:2111.0102
FAST CONVEX OPTIMIZATION VIA A THIRD-ORDER IN TIME EVOLUTION EQUATION
In a Hilbert space H, we develop fast convex optimization methods, which are based on a third order in time evolution system. The function to minimize f : H → R is convex, continuously differentiable, with argmin f = ∅, and enters the dynamic via its gradient. On the basis of Lyapunov's analysis and temporal scaling techniques, we show a convergence rate of the values of the order 1/t 3 , and obtain the convergence of the trajectories towards optimal solutions. When f is strongly convex, an exponential rate of convergence is obtained. We complete the study of the continuous dynamic by introducing a damping term induced by the Hessian of f. This allows the oscillations to be controlled and attenuated. Then, we analyze the convergence of the proximal-based algorithms obtained by temporal discretization of this system, and obtain similar convergence rates. The algorithmic results are valid for a general convex, lower semicontinuous, and proper function f : H → R ∪ {+∞}
Information Recovery In Behavioral Networks
In the context of agent based modeling and network theory, we focus on the
problem of recovering behavior-related choice information from
origin-destination type data, a topic also known under the name of network
tomography. As a basis for predicting agents' choices we emphasize the
connection between adaptive intelligent behavior, causal entropy maximization
and self-organized behavior in an open dynamic system. We cast this problem in
the form of binary and weighted networks and suggest information theoretic
entropy-driven methods to recover estimates of the unknown behavioral flow
parameters. Our objective is to recover the unknown behavioral values across
the ensemble analytically, without explicitly sampling the configuration space.
In order to do so, we consider the Cressie-Read family of entropic functionals,
enlarging the set of estimators commonly employed to make optimal use of the
available information. More specifically, we explicitly work out two cases of
particular interest: Shannon functional and the likelihood functional. We then
employ them for the analysis of both univariate and bivariate data sets,
comparing their accuracy in reproducing the observed trends.Comment: 14 pages, 6 figures, 4 table
Optimization as a design strategy. Considerations based on building simulation-assisted experiments about problem decomposition
In this article the most fundamental decomposition-based optimization method
- block coordinate search, based on the sequential decomposition of problems in
subproblems - and building performance simulation programs are used to reason
about a building design process at micro-urban scale and strategies are defined
to make the search more efficient. Cyclic overlapping block coordinate search
is here considered in its double nature of optimization method and surrogate
model (and metaphore) of a sequential design process. Heuristic indicators apt
to support the design of search structures suited to that method are developed
from building-simulation-assisted computational experiments, aimed to choose
the form and position of a small building in a plot. Those indicators link the
sharing of structure between subspaces ("commonality") to recursive
recombination, measured as freshness of the search wake and novelty of the
search moves. The aim of these indicators is to measure the relative
effectiveness of decomposition-based design moves and create efficient block
searches. Implications of a possible use of these indicators in genetic
algorithms are also highlighted.Comment: 48 pages. 12 figures, 3 table
- …