3,315 research outputs found
Logistic Regression: Tight Bounds for Stochastic and Online Optimization
The logistic loss function is often advocated in machine learning and
statistics as a smooth and strictly convex surrogate for the 0-1 loss. In this
paper we investigate the question of whether these smoothness and convexity
properties make the logistic loss preferable to other widely considered options
such as the hinge loss. We show that in contrast to known asymptotic bounds, as
long as the number of prediction/optimization iterations is sub exponential,
the logistic loss provides no improvement over a generic non-smooth loss
function such as the hinge loss. In particular we show that the convergence
rate of stochastic logistic optimization is bounded from below by a polynomial
in the diameter of the decision set and the number of prediction iterations,
and provide a matching tight upper bound. This resolves the COLT open problem
of McMahan and Streeter (2012)
Distance-Dependent Kronecker Graphs for Modeling Social Networks
This paper focuses on a generalization of stochastic
Kronecker graphs, introducing a Kronecker-like operator and
defining a family of generator matrices H dependent on distances
between nodes in a specified graph embedding. We prove
that any lattice-based network model with sufficiently small
distance-dependent connection probability will have a Poisson
degree distribution and provide a general framework to prove
searchability for such a network. Using this framework, we focus
on a specific example of an expanding hypercube and discuss
the similarities and differences of such a model with recently
proposed network models based on a hidden metric space. We
also prove that a greedy forwarding algorithm can find very short
paths of length O((log log n)^2) on the hypercube with n nodes,
demonstrating that distance-dependent Kronecker graphs can
generate searchable network models
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