1,193 research outputs found
Sample average approximation with heavier tails II: localization in stochastic convex optimization and persistence results for the Lasso
We present exponential finite-sample nonasymptotic deviation inequalities for
the SAA estimator's near-optimal solution set over the class of stochastic
optimization problems with heavy-tailed random \emph{convex} functions in the
objective and constraints. Such setting is better suited for problems where a
sub-Gaussian data generating distribution is less expected, e.g., in stochastic
portfolio optimization. One of our contributions is to exploit \emph{convexity}
of the perturbed objective and the perturbed constraints as a property which
entails \emph{localized} deviation inequalities for joint feasibility and
optimality guarantees. This means that our bounds are significantly tighter in
terms of diameter and metric entropy since they depend only on the near-optimal
solution set but not on the whole feasible set. As a result, we obtain a much
sharper sample complexity estimate when compared to a general nonconvex
problem. In our analysis, we derive some localized deterministic perturbation
error bounds for convex optimization problems which are of independent
interest. To obtain our results, we only assume a metric regular convex
feasible set, possibly not satisfying the Slater condition and not having a
metric regular solution set. In this general setting, joint near feasibility
and near optimality are guaranteed. If in addition the set satisfies the Slater
condition, we obtain finite-sample simultaneous \emph{exact} feasibility and
near optimality guarantees (for a sufficiently small tolerance). Another
contribution of our work is to present, as a proof of concept of our localized
techniques, a persistent result for a variant of the LASSO estimator under very
weak assumptions on the data generating distribution.Comment: 34 pages. Some correction
Estimating graph parameters with random walks
An algorithm observes the trajectories of random walks over an unknown graph
, starting from the same vertex , as well as the degrees along the
trajectories. For all finite connected graphs, one can estimate the number of
edges up to a bounded factor in
steps, where
is the relaxation time of the lazy random walk on and
is the minimum degree in . Alternatively, can be estimated in
, where is
the number of vertices and is the uniform mixing time on
. The number of vertices can then be estimated up to a bounded factor in
an additional steps. Our
algorithms are based on counting the number of intersections of random walk
paths , i.e. the number of pairs such that . This
improves on previous estimates which only consider collisions (i.e., times
with ). We also show that the complexity of our algorithms is optimal,
even when restricting to graphs with a prescribed relaxation time. Finally, we
show that, given either or the mixing time of , we can compute the
"other parameter" with a self-stopping algorithm
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