40,791 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
Width and extremal height distributions of fluctuating interfaces with window boundary conditions
We present a detailed study of squared local roughness (SLRDs) and local
extremal height distributions (LEHDs), calculated in windows of lateral size
, for interfaces in several universality classes, in substrate dimensions
and . We show that their cumulants follow a Family-Vicsek
type scaling, and, at early times, when ( is the correlation
length), the rescaled SLRDs are given by log-normal distributions, with their
th cumulant scaling as . This give rise to an
interesting temporal scaling for such cumulants , with . This scaling is analytically
proved for the Edwards-Wilkinson (EW) and Random Deposition interfaces, and
numerically confirmed for other classes. In general, it is featured by small
corrections and, thus, it yields exponents 's (and, consequently,
, and ) in nice agreement with their respective universality
class. Thus, it is an useful framework for numerical and experimental
investigations, where it is, usually, hard to estimate the dynamic and
mainly the (global) roughness exponents. The stationary (for ) SLRDs and LEHDs of Kardar-Parisi-Zhang (KPZ) class are also investigated
and, for some models, strong finite-size corrections are found. However, we
demonstrate that good evidences of their universality can be obtained through
successive extrapolations of their cumulant ratios for long times and large
's. We also show that SLRDs and LEHDs are the same for flat and curved KPZ
interfaces.Comment: 11 pages, 10 figures, 4 table
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