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 $l$, for interfaces in several universality classes, in substrate dimensions $d_s = 1$ and $d_s = 2$. We show that their cumulants follow a Family-Vicsek type scaling, and, at early times, when $\xi \ll l$ ($\xi$ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their $n$th cumulant scaling as $(\xi/l)^{(n-1)d_s}$. This give rise to an interesting temporal scaling for such cumulants $\left\langle w_n \right\rangle_c \sim t^{\gamma_n}$, with $\gamma_n = 2 n \beta + {(n-1)d_s}/{z} = \left[ 2 n + {(n-1)d_s}/{\alpha} \right] \beta$. 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 $\gamma_n$'s (and, consequently, $\alpha$, $\beta$ and $z$) 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 $z$ and mainly the (global) roughness $\alpha$ exponents. The stationary (for $\xi \gg l$) 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 $l$'s. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.Comment: 11 pages, 10 figures, 4 table