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

Necessary Conditions for Extended Noncontextuality in General Sets of Random Variables

We explore the graph approach to contextuality to restate the extended definition of noncontextuality as given by J. Kujala et. al. [Phys. Rev. Lett. 115, 150401 (2015)] using graph-theoretical terms. This extended definition avoids the assumption of the pre-sheaf or non-disturbance condition, which states that if two contexts overlap, then the marginal distribution obtained for the intersection must be the same, a restriction that will never be perfectly satisfied in real experiments. With this we are able to derive necessary conditions for extended noncontextuality for any set of random variables based on the geometrical aspects of the graph approach, which can be tested directly with experimental data in any contextuality experiment and which reduce to traditional necessary conditions for noncontextuality if the non-disturbance condition is satisfied

Estimating graph parameters with random walks

An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up to a bounded factor in $O\left(t_{\mathrm{rel}}^{3/4}\sqrt{m/d}\right)$ steps, where $t_{\mathrm{rel}}$ is the relaxation time of the lazy random walk on $G$ and $d$ is the minimum degree in $G$. Alternatively, $m$ can be estimated in $O\left(t_{\mathrm{unif}} +t_{\mathrm{rel}}^{5/6}\sqrt{n}\right)$, where $n$ is the number of vertices and $t_{\mathrm{unif}}$ is the uniform mixing time on $G$. The number of vertices $n$ can then be estimated up to a bounded factor in an additional $O\left(t_{\mathrm{unif}}\frac{m}{n}\right)$ steps. Our algorithms are based on counting the number of intersections of random walk paths $X,Y$, i.e. the number of pairs $(t,s)$ such that $X_t=Y_s$. This improves on previous estimates which only consider collisions (i.e., times $t$ with $X_t=Y_t$). 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 $m$ or the mixing time of $G$, we can compute the "other parameter" with a self-stopping algorithm