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

Stable pair compactification of moduli of K3 surfaces of degree 2

We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,\epsilon R)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.Comment: v2: Updated reference

Outlier-robust sparse/low-rank least-squares regression and robust matrix completion

We consider high-dimensional least-squares regression when a fraction $\epsilon$ of the labels are contaminated by an arbitrary adversary. We analyze such problem in the statistical learning framework with a subgaussian distribution and linear hypothesis class on the space of $d_1\times d_2$ matrices. As such, we allow the noise to be heterogeneous. This framework includes sparse linear regression and low-rank trace-regression. For a $p$-dimensional $s$-sparse parameter, we show that a convex regularized $M$-estimator using a sorted Huber-type loss achieves the near-optimal subgaussian rate $$\sqrt{s\log(ep/s)}+\sqrt{\log(1/\delta)/n}+\epsilon\log(1/\epsilon),$$ with probability at least $1-\delta$. For a $(d_1\times d_2)$-dimensional parameter with rank $r$, a nuclear-norm regularized $M$-estimator using the same sorted Huber-type loss achieves the subgaussian rate $$\sqrt{rd_1/n}+\sqrt{rd_2/n}+\sqrt{\log(1/\delta)/n}+\epsilon\log(1/\epsilon),$$ again optimal up to a log factor. In a second part, we study the trace-regression problem when the parameter is the sum of a matrix with rank $r$ plus a $s$-sparse matrix assuming the "low-spikeness" condition. Unlike multivariate regression studied in previous work, the design in trace-regression lacks positive-definiteness in high-dimensions. Still, we show that a regularized least-squares estimator achieves the subgaussian rate $$\sqrt{rd_1/n}+\sqrt{rd_2/n}+\sqrt{s\log(d_1d_2)/n} +\sqrt{\log(1/\delta)/n}.$$ Lastly, we consider noisy matrix completion with non-uniform sampling when a fraction $\epsilon$ of the sampled low-rank matrix is corrupted by outliers. If only the low-rank matrix is of interest, we show that a nuclear-norm regularized Huber-type estimator achieves, up to log factors, the optimal rate adaptively to the corruption level. The above mentioned rates require no information on $(s,r,\epsilon)$

Imagining Agbogbloshie: Issues of Electronic Waste and Representation

This report will focus on how artists, particularly new media artists, have represented the issues of electronic waste and recycling through their works and the implications of their output and processes. The research will specifically analyse electronic waste and its status as an object, its movement around the world, and will focus in on the specific electronic waste dump of Agbogbloshie, in Accra Ghana, as an important site for formulating ethical and environmental questions about how we represent global environmental issues through a local lens. In particular the research will investigate how the art world and electronic waste are intertwined, analysing the image economy that exploits the global south, and ask how art works can help to bring about actual and realistic change. It will also look at how new media art work can use its format to self reflexively document and enact issues of digital materiality. This takes the form of artworks produced in collaboration with people directly affected by electronic waste, as well as with activists who are working to improve the situation. Ultimately the research analyses the role of the electronic waste dump on a global and local scale. Investigating the role that the art world already plays in these areas and how it can take a better, more involved approach - both through working with activist groups, charities, and the people affected, but also through the use of new media and technology itself