The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years in both machine learning and statistics communities. While it seems to defy conventional wisdom that overfitting leads to poor prediction, recent theoretical research on its $\ell_2$-type risks reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. This paper takes a further step that aims at understanding its precise stochastic behavior as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which provides a precise quantification of the prescribed implicit regularization in the most general distributional sense. Our distributional characterizations hold for general non-Gaussian random designs and extend uniformly to positively regularized Ridge estimators. As a direct application, we obtain a complete characterization for a general class of weighted $\ell_q$ risks of the Ridge(less) estimators that are previously only known for $q=2$ by random matrix methods. These weighted $\ell_q$ risks not only include the standard prediction and estimation errors, but also include the non-standard covariate shift settings. Our uniform characterizations further reveal a surprising feature of the commonly used generalized and $k$-fold cross-validation schemes: tuning the estimated $\ell_2$ prediction risk by these methods alone lead to simultaneous optimal $\ell_2$ in-sample, prediction and estimation risks, as well as the optimal length of debiased confidence intervals.

In this paper, we study the behavior of the Upper Confidence Bound-Variance (UCB-V) algorithm for the Multi-Armed Bandit (MAB) problems, a variant of the canonical Upper Confidence Bound (UCB) algorithm that incorporates variance estimates into its decision-making process. More precisely, we provide an asymptotic characterization of the arm-pulling rates for UCB-V, extending recent results for the canonical UCB in Kalvit and Zeevi (2021) and Khamaru and Zhang (2024). In an interesting contrast to the canonical UCB, our analysis reveals that the behavior of UCB-V can exhibit instability, meaning that the arm-pulling rates may not always be asymptotically deterministic. Besides the asymptotic characterization, we also provide non-asymptotic bounds for the arm-pulling rates in the high probability regime, offering insights into the regret analysis. As an application of this high probability result, we establish that UCB-V can achieve a more refined regret bound, previously unknown even for more complicate and advanced variance-aware online decision-making algorithms. A matching regret lower bound is also established, demonstrating the optimality of our result.

Current global snow retrieval algorithms based on spaceborne microwave measurements inherit noticeable biases and uncertainties regarding spatial distribution and temporal variations. In this article, we present an improved spatiotemporally dynamic global snow depth retrieval algorithm to account for the heterogeneity of snowpacks in different seasons worldwide. The proposed model adopts nonlinear machine learning to retrieve snow depths from passive microwave measurements and other auxiliary information. We indirectly characterized the variation in snow grain size using the daily profiles of the temperature gradient within the snowpack. In addition, a zoning and multitemporal modeling strategy was employed to reduce the bias and uncertainty caused by snow heterogeneity across different ecoregions and seasons. The proposed model was implemented to retrieve the global daily snow depth from 2001 to 2010. The results were validated byin situobservations and compared with the NASA Advanced Microwave Scanning Radiometer for EOS (AMSR-E) snow water equivalent product (AE_DySno). Satisfactory accuracy was achieved for different ecoregions with regard to daily, monthly, and yearly validations (the root-mean-square error (RMSE) varied from ~7.5 to ~12 cm; the Pearson correlation coefficient$R$ranged from 0.75 to 0.85). The results of ten trials indicated the promising stability of the proposed model in different ecoregions with small variations in RMSE and$R$values. Compared with the AE_DySno products, the estimation results did not exhibit the overestimation problem and provided snow depth patterns with greater spatial heterogeneity, showing RMSEs ~5 cm lower and$R$values ~0.3 higher than those of the AE_DySno products.

Since the late 1990s, there are growing studies on the development of cellular automata (CA) as a simulation tool for assisting urban and regional planning in China. Rapid urban development is the main reason that this country has become one of the best places to test the methodology of CA and analyze the effectiveness of using these models. This paper attempts to summarize the experiences and issues of using CA to solve various environmental and planning problems in China. The analysis is based on the literature review using the search engines of ISI Web of Science and Google Scholar. These experiences could be important for those who want to apply CA in developing countries. For example, which environmental and ecological problems can be solved by using this bottom-up approach? What are the data inputs to these models and how can they be calibrated? Our analyses indicate that CA have the great potential to support land-use planning and policy analysis for fast-growing regions. Some specific features of using CA in China are also identified in the literature review, including delineation of urban growth boundary, prevention of illegal development and formulating zoning schemes. The CA studies in this fast-growing country provided valuable experiences for other developing countries to solve a series of simulation and planning problems by using this bottom-up approach.