Search for: All records

The softmax function is a ubiquitous component at the output of neural networks and increasingly in intermediate layers as well. This paper provides convex lower bounds and concave upper bounds on the softmax function, which are compatible with convex optimization formulations for characterizing neural networks and other ML models. We derive bounds using both a natural exponential-reciprocal decomposition of the softmax as well as an alternative decomposition in terms of the log-sum-exp function. The new bounds are provably and/or numerically tighter than linear bounds obtained in previous work on robustness verification of transformers. As illustrations of the utility of the bounds, we apply them to verification of transformers as well as of the robustness of predictive uncertainty estimates of deep ensembles. 

Shanghai has experienced a rapid process of urbanization and urban expansion, which increases travel costs and limits job accessibility for the economically disadvantaged population. This paper investigates the jobs-housing imbalance problem in Shanghai at the subdistrict-level (census-level) and reaches the following conclusions. First, the jobs-housing imbalance shows a ring pattern and is evident mainly in the suburban areas and periphery of the Shanghai metropolitan area because job opportunities are highly concentrated while residential areas are sprawling. Second, structural factors such as high housing prices and sprawling development significantly contribute to the jobs-housing imbalance. Third, regional planning policies such as development zones contribute to jobs-housing imbalance due to the specialized industrial structure and limited availability of housing. However, geographically weighted regression reveals the development zones in the traditional Pudong district are exceptional insofar as government policy has created spatial heterogeneity there. In addition, the multilevel model used in this study suggests regions with jobs-housing imbalance usually have well-connected streets, and this represents the local government’s efforts to reduce excessive commuting times created by jobs-housing imbalance.