Task: Language Identification
Input (Spanish, English): This paper shows:    1. Easy, constructive proofs to derive e-error upper-bounds on neural networks with O(log 1/e) layers and O(log 1/e) ReLU units.   2. Extensions of the previous results to more general function classes, such as smooth or vector-valued functions.   3. Lower bounds on the neural network size, as a function of its number of layers. The lower bound reveals the need of exponentially many more units to approximate functions using shallow architectures.  The paper is well written and easy to follow. The technical content, including the proofs in the Appendix, look correct. Although the proof techniques are simple (and are sometimes modifications of arguments by Gil, Telgarsky, or Dasgupta), they are brought together in a coherent manner to produce sharp results. Therefore, I am leaning toward acceptance.
Output (English): en