The tenth and final roundtable of the quarter was held after Krishna Balasubramanian’s Statistics seminar talk, “Normal Approximations for Stochastic Iterative Estimators (and Martingales)”. His talk slides can be found here.
The first part of the roundtable followed a question and answer format, so we recreate that here.
Q: What are the practical ramifications of this fairly abstract talk?
A: The results presented apply to bootstrap algorithms used widely in practice to obtain confidence sets. The main insight is the validity of the obtained confidence set is quantified by the rates of central limit theorems. If one is solving a 100 dimension stochastic convex optimization problem with 1000 samples/iterations, this result suggests that one cannot invoke the central limit theorem.
Q: What are the major current challenges in uncertainty quantification?
A: The primary challenge is determining how well rates of central limit theorem scales with dimension, as it has direct consequences for understanding the practically used bootstrap algorithm. There are results in the literature, including the presented one which argue for a poor scaling (d2), and it is unlikely these rates could be improved. One way around this issues, is to leverage the structure in the specific problem under consideration, which often times lead to significantly better scaling.
Q: Why does one choose the step size to be varying as a function of training time?
A: If we don’t do this, the resulting estimator is biased away from zero, so if one wants to train with a fixed step size, one needs to subtract off this bias term.
Q: How should we present uncertainty results to users when the dimension increases?
A: In one dimension, the shape of the quantiles is just intervals. In higher dimensions, there are results on the normality of coordinate-wise confidence intervals, as well as hyper-rectangles, since these correspond to the max-statistic. At present there are few results for more general sets.
The second part of the roundtable discussion was about the UCD4IDS plan for the winter quarter. The host presented the list of speakers for both the MADDD and Statistics seminars in Winter 2020. His slides are available here. However, he also asked the audience whether the roundtable discussions should be continued as in the fall quarter. Some of the audience suggested that it would be better to have two or three roundtable discussions per quarter instead of doing it every week, and move the roundtable discussion schedule earlier in the day (e.g., as brown bag discussions or discussion with coffee breaks). The UCD4IDS Steering Committee will discuss this issue further.
[Scribe: David Weber (GGAM)]