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SUMMARY:Approximate Cross Validation for Large Data and High Dimensions - 
 Tamara Broderick\, Massachusetts Institute of Technology
DTSTART:20200117T140000Z
DTEND:20200117T150000Z
UID:TALK136060@talks.cam.ac.uk
CONTACT:Dr Sergio Bacallado
DESCRIPTION:The error or variability of statistical and machine learning a
 lgorithms is often assessed by repeatedly re-fitting a model with differen
 t weighted versions of the observed data. The ubiquitous tools of cross-va
 lidation (CV) and the bootstrap are examples of this technique. These meth
 ods are powerful in large part due to their model agnosticism but can be s
 low to run on modern\, large data sets due to the need to repeatedly re-fi
 t the model. We use a linear approximation to the dependence of the fittin
 g procedure on the weights\, producing results that can be faster than rep
 eated re-fitting by orders of\nmagnitude. This linear approximation is som
 etimes known as the "infinitesimal jackknife" (IJ) in the statistics liter
 ature\, where it has mostly been used as a theoretical tool to prove asymp
 totic results. We provide explicit finite-sample error bounds for the infi
 nitesimal jackknife in terms of a small number of simple\, verifiable assu
 mptions. Without further modification\, though\, we note that the IJ deter
 iorates in accuracy in high dimensions and incurs a running time roughly c
 ubic in dimension. We additionally show\, then\, how dimensionality reduct
 ion can be used to successfully run the IJ in high dimensions in the case 
 of leave-one-out cross validation (LOOCV). Specifically\, we consider L1 r
 egularization for generalized linear models. We prove that\, under mild co
 nditions\, the resulting LOOCV approximation exhibits computation time and
  accuracy that depend on the recovered support size rather than the full d
 imension D. Simulated and real-data experiments support our theory.
LOCATION:MR12
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