convex FindMinimum for a large vector

*To*: mathgroup at smc.vnet.net*Subject*: [mg85712] convex FindMinimum for a large vector*From*: Art <grenander at gmail.com>*Date*: Tue, 19 Feb 2008 07:10:27 -0500 (EST)

What is the best way to formulate the following minimization. I am trying to estimate the parameters of a Poisson generalized linear model (GLM) by minimizing the negative loglikelihood. I think I am not formulating the FindMinimum call for a vector in a good way. Any help would be appreciated. n = 10000; (* number of data points *) d = 500; (* dimensionality of parameter being estimated *) f[x_] := E^x (* convex, log concave function from reals to reals *) (* data = a list of non-negative integers of length n *) (* input = a list of {real vectors of dimensionality d} of length n*) theta = Array[k, d]; (* would like to solve for vector k[i] *) logl = Sum[ data[[i]] Log[ f[theta . input[[i]]] ] - f[theta . input[[i]]] , {i,1,n} ] logl = logl /. Log[E^x_] ->x (* Mathematica doesn't simplify these *) FindMinimum[-logl, theta] This function surprisingly works for n < 1000 (not Method->"Newton" though), but otherwise doesn't return for me and uses a lot of memory. Is there a better way to do the above so that it can handle larger n? In principle, it's the size of d that should cause problems, but not n. FindFit has no problem with optimizations several times larger and is extremely fast. I think computing symbolic derivatives and so forth must balloon the number of terms involved here. Thanks! Art