Adaptive Regression by Mixing

Abstract

Adaptation over different procedures is of practical importance. Different procedures perform well under different conditions. In many practical situations, it is rather hard to assess which conditions are (approximately) satisfied so as to identify the best procedure for the data at hand. Thus automatic adaptation over various scenarios is desirable. A practically feasible method, named adaptive regression by mixing (ARM), is proposed to convexly combine general candidate regression procedures. Under mild conditions, the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best. Simulations are conducted in several settings, including comparing a parametric model with nonparametric alternatives, comparing a neural network with a projection pursuit in multidimensional regression, and combining bandwidths in kernel regression. The results clearly support the theoretical property of ARM. The ARM algorithm assigns weights on the candidate modelsprocedures via proper assessment of performance of the estimators. The data are split into two parts, one for estimation and the other for measuring behavior in prediction. Although there are many plausible ways to assign the weights, ARM has a connection with information theory, which ensures the desired adaptation capability. Indeed, under mild conditions, we show that the squared L 2 risk of the estimator based on ARM is basically bounded above by the risk of each candidate procedure plus a small penalty term of order 1/n. Minimizing over the procedures gives the automatically optimal rate of convergence for ARM. Model selection often induces unnecessarily large variability in estimation. Alternatively, a proper weighting of the candidate models can be more stable, resulting in a smaller risk. Simulations suggest that ARM works better than model selection using Akaike or Bayesian information criteria when the error variance is not very small.

Aggregation by exponential weighting, sharp PAC-Bayesian bounds and sparsity

“…normally distributed random variables with mean 0 and variance β/2. The idea of mixing with exponential weights has been discussed by many authors apparently since 1970-ies (see Yang 2001 for an overview of the subject). Most of the work has been focused on the important particular case where the set of estimators is finite, i.e., w.l.o.g.…”

Jackknife model averaging

“…There is also a rapidly-growing literature on frequentist methods for model averaging, including Buckland et al (1997), Juditsky and Nemirovski (2000), Yang (2001Yang ( , 2004, Hansen (2007), Goldenshluger (2009) and Wan et al (2010). Most of these methods involve sample splitting, which can be inefficient, and all exclude heteroskedasticity, which limits their applicability.…”

Markov-Switching Model Selection Using Kullback-Leibler Divergence

“…269-274) Raftery (1996, p. 252) for guidelines when using Bayes factors). Finally, alternative approaches for incorporating model uncertainty include forecast combinations (Timmermann 2005), Bayesian model averaging (e.g., Hoeting et al 1999), frequentist model averaging (Hjort and Claeskens 2003), and adaptive mixing of methods (Yang 2001). …”