ZVCV (version 2.1.2)

ZVCV_package: Zero-Variance Control Variates

Description

This package can be used to perform post-hoc variance reduction of Monte Carlo estimators when the derivatives of the log target are available. The main functionality is available through the following functions. All of these use a set of \(N\) \(d\)-dimensional samples along with the associated derivatives of the log target. You can evaluate posterior expectations of \(k\) functions.

• zvcv: For estimating expectations using (regularised) zero-variance control variates (ZV-CV, Mira et al, 2013; South et al, 2018). This function can also be used to choose between various versions of ZV-CV using cross-validation.

• CF: For estimating expectations using control functionals (CF, Oates et al, 2017).

• SECF: For estimating expectations using semi-exact control functionals (SECF, South et al, 2020).

• aSECF: For estimating expectations using approximate semi-exact control functionals (aSECF, South et al, 2020).

• CF_crossval: CF with cross-validation tuning.

• SECF_crossval: SECF with cross-validation tuning.

• aSECF_crossval: aSECF with cross-validation tuning.

ZV-CV is exact for polynomials of order at most polyorder under Gaussian targets and is fast for large \(N\) (although setting a limit on polyorder through polyorder_max is recommended for large \(N\)). CF is a non-parametric approach that offers better than the standard Monte Carlo convergence rates. SECF has both a parametric and a non-parametric component and it offers the advantages of both for an additional computational cost. The cost of SECF is reduced in aSECF using nystrom approximations and conjugate gradient.

Helper functions

• getX: Calculates the design matrix for ZV-CV (without the column of 1's for the intercept)

• medianTune: Calculates the median heuristic for use in e.g. the Gaussian, Matern and rational quadratic kernels. Using the median heuristic is an alternative to cross-validation.

• K0_fn: Calculates the \(K_0\) matrix. The output of this function can be used as an argument to CF, CF_crossval, SECF, SECF_crossval, aSECF and aSECF_crossval. The kernel matrix is automatically computed in all of the above methods, but it is faster to calculate in advance when using more than one of the above functions and when using any of the crossval functions.

• Phi_fn: Calculates the Phi matrix for SECF and aSECF (similar to getX but with different arguments and it includes the column of 1's)

• squareNorm: Gets the matrix of square norms which is needed for all kernels. Calculating this can help to save time if you are also interested in calculating the median heuristic, handling multiple tuning parameters or trying other kernels.

• nearPD: Finds the nearest symmetric positive definite matrix to the given matrix, for handling numerical issues.

• logsumexp: Performs stable computation of the log sum of exponential (useful when handling the sum of weights)

Evidence estimation

The following functions are used to estimate the evidence (the normalisiing constant of the posterior) as described in South et al (2018). They are relevant when sequential Monte Carlo with an annealing schedule has been used to collect the samples, and therefore are not of interest to those who are interested in variance reduction based on vanilla MCMC.

• evidence_CTI and evidence_CTI_CF: Functions to estimate the evidence using thermodynamic integration (TI) with ZV-CV and CF, respectively

• evidence_SMC and evidence_SMC_CF: Function to estimate the evidence using the SMC evidence identity with ZV-CV and CF, respectively.

The function Expand_Temperatures can be used to adjust the temperature schedule so that it is more (or less) strict than the original schedule of \(T\) temperatures.

Author

Leah F. South

References

Mira, A., Solgi, R., & Imparato, D. (2013). Zero variance Markov chain Monte Carlo for Bayesian estimators. Statistics and Computing, 23(5), 653-662.

South, L. F., Karvonen, T., Nemeth, C., Girolami, M. and Oates, C. J. (2020). Semi-Exact Control Functionals From Sard's Method. https://arxiv.org/abs/2002.00033

South, L. F., Oates, C. J., Mira, A., & Drovandi, C. (2018). Regularised zero-variance control variates for high-dimensional variance reduction. https://arxiv.org/abs/1811.05073

See Also

Useful links:

• Report bugs at https://github.com/LeahPrice/ZVCV/issues

Examples

# A real data example using ZV-CV is available at \link{VDP}.
# This involves estimating posterior expectations and the evidence from SMC samples.

# The remainder of this section is duplicating (albeit with a different random
# seed) Figure 2a of South et al. (2020).

N_repeats <- 2 # For speed, the actual code uses 100 
N_all <- 25 # For speed, the actual code uses c(10,25,50,100,250,500,1000) 
sigma_list <- list(10^(-1.5),10^(-1),10^(-0.5),1,10^(0.5),10)
nfolds <- 4 # For speed, the actual code uses 10
folds <- 2 # For speed, the actual code uses 5
d <- 4

integrand_fn <- function(x){
  return (1 + x[,2] + 0.1*x[,1]*x[,2]*x[,3] + sin(x[,1])*exp(-(x[,2]*x[,3])^2))
}

results <- data.frame()
for (N in N_all){

  # identify the largest polynomial order that can be fit without regularisation for auto ZV-CV
  max_r <- 0
  while (choose(d + max_r + 1,d)<((folds-1)/folds*N)){
  	max_r <- max_r + 1
  }

  MC <- ZV1 <- ZV2 <- ZVchoose <- rep(NaN,N_repeats)
  CF <- SECF1 <- aSECF1 <- SECF2 <- aSECF2 <- rep(NaN,N_repeats)
  CF_medHeur <- SECF1_medHeur <- aSECF1_medHeur <- rep(NaN,N_repeats)
  SECF2_medHeur <- aSECF2_medHeur <- rep(NaN,N_repeats)
  for (i in 1:N_repeats){     
    x <- matrix(rnorm(N*d),ncol=d)
    u <- -x
    f <- integrand_fn(x)
    
    MC[i] <- mean(f)
    ZV1[i] <- zvcv(f,x,u,options=list(polyorder=1,regul_reg=FALSE))$expectation
    # Checking if the sample size is large enough to accommodation a second order polynomial
    if (N > choose(d+2,d)){
      ZV2[i] <- zvcv(f,x,u,options=list(polyorder=2,regul_reg=FALSE))$expectation
    }
    myopts <- list(list(polyorder=Inf,regul_reg=FALSE,polyorder_max=max_r),
        list(polyorder=Inf,nfolds=nfolds))
    ZVchoose[i] <- zvcv(f,x,u,options=myopts,folds = folds)$expectation
    
    # Calculating the kernel matrix in advance for CF and SECF
    K0_list <- list()
    for (j in 1:length(sigma_list)){
      K0_list[[j]] <- K0_fn(x,u,sigma_list[[j]],steinOrder=2,kernel_function="RQ")
    }
    
    CF[i] <- CF_crossval(f,x,u,K0_list=K0_list,folds = folds)$expectation
    SECF1[i] <- SECF_crossval(f,x,u,K0_list=K0_list,folds = folds)$expectation
    aSECF1[i] <- aSECF_crossval(f,x,u,steinOrder=2,kernel_function="RQ",
        sigma_list=sigma_list,reltol=1e-05,folds = folds)$expectation
    if (max_r>=2){
      SECF2[i] <- SECF_crossval(f,x,u,polyorder=2,K0_list=K0_list,folds = folds)$expectation
      aSECF2[i] <- aSECF_crossval(f,x,u,polyorder=2,steinOrder=2,kernel_function="RQ",
          sigma_list=sigma_list,reltol=1e-05,folds = folds)$expectation
    }

    medHeur <- medianTune(x)
    K0_medHeur <- K0_fn(x,u,medHeur,steinOrder=2,kernel_function="RQ")
    CF_medHeur[i] <- CF(f,x,u,K0=K0_medHeur)$expectation
    SECF1_medHeur[i] <- SECF(f,x,u,K0=K0_medHeur)$expectation
    aSECF1_medHeur[i] <- aSECF(f,x,u,steinOrder=2,kernel_function="RQ",
          sigma=medHeur,reltol=1e-05)$expectation
    if (max_r>=2){
      SECF2_medHeur[i] <- SECF(f,x,u,polyorder=2,K0=K0_medHeur)$expectation
      aSECF2_medHeur[i] <- aSECF(f,x,u,polyorder=2,steinOrder=2,kernel_function="RQ",
          sigma=medHeur,reltol=1e-05)$expectation
    }
    
    # print(sprintf("--%d",i))
  }
  # Adding the results to a data frame
  MSE_crude <- mean((MC - 1)^2)
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = 1, type = "MC")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((ZV1 - 1)^2), type = "ZV")) 
  results <- rbind(results,data.frame(N=N, order = "2",
      efficiency = MSE_crude/mean((ZV2 - 1)^2), type = "ZV")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((ZVchoose - 1)^2), type = "ZVchoose")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((CF - 1)^2), type = "CF")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((SECF1 - 1)^2), type = "SECF")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((aSECF1 - 1)^2), type = "aSECF")) 
  if (((folds-1)/folds*N) > choose(d+2,d)){
    results <- rbind(results,data.frame(N=N, order = "2",
      efficiency = MSE_crude/mean((SECF2 - 1)^2), type = "SECF")) 
    results <- rbind(results,data.frame(N=N, order = "2",
      efficiency = MSE_crude/mean((aSECF2 - 1)^2), type = "aSECF")) 
  }

  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((CF_medHeur - 1)^2), type = "CF_medHeur")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((SECF1_medHeur - 1)^2), type = "SECF_medHeur")) 
  results <- rbind(results,data.frame(N=N, order = "1 or NA",
      efficiency = MSE_crude/mean((aSECF1_medHeur - 1)^2), type = "aSECF_medHeur")) 
  if (((folds-1)/folds*N) > choose(d+2,d)){
    results <- rbind(results,data.frame(N=N, order = "2",
      efficiency = MSE_crude/mean((SECF2_medHeur - 1)^2), type = "SECF_medHeur")) 
    results <- rbind(results,data.frame(N=N, order = "2",
      efficiency = MSE_crude/mean((aSECF2_medHeur - 1)^2), type = "aSECF_medHeur")) 
  }
  # print(N)
}


if (FALSE) {
# Plotting results where cross-validation is used for kernel methods
require(ggplot2)
require(ggthemes)
a <- ggplot(data=subset(results,!(type %in% c("CF_medHeur","SECF_medHeur",
  "aSECF_medHeur","SECF_medHeur","aSECF_medHeur"))),
  aes(x=N,y=efficiency,col=type,linetype=order)) + scale_color_pander() + 
  ggtitle("") + geom_line(size=1.5) + scale_x_log10() + scale_y_log10() + 
  annotation_logticks(base=10) + labs(x="N",y="Efficiency",color="Method",
  linetype="Polynomial Order") + theme_minimal(base_size = 15) +
  theme(legend.key.size = unit(0.5, "cm"),legend.key.width =  unit(1, "cm")) +
  guides(linetype = guide_legend(override.aes = list(size=1),title.position = "top"),
  color = guide_legend(override.aes = list(size=1),title.position = "top"))
print(a)


# Plotting results where the median heuristic is used for kernel methods
b <- ggplot(data=subset(results,!(type %in% c("CF","SECF","aSECF","SECF","aSECF"))),
            aes(x=N,y=efficiency,col=type,linetype=order)) + scale_color_pander() + 
  ggtitle("") + geom_line(size=1.5) + scale_x_log10() + scale_y_log10() + 
  annotation_logticks(base=10) + labs(x="N",y="Efficiency",color="Method",
  linetype="Polynomial Order") + theme_minimal(base_size = 15) +
  theme(legend.key.size = unit(0.5, "cm"),legend.key.width =  unit(1, "cm")) +
  guides(linetype = guide_legend(override.aes = list(size=1),title.position = "top"),
  color = guide_legend(override.aes = list(size=1),title.position = "top"))
print(b)
}