#### 2020-11-04

Implements the forward-mode automatic differentiation for multivariate functions using the matrix-calculus notation from Magnus and Neudecker (1988). Two key features of the package are: (i) the package incorporates various optimisation strategies to improve performance; this includes applying memoisation to cut down object construction time, using sparse matrix representation to save derivative calculation, and creating specialised matrix operations with Rcpp to reduce computation time; (ii) the package supports differentiating random variable with respect to their parameters, targeting MCMC (and in general simulation-based) applications.

### Installation

install.packages("ADtools")  # stable version
devtools::install_github("kcf-jackson/ADtools")  # development version

### Notation

Given a function $$f: X \mapsto Y = f(X)$$, where $$X \in R^{m \times n}, Y \in R^{h \times k}$$, the Jacobina matrix of $$f$$ w.r.t. $$X$$ is given by $\dfrac{\partial f(X)}{\partial X}:=\dfrac{\partial\,\text{vec}\, f(X)}{\partial\, (\text{vec}X)^T} = \dfrac{\partial\,\text{vec}\,Y}{\partial\,(\text{vec}X)^T}\in R^{mn \times hk}.$

### Example 1. Matrix multiplication

#### Function definition

Consider $$f(X, y) = X y$$ where $$X$$ is a matrix, and $$y$$ is a vector.

library(ADtools)
f <- function(X, y) X %*% y
X <- randn(2, 2)
y <- matrix(c(1, 1))
print(list(X = X, y = y, f = f(X, y)))
#> $X #> [,1] [,2] #> [1,] -0.07762707 0.7239523 #> [2,] -1.98844284 -0.2535326 #> #>$y
#>      [,1]
#> [1,]    1
#> [2,]    1
#>
#> $f #> [,1] #> [1,] 0.6463253 #> [2,] -2.2419754 #### Automatic differentiation Since $$X$$ has dimension (2, 2) and $$y$$ has dimension (2, 1), the input space has dimension $$2 \times 2 + 2 \times 1 = 6$$, and the output has dimension $$2$$, i.e. $$f$$ maps $$R^6$$ to $$R^2$$ and the Jacobian of $$f$$ should be a $$2 \times 6$$ matrix. # Full Jacobian matrix f_AD <- auto_diff(f, at = list(X = X, y = y)) f_AD@dx # returns a Jacobian matrix #> d_X1 d_X2 d_X3 d_X4 d_y1 d_y2 #> d_output_1 1 0 1 0 -0.07762707 0.7239523 #> d_output_2 0 1 0 1 -1.98844284 -0.2535326 auto_diff also supports computing a partial Jacobian matrix. For instance, suppose we are only interested in the derivative w.r.t. y, then we can run f_AD <- auto_diff(f, at = list(X = X, y = y), wrt = "y") f_AD@dx # returns a partial Jacobian matrix #> d_y1 d_y2 #> d_output_1 -0.07762707 0.7239523 #> d_output_2 -1.98844284 -0.2535326 #### Finite-differencing It is good practice to always check the result with finite-differencing. This can be done by calling finite_diff which has the same interface as auto_diff. f_FD <- finite_diff(f, at = list(X = X, y = y)) f_FD #> d_X1 d_X2 d_X3 d_X4 d_y1 d_y2 #> d_output_1 1 0 1 0 -0.07762706 0.7239523 #> d_output_2 0 1 0 1 -1.98844283 -0.2535326 ### Example 2. Estimating a linear regression model #### Simulate data from $$\quad y_i = X_i \beta + \epsilon_i, \quad \epsilon_i \sim N(0, 1)$$ set.seed(123) n <- 1000 p <- 3 X <- randn(n, p) beta <- randn(p, 1) y <- X %*% beta + rnorm(n) #### Inference with gradient descent gradient_descent <- function(f, vary, fix, learning_rate = 0.01, tol = 1e-6, show = F) { repeat { df <- auto_diff(f, at = append(vary, fix), wrt = names(vary)) if (show) print(df@x) delta <- learning_rate * as.numeric(df@dx) vary <- relist(unlist(vary) - delta, vary) if (max(abs(delta)) < tol) break } vary } lm_loss <- function(y, X, beta) sum((y - X %*% beta)^2) # Estimate gradient_descent( f = lm_loss, vary = list(beta = rnorm(p, 1)), fix = list(y = y, X = X), learning_rate = 1e-4 ) #>$beta
#> [1] -0.1417494 -0.3345771 -1.4484226
# Truth
t(beta)
#>            [,1]       [,2]      [,3]
#> [1,] -0.1503075 -0.3277571 -1.448165

### Example 3. Sensitivity analysis of MCMC algorithms

#### Simulate data from $$\quad y_i = X_i \beta + \epsilon_i, \quad \epsilon_i \sim N(0, 1)$$

set.seed(123)
n <- 30  # small data
p <- 10
X <- randn(n, p)
beta <- randn(p, 1)
y <- X %*% beta + rnorm(n)

#### Estimating a Bayesian linear regression model

$y \sim N(X\beta, \sigma^2), \quad \beta \sim N(\mathbf{b_0}, \mathbf{B_0}), \quad \sigma^2 \sim IG\left(\dfrac{\alpha_0}{2}, \dfrac{\delta_0}{2}\right)$

#### Inference using Gibbs sampler

gibbs_gaussian <- function(X, y, b_0, B_0, alpha_0, delta_0, num_steps = 1e4, burn_ins = ceiling(num_steps / 10)) {
# Initialisation
init_sigma <- 1 / sqrt(rgamma0(1, alpha_0 / 2, scale = 2 / delta_0))

n <- length(y)
alpha_1 <- alpha_0 + n
sigma_g <- init_sigma
inv_B_0 <- solve(B_0)
inv_B_0_times_b_0 <- inv_B_0 %*% b_0
XTX <- crossprod(X)
XTy <- crossprod(X, y)
beta_res <- vector("list", num_steps)
sigma_res <- vector("list", num_steps)

pb <- txtProgressBar(1, num_steps, style = 3)
for (i in 1:num_steps) {
# Update beta
B_g <- solve(sigma_g^(-2) * XTX + inv_B_0)
b_g <- B_g %*% (sigma_g^(-2) * XTy + inv_B_0_times_b_0)
beta_g <- t(rmvnorm0(1, b_g, B_g))

# Update sigma
delta_g <- delta_0 + sum((y - X %*% beta_g)^2)
sigma_g <- 1 / sqrt(rgamma0(1, alpha_1 / 2, scale = 2 / delta_g))

# Keep track
beta_res[[i]] <- beta_g
sigma_res[[i]] <- sigma_g
setTxtProgressBar(pb, i)
}

# Compute and return the posterior mean
sample_ids <- (burn_ins + 1):num_steps
beta_pmean <- Reduce(+, beta_res[sample_ids]) / length(sample_ids)
sigma_pmean <- Reduce(+, sigma_res[sample_ids]) / length(sample_ids)
list(sigma = sigma_pmean, beta = beta_pmean)
}

#### Automatic differentiation

gibbs_deriv <- auto_diff(
gibbs_gaussian,
at = list(
b_0 = numeric(p), B_0 = diag(p), alpha_0 = 4, delta_0 = 4,
X = X, y = y, num_steps = 50, burn_ins = 5 # Numbers are reduced for CRAN
),
wrt = c("b_0", "B_0", "alpha_0", "delta_0")
)