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To implement gradient descent in R, define a scalar objective and a matching gradient, then repeatedly update the parameter vector with par <- par - learning_rate * grad_f(par). Recalculate the gradient after each update, record the objective, and stop using a stated convergence rule or a maximum iteration limit.
What the gradient descent update does
Given an objective function f(par), gradient descent seeks parameter values that reduce the objective. The gradient points toward the direction of greatest local increase, so the algorithm moves in the opposite direction. The learning rate controls how far each update moves.
For a parameter vector par, the update is:
par_new = par - learning_rate * grad_f(par)
The gradient must contain one partial derivative for each parameter, in the same order as the parameter vector. After updating, calculate the gradient again at the new parameter values.
Write a basic gradient descent loop
This template leaves the objective, gradient, starting values, learning rate, and stopping tolerance to be supplied for the problem at hand. It records objective values and limits the number of iterations; it does not assert that any particular objective will converge.
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gradient_descent <- function(par, f, grad_f, learning_rate,
tol = 1e-6, maxit = 10000) {
if (!is.numeric(par) || length(par) == 0L || any(!is.finite(par))) {
stop("par must be a non-empty finite numeric vector")
}
if (length(learning_rate) != 1L || !is.finite(learning_rate) ||
learning_rate <= 0) {
stop("learning_rate must be a positive finite number")
}
if (length(tol) != 1L || !is.finite(tol) || tol <= 0) {
stop("tol must be a positive finite number")
}
if (length(maxit) != 1L || maxit < 1L) {
stop("maxit must be at least 1")
}
value <- f(par)
if (length(value) != 1L || !is.finite(value)) {
stop("f(par) must return one finite number")
}
history <- data.frame(iteration = 0L, objective = value)
converged <- FALSE
for (i in seq_len(maxit)) {
gradient <- grad_f(par)
if (!is.numeric(gradient) || length(gradient) != length(par) ||
any(!is.finite(gradient))) {
stop("grad_f(par) must return a finite numeric vector matching par")
}
if (sqrt(sum(gradient^2)) <= tol) {
converged <- TRUE
break
}
par <- par - learning_rate * gradient
value <- f(par)
if (length(value) != 1L || !is.finite(value)) {
stop("f(par) must return one finite number at every update")
}
history <- rbind(history, data.frame(iteration = i, objective = value))
}
list(par = par, value = value, history = history,
converged = converged, iterations = nrow(history) - 1L)
}For each application, define f to accept the parameter vector and return one scalar. Define grad_f to accept that same vector and return its derivatives in matching order. Supply a finite numeric starting vector and a positive learning rate. The function’s converged result is based on the gradient norm criterion shown in the loop; reaching maxit without meeting that criterion is not convergence.
Choose and assess a stopping rule
A small gradient norm is one possible stopping criterion: it indicates that the local slope is small at the current iterate. Other useful criteria include a small change in parameters or a small change in objective between updates. Whichever rule you choose, state it and retain a maximum iteration count as a safeguard.
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- Inspect the objective history rather than relying only on the returned parameter values.
- If objective values increase sharply or become non-finite, the step may be unstable; reduce the learning rate and try again.
- If progress is consistently very slow, the learning rate may be too small, or the objective may need a different optimization approach.
- A stopping rule reports what condition was met, not proof that the result is a global minimum.
Use R’s built-in optimizer when a custom loop is not needed
Base R’s stats::optim() is a general-purpose optimization function. The R Project describes it as “General-purpose optimization based on Nelder–Mead, quasi-Newton and conjugate-gradient algorithms.” Its default method is Nelder–Mead, which does not use a supplied gradient. BFGS, CG, and L-BFGS-B can use a gradient provided through gr; when gr is omitted for those methods, finite differences are used. See the R reference for optim().
For example, this call explicitly selects BFGS and supplies the gradient:
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result <- stats::optim(
par = initial_par,
fn = f,
gr = grad_f,
method = "BFGS"
)Here, initial_par is the numeric starting vector, f returns one scalar objective, and grad_f returns a gradient of matching length. This is a gradient-aware quasi-Newton method, not the plain steepest-descent update loop above. Choose a method intentionally rather than describing optim()‘s default behavior as gradient descent.
When to use a gradient-oriented package
The CRAN package optimg documents gradient-based STGD and ADAM methods. Its interface accepts either a user-supplied gradient or a finite-difference approximation and provides controls including maxit and relative tolerance. These controls belong to the package’s interface; they are not universal settings for every implementation of gradient descent. Consult the optimg documentation for its method-specific arguments.
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For broader solver comparisons, optimx can invoke optim() and other R optimization tools. Its results can include parameters, objective value, function and gradient evaluation counts, iteration count when available, and a convergence code; its documentation defines code 0 as successful convergence. Interpret that code alongside the selected method and the objective, not as a substitute for checking the result. See the optimx documentation.
Plain gradient descent is not the only practical gradient-based approach. The Rvmmin documentation describes a variable-metric method that uses an approximate inverse Hessian to choose a direction, applies a backtracking line search, and updates the matrix with a BFGS formula. It discourages numerical gradients for this method. Details are in the Rvmmin documentation.
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Choose an approach by the job
| Approach | What it does | Gradient and diagnostics |
|---|---|---|
| Hand-written loop | Implements the direct steepest-descent update; each step is easy to inspect. | You provide the gradient and stopping logic. You control what history to record. |
stats::optim() |
Offers several optimization methods; the default is Nelder–Mead, while BFGS, CG, and L-BFGS-B can use gradient information. | For BFGS, CG, and L-BFGS-B, provide gr or allow finite-difference estimation. Check the returned objective and convergence information. |
optimg |
Documents the gradient-based STGD and ADAM methods. | Accepts a supplied gradient or finite differences and exposes package-specific iteration and tolerance controls. |
optimx |
Wraps multiple optimization tools, including optim(). |
Can report evaluation counts, iterations when available, and a convergence code; the code should be read in method context. |
There is no universally best choice on the available evidence: performance depends on the objective, parameterization, gradient, and stopping criteria. A custom loop is useful for learning or exposing every update; a solver package is more suitable when its methods and diagnostics match the problem. Compare methods on the same defined objective and inspect convergence evidence rather than assuming one is faster or more accurate.
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