Alternating optimization

The ao R package implements the alternating optimization (AO) approach. This vignette provides an overview of the package. For theoretical results on AO, refer to:

Bezdek and Hathaway (2002), who explain how AO can avoid getting stuck in local optima
Hu and Hathaway (2002), who show that AO can speed up optimization
Bezdek and Hathaway (2003), who provide more details on the convergence speed of AO

What actually is alternating optimization?

Alternating optimization (AO) is an iterative procedure used to optimize a multivariate function by breaking it down into simpler sub-problems. It involves optimizing over one block of function parameters while keeping the others fixed, and then alternating this process among the parameter blocks. AO is particularly useful when the sub-problems are easier to solve than the original joint optimization problem, or when there is a natural partition of the parameters.

Mathematically, consider a real-valued objective function $f(\mathbf{x}, \mathbf{y})$ where $\mathbf{x}$ and $\mathbf{y}$ are two blocks of function parameters, namely a partition of the parameters. The AO procedure can be described as follows:

Initialization: Start with initial guesses $\mathbf{x}^{(0)}$ and $\mathbf{y}^{(0)}$ .
Iterative Steps: For $k = 0, 1, 2, \dots$
- Step 1: Fix $\mathbf{y} = \mathbf{y}^{(k)}$ and solve the sub-problem $\mathbf{x}^{(k+1)} = \arg \min_{\mathbf{x}} f(\mathbf{x}, \mathbf{y}^{(k)}).$
- Step 2: Fix $\mathbf{x} = \mathbf{x}^{(k+1)}$ and solve the sub-problem $\mathbf{y}^{(k+1)} = \arg \min_{\mathbf{y}} f(\mathbf{x}^{(k+1)}, \mathbf{y}).$
Convergence: Repeat the iterative steps until a convergence criterion is met, such as when the change in the objective function or the parameters falls below a specified threshold, or when a pre-defined iteration limit is reached.

The AO procedure can be

viewed as a generalization of joint optimization, where the parameter partition is trivial, consisting of the entire parameter vector as a single block,
also used for maximization problems by simply replacing $\arg \min$ by $\arg \max$ above,
generalized to more than two parameter blocks, i.e., for $f(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n)$ , the procedure involves cycling through each parameter block $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$ and solving the corresponding sub-problems iteratively (the parameter blocks do not necessarily have to be disjoint),
randomized by changing the parameter partition randomly after each iteration, which can further improve the convergence rate and help avoid getting trapped in local optima (Chib and Ramamurthy 2010),
run in multiple threads for different initial values, parameter partitions, and/or base optimizers.

Now how to use the `{ao}` package?

The ao package offers the function ao(), which can be used to perform different variants of alternating optimization.

The function call

The ao() function call with the default arguments looks as follows:

ao(
  f,
  initial,
  target = NULL,
  npar = NULL,
  gradient = NULL,
  ...,
  partition = "sequential",
  new_block_probability = 0.5,
  minimum_block_number = 2,
  minimize = TRUE,
  lower = -Inf,
  upper = Inf,
  iteration_limit = Inf,
  seconds_limit = Inf,
  tolerance_value = 1e-6,
  tolerance_parameter = 1e-6,
  tolerance_parameter_norm = function(x, y) sqrt(sum((x - y)^2)),
  tolerance_history = 1,
  base_optimizer = Optimizer$new("stats::optim", method = "L-BFGS-B"),
  verbose = FALSE,
  hide_warnings = TRUE
)

The arguments have the following meaning:

f: The objective function to be optimized. By default, f is optimized over its first argument. If optimization should target a different argument or multiple arguments, use npar and target, see below. Additional arguments for f can be passed via the ... argument as usual.
initial: Initial values for the parameters used in the AO procedure.
gradient: Optional argument to specify the analytical gradient of f. If not provided, a finite-difference approximation will be used.
partition: Specifies how parameters are partitioned for optimization. Can be one of the following:
- "sequential": Optimizes each parameter block sequentially.
- "random": Randomly partitions parameters in each iteration.
- "none": No partitioning; equivalent to joint optimization.
- Custom partition can be defined using a list of vectors of parameter indices.
new_block_probability and minimum_block_number are only relevant if partition = "random". In this case, the former controls the probability for creating a new block, and the latter defines the minimum number of parameter blocks.
minimize: Set to TRUE for minimization problems (default), or FALSE for maximization.
lower and upper: Lower and upper limits for constrained optimization.
iteration_limit is the maximum number of AO iterations before termination, while seconds_limit is the time limit in seconds. tolerance_value and tolerance_parameter (in combination with tolerance_parameter_norm) specify two other stopping criteria, namely when the difference between the current function value or the current parameter vector and the one before tolerance_history iterations, respectively, becomes smaller than these thresholds.
base_optimizer: Numerical optimizer used for solving sub-problems.
Set verbose to TRUE to print status messages, and hide_warnings to FALSE to show warning messages during the AO process.

A simple first example

The following is an implementation of the Himmelblau’s function $f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2:$

himmelblau <- function(x) (x[1]^2 + x[2] - 11)^2 + (x[1] + x[2]^2 - 7)^2

This function has four identical local minima, for example in $x = 3$ and $y = 2$ :

himmelblau(c(3, 2))
#> [1] 0

Minimizing Himmelblau’s function through alternating minimization for $\mathbf{x}$ and $\mathbf{y}$ with initial values $\mathbf{x}^{(0)} = \mathbf{y}^{(0)} = 0$ can be accomplished as follows:

ao(f = himmelblau, initial = c(0, 0))
#> $estimate
#> [1]  3.584428 -1.848126
#> 
#> $value
#> [1] 9.606386e-12
#> 
#> $details
#>    iteration        value       p1        p2 b1 b2     seconds
#> 1          0 1.700000e+02 0.000000  0.000000  0  0 0.000000000
#> 2          1 1.327270e+01 3.395691  0.000000  1  0 0.017858982
#> 3          1 1.743664e+00 3.395691 -1.803183  0  1 0.003690958
#> 4          2 2.847290e-02 3.581412 -1.803183  1  0 0.003906012
#> 5          2 4.687468e-04 3.581412 -1.847412  0  1 0.003304958
#> 6          3 7.368057e-06 3.584381 -1.847412  1  0 0.002804279
#> 7          3 1.164202e-07 3.584381 -1.848115  0  1 0.022371292
#> 8          4 1.893311e-09 3.584427 -1.848115  1  0 0.002305508
#> 9          4 9.153860e-11 3.584427 -1.848124  0  1 0.001745701
#> 10         5 6.347425e-11 3.584428 -1.848124  1  0 0.001754045
#> 11         5 9.606386e-12 3.584428 -1.848126  0  1 0.001746655
#> 
#> $seconds
#> [1] 0.06148839
#> 
#> $stopping_reason
#> [1] "change in function value between 1 iteration is < 1e-06"

Here, we see the output of the alternating optimization procedure, which is a list that contains the following elements:

estimate is the parameter vector at termination.
value is the function value at termination.
details is a data.frame with full information about the procedure: For each iteration (column iteration) it contains the function value (column value), parameter values (columns starting with p followed by the parameter index), the active parameter block (columns starting with b followed by the parameter index, where 1 stands for a parameter contained in the active parameter block and 0 if not), and computation times in seconds (column seconds).
seconds is the overall computation time in seconds.
stopping_reason is a message why the procedure has terminated.

Using the analytical gradient

For the Himmelblau’s function, it is straightforward to define the analytical gradient as follows:

gradient <- function(x) {
  c(
    4 * x[1] * (x[1]^2 + x[2] - 11) + 2 * (x[1] + x[2]^2 - 7),
    2 * (x[1]^2 + x[2] - 11) + 4 * x[2] * (x[1] + x[2]^2 - 7)
  )
}

The gradient function will be used by ao() if defined through the gradient argument as follows:

ao(f = himmelblau, initial = c(0, 0), gradient = gradient)

The output is not shown here because it closely resembles the previous example, where the gradient was not specified and thus a finite-difference approximation was employed. However, in scenarios involving higher dimensions, utilizing the analytical gradient can notably improve both the speed and stability of the process.

Random parameter partitions

Another version of the AO procedure involves using a new, random partition of the parameters in every iteration. This approach can enhance the convergence rate and prevent being stuck in local optima. It is activated by setting partition = "random". The randomness can be adjusted using two parameters:

new_block_probability determines the probability for creating a new block when building a new partition. Its value ranges from 0 (no blocks are created) to 1 (each parameter is a single block).
minimum_block_number sets the minimum number of parameter blocks for random partitions. By default, it is configured to 2 to avoid generating trivial partitions.

The random partitions are build as follows:¹

procedure <- ao:::Procedure$new(
  npar = 10,
  partition = "random",
  new_block_probability = 0.5,
  minimum_block_number = 2
)
procedure$get_partition()
#> [[1]]
#> [1] 5
#> 
#> [[2]]
#> [1] 1 6 9
#> 
#> [[3]]
#> [1] 10
#> 
#> [[4]]
#> [1] 7
#> 
#> [[5]]
#> [1] 4 8
#> 
#> [[6]]
#> [1] 2 3
procedure$get_partition()
#> [[1]]
#> [1] 1 7 8
#> 
#> [[2]]
#> [1]  6 10
#> 
#> [[3]]
#> [1] 3 4
#> 
#> [[4]]
#> [1] 2
#> 
#> [[5]]
#> [1] 9
#> 
#> [[6]]
#> [1] 5

As an example of AO with random partitions, consider fitting a two-class Gaussian mixture model via maximizing the model’s log-likelihood function

$\ell(\boldsymbol{\theta}) = \sum_{i=1}^n \log\Big( \lambda \phi_{\mu_1, \sigma_1^2}(x_i) + (1-\lambda)\phi_{\mu_2,\sigma_2^2} (x_i) \Big),$

where the sum goes over all observations $1, \dots, n$ , $\phi_{\mu_1, \sigma_1^2}$ and $\phi_{\mu_2, \sigma_2^2}$ denote the normal density for the first and second cluster, respectively, and $\lambda$ is the mixing proportion. The parameter vector to be estimated is thus $\boldsymbol{\theta} = (\mu_1, \mu_2, \sigma_1, \sigma_2, \lambda)$ . As there exists no closed-form solution for the maximum likelihood estimator $\boldsymbol{\theta}^* = \arg\max_{\boldsymbol{\theta}} \ell(\boldsymbol{\theta})$ , we need numerical optimization for finding the function optimum. The model is fitted to the following data:²

The following function calculates the log-likelihood value given the parameter vector theta and the observation vector data:³

normal_mixture_llk <- function(theta, data) {
  mu <- theta[1:2]
  sd <- exp(theta[3:4])
  lambda <- plogis(theta[5])
  c1 <- lambda * dnorm(data, mu[1], sd[1])
  c2 <- (1 - lambda) * dnorm(data, mu[2], sd[2])
  sum(log(c1 + c2))
}

The ao() call for performing alternating maximization with random partitions looks as follows, where we simplified the output for brevity:

out <- ao(
  f = normal_mixture_llk,
  initial = runif(5),
  data = datasets::faithful$eruptions,
  partition = "random",
  minimize = FALSE
)
round(out$details, 2)
#>    iteration   value   p1   p2    p3    p4    p5 b1 b2 b3 b4 b5 seconds
#> 1          0 -713.98 0.94 0.79  0.97  0.35  0.50  0  0  0  0  0    0.00
#> 2          1 -541.18 0.94 3.81  0.97  0.35  0.50  0  1  0  0  0    0.00
#> 3          1 -512.65 0.94 3.81  0.66 -0.30  0.50  0  0  1  1  0    0.01
#> 4          1 -447.85 3.08 3.81  0.66 -0.30  0.50  1  0  0  0  0    0.00
#> 5          1 -445.29 3.08 3.81  0.66 -0.30 -0.04  0  0  0  0  1    0.00
#> 6          2 -432.41 3.08 4.23  0.66 -0.30 -0.04  0  1  0  0  0    0.00
#> 7          2 -277.05 2.02 4.23 -1.46 -0.81 -0.63  1  0  1  1  1    0.03
#> 8          3 -276.39 2.02 4.27 -1.46 -0.81 -0.63  0  1  0  0  1    0.01
#> 9          3 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  0  1  1  0    0.01
#> 10         4 -276.36 2.02 4.27 -1.45 -0.83 -0.63  0  0  1  0  0    0.01
#> 11         4 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  1  0  1  1    0.01
#> 12         5 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  0  0  0  0    0.00
#> 13         5 -276.36 2.02 4.27 -1.45 -0.83 -0.63  0  1  1  1  1    0.01
#> 14         6 -276.36 2.02 4.27 -1.45 -0.83 -0.63  0  0  0  0  1    0.00
#> 15         6 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  1  1  1  0    0.01
#> 16         7 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  1  0  1  1    0.01
#> 17         7 -276.36 2.02 4.27 -1.45 -0.83 -0.63  0  0  1  0  0    0.01
#> 18         8 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  1  0  1  1    0.00
#> 19         8 -276.36 2.02 4.27 -1.45 -0.83 -0.63  0  0  1  0  0    0.00

More flexibility

The ao package offers some flexibility for performing AO.⁴

Generalized objective functions

Optimizers in R generally require that the objective function has a single target argument which must be in the first position. ao allows for optimization over an argument other than the first, or more than one argument. For example, say, the normal_mixture_llk function above has the following form and is supposed to be optimized over the parameters mu, sd, and lambda:

normal_mixture_llk <- function(data, mu, sd, lambda) {
  sd <- exp(sd)
  lambda <- plogis(lambda)
  c1 <- lambda * dnorm(data, mu[1], sd[1])
  c2 <- (1 - lambda) * dnorm(data, mu[2], sd[2])
  sum(log(c1 + c2))
}

In ao(), this scenario can be specified by setting

target = c("mu", "sd", "lambda") (the names of the target arguments)
and npar = c(2, 2, 1) (the lengths of the target arguments):

ao(
  f = normal_mixture_llk,
  initial = runif(5),
  target = c("mu", "sd", "lambda"),
  npar = c(2, 2, 1),
  data = datasets::faithful$eruptions,
  partition = "random",
  minimize = FALSE
)

Parameter bounds

Instead of using parameter transformations in the normal_mixture_llk() function above, parameter bounds can be directly specified in ao() via the arguments lower and upper, where both can either be a single number (a common bound for all parameters) or a vector of specific bounds per parameter. Therefore, an more straightforward implementation of the mixture example would be:

normal_mixture_llk <- function(mu, sd, lambda, data) {
  c1 <- lambda * dnorm(data, mu[1], sd[1])
  c2 <- (1 - lambda) * dnorm(data, mu[2], sd[2])
  sum(log(c1 + c2))
}
ao(
  f = normal_mixture_llk,
  initial = runif(5),
  target = c("mu", "sd", "lambda"),
  npar = c(2, 2, 1),
  data = datasets::faithful$eruptions,
  partition = "random",
  minimize = FALSE,
  lower = c(-Inf, -Inf, 0, 0, 0),
  upper = c(Inf, Inf, Inf, Inf, 1)
)

Custom parameter partition

ao allows for the specification of custom parameter partitions. For example, say, the parameters of the Gaussian mixture model are supposed to be grouped by type:

$\mathbf{x}_1 = (\mu_1, \mu_2),\ \mathbf{x}_2 = (\sigma_1, \sigma_2),\ \mathbf{x}_3 = (\lambda).$

In ao(), custom parameter partitions can be specified by setting partition = list(1:2, 3:4, 5), i.e. by defining a list where each element corresponds to a parameter block, containing a vector of parameter indices. Parameter indices can be members of any number of blocks.

Stopping criteria

Currently, four different stopping criteria for the AO procedure are implemented:

a predefined iteration limit is exceeded (via the iteration_limit argument)
a predefined time limit is exceeded (via the seconds_limit argument)
the absolute change in the function value in comparison to the last iteration falls below a predefined threshold (via the tolerance_value argument)
the change in parameters in comparison to the last iteration falls below a predefined threshold (via the tolerance_parameter argument, where the parameter distance is computed via the norm specified as tolerance_parameter_norm)

Any number of stopping criteria can be activated or deactivated⁵, and the final output contains information about the criterium that caused termination.

Optimizer for solving the sub-problems

By default, the L-BFGS-B algorithm (Byrd et al. 1995) implemented in stats::optim is used. for solving the sub-problems numerically. However, any other optimizer can be selected by specifying the base_optimizer argument. Such an optimizer must be defined through the framework provided by the optimizeR package, please see its documentation for details. For example, the stats::nlm optimizer can be selected by setting base_optimizer = Optimizer$new("stats::nlm").

Multiple threads

Alternating optimization can suffer from local optima. To increase the likelihood of reaching the global optimum, users can specify

multiple starting parameters,
multiple parameter partitions,
multiple base optimizers.

Use the initial, partition, and/or base_optimizer arguments to provide a list of possible values for each parameter. Each combination of initial values, parameter partitions, and base optimizers will create a separate alternating optimization thread:

normal_mixture_llk <- function(mu, sd, lambda, data) {
  c1 <- lambda * dnorm(data, mu[1], sd[1])
  c2 <- (1 - lambda) * dnorm(data, mu[2], sd[2])
  sum(log(c1 + c2))
}
out <- ao(
  f = normal_mixture_llk,
  initial = list(runif(5), runif(5)),
  target = c("mu", "sd", "lambda"),
  npar = c(2, 2, 1),
  data = datasets::faithful$eruptions,
  partition = list("random", "random", "random"),
  minimize = FALSE,
  lower = c(-Inf, -Inf, 0, 0, 0),
  upper = c(Inf, Inf, Inf, Inf, 1)
)
out$values
#> [[1]]
#> [1] -607.0916
#> 
#> [[2]]
#> [1] -421.417
#> 
#> [[3]]
#> [1] -607.2776
#> 
#> [[4]]
#> [1] -3102.617
#> 
#> [[5]]
#> [1] -276.36
#> 
#> [[6]]
#> [1] -421.417

In the case of multiple threads, the output changes slightly in comparison to the standard case. It is a list with the following elements:

estimate is the optimal parameter vector over all threads.
estimates is a list of optimal parameters in each thread.
value is the optimal function value over all threads.
values is a list of optimal function values in each thread.
details combines details of the single threads and has an additional column thread with an index for the different threads.
seconds gives the computation time in seconds for each thread.
stopping_reason gives the termination message for each thread.
threads give details how the different threads were specified.

By default, threads run sequentially. However, since they are independent, they can be parallelized. To enable parallel computation, use the {future} framework. For example, run the following before the ao() call:

future::plan(future::multisession, workers = 4)

When using multiple threads, setting verbose = TRUE to print tracing details during alternating optimization is not supported. However, progress of threads can still be tracked using the {progressr} framework. For example, run the following before the ao() call:

progressr::handlers(global = TRUE)
progressr::handlers(
  progressr::handler_progress(":percent :eta :message")
)

References

Bezdek, J, and R Hathaway. 2002. “Some Notes on Alternating Optimization.” Proceedings of the 2002 AFSS International Conference on Fuzzy Systems. Calcutta: Advances in Soft Computing. https://doi.org/10.1007/3-540-45631-7_39.

———. 2003. “Convergence of Alternating Optimization.” Neural, Parallel and Scientific Computations 11 (December): 351–68.

Byrd, Richard H., Peihuang Lu, Jorge Nocedal, and Ciyou Zhu. 1995. “A Limited Memory Algorithm for Bound Constrained Optimization.” SIAM Journal on Scientific Computing 16 (5): 1190–1208. https://doi.org/10.1137/0916069.

Chang, Winston. 2022. R6: Encapsulated Classes with Reference Semantics.

Chib, Siddhartha, and Srikanth Ramamurthy. 2010. “Tailored Randomized Block MCMC Methods with Application to DSGE Models.” Journal of Econometrics 155 (1): 19–38.

Hu, Y, and R Hathaway. 2002. “On Efficiency of Optimization in Fuzzy c-Means.” Neural, Parallel and Scientific Computations 10.