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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:

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:

  1. Initialization: Start with initial guesses \(\mathbf{x}^{(0)}\) and \(\mathbf{y}^{(0)}\).

  2. 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 \[\mathbf{y}^{(k+1)} = \arg \min_{\mathbf{y}} f(\mathbf{x}^{(k+1)}, \mathbf{y}).\]

  3. 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

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:

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 0.0000000000
#> 2          1 1.327270e+01 3.395691  0.000000  1  0 0.0125110149
#> 3          1 1.743664e+00 3.395691 -1.803183  0  1 0.0019571781
#> 4          2 2.847290e-02 3.581412 -1.803183  1  0 0.0015733242
#> 5          2 4.687468e-04 3.581412 -1.847412  0  1 0.0013482571
#> 6          3 7.368057e-06 3.584381 -1.847412  1  0 0.0011465549
#> 7          3 1.164202e-07 3.584381 -1.848115  0  1 0.0087573528
#> 8          4 1.893311e-09 3.584427 -1.848115  1  0 0.0009469986
#> 9          4 9.153860e-11 3.584427 -1.848124  0  1 0.0007493496
#> 10         5 6.347425e-11 3.584428 -1.848124  1  0 0.0007514954
#> 11         5 9.606386e-12 3.584428 -1.848126  0  1 0.0007536411
#> 
#> $seconds
#> [1] 0.03049517
#> 
#> $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:

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:

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    0.00
#> 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    0.02
#> 8          3 -276.39 2.02 4.27 -1.46 -0.81 -0.63  0  1  0  0  1    0.00
#> 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    0.00
#> 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    0.00
#> 16         7 -276.36 2.02 4.27 -1.45 -0.83 -0.63  1  1  0  1  1    0.00
#> 17         7 -276.36 2.02 4.27 -1.45 -0.83 -0.63  0  0  1  0  0    0.00
#> 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:

  1. a predefined iteration limit is exceeded (via the iteration_limit argument)

  2. a predefined time limit is exceeded (via the seconds_limit argument)

  3. the absolute change in the function value in comparison to the last iteration falls below a predefined threshold (via the tolerance_value argument)

  4. 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.

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

  • estimate is the optimal parameter vector over all threads.

  • value is the optimal function value over all threads.

  • 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.

  1. Procedure is an internal R6 object (Chang 2022).↩︎

  2. The faithful data set contains information about eruption times (eruptions) of the Old Faithful geyser in Yellowstone National Park, Wyoming, USA. The data histogram hints at two clusters with short and long eruption times, respectively. For both clusters, we assume a normal distribution, such that we consider a mixture of two Gaussian densities for modeling the overall eruption times.↩︎

  3. We restrict the standard deviations sd to be positive (via the exponential transformation) and lambda to be between 0 and 1 (via the logit transformation).↩︎

  4. Do you miss a functionality? Please let us know via an issue on GitHub.↩︎

  5. Stopping criteria of the AO procedure can be deactivated, e.g., by setting iteration_limit = Inf, seconds_limit = Inf, tolerance_value = 0, or tolerance_parameter = 0.↩︎

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