MultitargetSRRegressor

MultitargetSRRegressor

A model type for constructing a Multi-Target Symbolic Regression via Evolutionary Search, based on SymbolicRegression.jl, and implementing the MLJ model interface.

From MLJ, the type can be imported using

MultitargetSRRegressor = @load MultitargetSRRegressor pkg=SymbolicRegression

Do model = MultitargetSRRegressor() to construct an instance with default hyper-parameters. Provide keyword arguments to override hyper-parameter defaults, as in MultitargetSRRegressor(binary_operators=...).

Multi-target Symbolic Regression regressor (MultitargetSRRegressor) conducts several searches for expressions that predict each target variable from a set of input variables. All data is assumed to be Continuous. The search is performed using an evolutionary algorithm. This algorithm is described in the paper https://arxiv.org/abs/2305.01582.

Training data

In MLJ or MLJBase, bind an instance model to data with

mach = machine(model, X, y)

mach = machine(model, X, y, w)

Here:

X is any table of input features (eg, a DataFrame) whose columns are of scitype

Continuous; check column scitypes with schema(X). Variable names in discovered expressions will be taken from the column names of X, if available. Units in columns of X (use DynamicQuantities for units) will trigger dimensional analysis to be used.

y is the target, which can be any table of target variables whose element scitype is Continuous; check the scitype with schema(y). Units in columns of y (use DynamicQuantities for units) will trigger dimensional analysis to be used.
w is the observation weights which can either be nothing (default) or an AbstractVector whoose element scitype is Count or Continuous. The same weights are used for all targets.

Train the machine using fit!(mach), inspect the discovered expressions with report(mach), and predict on new data with predict(mach, Xnew). Note that unlike other regressors, symbolic regression stores a list of lists of trained models. The models chosen from each of these lists is defined by the function selection_method keyword argument, which by default balances accuracy and complexity. You can override this at prediction time by passing a named tuple with keys data and idx.

Hyper-parameters

binary_operators: Vector of binary operators (functions) to use. Each operator should be defined for two input scalars, and one output scalar. All operators need to be defined over the entire real line (excluding infinity - these are stopped before they are input), or return NaN where not defined. For speed, define it so it takes two reals of the same type as input, and outputs the same type. For the SymbolicUtils simplification backend, you will need to define a generic method of the operator so it takes arbitrary types.
unary_operators: Same, but for unary operators (one input scalar, gives an output scalar).
constraints: Array of pairs specifying size constraints for each operator. The constraints for a binary operator should be a 2-tuple (e.g., (-1, -1)) and the constraints for a unary operator should be an Int. A size constraint is a limit to the size of the subtree in each argument of an operator. e.g., [(^)=>(-1, 3)] means that the ^ operator can have arbitrary size (-1) in its left argument, but a maximum size of 3 in its right argument. Default is no constraints.
batching: Whether to evolve based on small mini-batches of data, rather than the entire dataset.
batch_size: What batch size to use if using batching.
elementwise_loss: What elementwise loss function to use. Can be one of the following losses, or any other loss of type SupervisedLoss. You can also pass a function that takes a scalar target (left argument), and scalar predicted (right argument), and returns a scalar. This will be averaged over the predicted data. If weights are supplied, your function should take a third argument for the weight scalar. Included losses: Regression: - LPDistLoss{P}(), - L1DistLoss(), - L2DistLoss() (mean square), - LogitDistLoss(), - HuberLoss(d), - L1EpsilonInsLoss(ϵ), - L2EpsilonInsLoss(ϵ), - PeriodicLoss(c), - QuantileLoss(τ), Classification: - ZeroOneLoss(), - PerceptronLoss(), - L1HingeLoss(), - SmoothedL1HingeLoss(γ), - ModifiedHuberLoss(), - L2MarginLoss(), - ExpLoss(), - SigmoidLoss(), - DWDMarginLoss(q).
loss_function: Alternatively, you may redefine the loss used as any function of tree::Node{T}, dataset::Dataset{T}, and options::Options, so long as you output a non-negative scalar of type T. This is useful if you want to use a loss that takes into account derivatives, or correlations across the dataset. This also means you could use a custom evaluation for a particular expression. If you are using batching=true, then your function should accept a fourth argument idx, which is either nothing (indicating that the full dataset should be used), or a vector of indices to use for the batch. For example,
```
  function my_loss(tree, dataset::Dataset{T,L}, options)::L where {T,L}
      prediction, flag = eval_tree_array(tree, dataset.X, options)
      if !flag
          return L(Inf)
      end
      return sum((prediction .- dataset.y) .^ 2) / dataset.n
  end
```
populations: How many populations of equations to use.
population_size: How many equations in each population.
ncycles_per_iteration: How many generations to consider per iteration.
tournament_selection_n: Number of expressions considered in each tournament.
tournament_selection_p: The fittest expression in a tournament is to be selected with probability p, the next fittest with probability p*(1-p), and so forth.
topn: Number of equations to return to the host process, and to consider for the hall of fame.
complexity_of_operators: What complexity should be assigned to each operator, and the occurrence of a constant or variable. By default, this is 1 for all operators. Can be a real number as well, in which case the complexity of an expression will be rounded to the nearest integer. Input this in the form of, e.g., [(^) => 3, sin => 2].
complexity_of_constants: What complexity should be assigned to use of a constant. By default, this is 1.
complexity_of_variables: What complexity should be assigned to each variable. By default, this is 1.
alpha: The probability of accepting an equation mutation during regularized evolution is given by exp(-delta_loss/(alpha * T)), where T goes from 1 to 0. Thus, alpha=infinite is the same as no annealing.
maxsize: Maximum size of equations during the search.
maxdepth: Maximum depth of equations during the search, by default this is set equal to the maxsize.
parsimony: A multiplicative factor for how much complexity is punished.
dimensional_constraint_penalty: An additive factor if the dimensional constraint is violated.
use_frequency: Whether to use a parsimony that adapts to the relative proportion of equations at each complexity; this will ensure that there are a balanced number of equations considered for every complexity.
use_frequency_in_tournament: Whether to use the adaptive parsimony described above inside the score, rather than just at the mutation accept/reject stage.
adaptive_parsimony_scaling: How much to scale the adaptive parsimony term in the loss. Increase this if the search is spending too much time optimizing the most complex equations.
turbo: Whether to use LoopVectorization.@turbo to evaluate expressions. This can be significantly faster, but is only compatible with certain operators. Experimental!
migration: Whether to migrate equations between processes.
hof_migration: Whether to migrate equations from the hall of fame to processes.
fraction_replaced: What fraction of each population to replace with migrated equations at the end of each cycle.
fraction_replaced_hof: What fraction to replace with hall of fame equations at the end of each cycle.
should_simplify: Whether to simplify equations. If you pass a custom objective, this will be set to false.
should_optimize_constants: Whether to use an optimization algorithm to periodically optimize constants in equations.
optimizer_nrestarts: How many different random starting positions to consider for optimization of constants.
optimizer_algorithm: Select algorithm to use for optimizing constants. Default is "BFGS", but "NelderMead" is also supported.
optimizer_options: General options for the constant optimization. For details we refer to the documentation on Optim.Options from the Optim.jl package. Options can be provided here as NamedTuple, e.g. (iterations=16,), as a Dict, e.g. Dict(:x_tol => 1.0e-32,), or as an Optim.Options instance.
output_file: What file to store equations to, as a backup.
perturbation_factor: When mutating a constant, either multiply or divide by (1+perturbation_factor)^(rand()+1).
probability_negate_constant: Probability of negating a constant in the equation when mutating it.
mutation_weights: Relative probabilities of the mutations. The struct MutationWeights should be passed to these options. See its documentation on MutationWeights for the different weights.
crossover_probability: Probability of performing crossover.
annealing: Whether to use simulated annealing.
warmup_maxsize_by: Whether to slowly increase the max size from 5 up to maxsize. If nonzero, specifies the fraction through the search at which the maxsize should be reached.
verbosity: Whether to print debugging statements or not.
print_precision: How many digits to print when printing equations. By default, this is 5.
save_to_file: Whether to save equations to a file during the search.
bin_constraints: See constraints. This is the same, but specified for binary operators only (for example, if you have an operator that is both a binary and unary operator).
una_constraints: Likewise, for unary operators.
seed: What random seed to use. nothing uses no seed.
progress: Whether to use a progress bar output (verbosity will have no effect).
early_stop_condition: Float - whether to stop early if the mean loss gets below this value. Function - a function taking (loss, complexity) as arguments and returning true or false.
timeout_in_seconds: Float64 - the time in seconds after which to exit (as an alternative to the number of iterations).
max_evals: Int (or Nothing) - the maximum number of evaluations of expressions to perform.
skip_mutation_failures: Whether to simply skip over mutations that fail or are rejected, rather than to replace the mutated expression with the original expression and proceed normally.
enable_autodiff: Whether to enable automatic differentiation functionality. This is turned off by default. If turned on, this will be turned off if one of the operators does not have well-defined gradients.
nested_constraints: Specifies how many times a combination of operators can be nested. For example, [sin => [cos => 0], cos => [cos => 2]] specifies that cos may never appear within a sin, but sin can be nested with itself an unlimited number of times. The second term specifies that cos can be nested up to 2 times within a cos, so that cos(cos(cos(x))) is allowed (as well as any combination of + or - within it), but cos(cos(cos(cos(x)))) is not allowed. When an operator is not specified, it is assumed that it can be nested an unlimited number of times. This requires that there is no operator which is used both in the unary operators and the binary operators (e.g., - could be both subtract, and negation). For binary operators, both arguments are treated the same way, and the max of each argument is constrained.
deterministic: Use a global counter for the birth time, rather than calls to time(). This gives perfect resolution, and is therefore deterministic. However, it is not thread safe, and must be used in serial mode.
define_helper_functions: Whether to define helper functions for constructing and evaluating trees.
niterations::Int=10: The number of iterations to perform the search. More iterations will improve the results.
parallelism=:multithreading: What parallelism mode to use. The options are :multithreading, :multiprocessing, and :serial. By default, multithreading will be used. Multithreading uses less memory, but multiprocessing can handle multi-node compute. If using :multithreading mode, the number of threads available to julia are used. If using :multiprocessing, numprocs processes will be created dynamically if procs is unset. If you have already allocated processes, pass them to the procs argument and they will be used. You may also pass a string instead of a symbol, like "multithreading".
numprocs::Union{Int, Nothing}=nothing: The number of processes to use, if you want equation_search to set this up automatically. By default this will be 4, but can be any number (you should pick a number <= the number of cores available).
procs::Union{Vector{Int}, Nothing}=nothing: If you have set up a distributed run manually with procs = addprocs() and @everywhere, pass the procs to this keyword argument.
addprocs_function::Union{Function, Nothing}=nothing: If using multiprocessing (parallelism=:multithreading), and are not passing procs manually, then they will be allocated dynamically using addprocs. However, you may also pass a custom function to use instead of addprocs. This function should take a single positional argument, which is the number of processes to use, as well as the lazy keyword argument. For example, if set up on a slurm cluster, you could pass addprocs_function = addprocs_slurm, which will set up slurm processes.
heap_size_hint_in_bytes::Union{Int,Nothing}=nothing: On Julia 1.9+, you may set the --heap-size-hint flag on Julia processes, recommending garbage collection once a process is close to the recommended size. This is important for long-running distributed jobs where each process has an independent memory, and can help avoid out-of-memory errors. By default, this is set to Sys.free_memory() / numprocs.
runtests::Bool=true: Whether to run (quick) tests before starting the search, to see if there will be any problems during the equation search related to the host environment.
loss_type::Type=Nothing: If you would like to use a different type for the loss than for the data you passed, specify the type here. Note that if you pass complex data ::Complex{L}, then the loss type will automatically be set to L.
selection_method::Function: Function to selection expression from the Pareto frontier for use in predict. See SymbolicRegression.MLJInterfaceModule.choose_best for an example. This function should return a single integer specifying the index of the expression to use. By default, this maximizes the score (a pound-for-pound rating) of expressions reaching the threshold of 1.5x the minimum loss. To override this at prediction time, you can pass a named tuple with keys data and idx to predict. See the Operations section for details.
dimensions_type::AbstractDimensions: The type of dimensions to use when storing the units of the data. By default this is DynamicQuantities.SymbolicDimensions.

Operations

predict(mach, Xnew): Return predictions of the target given features Xnew, which should have same scitype as X above. The expression used for prediction is defined by the selection_method function, which can be seen by viewing report(mach).best_idx.
predict(mach, (data=Xnew, idx=i)): Return predictions of the target given features Xnew, which should have same scitype as X above. By passing a named tuple with keys data and idx, you are able to specify the equation you wish to evaluate in idx.

Fitted parameters

The fields of fitted_params(mach) are:

best_idx::Vector{Int}: The index of the best expression in each Pareto frontier, as determined by the selection_method function. Override in predict by passing a named tuple with keys data and idx.
equations::Vector{Vector{Node{T}}}: The expressions discovered by the search, represented in a dominating Pareto frontier (i.e., the best expressions found for each complexity). The outer vector is indexed by target variable, and the inner vector is ordered by increasing complexity. T is equal to the element type of the passed data.
equation_strings::Vector{Vector{String}}: The expressions discovered by the search, represented as strings for easy inspection.

Report

The fields of report(mach) are:

best_idx::Vector{Int}: The index of the best expression in each Pareto frontier, as determined by the selection_method function. Override in predict by passing a named tuple with keys data and idx.
equations::Vector{Vector{Node{T}}}: The expressions discovered by the search, represented in a dominating Pareto frontier (i.e., the best expressions found for each complexity). The outer vector is indexed by target variable, and the inner vector is ordered by increasing complexity.
equation_strings::Vector{Vector{String}}: The expressions discovered by the search, represented as strings for easy inspection.
complexities::Vector{Vector{Int}}: The complexity of each expression in each Pareto frontier.
losses::Vector{Vector{L}}: The loss of each expression in each Pareto frontier, according to the loss function specified in the model. The type L is the loss type, which is usually the same as the element type of data passed (i.e., T), but can differ if complex data types are passed.
scores::Vector{Vector{L}}: A metric which considers both the complexity and loss of an expression, equal to the change in the log-loss divided by the change in complexity, relative to the previous expression along the Pareto frontier. A larger score aims to indicate an expression is more likely to be the true expression generating the data, but this is very problem-dependent and generally several other factors should be considered.

Examples

using MLJ
MultitargetSRRegressor = @load MultitargetSRRegressor pkg=SymbolicRegression
X = (a=rand(100), b=rand(100), c=rand(100))
Y = (y1=(@. cos(X.c) * 2.1 - 0.9), y2=(@. X.a * X.b + X.c))
model = MultitargetSRRegressor(binary_operators=[+, -, *], unary_operators=[exp], niterations=100)
mach = machine(model, X, Y)
fit!(mach)
y_hat = predict(mach, X)
## View the equations used:
r = report(mach)
for (output_index, (eq, i)) in enumerate(zip(r.equation_strings, r.best_idx))
    println("Equation used for ", output_index, ": ", eq[i])
end