Visualization of Objective Functions
objective.RmdThe Visualizer1DObjective and
Visualizer2DObjective classes visualize 1 and 2-dimensional
objective functions. The package contains predefined objective
functions.
as.data.table(dict_objective)
#> Key: <key>
#> key label xdim limits_lower limits_upper
#> <char> <char> <int> <list> <list>
#> 1: TF_Gfunction Gfunction NA NA NA
#> 2: TF_GoldsteinPrice GoldsteinPrice 2 0,0 1,1
#> 3: TF_GoldsteinPriceLog GoldsteinPriceLog 2 0,0 1,1
#> 4: TF_OTL_Circuit OTL_Circuit 6 NA NA
#> 5: TF_RoosArnold RoosArnold NA NA NA
#> 6: TF_ackley ackley 2 0,0 1,1
#> 7: TF_banana banana 2 0,0 1,1
#> 8: TF_beale beale 2 0,0 1,1
#> 9: TF_borehole borehole 2 0,0 1.5,1.0
#> 10: TF_branin branin 2 -2,-2 3,3
#> 11: TF_currin1991 currin1991 2 0,0 1,1
#> 12: TF_easom easom 2 0,0 1,1
#> 13: TF_franke franke 2 -0.5,-0.5 1,1
#> 14: TF_gaussian1 gaussian1 NA NA NA
#> 15: TF_griewank griewank NA NA NA
#> 16: TF_hartmann hartmann 6 NA NA
#> 17: TF_hump hump 2 0,0 1,1
#> 18: TF_levy levy NA NA NA
#> 19: TF_linkletter_nosignal linkletter_nosignal NA NA NA
#> 20: TF_michalewicz michalewicz NA NA NA
#> 21: TF_piston piston 7 NA NA
#> 22: TF_powsin powsin NA NA NA
#> 23: TF_quad_peaks quad_peaks 2 0,0 1,1
#> 24: TF_quad_peaks_slant quad_peaks_slant 2 0,0 1,1
#> 25: TF_rastrigin rastrigin NA NA NA
#> 26: TF_robotarm robotarm 8 NA NA
#> 27: TF_sinumoid sinumoid 2 0,0 1,1
#> 28: TF_sqrtsin sqrtsin NA NA NA
#> 29: TF_waterfall waterfall 2 0,0 1,1
#> 30: TF_wingweight wingweight 10 NA NA
#> 31: TF_zhou1998 zhou1998 2 0,0 1,1
#> key label xdim limits_lower limits_upperTo get an objective function from the dictionary, use the
obj() function.
obj_branin = obj("TF_branin")We can evaluate the objective function at a point.
x = c(0.9, 1)
obj_branin$eval(x)
#> [1] 178.3166The gradient and Hessian at point x can be extracted.
obj_branin$grad(x)
#> [1] -354.3258 395.8377
obj_branin$hess(x)
#> [,1] [,2]
#> [1,] -1989.5197 -284.2171
#> [2,] -284.2171 2162.1162We create a Visualizer2DObjective to visualize the
objective function.
viz = as_visualizer(obj_branin)The objective can be visualized with a surface
viz$init_layer_surface()
viz$plot()and contour plot.
viz$init_layer_contour()
viz$plot()Custom Objectives
Let’s define a loss for a linear model on the iris data with target
Sepal.Width and feature Petal.Width. First, an
Objective requires a function for evaluation:
# Define the linear model loss function as SSE:
l2norm = function(x) sqrt(sum(crossprod(x)))
mylm = function(x, Xmat, y) {
l2norm(y - Xmat %*% x)
}To fix the loss for the data, the Ojbective$new() call
allows to pass custom arguments that are stored and reused in every call
to $eval() to evaluate fun. So, calling
$eval(x) internally calls fun(x, ...). These
arguments must be specified just once:
# Use the iris dataset with response `Sepal.Width` and feature `Petal.Width`:
Xmat = model.matrix(~ Petal.Width, data = iris)
y = iris$Sepal.Width
# Create a new object:
obj_lm = Objective$new(id = "iris LM", fun = mylm, xdim = 2, Xmat = Xmat, y = y, minimize = TRUE)
obj_lm$evalStore(c(1, 2))
obj_lm$evalStore(c(2, 3))
obj_lm$evalStore(coef(lm(Sepal.Width ~ Petal.Width, data = iris)))
obj_lm$archive
#> x fval grad gnorm
#> <list> <num> <list> <num>
#> 1: 1,2 21.553654 2.375467,11.722838 1.196109e+01
#> 2: 2,3 43.410022 8.779078,16.929270 1.907020e+01
#> 3: 3.3084256,-0.2093598 4.951004 4.832272e-07,2.664535e-07 5.518206e-07Visualize lm Objective:
viz_lm = as_visualizer(obj_lm, x1_limits = c(-0.5, 5), x2_limits = c(-3.2, 2.8))
viz_lm$plot()More advanced (see More advanced) is to
add points to the plotly object:
Optimizer
The optimizer class defines the optimization strategy and is initialized by taking an objective function, start value, and learning rate. Available optimizer are:
- Gradient descent with
OptimizerGD - Momentum with
OptimizerMomentum - Nesterovs momentum with
OptimizerNAG
Creating an optimizer is done by (let’s use an x value that works well):
obj_banana = obj("TF_banana")
opt = OptimizerGD$new(obj_banana, x_start = c(0.8, 0.6), lr = 0.01)With these value set, optimization is done by calling
$optimize() with the number of steps as argument:
opt$optimize(10L)
#> TF_banana: Batch 1 step 1: f(x) = 0.1572, x = c(0.7352, 0.5778)
#> TF_banana: Batch 1 step 2: f(x) = 0.4955, x = c(0.7888, 0.6174)
#> TF_banana: Batch 1 step 3: f(x) = 0.2118, x = c(0.7296, 0.5977)
#> TF_banana: Batch 1 step 4: f(x) = 0.3741, x = c(0.792, 0.6455)
#> TF_banana: Batch 1 step 5: f(x) = 0.069, x = c(0.6932, 0.602)
#> TF_banana: Batch 1 step 6: f(x) = 0.2809, x = c(0.7189, 0.6245)
#> TF_banana: Batch 1 step 7: f(x) = 0.2565, x = c(0.7877, 0.6806)
#> TF_banana: Batch 1 step 8: f(x) = 0.1753, x = c(0.6922, 0.6351)
#> TF_banana: Batch 1 step 9: f(x) = 0.6265, x = c(0.7418, 0.6799)
#> TF_banana: Batch 1 step 10: f(x) = 0.5908, x = c(0.717, 0.6795)Calling $optimize() also writes into the archive of the
optimizer and also calls $evalStore() of the objective.
Therefore, $optimize() writes into two archives:
opt$archive
#> x_out x_in update
#> <list> <list> <list>
#> 1: 0.7351819,0.5778020 0.8,0.6 -0.06481810,-0.02219798
#> 2: 0.7888135,0.6173630 0.7351819,0.5778020 0.05363157,0.03956093
#> 3: 0.7296113,0.5977212 0.7888135,0.6173630 -0.05920220,-0.01964177
#> 4: 0.7920259,0.6454892 0.7296113,0.5977212 0.06241458,0.04776799
#> 5: 0.6931781,0.6019537 0.7920259,0.6454892 -0.09884780,-0.04353545
#> 6: 0.7189369,0.6245137 0.6931781,0.6019537 0.02575889,0.02256001
#> 7: 0.7876975,0.6806080 0.7189369,0.6245137 0.06876058,0.05609425
#> 8: 0.6921665,0.6351359 0.7876975,0.6806080 -0.09553100,-0.04547204
#> 9: 0.7417730,0.6798539 0.6921665,0.6351359 0.04960652,0.04471794
#> 10: 0.7170136,0.6795146 0.7417730,0.6798539 -0.0247594837,-0.0003392663
#> fval_out fval_in lr step_size
#> <num> <num> <num> <num>
#> 1: 0.15716889 0.46245779 0.01 1
#> 2: 0.49548040 0.15716889 0.01 1
#> 3: 0.21179976 0.49548040 0.01 1
#> 4: 0.37414391 0.21179976 0.01 1
#> 5: 0.06900854 0.37414391 0.01 1
#> 6: 0.28085919 0.06900854 0.01 1
#> 7: 0.25645358 0.28085919 0.01 1
#> 8: 0.17532134 0.25645358 0.01 1
#> 9: 0.62647891 0.17532134 0.01 1
#> 10: 0.59083776 0.62647891 0.01 1
#> objective
#> <list>
#> 1: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 2: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 3: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 4: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 5: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 6: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 7: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 8: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 9: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 10: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> momentum step batch
#> <num> <int> <num>
#> 1: 0 1 1
#> 2: 0 2 1
#> 3: 0 3 1
#> 4: 0 4 1
#> 5: 0 5 1
#> 6: 0 6 1
#> 7: 0 7 1
#> 8: 0 8 1
#> 9: 0 9 1
#> 10: 0 10 1
opt$objective$archive
#> x fval grad gnorm
#> <list> <num> <list> <num>
#> 1: 0.8,0.6 0.46245779 -6.481810,-2.219798 6.851377
#> 2: 0.7351819,0.5778020 0.15716889 5.363157,3.956093 6.664393
#> 3: 0.7888135,0.6173630 0.49548040 -5.920220,-1.964177 6.237547
#> 4: 0.7296113,0.5977212 0.21179976 6.241458,4.776799 7.859618
#> 5: 0.7920259,0.6454892 0.37414391 -9.884780,-4.353545 10.801029
#> 6: 0.6931781,0.6019537 0.06900854 2.575889,2.256001 3.424142
#> 7: 0.7189369,0.6245137 0.28085919 6.876058,5.609425 8.873884
#> 8: 0.7876975,0.6806080 0.25645358 -9.553100,-4.547204 10.580113
#> 9: 0.6921665,0.6351359 0.17532134 4.960652,4.471794 6.678699
#> 10: 0.7417730,0.6798539 0.62647891 -2.47594837,-0.03392663 2.476181We can let the algorithm run for another 10 iterations in a second batch:
opt$optimize(10L)
#> TF_banana: Batch 2 step 1: f(x) = 0.2592, x = c(0.7638, 0.728)
#> TF_banana: Batch 2 step 2: f(x) = 0.2873, x = c(0.6702, 0.679)
#> TF_banana: Batch 2 step 3: f(x) = 0.5391, x = c(0.7292, 0.7404)
#> TF_banana: Batch 2 step 4: f(x) = 0.1765, x = c(0.6349, 0.6896)
#> TF_banana: Batch 2 step 5: f(x) = 0.6858, x = c(0.6718, 0.7368)
#> TF_banana: Batch 2 step 6: f(x) = 0.401, x = c(0.713, 0.7914)
#> TF_banana: Batch 2 step 7: f(x) = 0.3109, x = c(0.6133, 0.7283)
#> TF_banana: Batch 2 step 8: f(x) = 0.8142, x = c(0.657, 0.7963)
#> TF_banana: Batch 2 step 9: f(x) = 0.7474, x = c(0.6208, 0.7806)
#> TF_banana: Batch 2 step 10: f(x) = 0.5501, x = c(0.6575, 0.8468)Still not very satisfying.
Visualize Optimization Traces
A layer of the Visualizer class is
$add_optimization_trace() that gets the optimizer as
argument and adds the optimization trace to the plot:
viz = as_visualizer(obj_banana)
viz$add_optimization_trace(opt)
viz$plot()Step size control
When calling $optimize(), the second argument is
stepSizeControl that allows to expand or compress the
update added to the old value of \(x\).
For example, for GD with \(x_{\text{new}} =
x_{\text{old}} + lr * \Delta_f(x_{\text{old}})\) the update \(u = lr * \Delta_f(x_{\text{old}})\) is
multiplied with the return value of stepSizeControl().
There are a few pre-implemented control functions like line search or
various decaying methods:
-
stepSizeControlLineSearch(lower, upper): Conduct a line search for \(a\) in \(x_{\text{new}} = x_{\text{old}} + a * lr * \Delta_f(x_{\text{old}})\)`. -
stepSizeControlDecayTime(decay): Lower the updates by \((1 + decay * iteration)^{-1}\). -
stepSizeControlDecayExp(decay): Lower the updates by \(exp(-decay * iteration)\). -
stepSizeControlDecayLinear(iter_zero): Lower the updates untiliter_zerois reached. Updates withiter > iter_zeroare 0. -
stepSizeControlDecaySteps(drop_rate, every_iter): Lower the updatesevery_iterbydrop_rate.
Note that these functions return a function that contains a function with the required signature:
stepSizeControlDecayTime()
#> function (x, u, obj, opt)
#> {
#> assertStepSizeControl(x, u, obj, opt)
#> epoch = nrow(obj$archive)
#> return(1/(1 + decay * epoch))
#> }
#> <bytecode: 0x55f5b7e92720>
#> <environment: 0x55f5b7e8f2c8>Let’s define multiple gradient descent optimizers and optimize 100 steps with a step size control:
x0 = c(0.8, 0.6)
lr = 0.01
obj_banana = obj("TF_banana")
oo1 = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD without LR Control", print_trace = FALSE)
oo2 = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD with Line Search", print_trace = FALSE)
oo3 = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD with Time Decay", print_trace = FALSE)
oo4 = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD with Exp Decay", print_trace = FALSE)
oo5 = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD with Linear Decay", print_trace = FALSE)
oo6 = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD with Step Decay", print_trace = FALSE)
oo1$optimize(steps = 100)
oo2$optimize(steps = 100, stepSizeControlLineSearch())
oo3$optimize(steps = 100, stepSizeControlDecayTime())
oo4$optimize(steps = 100, stepSizeControlDecayExp())
oo5$optimize(steps = 100, stepSizeControlDecayLinear())
oo6$optimize(steps = 100, stepSizeControlDecaySteps())For now we don’t know how well it worked. Let’s collect all archives
with mergeOptimArchives() and visualize the step sizes and
function values with patchwork magic:
arx = mergeOptimArchives(oo1, oo2, oo3, oo4, oo5, oo6)
library(patchwork)
gg1 = ggplot(arx, aes(x = iteration, y = step_size, color = optim_id))
gg2 = ggplot(arx, aes(x = iteration, y = fval_out, color = optim_id))
(gg1 + ggtitle("Step sizes") |
gg1 + ylim(0, 1) + ggtitle("Step sizes (zoomed)") |
gg2 + ggtitle("Objective")) +
plot_layout(guides = "collect") &
geom_line() &
theme_minimal() &
theme(legend.position = "bottom") &
ggsci::scale_color_simpsons()
Visualizing the traces is done as before by adding optimization trace
layer. We can do this for all optimizers to add multiple traces to the
plot (colors are picked randomly, see the Visualizer section for more details
about plotting):
viz = as_visualizer(obj_banana)
viz$add_optimization_trace(oo1)
viz$add_optimization_trace(oo2)
viz$add_optimization_trace(oo3)
viz$add_optimization_trace(oo4)
viz$add_optimization_trace(oo5)
viz$add_optimization_trace(oo6)
viz$plot()Practically, it should be no issue to also combine multiple control
functions. The important thing is to keep the signature of the function
by allowing the function to get the arguments x (current
value), u (current update), obj
(Objective object), and opt
(Optimizer object):
myStepSizeControl = function(x, u, obj, opt) {
sc1 = stepSizeControlLineSearch(0, 10)
sc2 = stepSizeControlDecayTime(0.1)
return(sc1(x, u, obj, opt) * sc2(x, u, obj, opt))
}
my_oo = OptimizerGD$new(obj_banana, x_start = x0, lr = lr, id = "GD without LR Control", print_trace = FALSE)
my_oo$optimize(100, myStepSizeControl)
tail(my_oo$archive)
#> x_out x_in update fval_out
#> <list> <list> <list> <num>
#> 1: 0.5000029,0.8666667 0.5000032,0.8666667 -5.170309e-07, 1.110223e-10 1
#> 2: 0.5000026,0.8666667 0.5000029,0.8666667 -4.67848e-07, 0.00000e+00 1
#> 3: 0.5000024,0.8666667 0.5000026,0.8666667 -4.238832e-07,-1.110223e-10 1
#> 4: 0.5000022,0.8666667 0.5000024,0.8666667 -3.843592e-07, 1.110223e-10 1
#> 5: 0.5000020,0.8666667 0.5000022,0.8666667 -3.487211e-07, 0.000000e+00 1
#> 6: 0.5000018,0.8666667 0.5000020,0.8666667 -3.168577e-07, 0.000000e+00 1
#> fval_in lr step_size
#> <num> <num> <num>
#> 1: 1 0.01 0.5942504
#> 2: 1 0.01 0.5888878
#> 3: 1 0.01 0.5829326
#> 4: 1 0.01 0.5774413
#> 5: 1 0.01 0.5722183
#> 6: 1 0.01 0.5667869
#> objective
#> <list>
#> 1: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 2: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 3: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 4: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 5: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> 6: <Objective>\n Public:\n addLogFun: function (l, label) \n archive: active binding\n assertX: function (x, ...) \n clearArchive: function () \n clone: function (deep = FALSE) \n eval: function (x) \n evalStore: function (x) \n grad: function (x) \n hess: function (x) \n id: TF_banana\n initialize: function (id, fun, label = "f", xdim, limits_lower = NA, limits_upper = NA, \n label: banana\n limits_lower: 0 0\n limits_upper: 1 1\n log_funs: active binding\n minimize: FALSE\n xdim: active binding\n Private:\n p_archive: data.table, data.frame\n p_fargs: list\n p_fun: function (x, scale_it = T, scale_low = c(-20, -10), scale_high = c(20, \n p_gradient: NULL\n p_gradientFallback: function (x, ...) \n p_hessian: NULL\n p_hessianFallback: function (x, ...) \n p_log_funs: list\n p_xdim: 2\n p_xtest: 0 0
#> momentum step batch
#> <num> <int> <num>
#> 1: 0 95 1
#> 2: 0 96 1
#> 3: 0 97 1
#> 4: 0 98 1
#> 5: 0 99 1
#> 6: 0 100 1Visualizer
Visualizer class
The Visualizer class is initialized by calling
as_visualizer(obj) for a given Objective.
Furhter arguments are x1limits and x2limits as
well as padding to stretch the limits by a factor and
npoints to specify the number of generated point per
dimension, hence npoints^2 points are evaluated to create
the initial layers.
Initial layers
An initial layer is always required to which other layers (such as
optimization traces) are added step by step. The two available base
layers are contour lines $init_layer_contour() and
$init_layer_surface().
For both, the first argument tis the opacity and the second a colorscale:
viz = as_visualizer(obj("TF_franke"))
viz$init_layer_contour(opacity = 1, colorscale = list(c(0, 1), c("rgb(176,196,222)", "rgb(160,82,45)")))
viz$plot()
viz = as_visualizer(obj("TF_franke"))
viz$init_layer_contour(colorscale = list(c(0, 1), c("white", "blue")))
viz$plot()
viz = as_visualizer(obj("TF_franke"))
viz$init_layer_surface(opacity = 1, colorscale = list(c(0, 1), c("white", "black")))
viz$plot()Internally, add_trace() from plotly is
called. Further arguments are directly passed to it, e.g. by adding
contour lines:
viz = as_visualizer(obj("TF_franke"))
# Add 10 grid lines per dim, limits (sometimes) can be obtained form the
# objective:
llower = viz$objective$limits_lower
lupper = viz$objective$limits_upper
ssize = (lupper - llower) / 10
viz$init_layer_surface(opacity = 1, colorscale = list(c(0, 1), c("white", "black")),
contours = list(
x = list(show = TRUE, start = llower[1], end = lupper[1], size = ssize[1], color = "black"),
y = list(show = TRUE, start = llower[2], end = lupper[2], size = ssize[2], color = "black")))
viz$plot()Optimization traces
As shown previously, optimization traces can be added by
$add_optimization_trace. Let’s optimize our custom linear
model objective for the three available optimizers:
# Reset the archive to have an empty objective:
obj_lm$clearArchive()
oo1 = OptimizerGD$new(obj_lm, x_start = c(0, -0.05), lr = 0.001, print_trace = FALSE)
oo2 = OptimizerMomentum$new(obj_lm, x_start = c(-0.05, 0), lr = 0.001, print_trace = FALSE)
oo3 = OptimizerNAG$new(obj_lm, x_start = c(0, 0), lr = 0.001, print_trace = FALSE)
oo1$optimize(steps = 100)
oo2$optimize(steps = 100)
oo3$optimize(steps = 100)
viz = as_visualizer(obj_lm, x1_limits = c(-0.5, 5), x2_limits = c(-3.2, 2.8))
viz$init_layer_contour()
viz$add_optimization_trace(oo1, add_marker_at = round(seq(1, 100, len = 10L)))
viz$add_optimization_trace(oo2, add_marker_at = c(1, 50, 90), marker_shape = c("square", "star-triangle-down", "cross"))
viz$add_optimization_trace(oo3, add_marker_at = 100, marker_shape = "star")
viz$plot()Setting the layout and scene (TODO)
viz$set_layout(legend = list(orientation = "h", xanchor = "center", x = 0.5))
viz$plot()Overlaying layers (TODO)
obj = obj("TF_banana")
viz = as_visualizer(obj)
viz$init_layer_surface()
x0 = c(0.85, 0.47)
viz$add_layer_taylor(x0, npoints_per_dim = 5, degree = 1, x1margin = 0.3, x2margin = 0.3, contours = list(
x = list(show = TRUE, start = 0, end = 1, size = 0.03, color = "black"),
y = list(show = TRUE, start = 0, end = 1, size = 0.03, color = "black")))
viz$add_layer_hessian(x0)
viz$plot()Manual layers (TODO)
obj = obj("TF_banana")
viz = as_visualizer(obj)
viz$init_layer_surface(opacity = 0.5)
p = viz$plot()
class(p)
#> [1] "plotly" "htmlwidget"
nsim = 100
grid = data.frame(x = runif(nsim), y = runif(nsim))
grid$z = apply(grid, 1, viz$objective$eval) + rnorm(nsim, sd = 0.05)
p %>% add_trace(data = grid, x = ~x, y = ~y, z = ~z, mode = "markers",
type = "scatter3d", marker = list(symbol = "cross"))
