Improved Rprop- (iRprop-)

class pints.IRPropMin(x0, sigma0=0.1, boundaries=None)

iRprop- algorithm, as described in Figure 3 of [1].

The name “iRprop-” was introduced by [1], and is a variation on the “Resilient backpropagation (Rprop)” optimiser introduced in [2].

This is a local optimiser that requires gradient information, although it uses only the direction (sign) of the gradient in each dimension and ignores the magnitude. Instead, it maintains a separate step size for each dimension which grows when the sign of the gradient stays the same and shrinks when it changes.

Pseudo-code is given below. Here p_j[i] denotes the j-th parameter at iteration i, and df_j[i] is the corresponding derivative of the objective function (so both are scalars):

if df_j[i] * df_j[i - 1] > 0:
    step_size_j[i] = 1.2 * step_size_j[i - 1]
elif df_j[i] * df_j[i - 1] < 0:
    step_size_j[i] = 0.5 * step_size_j[i - 1]
    df_j[i] = 0
step_size_j[i] = min(max(step_size_j[i], min_step_size), max_step_size)
p_j[i] = p_j[i] - sign(df_j[i]) * step_size_j[i]

The line df_j[i] = 0 has two effects:

1. It sets the update at this iteration to zero (using sign(df_j[i]) * step_size_j[i] = 0 * step_size_j[i]).

2. It ensures that the next iteration is performed (since df_j[i] * df_j[i - 1] == 0 so neither if-statement holds).

In this implementation, the initial step_size is set to sigma0, the default minimum step size is set as 1e-3 * min(sigma0), and no default maximum step size is set. Minimum and maximum step sizes can be changed with set_min_step_size() and set_max_step_size() or through the hyper-parameter interface.

If boundaries are provided, an extra step is added at the end of the algorithm that reduces the step size where boundary constraints are not met. For RectangularBoundaries this works on a per-dimension basis:

while p_j[i] < lower or p_j[i] >= upper:
    step_size_j[i] *= 0.5
    p_j[i] = p_j[i] - sign(df_j[i]) * step_size_j[i]

For general boundaries a more complex heuristic is used: First, the step size in all dimensions is reduced until the boundary constraints are met. Next, this reduction is undone for each dimension in turn: if the constraint is still met without the reduction the step size in this dimension is left unchanged.

The numbers 0.5 and 1.2 shown in the (main and boundary) pseudo-code are technically hyper-parameters, but are fixed in this implementation.

References

[1] (1,2)

Empirical Evaluation of the Improved Rprop Learning Algorithms. Igel and Hüsken, 2003, Neurocomputing https://doi.org/10.1016/S0925-2312(01)00700-7

[2]

A direct adaptive method for faster backpropagation learning: the RPROP algorithm. Riedmiller and Braun, 1993. https://doi.org/10.1109/ICNN.1993.298623

ask()

See Optimiser.ask().

f_best()

See Optimiser.f_best().

f_guessed()

See Optimiser.f_guessed().

fbest()

Deprecated alias of f_best().

max_step_size()

Returns the maximum step size (or None if not set).

min_step_size()

Returns the minimum step size (or None if not set).

n_hyper_parameters()

See pints.TunableMethod.n_hyper_parameters().

name()

See Optimiser.name().

needs_sensitivities()

See Optimiser.needs_sensitivities().

running()

See Optimiser.running().

set_hyper_parameters(x)

See pints.TunableMethod.set_hyper_parameters().

The hyper-parameter vector is [min_step_size, max_step_size].

set_max_step_size(step_size)

Sets the maximum step size (use None to let step sizes grow indefinitely).

set_min_step_size(step_size)

Sets the minimum step size (use None to let step sizes shrink indefinitely).

stop()

Checks if this method has run into trouble and should terminate. Returns False if everything’s fine, or a short message (e.g. “Ill-conditioned matrix.”) if the method should terminate.

tell(reply)

See Optimiser.tell().

x_best()

See Optimiser.x_best().

x_guessed()

See Optimiser.x_guessed().

xbest()

Deprecated alias of x_best().