Single Image

find_adversarial_example(
    nn,
    input,
    target_selection,
    optimizer,
    main_solve_options;
    invert_target_selection,
    pp,
    norm_order,
    adversarial_example_objective,
    tightening_algorithm,
    tightening_options,
    solve_if_predicted_in_targeted
)

Perturbs input such that the network nn classifies the perturbed image in one of the categories identified by the indexes in target_selection.

IMPORTANT:

1. target_selection can include the correct label for input.
2. It is possible (particularly with the closest objective) to see 'ties' – that is, the perturbed input produces an output with two logits (one corresponding to a target category, and one corresponding to a non-target category) taking on the same maximal value. See the formal definition below for more; in particular, note that '≥' sign.

optimizer is used to build and solve the MIP problem.

The output dictionary has keys :Model, :PerturbationFamily, :TargetIndexes, :SolveStatus, :Perturbation, :PerturbedInput, :Output. See the tutorial on what individual dictionary entries correspond to.

Formal Definition: If there are a total of n categories, the (perturbed) output vector y=d[:Output]=d[:PerturbedInput] |> nn has length n. If :SolveStatus is feasible, we guarantee that y[j] - y[i] ≥ 0 for some j ∈ target_selection and for all i ∉ target_selection.

Keyword Arguments:

• invert_target_selection: Defaults to false. If true, sets target_selection to be its complement.
• pp: Defaults to UnrestrictedPerturbationFamily(). Determines the search space for adversarial examples.
• norm_order: Defaults to 1. Determines the distance norm used to determine the distance from the perturbed image to the original. Allowed options are 1 and Inf, and 2 if the optimizer can solve MIQPs.
• adversarial_example_objective: Defaults to closest. Allowed options are closest or worst.
  • closest finds the closest adversarial example, as measured by the norm_order norm.
  • worst finds the adversarial example with the largest gap between max(y[j) for j ∈ target_selection and max(y[i]) for all i ∉ target_selection.
• tightening_algorithm: Defaults to mip. Determines how we determine the upper and lower bounds on input to each nonlinear unit. Allowed options are interval_arithmetic, lp, mip.
  • interval_arithmetic looks at the bounds on the output to the previous layer.
  • lp solves an lp corresponding to the mip formulation, but with any integer constraints relaxed.
  • mip solves the full mip formulation.
• tightening_options: Solver-specific options passed to optimizer when used to determine upper and lower bounds for input to nonlinear units. Note that these are only used if the tightening_algorithm is lp or mip (no solver is used when interval_arithmetic is used to compute the bounds). Defaults for Gurobi and HiGHS to a time limit of 20s per solve, with output suppressed.
• solve_if_predicted_in_targeted: Defaults to true. The prediction that nn makes for the unperturbed input can be determined efficiently. If the predicted index is one of the indexes in target_selection, we can skip the relatively costly process of building the model for the MIP problem since we already have an "adversarial example" –- namely, the input itself. We continue build the model and solve the (trivial) MIP problem if and only if solve_if_predicted_in_targeted is true.

frac_correct(nn, dataset, num_samples)

Returns the fraction of items the neural network correctly classifies of the first num_samples of the provided dataset. If there are fewer than num_samples items, we use all of the available samples.

Named Arguments:

• nn::NeuralNet: The parameters of the neural network.
• dataset::LabelledDataset:
• num_samples::Integer: Number of samples to use.