16.3 Shell Bound Calculation

16.3. Shell Bound Calculation

The program ’bound’ can calculate a shell bound or a sampling shell bound. To do these calculations, two extra parameters must be specified:

error_log_count <int> <float>: <float> should be the log (base e) of the number of hypotheses with <int> empirical error.
sample_space_log_size <float>: <float> should be the size of space of hypotheses uniformly sampled from if the evaluation is inexact (and not passed if the evaluation is exact)

Suppose wanted to use the shell bound and knew that $e^{5}$ hypotheses had $25$ training error and $e^{30}$ had $50$ training errors. Then, we might apply the bound as:

12:07AM z-28: cat shell_error
train_examples 100
train_errors 3
error_log_count 3 1
error_log_count 25 5
error_log_count 50 30
delta 0.3
12:07AM z-29: bound shell_error
Applying varying approximation tail bound
Applying shell bound
delta 0.3
lower_delta 0.5
train_examples 100
error_log_count 3 1
error_log_count 25 5
error_log_count 50 30
approximation automatic
true_error = 0.03 0.144696196541 0.0266506755725

Now, suppose that we wanted to use the sampling shell bound and sampled $e^{5} + e^{10}$ times observing $e^{5}$ hypotheses with training error $25$ and $e^{10}$ with training error $50$ . Then, we might apply ’bound’ as follows:

12:23AM z-36: cat sampling_shell_error
train_examples 100
train_errors 3
error_log_count 3 1
error_log_count 25 5
error_log_count 50 10
sample_space_log_size 30
delta 0.3
12:23AM z-37: bound sampling_shell_error
Applying varying approximation tail bound
Applying shell bound
delta 0.3
lower_delta 0.5
train_examples 100
sample_space_log_size 30
error_log_count 3 1
error_log_count 25 5
error_log_count 50 10
approximation automatic
true_error = 0.03 0.405203092843 0.0266506755725

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