Looseness and Limitations of Bounds
Generalization bounds are designed to provide guarantees on how well a hypothesis learned from training data will perform on unseen data. However, these bounds are often much looser than the actual gap between training and test error observed in practice. There are several key sources of this looseness. First, most generalization bounds rely on worst-case analysis: they are constructed to hold for any possible dataset, not just the one you happen to observe. This means the bound must account for even the most unlikely data configurations, inflating the bound far above what usually occurs. Second, union bounds are often used to extend guarantees from a single hypothesis to an entire hypothesis class. This technique, while mathematically convenient, can drastically increase the bound, especially as the size or complexity of the hypothesis class grows. Third, many bounds involve unknown constants or quantities that are difficult to estimate tightly, such as the VC dimension or Rademacher complexity, or even hidden multiplicative factors that arise from inequalities used in the derivation. These factors can make the bound much larger than the actual generalization error.
Suppose you are working with a simple class of threshold classifiers on the real line. A classical VC-based generalization bound might state that, with high probability, the true error of your chosen classifier is at most the empirical error plus a term like nlogn, where n is the number of samples. In practice, you might observe that the empirical error is 0% and the test error is 2%, while the bound allows for a true error as high as 20%. This large gap demonstrates how the bound, though mathematically correct, can be very loose compared to what you actually see.
Generalization bounds must work for all possible datasets, including rare or pathological ones. In practice, most datasets are not adversarial, so the actual generalization gap is much smaller than the worst-case scenario.
The use of union bounds and other inequalities in the derivation of generalization bounds introduces extra slack. These steps are mathematically necessary to guarantee uniformity over the hypothesis class, but they often lead to significant overestimation.
Many bounds involve constants or complexity measures that are either unknown or estimated pessimistically. This further increases the gap between the bound and the observed error.
12345678910111213141516171819202122232425262728293031import numpy as np import matplotlib.pyplot as plt # Simulate a toy hypothesis class: threshold classifiers np.random.seed(42) n_samples = 100 X = np.random.uniform(-1, 1, n_samples) y = (X > 0).astype(int) # Empirical risk minimization: pick threshold minimizing training error thresholds = np.linspace(-1, 1, 50) empirical_errors = [np.mean((X > t) != y) for t in thresholds] best_t = thresholds[np.argmin(empirical_errors)] empirical_risk = np.min(empirical_errors) # Simulate a large test set to estimate true error X_test = np.random.uniform(-1, 1, 10000) y_test = (X_test > 0).astype(int) test_error = np.mean((X_test > best_t) != y_test) # Compute a VC-based generalization bound (for illustration) from math import log, sqrt vc_dim = 2 # threshold classifiers on R delta = 0.05 bound = empirical_risk + sqrt((8 * log(2 * n_samples) + 8 * vc_dim * log(2 * np.e * n_samples / vc_dim) + log(4/delta)) / n_samples) plt.bar(['Empirical Risk', 'Test Error', 'VC Bound'], [empirical_risk, test_error, bound], color=['C0', 'C1', 'C2']) plt.ylabel('Error') plt.title('Empirical Risk, Test Error, and Generalization Bound') plt.ylim(0, max(bound, test_error) + 0.1) plt.show()
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Can you explain why the VC bound is so much higher than the test error in this example?
What are some ways to obtain tighter generalization bounds in practice?
Could you walk me through the code and explain each step?
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Generalization bounds are designed to provide guarantees on how well a hypothesis learned from training data will perform on unseen data. However, these bounds are often much looser than the actual gap between training and test error observed in practice. There are several key sources of this looseness. First, most generalization bounds rely on worst-case analysis: they are constructed to hold for any possible dataset, not just the one you happen to observe. This means the bound must account for even the most unlikely data configurations, inflating the bound far above what usually occurs. Second, union bounds are often used to extend guarantees from a single hypothesis to an entire hypothesis class. This technique, while mathematically convenient, can drastically increase the bound, especially as the size or complexity of the hypothesis class grows. Third, many bounds involve unknown constants or quantities that are difficult to estimate tightly, such as the VC dimension or Rademacher complexity, or even hidden multiplicative factors that arise from inequalities used in the derivation. These factors can make the bound much larger than the actual generalization error.
Suppose you are working with a simple class of threshold classifiers on the real line. A classical VC-based generalization bound might state that, with high probability, the true error of your chosen classifier is at most the empirical error plus a term like nlogn, where n is the number of samples. In practice, you might observe that the empirical error is 0% and the test error is 2%, while the bound allows for a true error as high as 20%. This large gap demonstrates how the bound, though mathematically correct, can be very loose compared to what you actually see.
Generalization bounds must work for all possible datasets, including rare or pathological ones. In practice, most datasets are not adversarial, so the actual generalization gap is much smaller than the worst-case scenario.
The use of union bounds and other inequalities in the derivation of generalization bounds introduces extra slack. These steps are mathematically necessary to guarantee uniformity over the hypothesis class, but they often lead to significant overestimation.
Many bounds involve constants or complexity measures that are either unknown or estimated pessimistically. This further increases the gap between the bound and the observed error.
12345678910111213141516171819202122232425262728293031import numpy as np import matplotlib.pyplot as plt # Simulate a toy hypothesis class: threshold classifiers np.random.seed(42) n_samples = 100 X = np.random.uniform(-1, 1, n_samples) y = (X > 0).astype(int) # Empirical risk minimization: pick threshold minimizing training error thresholds = np.linspace(-1, 1, 50) empirical_errors = [np.mean((X > t) != y) for t in thresholds] best_t = thresholds[np.argmin(empirical_errors)] empirical_risk = np.min(empirical_errors) # Simulate a large test set to estimate true error X_test = np.random.uniform(-1, 1, 10000) y_test = (X_test > 0).astype(int) test_error = np.mean((X_test > best_t) != y_test) # Compute a VC-based generalization bound (for illustration) from math import log, sqrt vc_dim = 2 # threshold classifiers on R delta = 0.05 bound = empirical_risk + sqrt((8 * log(2 * n_samples) + 8 * vc_dim * log(2 * np.e * n_samples / vc_dim) + log(4/delta)) / n_samples) plt.bar(['Empirical Risk', 'Test Error', 'VC Bound'], [empirical_risk, test_error, bound], color=['C0', 'C1', 'C2']) plt.ylabel('Error') plt.title('Empirical Risk, Test Error, and Generalization Bound') plt.ylim(0, max(bound, test_error) + 0.1) plt.show()
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