The test cases for the Pegasos Kernel SVM Implementation problem use inconsistent alpha conventions.
The problem statement describes the deterministic Pegasos update using the decision function:
f(x_i) = sum(alpha[j] * y[j] * K(x_j, x_i)) + b
Under this convention, alpha[j] is an unsigned coefficient, and the label sign is applied separately through y[j].
So when the margin is violated, the update should be:
alpha[i] = (1 - eta * lambda_val) * alpha[i] + eta
bias = bias + eta * y_i
and when the margin is satisfied:
alpha[i] = (1 - eta * lambda_val) * alpha[i]
However, one of the hidden/torch tests expects signed alpha values:
[100.0, 0.0, -100.0, -100.0]
This implies the implementation should update alpha using:
alpha[i] = (1 - eta * lambda_val) * alpha[i] + eta * y_i
...which conflicts with the stated decision function, which already multiplies by y[j]. Using both signed alphas and alpha[j] * y[j] double-counts the label sign.
Reproduction:
Using the prompt’s stated algorithm/convention:
alphas, b = pegasos_kernel_svm(
np.array([[1, 2], [2, 3], [3, 1], [4, 1]]),
np.array([1, 1, -1, -1]),
kernel='linear',
lambda_val=0.01,
iterations=100
)
print(([round(a, 4) for a in alphas], round(b, 4)))Expected output according to the problem statement:
([2.0, 2.0, 6.0, 1.0], 36.2027)
But the hidden/torch test expects:
([100.0, 0.0, -100.0, -100.0], -937.4755)
These two outputs correspond to different interpretations of what alpha represents.
Suggested fix:
The tests should use the unsigned-alpha convention in the problem statement:
decision_value += alpha[j] * y[j] * K(x_j, x_i)
with update:
alpha[i] = (1 - eta * lambda_val) * alpha[i] + eta
and with regularization decay in the non-violating case:
alpha[i] = (1 - eta * lambda_val) * alpha[i]
The corrected test cases could be:
[
{
"test": "alphas, b = pegasos_kernel_svm(np.array([[1, 2], [2, 3], [3, 1], [4, 1]]), np.array([1, 1, -1, -1]), kernel='linear', lambda_val=0.01, iterations=100)\nprint(([round(a, 4) for a in alphas], round(b, 4)))",
"expected_output": "([2.0, 2.0, 6.0, 1.0], 36.2027)"
},
{
"test": "alphas, b = pegasos_kernel_svm(np.array([[1, 2], [2, 3], [3, 1], [4, 1]]), np.array([1, 1, -1, -1]), kernel='rbf', lambda_val=0.01, iterations=100, sigma=0.5)\nprint(([round(a, 4) for a in alphas], round(b, 4)))",
"expected_output": "([1.0, 1.0, 1.0, 1.0], 0.0)"
}
]
The test cases for the Pegasos Kernel SVM Implementation problem use inconsistent alpha conventions.
The problem statement describes the deterministic Pegasos update using the decision function:
Under this convention,
alpha[j]is an unsigned coefficient, and the label sign is applied separately throughy[j].So when the margin is violated, the update should be:
and when the margin is satisfied:
However, one of the hidden/torch tests expects signed alpha values:
This implies the implementation should update alpha using:
...which conflicts with the stated decision function, which already multiplies by
y[j]. Using both signed alphas andalpha[j] * y[j]double-counts the label sign.Reproduction:
Using the prompt’s stated algorithm/convention:
Expected output according to the problem statement:
But the hidden/torch test expects:
These two outputs correspond to different interpretations of what
alpharepresents.Suggested fix:
The tests should use the unsigned-alpha convention in the problem statement:
with update:
and with regularization decay in the non-violating case:
The corrected test cases could be:
[ { "test": "alphas, b = pegasos_kernel_svm(np.array([[1, 2], [2, 3], [3, 1], [4, 1]]), np.array([1, 1, -1, -1]), kernel='linear', lambda_val=0.01, iterations=100)\nprint(([round(a, 4) for a in alphas], round(b, 4)))", "expected_output": "([2.0, 2.0, 6.0, 1.0], 36.2027)" }, { "test": "alphas, b = pegasos_kernel_svm(np.array([[1, 2], [2, 3], [3, 1], [4, 1]]), np.array([1, 1, -1, -1]), kernel='rbf', lambda_val=0.01, iterations=100, sigma=0.5)\nprint(([round(a, 4) for a in alphas], round(b, 4)))", "expected_output": "([1.0, 1.0, 1.0, 1.0], 0.0)" } ]