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Accuracy of my regularized model is higher for training set than for validation set.

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The situation improves when regularization coeefficient is reduced: enter image description here

What does this really imply?

From my understanding, this seems to suggest that regularization is actually resulting in the model overfitting training set, which is the opposite of the intended outcome

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It implies that your regularization effects are too much, and prevent the model from learning from data. Also, at such a low accuracy (~10%), we can't really talk about overfitting.

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