We’ve established in the previous article that there is still hope of generalization even in hypotheses’ spaces that are infinite in dimension. The formulation of the generalization inequality reveals a main reason why we need to consider all the hypothesis in $\mathcal{H}$. The answer is very simple; we consider a hypothesis to be a new effective one if it produces new labels/values on the dataset samples, then the maximum number of distinct hypothesis (a.k.a the maximum number of the restricted space) is the maximum number of distinct labels/values the dataset points can take. Assumptions are common practice in theoretical work. This explains why the memorization hypothesis form last time, which theoretically has $|\mathcal{H}| = \infty$, fails miserably as a solution to the learning problem despite having $R_\text{emp} = 0$; because for the memorization hypothesis $h_\text{mem}$: But wait a second! B(N,k) corresponds to the number of rows in the following table: Let α be the count of rows in the S1 group. (2006). Looking at the above plot of binary classification problem, it’s clear that this rainbow of hypothesis produces the same classification on the data points, so all of them have the same empirical risk. B(N,k) = α + 2β ≤ B(N-1, k) + B(N-1, k-1) (*). 1Introduction Neural network learning has become a key machine learning approach and has achieved remarkable success in a wide range of real-world domains, such as computer vision, speech recognition, and game playing [25, 26, 30, 41]. Learning and Generalization provides a formal mathematical theory for addressing intuitive questions such as: • How does a machine learn a new concept on the basis of examples? This works because we assume that this test set is drawn i.i.d. Now that we’ve established that we do need to consider every single hypothesis in $\mathcal{H}$, we can ask ourselves: are the events of each hypothesis having a big generalization gap are likely to be independent? Up until this point, all our analysis was for the case of binary classification. ! In this post, you will discover […] We’ve seen before that the union bound was added to Hoeffding’s inequality to take into account the multiple testing problem that occurs when we go through a hypotheses’ set searching for the best hypothesis. We build models on existing data, … Generalization is the concept that humans and other animals use past learning in present situations of learning if the conditions in the situations are regarded as similar. Foundations of machine learning. The following animation shows how many ways a linear classifier in 2D can label 3 points (on the left) and 4 points (on the right). This paper provides theoretical insights into why and how deep learning can generalize well, despite its large capacity, complexity, possible algorithmic instability, nonrobustness, and sharp minima, responding to an open question in the literature. Intriguingly our theory also reveals the existence of a learning algorithm that proveably out-performs neural network training through gradient descent. That machine learning algorithms all seek to learn a mapping from inputs to outputs. The same argument can be made for many different regions in the $\mathcal{X \times Y}$ space with different degrees of certainty as in the following figure. The ultimate goal of machine learning is to find statistical patterns in a training set that generalize to data outside the training set. Abu-Mostafa, Y. S., Magdon-Ismail, M., & Lin, H. (2012). This is not convenient since we’ve built our argument on dichotomies and not hypotheses. But can any hypothesis space shatter any dataset of any size? 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