then for a small positive non-zero value $\epsilon$: This version of the law is called the weak law of large numbers. We need it to start using the tools form probability theory to investigate our generalization probability, and it’s a very reasonable assumption because: So we can build upon that assumption with no fear. 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. 81–88. Before we continue I’d like to remind you that if k is a break point, then for any k points in our data, it is impossible to get all possible combinations (2^k). I'm writing a book, check it out here. Up until this point, all our analysis was for the case of binary classification. The fact that $d_\mathrm{vc}$ is distribution-free comes with a price: by not exploiting the structure and the distribution of the data samples, the bound tends to get loose. This is theoretical motivation behind Support Vector Machines (SVMs) which attempts to classify data using the maximum margin hyperplane. Challenges of Generalization in Machine Learning. Learning theory: generalization and VC dimension Yifeng Tao School of Computer Science Carnegie Mellon University Slides adapted from Eric Xing Yifeng Tao Carnegie Mellon University 1 Introduction to Machine Learning This means that in any similar matrix, we have a group of unique rows that get their “uniqueness” via a certain xN (x3 in the example). Here, we use insights from machine learning to demonstrate that exemplar models can actually generalize very well. Can we do any better? Furthermore, this bound can be described in term of a quantity ($d_\mathrm{vc}$), that solely depends on the hypothesis space and not on the distribution of the data points! That machine learning algorithms all seek to learn a mapping from inputs to outputs. Because learning algorithms are evaluated on finite samples, the evaluation of a learning algorithm may be sensitive to sampling error. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The basic idea of the union bound is that it bounds the probability by the worst case possible, which is when all the events under union are mutually independent. This is a problem that faces any theoretical analysis of a real world phenomenon; because usually we can’t really capture all the messiness in mathematical terms, and even if we’re able to; we usually don’t have the tools to get any results from such a messy mathematical model. Take the following simple NLP problem: Say you want to predict a word in a sequence given its preceding words. The supremum in the inequality guarantees that there’s a very little chance that the biggest generalization gap possible is greater than $\epsilon$; this is a strong claim and if we omit a single hypothesis out of $\mathcal{H}$, we might miss that “biggest generalization gap possible” and lose that strength, and that’s something we cannot afford to lose. In machine learning, generalization usually refers to the ability of an algorithm to be effective across a range of inputs and applications. References and Additional Readings. We are interested in both experimental and theoretical approaches that advance our understanding. The most important theoretical result in machine learning. For simplicity, we’ll focus now on the case of binary classification, in which $\mathcal{Y}=\{-1, +1\}$. That means, a complex ML model will adapt to subtle patterns in your training set, which in some cases could be noise. [Courtesy of Yann LeCunn]. In the fourth line we extracted the N choose 0 (=1) from the sum. Assignments (only accessible for … Key topics include: generalization, over-parameterization, robustness, dynamics of SGD, and relations to kernel methods. Hence, if we are trying dichotomies instead of hypotheses, and are unlucky to get a false positive, this false positive includes all the false positives we could’ve fallen into if we tried every hypothesis that belongs to this dichotomy. But can any hypothesis space shatter any dataset of any size? And since B(N-1, k-1) is the maximum number of combinations for N-1 who have a break point of k-1, we conclude that β < B(N-1, k-1) : (2). MIT press, 2012. In this post, you will discover […] Exemplar theories of categorization depend on similarity for explaining subjects’ ability to generalize to new stimuli. During the last decade, deep learning has drawn increasing attention both in machine learning and statistics because of its superb empirical performance in various fields of application, including speech and image recognition, natural language processing, social network filtering, bioinformatics, drug design and board games (e.g. Using the union bound inequality, we get: We exactly know the bound on the probability under the summation from our analysis using the Heoffding’s inequality, so we end up with: Where $|\mathcal{H}|$ is the size of the hypothesis space. • How can a neural network, after sufficient training, correctly predict the outcome of a previously unseen input? Which will give us: α + β < B(N-1,k) : (2). 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? The formulation of the weak law lends itself naturally to use with our generalization probability. Abu-Mostafa, Y. S., Magdon-Ismail, M., & Lin, H. (2012). Based on this theory, a new regularization method in deep learning is derived and shown to outperform previous methods in CIFAR-10, CIFAR-100, and SVHN. In this post I try to list some of the "puzzles" of modern machine learning, from a theoretical perspective. open source implementation of a large number of machine learning algorithms; We offer theoretical and practical advice in machine learning and computational intelligence to other research groups and industrial partners. The Theory of Generalization. How can a neural network, after sufficient training, correctly predict the output of a previously unseen input? Shalev-Shwartz, Shai, and Shai Ben-David. This inequality basically says the generalization error can be decomposed into two parts: the empirical training error, and the complexity of the learning model. If we add the last row, the highlighted cells give us all 4 combinations of the points x2 & x3, which is not allowed by the break point. On the other hand, the strong version says that with very large sample size, the sample mean is almost surely equal to the true mean. The bound on the growth function provided by sauer’s lemma is indeed much better than the exponential one we already have, it’s actually polynomial! The result is a sub-table where all rows are different since the rows in S1 are inherently different without xN, and the rows in S2+ are different from the ones in S1 because if that wasn’t the case, the duplicate version of that row in S1 would get its “uniqueness” from xN and forcing it to leave S1 and join S2 (just like we’ve seen in the simple case example). By only choosing the distinct effective hypotheses on the dataset $S$, we restrict the hypothesis space $\mathcal{H}$ to a smaller subspace that depends on the dataset $\mathcal{H}_{|S}$. 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