Mohri, Mehryar, Afshin Rostamizadeh, and Ameet Talwalkar. So this model will not be a good predictor for new instances (not in the training set). 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. Most people when they were kids were fascinated by magicians and magic tricks, they were captivated by what appeared to be reality-defying and riddled with curiosity about how it was being done to the point they wished they become professional magicians as adults. It’s weak because it guarantees that as the sample size goes larger, the sample and true means will likely be very close to each other by a non-zero distance no greater than epsilon. It’s called the growth function because it’s value for a single hypothesis space $\mathcal{H}$ (aka the size of the restricted subspace $\mathcal{H_{|S}}$) grows as the size of the dataset grows. Shalev-Shwartz, Shai, and Shai Ben-David. However, the success of linear hypothesis can now be explained by the fact that they have a finite $d_\mathrm{vc} = n + 1$ in $\mathbb{R}^n$. Our theoretical result was able to account for some phenomena (the memorization hypothesis, and any finite hypothesis space) but not for others (the linear hypothesis, or other infinite hypothesis spaces that empirically work). I'm writing a book, check it out here. 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 $\mathcal{H}_{|S}$. cats vs. dogs), or predict future values of a time series (e.g. The question now is what is the maximum size of a restricted hypothesis space? Machine Learning Computational Learning Theory: The Theory of Generalization Slides based on material from Dan Roth, AvrimBlum, Tom Mitchell and others 1. By recalling that the empirical risk is actually the sample mean of the errors and the risk is the true mean, for a single hypothesis $h$ we can say that: Well, that’s a progress, A pretty small one, but still a progress! Consider for example the case of linear binary classifiers in a very higher n-dimensional feature space, using the distribution-free $d_\mathrm{vc} = n + 1$ means that the bound on the generalization error would be poor unless the size of the dataset $N$ is also very large to balance the effect of the large $d_\mathrm{vc}$. In this post, you will discover […] How can a neural network, after sufficient training, correctly predict the output of a previously unseen input? Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. If that’s not the case and the samples are dependent, then the dataset will suffer from a bias towards a specific direction in the distribution, and hence will fail to reflect the underlying distribution correctly. Let’s investigate that with the binary classification case and the $\mathcal{H}$ of linear classifiers $\mathrm{sign}(wx + b)$. Learning from data: a short course. Second, we need to verify if we’re allowed to replace the number of possible hypotheses M in the generalization bound with the growth function. This text treats the problem of machine learning in conjunction with the theory of empirical processes, the latter being a well-established branch of probability theory. We close this first part with the fact that if, for any hypotheses’ space H, a break point k exists, we have: This is true, because B(N,k) is the maximum number of possible combinations of N points independently of how many they are and also of the hypotheses’ space we’re studying (the growth function depends on the space). Generalization. This is the good old curse of dimensionality we all know and endure. In this tutorial, we will review the generalization theory for traditional machine learning methods. In this part we’ll start investigating that probability at depth and see if it indeed can be small, but before starting you should note that I skipped a lot of the mathematical proofs here. This is an instance of wildly known fact about probability that if we retried an experiment for a sufficiency large amount of times, the average outcome of these experiments (or, more formally, the sample mean) will be very close to the true mean of the underlying distribution. Outline • Learning Feasibility • VC Dimension • Theory of Generalization • Bayesian Concept Learning • Beta-Binomial Model ... • In Machine Learning we wish to learn an unknown target function f. A natural question arises: This can be expressed formally by stating that: Where $\bigcup$ denotes the union of the events, which also corresponds to the logical OR operator. Exemplar theories of categorization depend on similarity for explaining subjects’ ability to generalize to new stimuli. Challenges of Generalization in Machine Learning. Blaine Bateman. Finally we use the combinatorial Lemme and add 1 back to the sum. Last time we concluded by noticing that minimizing the empirical risk (or the training error) is not in itself a solution to the learning problem, it could only be considered a solution if we can guarantee that the difference between the training error and the generalization error (which is also called the generalization gap) is small enough. 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). The world can be a very messy place! We’ll focus more on the intuition of the theory with a sufficient amount of math to retain the rigor. For simplicity, we’ll focus now on the case of binary classification, in which $\mathcal{Y}=\{-1, +1\}$. 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. (2006). The latter might lead to a problem called overfitting whereby we memorize data instead of learning from it. Learning Theory & Theory of Generalization CS 446/546 . Because learning algorithms are evaluated on finite samples, the evaluation of a learning algorithm may be sensitive to sampling error. Machine Learning (Chinese Edition). I'm quite familiar with loss functions in machine learning, but am struggling to connect them to loss functions in statistical decision theory [1]. And the idea is, since both Ein and E’in are approximations of Eout, Ein will approximate E’in. In that case, $d_\mathrm{vc}$ would be a measure of the complexity or richness of the hypothesis space. Intriguingly our theory also reveals the existence of a learning algorithm that proveably out-performs neural network training through gradient descent. SGD theory of escaping saddle points. This is to make the post easier to read and to focus all the effort on the conceptual understanding of the subject. cats vs. dogs), or predict future values of a time series (e.g. We are interested in both experimental and theoretical approaches that advance our understanding. We build models on existing data, … The new resulting inequality is called the Vapnik-Chervonenkis (VC) Inequality and is as follows: http://www.cs.rpi.edu/~magdon/courses/LFD-Slides/SlidesLect06.pdf, Another Unreasonable Deep Dive into Project Euler Problem 1. k=1 and N=1, no one point can have all possible combinations, this means this point is either “+” or “-”. “Reconciling modern machine learning and the bias-variance trade-off.” arXiv preprint arXiv:1812.11118(2018). Finally, since we are not going to use Eout, we will be able to find a bound that has the growth function in it and that is legit to use. Browse other questions tagged machine-learning deep-neural-networks overfitting learning-theory generalization or ask your own question. In order to measure the accuracy of our model, we hold out a part of the training set to evaluate the model on after training, and we consider the model’s accuracy on this left out portion as an estimate for the generalization error. 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 means that: Our purpose of the following steps is to find recursive bound of B(N,k) (a bound defined by B on different values of N & k). In supervised learning applications in machine learning and statistical learning theory, generalization error is a measure of how accurately an algorithm is able to predict outcome values for previously unseen data. For this smaller version of the original table, if we suppose that k is it’s break point, then we can find k-1 points that exist in all possible combinations. ICML ’06: Proceedings of the 23rd International Conference on Machine learning, Pittsburgh, pp. If you noticed, all our analysis up till now was focusing on a single hypothesis $h$. Deep Learning is currently being used for a variety of different applications. In our case, for the bound to be tight and reasonable, we need the following to be true: For every two hypothesis $h_1, h_2 \in \mathcal{H}$ the two events $|R(h_1) - R_\text{emp}(h_1)| > \epsilon$ and $|R(h_2) - R_\text{emp}(h_2)| > \epsilon$ are likely to be independent. This may seem like a trivial question; as the answer is simply that because the learning algorithm can search the entire hypothesis space looking for its optimal solution. So the union bound and the independence assumption seem like the best approximation we can make,but it highly overestimates the probability and makes the bound very loose, and very pessimistic! 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. Exemplar theories of categorization depend on similarity for explaining subjects’ ability to generalize to new stimuli. As a result, measurements of prediction error on the current data … Follow. Therefore, we conclude that k-1 is in fact a break point for S2+. This is theoretical motivation behind Support Vector Machines (SVMs) which attempts to classify data using the maximum margin hyperplane. This works because we assume that this test set is drawn i.i.d. References and Additional Readings. samples of a random variable $X$ distributed by $P$. We’re not gonna go over the proof here, but using that ghost dataset one can actually prove that: where $R_\text{emp}’(h)$ is the empirical risk of hypothesis $h$ on the ghost dataset. Get this from a library! But it's clearly a bad idea: It over ts to the training data and doesn't generalize to unseen examples. Finally, for transfer learning , our theory reveals that knowledge transfer depends sensitively, but computably, on … For example, For data points that are linearly separable, contained in a ball of radius $R$, with a margin $\rho$ between the closest points in the two classes, one can prove that for a hyperplane classifier: It follows that the larger the margin, the lower the $d_\mathrm{vc}$ of the hypothesis. On the other hand, there’s a group of rows that are unique independently of xN, and they either occur with xN being “-” or “+” and not both. Assignments (only accessible for … It’s a clear and concise mathematical statement that the learning problem is solvable, and that for infinite hypotheses spaces there is a finite bound on the their generalization error! Machine learning (ML) is the study of computer algorithms that improve automatically through experience. We will call that group of rows S2 in what follows. Here, we use insights from machine learning to demonstrate that exemplar models can actually generalize very well. To get around this problem, instead of computing just one in sample error Ein, we apply our hypothesis on two different data sets of the same size, and get Ein and E’in. In case you wish to get your hands dirty with proofs, you can find all of them in the additional readings, or on the Internet of course! This is a set of inequalities that quantifies how much random variables (or function of them) deviate from their expected values (or, also, functions of them). We will also point out where deep learning method differ. Harvard Machine Learning Theory. To understand the concept of generalisation in ML, you need to understand the concept of “overfitting”. We can naturally apply this inequality to our generalization probability, assuming that our errors are bounded between 0 and 1 (which is a reasonable assumption, as we can get that using a 0/1 loss function or by squashing any other loss between 0 and 1) and get for a single hypothesis $h$: This means that the probability of the difference between the training and the generalization errors exceeding $\epsilon$ exponentially decays as the dataset size goes larger. Generalization. Maurer A Unsupervised slow subspace-learning from stationary processes Proceedings of the 17th international conference on Algorithmic Learning Theory, (363-377) Zou B, Li L and Xu J The generalization performance of learning machine with NA dependent sequence Proceedings of the First international conference on Rough Sets and Knowledge Technology, (568-573) Our theory reveals that deep networks progressively learn the most important task structure first, so that generalization error at the early stopping time primarily depends on task structure and is independent of network size. Foundations of machine learning. However, in the previous inequality, the generalization bound often goes to infinity, not only because most of hypotheses’ sets are infinite (M->∞), but also because the union bound assumes that the probabilities in Hoeffding’s inequality related to the different hypotheses do not overlap. 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. We’ve established in the previous article that there is still hope of generalization even in hypotheses’ spaces that are infinite in dimension. Assumptions are common practice in theoretical work. producing the same labels/values on the data points), we can safely choose one of them as a representative of the whole group, we’ll call that an effective hypothesis, and discard all the others. It’s more likely for a dataset used for inferring about an underlying probability distribution to be all sampled for that same distribution. 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. [23] Belkin, Mikhail, et al. In the theory of statistical machine learning, a generalization bound – or, more precisely, a generalization error bound – is a statement about the predictive performance of a learning algorithm or class of algorithms. Learned generalization or secondary generalization is an aspect of learning theory.In learning studies it can be shown that subjects, both animal and human will respond in the same way to different stimuli if they have similar properties established by a process of conditioning.This underpins the process by which subjects are able to perform newly acquired behaviours in new settings. Machine learning models based on deep neural networks have attained state-of-the-art performance across a dizzying array of tasks including vision (Cubuk et al., 2019), speech recognition (Park et al., 2019), machine translation (Bahdanau et al., 2014), chemical property prediction (Gilmer et al., 2017), diagnosing medical conditions (Raghu et al., 2019), and playing games (Silver et al., 2018). This was also proved by Vapnik and Chervonenkis. This group corresponds to S1 in the following section. Featured on Meta MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC… But the learning problem doesn’t know that single hypothesis beforehand, it needs to pick one out of an entire hypothesis space $\mathcal{H}$, so we need a generalization bound that reflects the challenge of choosing the right hypothesis. The most important theoretical result in machine learning. Now, in light of these results, is there’s any hope for the memorization hypothesis? This means that there’s still something missing from our theoretical model, and it’s time for us to revise our steps. We also discuss approaches to provide non-vacuous generalization guarantees for deep learning. Actually, no linear classifier in 2D can shatter any set of 4 points, not just that set; because there will always be two labellings that cannot be produced by a linear classifier which is depicted in the following figure. A good starting point is from the source of the problem itself, which is the infinity in $|\mathcal{H}|$. The cool thing about Rademacher’s complexity is that it’s flexible enough to be adapted to any learning problem, and it yields very similar generalization bounds to the other methods mentioned. From the decision boundary plot (on the right), it’s clear why no linear classifier can produce such labellings; as no linear classifier can divide the space in this way. Generalization refers to your model's ability to adapt properly to new, previously unseen data, drawn from the same distribution as the one used to … The most important theoretical result in machine learning. A Theory of Learning and Generalization provides a formal mathematical theory for addressing intuitive questions of the type: How does a machine learn a new concept on the basis of examples? Alpha go, Alpha zero). Statistical Machine Learning (Summer term 2020) Quick links (publically available): youtube channel for the videos Slides Course material Slides: Latest version, updated 2020-08-19: pdf Videos: The videos of the lecture can all be found on youtube. On Bayesian bounds. 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. With that, and by combining inequalities (1) and (2), the Vapnik-Chervonenkis theory follows: This can be re-expressed as a bound on the generalization error, just as we did earlier with the previous bound, to get the VC generalization bound: or, by using the bound on growth function in terms of $d_\mathrm{vc}$ as: Professor Vapnik standing in front of a white board that has a form of the VC-bound and the phrase “All your bayes are belong to us”, which is a play on the broken english phrase found in the classic video game Zero Wing in a claim that the VC framework of inference is superior to that of Bayesian inference. We build models on existing data, … If a hypothesis space can indeed produce all the possible labels on a set of data points, we say that the hypothesis space shatters that set. 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). 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? 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? 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). While this answer is correct, we need a more formal answer in light of the generalization inequality we’re studying. samples of a random variable $X$ distributed by $P$, and $a \leq x_i \leq b$ for every $i$, then for a small positive non-zero value $\epsilon$: You probably see why we specifically chose Heoffding’s inequality from among the others. In general, it can be proved that hyperplane classifiers (the higher-dimensional generalization of line classifiers) in $\mathbb{R}^n$ space has $d_\mathrm{vc} = n + 1$. This list is neither comprehensive nor … In predictive analytics, we want to predict classes for new data (e.g. That means, a complex ML model will adapt to subtle patterns in your training set, which in some cases could be noise. 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. This paper introduces a novel measure-theoretic theory for machine learning that does not require statistical assumptions. However, no matter what the exact form of the bound produced by any of these methods is, it always takes the form: where $C$ is a function of the hypothesis space complexity (or size, or richness), $N$ the size of the dataset, and the confidence $1 - \delta$ about the bound. Statistical Machine Learning (Summer term 2020) Quick links (publically available): youtube channel for the videos Slides Course material Slides: Latest version, updated 2020-08-19: pdf Videos: The videos of the lecture can all be found on youtube. This paper introduces a novel measure-theoretic theory for machine learning that does not require statistical assumptions. It turns out that we can do a similar thing mathematically, but instead of taking out a portion of our dataset $S$, we imagine that we have another dataset $S’$ with also size $m$, we call this the ghost dataset. In predictive analytics, we want to predict classes for new data (e.g. forecast sales for next month). Our goals now are, first, prove that if a break point exists, the growth function is going to be upper-bounded by a polynomial: If this is the case, then as N gets bigger we will get: Due to the fact that exponentials dominate polynomials. Let’s consider now a more general case, but first Lemme take a selfie ! Therefore, it makes sense to not try every possible hypothesis and instead try the possible dichotomies instead (all hypotheses in a dichotomy will classify the data the same way). 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. 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! The theory is now consistent with the empirical observations. A major criticism of exemplar theories concerns their lack of abstraction mechanisms and thus, seemingly, of generalization ability. It’s more likely that each sample in the dataset is chosen without considering any other sample that has been chosen before or will be chosen after. Take the following simple NLP problem: Say you want to predict a word in a sequence given its preceding words. The term ‘generalization’ refers to the model’s capability to adapt and react properly to previously unseen, new data, which has been drawn from the same distribution as the one used to build the model. If this is not the case, then the statistics we get from the dataset will be noisy and won’t correctly reflect the target underlying distribution. Thus we can use the VC-dimension as a proxy for growth function and, hence, for the size of the restricted space $\mathcal{H_{|S}}$. Mohri, Mehryar, Afshin Rostamizadeh, and Ameet Talwalkar. 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]. Assumptions are not bad in themselves, only bad assumptions are bad! Share. Cambridge University Press, 2014. We typically aim to minimize the non-computable expected risk R[f A(S)] by minimizing the com-putable empirical risk R S[f A( )] (i.e., empirical risk minimization).