This paper introduces a novel measure-theoretic theory for machine learning that does not require statistical assumptions. Checkpoint: The bigger picture •Supervised learning: instances, concepts, and hypotheses •Specific learners –Decision trees Blaine Bateman. Theory of Generalization - How an infinite model can learn from a finite sample. However, the success of linear hypothesis can now be explained by the fact that they have a finite $d_\mathrm{vc} = n + 1$ in $\mathbb{R}^n$. 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. Well, Not even close! This paper introduces a novel measure-theoretic learning theory to analyze generalization behaviors of practical interest. 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. Finally, for transfer learning , our theory reveals that knowledge transfer depends sensitively, but computably, on … from the same distribution of the training set (this is why we usually shuffle the whole dataset beforehand to break any correlation between the samples). Software: Shark We provide and maintain a fast, modular, open source C++ library for the design and optimization of adaptive systems. To our destination of ensuring that the training and generalization errors do not differ much, we need to know more info about the how the road down the law of large numbers look like. Challenges of Generalization in Machine Learning. 2015. Generalization. Browse other questions tagged machine-learning deep-neural-networks overfitting learning-theory generalization or ask your own question. B(N,k) = α + 2β ≤ B(N-1, k) + B(N-1, k-1) (*). Follow. Also, for a better understanding of this, I really advise you to watch the lecture at least starting from the 45th to the 60th minute. To understand the concept of generalisation in ML, you need to understand the concept of “overfitting”. In machine learning, … 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. You can find the full proof here. For the binary classification case, we can say that: But $2^m$ is exponential in $m$ and would grow too fast for large datasets, which makes the odds in our inequality go too bad too fast! The superpower of machine learning is generalization. Deep Learning is currently being used for a variety of different applications. One inequality of those is Heoffding’s inequality: If $x_1, x_2, …, x_m$ are $m$ i.i.d. 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? CBMM Memo No. Learning bounds are available for traditional machine learning methods (support vector machines (SVMs), and kernel methods), but not for deep neural networks. 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. 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. Is that the best bound we can get on that growth function? The most important theoretical result in machine learning. Featured on Meta MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC… 067 September 15, 2017 Theory of Deep Learning III: Generalization Properties of SGD by Chiyuan Zhang 1Qianli Liao Alexander Rakhlin2 Brando Miranda Noah Golowich Tomaso Poggio1 1Center for Brains, Minds, and Machines, McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, MA, 02139. A theory of learning and generalization : with applications to neural networks and control systems. We are a research group focused on building towards a theory of modern machine learning. This fact is formally captured into what we call The law of large numbers: If $x_1, x_2, …, x_m$ are $m$ i.i.d. Banerjee, A. These info are provided by what we call the concentration inequalities. This is to make the post easier to read and to focus all the effort on the conceptual understanding of the subject. 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. This means that there’s still something missing from our theoretical model, and it’s time for us to revise our steps. 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. cats vs. dogs), or predict future values of a time series (e.g. We also divide the group S2 into S2+ where xN is a “+” and S2- where xN is “-” and each of them have β rows. 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. Which will give us: α + β < B(N-1,k) : (2). Conference on Learning Theory. Here, we use insights from machine learning to demonstrate that exemplar models can actually generalize very well. 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! This implies that k is a break point for the smaller table too. Let’s think for a moment about something we do usually in machine learning practice. The formulation of the generalization inequality reveals a main reason why we need to consider all the hypothesis in $\mathcal{H}$. This works because we assume that this test set is drawn i.i.d. B(N,k) corresponds to the number of rows in the following table: Let α be the count of rows in the S1 group. Using algebraic manipulation, we can prove that: Where $O$ refers to the Big-O notation for functions asymptotic (near the limits) behavior, and $e$ is the mathematical constant. 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) So one might think, as they all have the same $R_\text{emp}$, why not choose one and omit the others?! 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. We need to be able to make that claim to ensure that the learning algorithm would never land on a hypothesis with a bigger generalization gap than $\epsilon$. However, what if somehow we can get a very good estimate of the risk $R(h)$ without needing to go over the whole of the $\mathcal{X \times Y}$ space, would there be any hope to get a better bound? Most of us, since we were kids, know that if we tossed a fair coin a large number of times, roughly half of the times we’re gonna get heads. For a linear hypothesis of the form $h(x) = wx + b$, we also have $|\mathcal{H}| = \infty$ as there is infinitely many lines that can be drawn. Learning Theory & Theory of Generalization CS 446/546 . That simpler skillful machine learning models are easier to understand and more robust. It is often said that "we don't understand deep learning" but it is not as often clarified what is it exactly that we don't understand. This paper introduces a novel measure-theoretic theory for machine learning that does not require statistical assumptions. So clearly machine learning can't be about just minimizing the training loss. For simplicity, we’ll focus now on the case of binary classification, in which $\mathcal{Y}=\{-1, +1\}$. This means that the probability of the largest generalization gap being bigger than $\epsilon$ is at most twice the probability that the empirical risk difference between $S, S’$ is larger than $\frac{\epsilon}{2}$. So it’s possible for a hypothesis space $\mathcal{H}$ to be unable to shatter all sizes. Assignments (only accessible for … However, the conceptual framework of VC (that is: shattering, growth function and dimension) generalizes very well to both multi-class classification and regression. The rote learning algorithm does a perfect job of minimizing the training loss. Let’s investigate that with the binary classification case and the $\mathcal{H}$ of linear classifiers $\mathrm{sign}(wx + b)$. 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. Lecture 6 of 18 of Caltech's Machine Learning Course - CS 156 by Professor Yaser Abu-Mostafa. That means, a complex ML model will adapt to subtle patterns in your training set, which in some cases could be noise. 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). 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. Foundations of machine learning. Can we do any better? This inequality basically says the generalization error can be decomposed into two parts: the empirical training error, and the complexity of the learning model. 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! Assignments (only accessible for … As a result, measurements of prediction error on the current data … We are a research group focused on building towards a theory of modern machine learning. This form of the inequality holds to any learning problem no matter the exact form of the bound, and this is the one we’re gonna use throughout the rest of the series to guide us through the process of 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 In predictive analytics, we want to predict classes for new data (e.g. We’ll define the maximum number of distinct labellings/values on a dataset $S$ of size $m$ by a hypothesis space $\mathcal{H}$ as the growth function of $\mathcal{H}$ given $m$, and we’ll denote that by $\Delta_\mathcal{H}(m)$. That machine learning is only suitable when the problem requires generalization. The conventional wisdom in machine learning offers the following about generalization: A model that is too simple will underfit the true patterns in the training data, and thus, it will predict poorly on new data. We are interested in both experimental and theoretical approaches that advance our understanding. 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. The Theory of Generalization. We therefore get: The growth function takes care of the redundancy of hypotheses that result in the same classification. [23] Belkin, Mikhail, et al. The proposed learning theory has the following abilities: 1) to utilize the qualities of each learned representation on the path from raw inputs to outputs in representation learning, 2) to guarantee good generalization errors possibly with arbitrarily rich hypothesis spaces (e.g., arbitrarily large capacity and Rademacher complexity) and non-stable/non-robust learning algorithms, and 3) to clearly distinguish … This fact can be used to get a better bound on the growth function, and this is done using Sauer’s lemma: If a hypothesis space $\mathcal{H}$ cannot shatter any dataset with size more than $k$, then: This was the other key part of Vapnik-Chervonenkis work (1971), but it’s named after another mathematician, Norbert Sauer; because it was independently proved by him around the same time (1972). 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