DOCSLIB.ORG

A Sample Complexity Separation Between Non-Convex and Convex Meta-Learning

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Sample Complexity Separation Between Non-Convex and Convex Meta-Learning A Sample Complexity Separation between Non-Convex and Convex Meta-Learning Nikunj Saunshi 1 Yi Zhang 1 Mikhail Khodak 2 Sanjeev Arora 13 Abstract complexity of an unseen but related test task. Although there is a long history of successful methods in meta-learning One popular trend in meta-learning is to learn and the related areas of multi-task and lifelong learning from many training tasks a common initialization (Evgeniou & Pontil, 2004; Ruvolo & Eaton, 2013), recent that a gradient-based method can use to solve a approaches have been developed with the diversity and scale new task with few samples. The theory of meta- of modern applications in mind. This has given rise to learning is still in its early stages, with several re- simple, model-agnostic methods that focus on learning a cent learning-theoretic analyses of methods such good initialization for some gradient-based method such as as Reptile (Nichol et al., 2018) being for convex stochastic gradient descent (SGD), to be run on samples models. This work shows that convex-case analy- from a new task (Finn et al., 2017; Nichol et al., 2018). sis might be insufficient to understand the success These methods have found widespread applications in a of meta-learning, and that even for non-convex variety of areas such as computer vision (Nichol et al., 2018), models it is important to look inside the optimiza- reinforcement learning (Finn et al., 2017), and federated tion black-box, specifically at properties of the learning (McMahan et al., 2017). optimization trajectory. We construct a simple meta-learning instance that captures the problem Inspired by their popularity, several recent learning-theoretic of one-dimensional subspace learning. For the analyses of meta-learning have followed suit, eschewing cus- convex formulation of linear regression on this tomization to specific hypothesis classes such as halfspaces instance, we show that the new task sample com- (Maurer & Pontil, 2013; Balcan et al., 2015) and instead plexity of any initialization-based meta-learning favoring the convex-case study of gradient-based algorithms algorithm is ⌦(d), where d is the input dimension. that could potentially be applied to deep neural networks In contrast, for the non-convex formulation of a (Denevi et al., 2019; Khodak et al., 2019). This has yielded two layer linear network on the same instance, results showing that meta-learning an initialization by using we show that both Reptile and multi-task rep- methods similar to Reptile (Nichol et al., 2018) for con- resentation learning can have new task sample vex models leads to a reduction in sample complexity of complexity of (1), demonstrating a separation unseen tasks. These benefits are shown using natural no- O from convex meta-learning. Crucially, analyses tions of task-similarity like the average distance between of the training dynamics of these methods reveal the risk minimizers of tasks drawn from an underlying meta- that they can meta-learn the correct subspace onto distribution. A good initialization in these models is one which the data should be projected. that is close to the population risk minimizers for tasks in this meta-distribution. In this paper we argue that, even in some simple settings, 1. Introduction such convex-case analyses are insufficient to understand We consider the problem of meta-learning, or learning-to- the success of initialization-based meta-learning algorithms. learn (Thrun & Pratt, 1998), in which the goal is to use For this purpose, we pose a simple instance for meta- the data from numerous training tasks to reduce the sample learning linear regressors that share a one-dimensional sub- space, for which we prove a sample complexity separa- 1Princeton University, Princeton, New Jersey, USA 2Carnegie tion between convex and non-convex methods. In the pro- 3 Mellon University, Pittsburgh, Pennsylvania, USA Institute for cess, we provide a first theoretical demonstration of how Advanced Study, Princeton, New Jersey, USA. Correspondence to: initialization-based meta-learning methods can learn good Nikunj Saunshi <[email protected]>. representations, as observed empirically (Raghu et al., 2019). Proceedings of the 37 th International Conference on Machine Specifically, our contributions are the following: Learning, Online, PMLR 119, 2020. Copyright 2020 by the au- thor(s). A Sample Complexity Separation between Non-Convex and Convex Meta-Learning We show, in the convex formulation of linear regression 2. Related Work • on this instance, a new task sample complexity lower bound of ⌦(d) for any initialization-based meta-learning There is a rich history of theoretical analysis of learning-to- algorithm. This suggests that no amount of meta-training learn (Baxter, 2000; Maurer, 2005; Maurer et al., 2016). Our data can yield an initialization that can be used by a focus is on a well-studied setting in which tasks such as half- common gradient-based within-task algorithms to solve space learning share a common low-dimensional subspace, a new task with fewer samples than if no meta-learning with the goal of obtaining sample complexity depending had been done; thus initialization-based meta-learning on this sparse structure rather than on the ambient dimen- in the convex formulation fails to learn the underlying sion (Maurer, 2009; Maurer & Pontil, 2013; Balcan et al., task-similarity. The lower bounds also holds for more 2015; Denevi et al., 2018; Bullins et al., 2019; Khodak et al., general meta-learning instances that have the property 2019). While these works derive specialized algorithms, we that the average prediction for an input across tasks is instead focus on learning an initialization for gradient-based independent of the input, as in true in many standard methods such as SGD or few steps of gradient descent (Finn meta-learning benchmarks. et al., 2017; Nichol et al., 2018). Some of these methods have recently been studied in the convex setting (Denevi We show for the same instance that formulating the model et al., 2019; Khodak et al., 2019; Zhou et al., 2019). Our • as a two-layer linear network – an over-parameterization results show that such convex-case analyses cannot hope of the same hypothesis class – allows a Reptile-like proce- to show adaptation to an underlying low-dimensional sub- dure to use training tasks from this meta-learning instance space leading to dimension-independent sample complexity and find an initialization for gradient descent that will bounds. On the other hand, we show that their guarantees have (1) sample complexity on a new task. To the best using distance-from-initialization are almost tight for the O of our knowledge, this is the first sample complexity anal- meta-learning of convex Lipschitz functions. ysis of initialization-based meta-learning algorithms in To get around the limitations of convexity for the prob- the non-convex setting. Additionally, this demonstrates lem of meta-learning a shared subspace, we instead study that initialization-based methods can capture the strong non-convex models. While the optimization properties of property of representation learning. gradient-based meta-learning algorithms have been recently Central to our proof is a trajectory-based analysis to ana- studied in the non-convex setting (Fallah et al., 2019; Ra- • lyze properties of the solution found by a specific proce- jeswaran et al., 2019; Zhou et al., 2019), these results only dures like Reptile or gradient descent on a representation provide stationary-point convergence guarantees and do not learning objective. For the latter, we show that looking show a reduction in sample complexity, the primary goal at the trajectory is crucial as not all minimizers can learn of meta-learning. Our theory is more closely related to the subspace structure. recent empirical work that tries to understand various in- herently non-convex properties of learning-to-learn. Most Finally, we revisit existing upper bounds for the convex notably, Arnold et al. (2019) hypothesize and show some • case. We show that our lower bound does not contradict experimental evidence that the success of gradient-based these upper bounds, since their task similarity measure meta-learning requires non-convexity, a view theoretically of average parameter distance is large in our case. We supported by our work. Meanwhile, Raghu et al. (2019) complement this observation by proving that the existing demonstrate that the success of the popular MAML algo- bounds are tight, in some sense, and going beyond them rithm (Finn et al., 2017) is likely due to its ability to learn will require additional structural assumptions. good data-representations rather than adapt quickly. Our upper bound, in fact, supports these empirical findings by Paper organization: We discuss related work in Sec- showing that Reptile can indeed learn a good representation tion 2. Section 3 sets up notation for the rest of the pa- at the penultimate layer, and learning just a final linear layer per, formalizes initialization-based meta-learning methods for a new task will reduce sample complexity for a new task. and defines the subspace meta-learning instance that we Our results draw upon work motivated by understanding are interested in. The lower bound for linear regression is deep learning that analyzes trajectories and implicit regu- stated in Section 4, while the corresponding upper bounds larization in deep linear neural networks (Saxe et al., 2014; for non-convex meta-learning with two-layer linear network Gunasekar et al., 2018; Saxe et al., 2019; Gidel et al., 2019). is provided in Section 5. While all proofs are provided in The analysis of solutions found by gradient flow in deep lin- the appendix, we give a sketch of the proofs for the up- ear networks by (Saxe et al., 2014; Gidel et al., 2019) form per bounds in Section 6 to highlight the key steps in the a core component of our analysis.
Load more

Recommended publications
  • Arxiv:1811.04918V6 [Cs.LG] 1 Jun 2020
    Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers Zeyuan Allen-Zhu Yuanzhi Li Yingyu Liang [email protected] [email protected] [email protected] Microsoft Research AI Stanford University University of Wisconsin-Madison November 12, 2018 (version 6)∗ Abstract The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn? Why doesn't the trained network overfit when it is overparameterized? In this work, we prove that overparameterized neural networks can learn some notable con- cept classes, including two and three-layer networks with fewer parameters and smooth activa- tions. Moreover, the learning can be simply done by SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. The sample complexity can also be almost independent of the number of parameters in the network. On the technique side, our analysis goes beyond the so-called NTK (neural tangent kernel) linearization of neural networks in prior works. We establish a new notion of quadratic ap- proximation of the neural network (that can be viewed as a second-order variant of NTK), and connect it to the SGD theory of escaping saddle points. arXiv:1811.04918v6 [cs.LG] 1 Jun 2020 ∗V1 appears on this date, V2/V3/V4 polish writing and parameters, V5 adds experiments, and V6 reflects our conference camera ready version. Authors sorted in alphabetical order. We would like to thank Greg Yang and Sebastien Bubeck for many enlightening conversations. Y. Liang was supported in part by FA9550-18-1-0166, and would also like to acknowledge that support for this research was provided by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
    [Show full text]
  • Analysis of Perceptron-Based Active Learning
    JournalofMachineLearningResearch10(2009)281-299 Submitted 12/07; 12/08; Published 2/09 Analysis of Perceptron-Based Active Learning Sanjoy Dasgupta [email protected] Department of Computer Science and Engineering University of California, San Diego La Jolla, CA 92093-0404, USA Adam Tauman Kalai [email protected] Microsoft Research Office 14063 One Memorial Drive Cambridge, MA 02142, USA Claire Monteleoni [email protected] Center for Computational Learning Systems Columbia University Suite 850, 475 Riverside Drive, MC 7717 New York, NY 10115, USA Editor: Manfred Warmuth Abstract Ω 1 We start by showing that in an active learning setting, the Perceptron algorithm needs ( ε2 ) labels to learn linear separators within generalization error ε. We then present a simple active learning algorithm for this problem, which combines a modification of the Perceptron update with an adap- tive filtering rule for deciding which points to query. For data distributed uniformly over the unit 1 sphere, we show that our algorithm reaches generalization error ε after asking for just O˜(d log ε ) labels. This exponential improvement over the usual sample complexity of supervised learning had previously been demonstrated only for the computationally more complex query-by-committee al- gorithm. Keywords: active learning, perceptron, label complexity bounds, online learning 1. Introduction In many machine learning applications, unlabeled data is abundant but labeling is expensive. This distinction is not captured in standard models of supervised learning, and has motivated the field of active learning, in which the labels of data points are initially hidden, and the learner must pay for each label it wishes revealed.
    [Show full text]
  • The Sample Complexity of Agnostic Learning Under Deterministic Labels
    JMLR: Workshop and Conference Proceedings vol 35:1–16, 2014 The sample complexity of agnostic learning under deterministic labels Shai Ben-David [email protected] Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, CANADA Ruth Urner [email protected] School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Abstract With the emergence of Machine Learning tools that allow handling data with a huge number of features, it becomes reasonable to assume that, over the full set of features, the true labeling is (almost) fully determined. That is, the labeling function is deterministic, but not necessarily a member of some known hypothesis class. However, agnostic learning of deterministic labels has so far received little research attention. We investigate this setting and show that it displays a behavior that is quite different from that of the fundamental results of the common (PAC) learning setups. First, we show that the sample complexity of learning a binary hypothesis class (with respect to deterministic labeling functions) is not fully determined by the VC-dimension of the class. For any d, we present classes of VC-dimension d that are learnable from O~(d/)- many samples and classes that require samples of size Ω(d/2). Furthermore, we show that in this setup, there are classes for which any proper learner has suboptimal sample complexity. While the class can be learned with sample complexity O~(d/), any proper (and therefore, any ERM) algorithm requires Ω(d/2) samples. We provide combinatorial characterizations of both phenomena, and further analyze the utility of unlabeled samples in this setting.
    [Show full text]
  • Sinkhorn Autoencoders
    Sinkhorn AutoEncoders Giorgio Patrini⇤• Rianne van den Berg⇤ Patrick Forr´e Marcello Carioni UvA Bosch Delta Lab University of Amsterdam† University of Amsterdam KFU Graz Samarth Bhargav Max Welling Tim Genewein Frank Nielsen ‡ University of Amsterdam University of Amsterdam Bosch Center for Ecole´ Polytechnique CIFAR Artificial Intelligence Sony CSL Abstract 1INTRODUCTION Optimal transport o↵ers an alternative to Unsupervised learning aims at finding the underlying maximum likelihood for learning generative rules that govern a given data distribution PX . It can autoencoding models. We show that mini- be approached by learning to mimic the data genera- mizing the p-Wasserstein distance between tion process, or by finding an adequate representation the generator and the true data distribution of the data. Generative Adversarial Networks (GAN) is equivalent to the unconstrained min-min (Goodfellow et al., 2014) belong to the former class, optimization of the p-Wasserstein distance by learning to transform noise into an implicit dis- between the encoder aggregated posterior tribution that matches the given one. AutoEncoders and the prior in latent space, plus a recon- (AE) (Hinton and Salakhutdinov, 2006) are of the struction error. We also identify the role of latter type, by learning a representation that maxi- its trade-o↵hyperparameter as the capac- mizes the mutual information between the data and ity of the generator: its Lipschitz constant. its reconstruction, subject to an information bottle- Moreover, we prove that optimizing the en- neck. Variational AutoEncoders (VAE) (Kingma and coder over any class of universal approxima- Welling, 2013; Rezende et al., 2014), provide both a tors, such as deterministic neural networks, generative model — i.e.
    [Show full text]
  • Data-Dependent Sample Complexity of Deep Neural Networks Via Lipschitz Augmentation
    Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation Colin Wei Tengyu Ma Computer Science Department Computer Science Department Stanford University Stanford University [email protected] [email protected] Abstract Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is unavoidable when no additional properties of the training data are considered. We suspect that this conundrum comes from the fact that these bounds depend on the training data only through the margin. In practice, many data-dependent techniques such as Batchnorm improve the generalization performance. For feedforward neural nets as well as RNNs, we obtain tighter Rademacher complexity bounds by considering additional data-dependent properties of the network: the norms of the hidden layers of the network, and the norms of the Jacobians of each layer with respect to all previous layers. Our bounds scale polynomially in depth when these empirical quantities are small, as is usually the case in practice. To obtain these bounds, we develop general tools for augmenting a sequence of functions to make their composition Lipschitz and then covering the augmented functions. Inspired by our theory, we directly regularize the network’s Jacobians during training and empirically demonstrate that this improves test performance. 1 Introduction Deep networks trained in
    [Show full text]
  • Nearly Tight Sample Complexity Bounds for Learning Mixtures of Gaussians Via Sample Compression Schemes∗
    Nearly tight sample complexity bounds for learning mixtures of Gaussians via sample compression schemes∗ Hassan Ashtiani Shai Ben-David Department of Computing and Software School of Computer Science McMaster University, and University of Waterloo, Vector Institute, ON, Canada Waterloo, ON, Canada [email protected] [email protected] Nicholas J. A. Harvey Christopher Liaw Department of Computer Science Department of Computer Science University of British Columbia University of British Columbia Vancouver, BC, Canada Vancouver, BC, Canada [email protected] [email protected] Abbas Mehrabian Yaniv Plan School of Computer Science Department of Mathematics McGill University University of British Columbia Montréal, QC, Canada Vancouver, BC, Canada [email protected] [email protected] Abstract We prove that Θ(e kd2="2) samples are necessary and sufficient for learning a mixture of k Gaussians in Rd, up to error " in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that Oe(kd="2) samples suffice, matching a known lower bound. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in Rd has a small-sized sample compression. 1 Introduction Estimating distributions from observed data is a fundamental task in statistics that has been studied for over a century.
    [Show full text]
  • Sample Complexity Results
    10-601 Machine Learning Maria-Florina Balcan Spring 2015 Generalization Abilities: Sample Complexity Results. The ability to generalize beyond what we have seen in the training phase is the essence of machine learning, essentially what makes machine learning, machine learning. In these notes we describe some basic concepts and the classic formalization that allows us to talk about these important concepts in a precise way. Distributional Learning The basic idea of the distributional learning setting is to assume that examples are being provided from a fixed (but perhaps unknown) distribution over the instance space. The assumption of a fixed distribution gives us hope that what we learn based on some training data will carry over to new test data we haven't seen yet. A nice feature of this assumption is that it provides us a well-defined notion of the error of a hypothesis with respect to target concept. Specifically, in the distributional learning setting (captured by the PAC model of Valiant and Sta- tistical Learning Theory framework of Vapnik) we assume that the input to the learning algorithm is a set of labeled examples S :(x1; y1);:::; (xm; ym) where xi are drawn i.i.d. from some fixed but unknown distribution D over the the instance space ∗ ∗ X and that they are labeled by some target concept c . So yi = c (xi). Here the goal is to do optimization over the given sample S in order to find a hypothesis h : X ! f0; 1g, that has small error over whole distribution D. The true error of h with respect to a target concept c∗ and the underlying distribution D is defined as err(h) = Pr (h(x) 6= c∗(x)): x∼D (Prx∼D(A) means the probability of event A given that x is selected according to distribution D.) We denote by m ∗ 1 X ∗ errS(h) = Pr (h(x) 6= c (x)) = I[h(xi) 6= c (xi)] x∼S m i=1 the empirical error of h over the sample S (that is the fraction of examples in S misclassified by h).
    [Show full text]
  • Benchmarking Model-Based Reinforcement Learning
    Benchmarking Model-Based Reinforcement Learning Tingwu Wang1;2, Xuchan Bao1;2, Ignasi Clavera3, Jerrick Hoang1;2, Yeming Wen1;2, Eric Langlois1;2, Shunshi Zhang1;2, Guodong Zhang1;2, Pieter Abbeel3& Jimmy Ba1;2 1University of Toronto 2 Vector Institute 3 UC Berkeley {tingwuwang,jhoang,ywen,edl,gdzhang}@cs.toronto.edu {xuchan.bao,matthew.zhang}@mail.utoronto.ca, [email protected], [email protected], [email protected] Abstract Model-based reinforcement learning (MBRL) is widely seen as having the potential to be significantly more sample efficient than model-free RL. However, research in model-based RL has not been very standardized. It is fairly common for authors to experiment with self-designed environments, and there are several separate lines of research, which are sometimes closed-sourced or not reproducible. Accordingly, it is an open question how these various existing MBRL algorithms perform relative to each other. To facilitate research in MBRL, in this paper we gather a wide collection of MBRL algorithms and propose over 18 benchmarking environments specially designed for MBRL. We benchmark these algorithms with unified problem settings, including noisy environments. Beyond cataloguing performance, we explore and unify the underlying algorithmic differences across MBRL algorithms. We characterize three key research challenges for future MBRL research: the dynamics bottleneck, the planning horizon dilemma, and the early-termination dilemma. Finally, to maximally facilitate future research on MBRL, we open-source our benchmark in http://www.cs.toronto.edu/~tingwuwang/mbrl.html. 1 Introduction Reinforcement learning (RL) algorithms are most commonly classified in two categories: model-free RL (MFRL), which directly learns a value function or a policy by interacting with the environment, and model-based RL (MBRL), which uses interactions with the environment to learn a model of it.
    [Show full text]
  • Theory of Deep Learning
    CONTRIBUTORS:RAMANARORA,SANJEEVARORA,JOANBRUNA,NADAVCOHEN,SIMONDU, RONGGE,SURIYAGUNASEKAR,CHIJIN,JASONLEE,TENGYUMA,BEHNAMNEYSHABUR, ZHAOSONG THEORYOFDEEP LEARNING Contents 1 Basic Setup and some math notions 13 1.1 List of useful math facts 14 1.1.1 Probability tools 14 1.1.2 Singular Value Decomposition 15 2 Basics of Optimization 17 2.1 Gradient descent 17 2.1.1 Formalizing the Taylor Expansion 18 2.1.2 Descent lemma for gradient descent 18 2.2 Stochastic gradient descent 19 2.3 Accelerated Gradient Descent 19 2.4 Local Runtime Analysis of GD 20 2.4.1 Pre-conditioners 21 3 Backpropagation and its Variants 23 3.1 Problem Setup 23 3.1.1 Multivariate Chain Rule 25 3.1.2 Naive feedforward algorithm (not efficient!) 26 3.2 Backpropagation (Linear Time) 26 3.3 Auto-differentiation 27 3.4 Notable Extensions 28 3.4.1 Hessian-vector product in linear time: Pearlmutter’s trick 29 4 4 Basics of generalization theory 31 4.0.1 Occam’s razor formalized for ML 31 4.0.2 Motivation for generalization theory 32 4.1 Some simple bounds on generalization error 32 4.2 Data dependent complexity measures 34 4.2.1 Rademacher Complexity 34 4.2.2 Alternative Interpretation: Ability to correlate with random labels 35 4.3 PAC-Bayes bounds 35 5 Advanced Optimization notions 39 6 Algorithmic Regularization 41 6.1 Linear models in regression: squared loss 42 6.1.1 Geometry induced by updates of local search algorithms 43 6.1.2 Geometry induced by parameterization of model class 46 6.2 Matrix factorization 47 6.3 Linear Models in Classification 47 6.3.1 Gradient Descent
    [Show full text]
  • B.Sc Computer Science with Specialization in Artificial Intelligence & Machine Learning
    B.Sc Computer Science with Specialization in Artificial Intelligence & Machine Learning Curriculum & Syllabus (Based on Choice Based Credit System) Effective from the Academic year 2020-2021 PROGRAMME EDUCATIONAL OBJECTIVES (PEO) PEO 1 : Graduates will have solid basics in Mathematics, Programming, Machine Learning, Artificial Intelligence fundamentals and advancements to solve technical problems. PEO 2 : Graduates will have the capability to apply their knowledge and skills acquired to solve the issues in real world Artificial Intelligence and Machine learning areas and to develop feasible and reliable systems. PEO 3 : Graduates will have the potential to participate in life-long learning through the successful completion of advanced degrees, continuing education, certifications and/or other professional developments. PEO 4 : Graduates will have the ability to apply the gained knowledge to improve the society ensuring ethical and moral values. PEO 5 : Graduates will have exposure to emerging cutting edge technologies and excellent training in the field of Artificial Intelligence & Machine learning PROGRAMME OUTCOMES (PO) PO 1 : Develop knowledge in the field of AI & ML courses necessary to qualify for the degree. PO 2 : Acquire a rich basket of value added courses and soft skill courses instilling self-confidence and moral values. PO 3 : Develop problem solving, decision making and communication skills. PO 4 : Demonstrate social responsibility through Ethics and values and Environmental Studies related activities in the campus and in the society. PO 5 : Strengthen the critical thinking skills and develop professionalism with the state of art ICT facilities. PO 6 : Quality for higher education, government services, industry needs and start up units through continuous practice of preparatory examinations.
    [Show full text]
  • Computational Learning Theory – Part 2
    Computational Learning Theory – Part 2 Reading: • Mitchell chapter 7 Suggested exercises: • 7.1, 7.2, 7.5, 7.7 Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University March 22, 2010 1 What it means [Haussler, 1988]: probability that the version space is not ε-exhausted after m training examples is at most Suppose we want this probability to be at most δ 1. How many training examples suffice? 2. If then with probability at least (1-δ): Agnostic Learning Result we proved: probability, after m training examples, that H contains a hypothesis h with zero training error, but true error greater than ε is bounded Agnostic case: don’t know whether H contains a perfect hypothesis overfitting 2 General Hoeffding Bounds • When estimating the mean θ inside [a,b] from m examples • When estimating a probability θ is inside [0,1], so • And if we’re interested in only one-sided error, then 3 Sufficient condition: Holds if L requires only a polynomial number of training examples, and processing per example is polynomial Sample Complexity based on VC dimension How many randomly drawn examples suffice to ε-exhaust VSH,D with probability at least (1-δ)? ie., to guarantee that any hypothesis that perfectly fits the training data is probably (1-δ) approximately (ε) correct Compare to our earlier results based on |H|: 4 VC(H)=3 VC dimension: examples What is VC dimension of lines in a plane? • H2 = { ((w0 + w1x1 + w2x2)>0 y=1) } 5 VC dimension: examples What is VC dimension of • H2 = { ((w0 + w1x1 + w2x2)>0 y=1) } – VC(H2)=3 • For Hn = linear separating hyperplanes in n dimensions, VC(Hn)=n+1 Can you give an upper bound on VC(H) in terms of |H|, for any hypothesis space H? (hint: yes) 6 More VC Dimension Examples to Think About • Logistic regression over n continuous features – Over n boolean features? • Linear SVM over n continuous features • Decision trees defined over n boolean features F: <X1, ..
    [Show full text]
  • The Sample Complexity of Pattern Classification with Neural Networks
    IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 2, MARCH 1998 525 The Sample Complexity of Pattern Classification with Neural Networks: The Size of the Weights is More Important than the Size of the Network Peter L. Bartlett, Member, IEEE Abstract—Sample complexity results from computational learn- VC dimension is a combinatorial complexity measure that is ing theory, when applied to neural network learning for pat- typically at least as large as the number of adjustable network tern classification problems, suggest that for good generalization parameters.) These results do not provide a satisfactory expla- performance the number of training examples should grow at least linearly with the number of adjustable parameters in the nation of the sample size requirements of neural networks for network. Results in this paper show that if a large neural pattern classification applications, for several reasons. First, network is used for a pattern classification problem and the neural networks often perform successfully with training sets learning algorithm finds a network with small weights that has that are considerably smaller than the number of network small squared error on the training patterns, then the general- parameters (see, for example, [29]). Second, the VC dimension ization performance depends on the size of the weights rather than the number of weights. For example, consider a two- of the class of functions computed by a network is sensitive to layer feedforward network of sigmoid units, in which the sum small perturbations of the computation unit transfer functions of the magnitudes of the weights associated with each unit is (to the extent that an arbitrarily small change can make the bounded by e and the input dimension is n.
    [Show full text]