Sample complexity: Difference between revisions

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{{Short description|Attribute of machine learning models}}

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The '''sample complexity''' of a [[machine learning]] algorithm represents the number of training-samples that it needs in order to successfully learn a target function.

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* The strong variant takes the worst-case sample complexity over all input-output distributions.

The [[No Freefree Lunchlunch theorem]], discussed below, proves that, in general, the strong sample complexity is infinite, i.e. that there is no algorithm that can learn the globally-optimal target function using a finite number of training samples.

However, if we are only interested in a particular class of target functions (e.g., only linear functions) then the sample complexity is finite, and it depends linearly on the [[VC dimension]] on the class of target functions.<ref name=":0" />

==Definition==

Let <math>X</math> be a space which we call the input space, and <math>Y</math> be a space which we call the output space, and let <math>Z</math> denote the product <math>X\times Y</math>. For example, in the setting of binary classification, <math>X</math> is typically a finite-dimensional vector space and <math>Y</math> is the set <math>\{-1,1\}</math>.

Fix a hypothesis space <math>\mathcal H</math> of functions <math>h\colon X\to Y</math>. A learning algorithm over <math>\mathcal H</math> is a computable map from <math>Z^*</math> to <math>\mathcal H</math>. In other words, it is an algorithm that takes as input a finite sequence of training samples and outputs a function from <math>X</math> to <math>Y</math>. Typical learning algorithms include [[empirical risk minimization]], without or with [[Tikhonov regularization]].

Fix a loss function <math>\mathcal{L}\colon Y\times Y\to\R_{\geq 0}</math>, for example, the square loss <math>\mathcal{L}(y, y') = (y - y')^2</math>, where <math>h(x) = y'</math>. For a given distribution <math>\rho</math> on <math>X\times Y</math>, the '''expected risk''' of a hypothesis (a function) <math>h\in\mathcal H</math> is

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In our setting, we have <math>h=\mathcal{A}(S_n)</math>, where <math>\mathcal{A}</math> is a learning algorithm and <math>S_n = ((x_1,y_1),\ldots,(x_n,y_n))\sim \rho^n</math> is a sequence of vectors which are all drawn independently from <math>\rho</math>. Define the optimal risk<math display="block">

\mathcal E^*_\mathcal{H} = \underset{h \in \mathcal H}{\inf}\mathcal E(h).

</math>Set <math>h_n=\mathcal{A}(S_n)</math>, for each [[sample size]] <math>n</math>. Note that <math>h_n</math> is a [[random variable]] and depends on the random variable <math>S_n</math>, which is drawn from the distribution <math>\rho^n</math>. The algorithm <math>\mathcal{A}</math> is called '''consistent''' if <math> \mathcal E(h_n) </math> probabilistically converges to <math> \mathcal E_\mathcal H^*</math>. In other words, for all <math>\epsilon, \delta > 0</math>, there exists a positive integer <math>N</math>, such that, for all sample sizes <math>n \geq N</math>, we have

<math display="block">

\Pr_{\rho^n}[\mathcal E(h_n) - \mathcal E^*_\mathcal{H}\geq\varepsilon]<\delta.

</math>The '''sample complexity''' of <math>\mathcal{A}</math> is then the minimum <math>N</math> for which this holds, as a function of <math>\rho, \epsilon</math>, and <math>\delta</math>. We write the sample complexity as <math>N(\rho, \epsilon, \delta)</math> to emphasize that this value of <math>N</math> depends on <math>\rho, \epsilon</math>, and <math>\delta</math>. If <math>\mathcal{A}</math> is '''not consistent''', then we set <math>N(\rho,\epsilon,\delta)=\infty</math>. If there exists an algorithm for which <math>N(\rho,\epsilon,\delta)</math> is finite, then we say that the hypothesis space <math> \mathcal H</math> is '''learnable'''.

</math>

The '''sample complexity''' of <math>\mathcal{A}</math> is then the minimum <math>N</math> for which this holds, as a function of <math>\rho, \epsilon</math>, and <math>\delta</math>. We write the sample complexity as <math>N(\rho, \epsilon, \delta)</math> to emphasize that this value of <math>N</math> depends on <math>\rho, \epsilon</math>, and <math>\delta</math>. If <math>\mathcal{A}</math> is '''not consistent''', then we set <math>N(\rho,\epsilon,\delta)=\infty</math>. If there exists an algorithm for which <math>N(\rho,\epsilon,\delta)</math> is finite, then we say that the hypothesis space <math> \mathcal H</math> is '''learnable'''.

In others words, the sample complexity <math>N(\rho,\epsilon,\delta)</math> defines the rate of consistency of the algorithm: given a desired accuracy <math>\epsilon</math> and confidence <math>\delta</math>, one needs to sample <math>N(\rho,\epsilon,\delta)</math> data points to guarantee that the risk of the output function is within <math>\epsilon</math> of the best possible, with probability at least <math>1 - \delta</math> .<ref name="Rosasco">{{citation |last = Rosasco | first = Lorenzo | title = Consistency, Learnability, and Regularization | series = Lecture Notes for MIT Course 9.520. | year = 2014 }}</ref>

In [[Probably approximately correct learning|probably approximately correct (PAC) learning]], one is concerned with whether the sample complexity is ''polynomial'', that is, whether <math>N(\rho,\epsilon,\delta)</math> is bounded by a polynomial in <math>1/\epsilon</math> and <math>1/\delta</math>. If <math>N(\rho,\epsilon,\delta)</math> is polynomial for some learning algorithm, then one says that the hypothesis space <math> \mathcal H </math> is '''PAC-learnable'''. Note that thisThis is a stronger notion than being learnable.

==Unrestricted hypothesis space: infinite sample complexity==

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=== An example of a PAC-learnable hypothesis space ===

<math>X = \R^d, Y = \{-1, 1\}</math>, and let <math>\mathcal H</math> be the space of affine functions on <math>X</math>, that is, functions of the form <math>x\mapsto \langle w,x\rangle+b</math> for some <math>w\in\R^d,b\in\R</math>. This is the linear classification with offset learning problem. Now, note that four coplanar points in a square cannot be shattered by any affine function, since no affine function can be positive on two diagonally opposite vertices and negative on the remaining two. Thus, the VC dimension of <math>\mathcal H</math> is <math>d + 1</math>, so it is finite. It follows by the above characterization of PAC-learnable classes that <math>\mathcal H</math> is PAC-learnable, and by extension, learnable.

=== Sample-complexity bounds ===

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Suppose <math>\mathcal H</math> is a class of binary functions (functions to <math>\{0,1\}</math>). Then, <math>\mathcal H</math> is <math>(\epsilon,\delta)</math>-PAC-learnable with a sample of size:

<ref>{{Cite journal|title=The optimal sample complexity of PAC learning

|journal=J. Mach. Learn. Res.|volume=17|issue=1|pages=1319–1333|author=Steve Hanneke|year=2016|arxiv=1507.00473|url=https://www.jmlr.org/papers/v17/15-389.html}}</ref>

<math display="block">

N = O\bigg(\frac{VC(\mathcal H) + \ln{1\over \delta}}{\epsilon}\bigg)

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Suppose <math>\mathcal H</math> is a class of real-valued functions with range in <math>[0,T]</math>. Then, <math>\mathcal H</math> is <math>(\epsilon,\delta)</math>-PAC-learnable with a sample of size:

<ref name=mr15>{{cite book|first1=Martin|last1=Anthony|first2=Peter L.|last2=Bartlett|title=Neural Network Learning: Theoretical Foundations|year=2009|isbn=9780521118620}}</ref><ref>{{cite conference|title=On the Pseudo-Dimension of Nearly Optimal Auctions|year=2015|conference=NIPS|url=http://papers.nips.cc/paper/5766-on-the-pseudo-dimension-of-nearly-optimal-auctions|arxiv=1506.03684|last1=Morgenstern|first1=Jamie|author1-link=Jamie Morgenstern|last2=Roughgarden|first2=Tim|pages=136–144|publisher=Curran Associates}}</ref>

<math display="block">

N = O\bigg(T^2\frac{PD(\mathcal H)\ln{T\over \epsilon} + \ln{1\over \delta}}{\epsilon^2}\bigg)

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where <math>PD(\mathcal H)</math> is [[VC dimension#Generalizations|Pollard's pseudo-dimension]] of <math>\mathcal H</math>.

==Other Settingssettings==

In addition to the supervised learning setting, sample complexity is relevant to [[semi-supervised learning]] problems including [[Active learning (machine learning)|active learning]],<ref name="Balcan">{{cite journal |doi = 10.1007/s10994-010-5174-y|title = The true sample complexity of active learning|journal = Machine Learning|date = 2010|volume = 80|issue = 2–3|pages = 111–139|last1=Balcan|first1=Maria-Florina|authorlink1= Maria-Florina Balcan|last2=Hanneke|first2=Steve|last3=Wortman Vaughan|first3=Jennifer|doi-access = free}}</ref> where the algorithm can ask for labels to specifically chosen inputs in order to reduce the cost of obtaining many labels. The concept of sample complexity also shows up in [[reinforcement learning]],<ref>{{citation |last = Kakade | first = Sham | title = On the Sample Complexity of Reinforcement Learning | place = University College London | publisher = Gatsby Computational Neuroscience Unit. | series = PhD Thesis | year = 2003 | url = http://www.ias.tu-darmstadt.de/uploads/Research/NIPS2006/SK.pdf}}</ref> [[online machine learning|online learning]], and unsupervised algorithms, e.g. for [[dictionary learning]].<ref>{{cite journal |last1 = Vainsencher | first1 = Daniel | last2 = Mannor | first2 = Shie | last3 = Bruckstein | first3 = Alfred | title = The Sample Complexity of Dictionary Learning | journal = Journal of Machine Learning Research | volume = 12 | pages = 3259–3281 | date = 2011 | url = http://www.jmlr.org/papers/volume12/vainsencher11a/vainsencher11a.pdf}}</ref>

==Efficiency in robotics==

A high sample complexity means, that many calculations are needed for running a [[Monte Carlo tree search]].<ref>{{cite conference |title=Monte-carlo tree search by best arm identification |author=Kaufmann, Emilie and Koolen, Wouter M |conference=Advances in Neural Information Processing Systems |pages=4897–4906 |year=2017 }}</ref> ItsIt equalis equivalent to a [[Model-free (reinforcement learning)|model -free]] brute force search in the state space. In contrast, a high -efficiency algorithm has a low sample complexity.<ref>{{cite conference |title=The chin pinch: A case study in skill learning on a legged robot |author=Fidelman, Peggy and Stone, Peter |conference=Robot Soccer World Cup |pages=59–71 |year=2006 |publisher=Springer }}</ref> Possible techniques for reducing the sample complexity are [[metric learning]]<ref>{{cite conference |title=Sample complexity of learning mahalanobis distance metrics |author=Verma, Nakul and Branson, Kristin |conference=Advances in neural information processing systems |pages=2584–2592 |year=2015 }}</ref> and model -based reinforcement learning.<ref>{{cite arxivarXiv |title=Model-ensemble trust-region policy optimization |author=Kurutach, Thanard and Clavera, Ignasi and Duan, Yan and Tamar, Aviv and Abbeel, Pieter |eprint=1802.10592 |year=2018 |class=cs.LG }}</ref>

== See also ==

* [[Active learning (machine learning)]]

==References==