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" />

<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>Loss\mathcal{L}\colon Y\times Y\to\R_{\geq 0}</math>, for example, the square loss <math> Loss\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

:<math>\mathcal E(h) :=\mathbb E_\rho[Loss\mathcal{L}(h(x),y)]=\int_{X\times Y} Loss\mathcal{L}(h(x),y)\,d\rho(x,y)</math>

In our setting, we have <math>h=ALG\mathcal{A}(S_n)</math>, where <math>ALG\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>. <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>Set <math>h_n=Alg(S_n)</math> for each <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>ALG</math> is called consistent if <math>

<math display="block">

\mathcal E(h_n)

</math> probabilistically converges to <math>

\mathcal E_\mathcal H^*

</math>, in other words, for all ''ε'', ''δ'' > 0, there exists a positive integer ''N'' such that for all ''n'' ≥ ''N'', 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>ALG\mathcal{A}</math> is then the minimum ''<math>N''</math> for which this holds, as a function of ''ρ''<math>\rho, ''ε''\epsilon</math>, and ''δ''<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 ''ρ''<math>\rho, ''ε''\epsilon</math>, and ''δ''<math>\delta</math>. If <math>ALG\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 ''ε''<math>\epsilon</math> and confidence ''δ''<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 ''ε''<math>\epsilon</math> of the best possible, with probability at least <math>1 - ''δ''\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|probabilisticallyprobably 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/''ε''\epsilon</math> and <math>1/''δ''\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'''. This is a stronger notion than being learnable.

<math> \mathcal H </math> is '''PAC-learnable'''. Note that this is a stronger notion than being learnable.

==Unrestricted hypothesis space: infinite sample complexity==

<span id='No Free Lunch Theorem'></span>

One can ask whether there exists a learning algorithm so that the sample complexity is finite in the strong sense, that is, there is a bound on the number of samples needed so that the algorithm can learn any distribution over the input-output space with a specified target error. More formally, one asks whether there exists a learning algorithm <math>ALG\mathcal{A}</math>, such that, for all ''ε''<math>\epsilon, ''δ''\delta > 0</math>, there exists a positive integer ''<math>N''</math> such that for all ''<math>n'' ≥\geq ''N''</math>, we have

<math display="block">

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

</math>

</math>where <math>h_n=ALG\mathcal{A}(S_n)</math>, with <math>S_n = ((x_1,y_1),\ldots,(x_n,y_n))\sim \rho^n</math> as above. The [[No free lunch in search and optimization|No Free Lunch Theorem]] says that without restrictions on the hypothesis space <math>\mathcal H</math>, this is not the case, i.e., there always exist "bad" distributions for which the sample complexity is arbitrarily large.<ref name=":0">{{citation |last = Vapnik | first = Vladimir | title = Statistical Learning Theory | place = New York | publisher = Wiley. | year = 1998}}</ref>

Thus, in order to make statements about the rate of convergence of the quantity

<math display="block">

\sup_\rho\left(\Pr_{\rho^n}[\mathcal E(f_nh_n) - \mathcal E^*_\mathcal{H}\geq\varepsilon]\right),

</math>

one must either

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

Let ''<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>, inso particularit 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 ===

<span id='bounds'></span>

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 OFof PAC learning

|journal=J. Mach. Learn. Res.|volume=17|issue=1|pages=1319–1333|author=Steve Hanneke|year=2016|arxiv=1507.00473|url=httphttps://dlwww.acmjmlr.org/citationpapers/v17/15-389.cfm?id=2946683html}}</ref>

<math display="block">

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

</math>

where <math>VC(\mathcal H)</math> is the [[VC dimension]] of <math>\mathcal H</math>.

Moreover, any <math>(\epsilon,\delta)</math>-PAC-learning algorithm for <math>\mathcal H</math> must have sample-complexity:<ref>{{Cite journal|doi=10.1016/0890-5401(89)90002-3|title=A general lower bound on the number of examples needed for learning|journal=Information and Computation|volume=82|issue=3|pages=247|year=1989|last1=Ehrenfeucht|first1=Andrzej|last2=Haussler|first2=David|last3=Kearns|first3=Michael|last4=Valiant|first4=Leslie|doi-access=free}}</ref>

<math display="block">

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

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Thus, the sample-complexity is a linear function of the [[VC dimension]] of the hypothesis space.

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|bibcodelast1=2015arXiv150603684MMorgenstern|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|first last1=Balcan|first1=Maria-Florina|authorlink1= Maria-Florina Balcan|lastlast2=Hanneke|first2=Steve|last3=Wortman Vaughan|first3=Jennifer|doi-access Balcan= 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> It is 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 arXiv |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==