Bayesian interpretation of regularization

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In statistical learning theory, a representer theorem is any of several related results stating that a minimizer $f^{*}$ of a regularized empirical risk function defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.

Formal Statement

The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola:

Theorem: Let ${\mathcal {X}}$ be a nonempty set and $k$ a positive-definite real-valued kernel on ${\mathcal {X}}\times {\mathcal {X}}$ with corresponding reproducing kernel Hilbert space $H_{k}$. Given a training sample $(x_{1},y_{1}),\dotsc ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R}$, a strictly monotonically increasing real-valued function $g\colon [0,\infty )\to \mathbb {R}$, and an arbitrary empirical risk function $E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{m}\to \mathbb {R} \cup \lbrace \infty \rbrace$, then for any $f^{*}\in H_{k}$ satisfying

$f^{*}=\operatorname {argmin} _{f\in H_{k}}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),...,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right)\right\rbrace ,\quad (*)$

$f^{*}$ admits a representation of the form:

$f^{*}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),$

where $\alpha _{i}\in \mathbb {R}$ for all $1\leq i\leq n$.

Proof: Define a mapping

${\begin{aligned}\varphi \colon {\mathcal {X}}&\to \mathbb {R} ^{\mathcal {X}}\\\varphi (x)&=k(\cdot ,x)\end{aligned}}$

(so that $\varphi (x)=k(\cdot ,x)$ is itself a map ${\mathcal {X}}\to \mathbb {R}$). Since $k$ is reproducing kernel, then

$\varphi (x)(x')=k(x',x)=\langle \varphi (x'),\varphi (x)\rangle ,$

where $\langle \cdot ,\cdot \rangle$ is the inner product on $H_{k}$.

Given any $x_{1},...,x_{n}$, one can use orthogonal projection to decompose any $f\in H_{k}$ into a sum of two function, one lying in $\operatorname {span} \left\lbrace \varphi (x_{1}),...,\varphi (x_{n})\right\rbrace$, and the other lying in the orthogonal complement:

$f=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,$

where $\langle v,\varphi (x_{i})\rangle =0$ for all $i$.

The above orthogonal decomposition and the reproducing property together show that applying $f$ to any training point $x_{j}$ produces

$f(x_{j})=\left\langle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,\varphi (x_{j})\right\rangle =\sum _{i=1}^{n}\alpha _{i}\langle \varphi (x_{i}),\varphi (x_{j})\rangle ,$

which we observe is independent of $v$. Consequently, the value of the empirical risk $E$ in (*) is likewise independent of $v$. For the second term (the regularization term), since $v$ is orthogonal to $\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})$ and $g$ is strictly monotonic, we have

${\begin{aligned}g\left(\lVert f\rVert \right)&=g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v\rVert \right)\\&=g\left({\sqrt {\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert ^{2}+\lVert v\rVert ^{2}}}\right)\\&\geq g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert \right).\end{aligned}}$

Therefore setting $v=0$ does not affect the first term of (*), while it strictly decreasing the second term. Consequently, any minimizer $f^{*}$ in (*) must have $v=0$, i.e., it must be of the form

$f^{*}(\cdot )=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),$

which is the desired result.

Generalizations: Variations on a theme by Kimeldorf and Wahba

The Theorem stated above is a particular example of a family of results that are collectively referred to as "Representer Theorems"; here we describe several such.

The first statement of a Representer Theorem was due to Kimeldorf and Wahba for the special case in which

${\begin{aligned}E\left((x_{1},y_{1},f(x_{1})),...,(x_{n},y_{n},f(x_{n}))\right)&={\frac {1}{n}}\sum _{i=1}^{n}(f(x_{i})-y_{i})^{2},\\g(\lVert f\rVert )&=\lambda \lVert f\rVert ^{2}\end{aligned}}$

for $\lambda >0$. Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function $g(\cdot )$ of the Hilbert space norm.

It is possible to generalize further by augmenting the regularized empirical risk function through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization

${\tilde {f}}^{*}=\operatorname {argmin} \left\lbrace E\left((x_{1},y_{1},{\tilde {f}}(x_{1})),...,(x_{n},y_{n},{\tilde {f}}(x_{n}))\right)+g\left(\lVert f\rVert \right)\mid {\tilde {f}}=f+h\in H_{k}\oplus \operatorname {span} \lbrace \psi _{p}\mid 1\leq p\leq M\rbrace \right\rbrace ,\quad (\dagger )$

i.e., we consider functions of the form ${\tilde {f}}=f+h$, where $f\in H_{k}$ and $h$ is an unpenalized function lying in the span of a finite set of real-valued functions $\lbrace \psi _{p}\colon {\mathcal {X}}\to \mathbb {R} \mid 1\leq p\leq M\rbrace$. Under the assumption that the $m\times M$ matrix $\left(\psi _{p}(x_{i})\right)_{ip}$ has rank $M$, they show that the minimizer ${\tilde {f}}^{*}$ in $(\dagger )$ admits a representation of the form

${\tilde {f}}^{*}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i})+\sum _{p=1}^{M}\beta _{p}\psi _{p}(\cdot )$

where $\alpha _{i},\beta _{p}\in \mathbb {R}$ and the $\beta _{p}$ are all uniquely determined.

The conditions under which a Representer Theorem exists were investigated by Argyriou, Miccheli, and Pontil, who proved the following:

Theorem: Let ${\mathcal {X}}$ be a nonempty set, $k$ a positive-definite real-valued kernel on ${\mathcal {X}}\times {\mathcal {X}}$ with corresponding reproducing kernel Hilbert space $H_{k}$, and let $R\colon H_{k}\to \mathbb {R}$ be a differentiable regularization function. Then given a training sample $(x_{1},y_{1}),...,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R}$ and an arbitrary empirical risk function $E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{m}\to \mathbb {R} \cup \lbrace \infty \rbrace$, a minimizer

$f^{*}=\operatorname {argmin} _{f\in H_{k}}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),...,(x_{n},y_{n},f(x_{n}))\right)+R(f)\right\rbrace \quad (\ddagger )$

of the regularized empirical risk minimization problem admits a representation of the form

$f^{*}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),$

where $\alpha _{i}\in \mathbb {R}$ for all $1\leq i\leq n$, if and only if there exists a nondecreasing function $h\colon [0,\infty )\to \mathbb {R}$ for which

$R(f)=h(\lVert f\rVert ).$

Effectively, this result provides a necessary and sufficient condition on a differentiable regularizer $R(\cdot )$ under which the corresponding regularized empirical risk minimization $(\ddagger )$ will have a Representer Theorem. In particular, this shows that a broad class of regularized risk minimizations (much broader than those originally considered by Kimeldorf and Wahba) have Representer Theorems.

Applications

Representer theorems are useful from a practical standpoint because they dramatically simplify the regularized empirical risk minimization problem $(\ddagger )$. In most interesting applications, the search domain $H_{k}$ for the minimization will be an infinite-dimensional subspace of $L^{2}({\mathcal {X}})$, and therefore the search (as written) does not admit implementation on finite-memory and finite-precision computers. In contrast, the representation of $f^{*}(\cdot )$ afforded by a representer theorem reduces the original (infinite-dimensional) minimization problem to a search for the optimal $n$-dimensional vector of coefficients $\alpha =(\alpha _{1},...,\alpha _{n})\in \mathbb {R} ^{n}$; $\alpha$ can then be obtained by applying any standard function minimization algorithm. Consequently, representer theorems provide the theoretical basis for the reduction of the general machine learning problem to algorithms that can actually be implemented on computers in practice.

See also

• Mercer's theorem

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang