SRM: Structural Risk Minimization

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Category: { Machine Learning::Theories }
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Summary: ERM ERM: Empirical Risk Minimization In a learning problem The Learning Problem The learning problem posed by Vapnik:1 Given a sample: $\{z_i\}$ in the probability space $Z$; Assuming a probability measure on the probability space $Z$; Assuming a set of functions $Q(z, \alpha)$ (e.g. loss functions), where $\alpha$ is a set of parameters; A risk functional to be minimized by tunning “the handles” $\alpha$, $R(\alpha)$. The risk functional is $$ R(\alpha) = \int Q(z, \alpha) \,\mathrm d F(z). $$ A learning problem is … may lead to overfitting since ERM only selects the model to fit the train data well.
Pages: 7

ERM: Empirical Risk Minimization

Published:
Category: { Machine Learning::Theories }
Tags:
Summary: In a learning problem The Learning Problem The learning problem posed by Vapnik:1 Given a sample: $\{z_i\}$ in the probability space $Z$; Assuming a probability measure on the probability space $Z$; Assuming a set of functions $Q(z, \alpha)$ (e.g. loss functions), where $\alpha$ is a set of parameters; A risk functional to be minimized by tunning “the handles” $\alpha$, $R(\alpha)$. The risk functional is $$ R(\alpha) = \int Q(z, \alpha) \,\mathrm d F(z). $$ A learning problem is the minimization of this risk. Vapnik2000 … , empirical risk $R$ is a measurement the goodness of fit based on empirical information.
Pages: 7

The Learning Problem

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Category: { Machine Learning::Theories }
Summary: The learning problem posed by Vapnik:1 Given a sample: $\{z_i\}$ in the probability space $Z$; Assuming a probability measure on the probability space $Z$; Assuming a set of functions $Q(z, \alpha)$ (e.g. loss functions), where $\alpha$ is a set of parameters; A risk functional to be minimized by tunning “the handles” $\alpha$, $R(\alpha)$. The risk functional is $$ R(\alpha) = \int Q(z, \alpha) \,\mathrm d F(z). $$ A learning problem is the minimization of this risk. Vapnik2000 Vladimir N. Vapnik. The Nature of Statistical Learning Theory. 2000. doi:10.1007/978-1-4757-3264-1  ↩︎
Pages: 7

Cross Validation

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Category: { Machine Learning::Theories }
Summary: Cross validation is a method to estimate the risk The Learning Problem The learning problem posed by Vapnik:1 Given a sample: $\{z_i\}$ in the probability space $Z$; Assuming a probability measure on the probability space $Z$; Assuming a set of functions $Q(z, \alpha)$ (e.g. loss functions), where $\alpha$ is a set of parameters; A risk functional to be minimized by tunning “the handles” $\alpha$, $R(\alpha)$. The risk functional is $$ R(\alpha) = \int Q(z, \alpha) \,\mathrm d F(z). $$ A learning problem is the minimization of this risk. Vapnik2000 … . To perform cross validation, we split the train dataset $\mathcal D$ into $k$ folds, with each fold denoted as $\mathcal D_k$.
Pages: 7

Noise Contrastive Estimation: NCE

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Category: { Machine Learning::Theories }
Summary: Noise contrastive estimation (NCE) objective function is1 $$ \mathcal L = \mathbb E_{x, x^{+}, x^{-}} \left[ - \ln \frac{ C(x, x^{+})}{ C(x,x^{+}) + C(x,x^{-}) } \right], $$ where $x^{+}$ represents data similar to $x$, $x^{-}$ represents data dissimilar to $x$, $C(\cdot, \cdot)$ is a function to compute the similarities. For example, we can use $$ C(x, x^{+}) = e^{ f(x)^T f(x^{+}) }, $$ so that the objective function becomes $$ \mathcal L = \mathbb E_{x, x^{+}, x^{-}} \left[ - \ln \frac{ e^{ f(x)^T f(x^{+}) } }{ e^{ f(x)^T f(x^{+}) } + e^{ f(x)^T f(x^{-}) } } \right]. $$
Pages: 7

Shatter

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Category: { Machine Learning::Theories }
Summary: Given a set $\mathcal S$, and a class (collection of sets) $\mathcal H$. For any subset of $\mathcal S$, denoted as $\mathcal s$, if we have an element of class $\mathcal H$, denoted as $\mathcal h$, that leads to1 $$ \mathcal h \cap \mathcal S = \mathcal s. $$ Since the power set of $\mathcal S$ ($P(\mathcal S)$) contains all the possible subsets of $\mathcal S$, we can also rephrase the concept using power set. If we can find the power set $P(\mathcal S)$ by looking into intersections of elements $\mathcal h$ of $\mathcal H$ ($\mathcal h\in \mathcal H$), then we say $\mathcal H$ shatters $\mathcal S$ 1.
Pages: 7