L1-norm and L2-norm regularization doc - #3586
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| This class use [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) to formulate the optimized problem built upon collected data. | ||
| If the training data does not contain enough data points (for example, to train a linear model in $n$-dimensional space, we at least need $n$ data points), | ||
| (overfitting)(https://en.wikipedia.org/wiki/Overfitting) may happen so the trained model is good at describing training data but may fail to predict correct results in unseen events. | ||
| [Regularization](https://en.wikipedia.org/wiki/Regularization_(mathematics)) is a common technique to alleviate such a phenomenon by penalizing the magnitude (usually measureed by [norm function](https://en.wikipedia.org/wiki/Norm_(mathematics))) of model parameters. |
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measureed [](start = 167, length = 9)
typo: measured #Resolved
| If the training data does not contain enough data points (for example, to train a linear model in $n$-dimensional space, we at least need $n$ data points), | ||
| (overfitting)(https://en.wikipedia.org/wiki/Overfitting) may happen so the trained model is good at describing training data but may fail to predict correct results in unseen events. | ||
| [Regularization](https://en.wikipedia.org/wiki/Regularization_(mathematics)) is a common technique to alleviate such a phenomenon by penalizing the magnitude (usually measureed by [norm function](https://en.wikipedia.org/wiki/Norm_(mathematics))) of model parameters. | ||
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. |
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penalizing [](start = 115, length = 10)
penalizes #Resolved
| /// Using L1-norm can increase sparsity of the trained $\textbf{w}_c$. | ||
| /// When working with high-dimensional data, it shrinks small weights of irrelevant features to 0 and therefore no resource will be spent on those bad features when making prediction. | ||
| /// L2-norm regularization is preferable for data that is not sparse and it largely penalizes the existence of large weights. | ||
| /// Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. |
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Togehter [](start = 8, length = 8)
typo #Resolved
| /// Using L1-norm can increase sparsity of the trained $\textbf{w}_c$. | ||
| /// When working with high-dimensional data, it shrinks small weights of irrelevant features to 0 and therefore no resource will be spent on those bad features when making prediction. | ||
| /// L2-norm regularization is preferable for data that is not sparse and it largely penalizes the existence of large weights. | ||
| /// Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. |
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Togehter [](start = 8, length = 8)
typo #Resolved
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
| Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. | ||
| For high-dimention and sparse data set, if user carefully select the coefficient of L1-norm, it is possible to achieve a good prediction quality with a model with a few of non-zeros (e.g., 1% values) in $\textbf{w}_1,\dots,\textbf{w}_m$ without affecting its . |
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. [](start = 258, length = 2)
missing a word #Resolved
| /// Using L1-norm can increase sparsity of the trained $\textbf{w}_c$. | ||
| /// When working with high-dimensional data, it shrinks small weights of irrelevant features to 0 and therefore no resource will be spent on those bad features when making prediction. | ||
| /// L2-norm regularization is preferable for data that is not sparse and it largely penalizes the existence of large weights. | ||
| /// Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. |
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Togehter [](start = 8, length = 8)
uses #Pending
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| /// An accurate model with extreme coefficient values would be penalized more, but a less accurate model with more conservative values would be penalized less. | ||
| /// | ||
| /// This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization): a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| /// This class use [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) to formulate the optimized problem built upon collected data. |
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optimized [](start = 129, length = 9)
optimization ? #Resolved
| /// This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization): a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| /// This class use [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) to formulate the optimized problem built upon collected data. | ||
| /// If the training data does not contain enough data points (for example, to train a linear model in $n$-dimensional space, we at least need $n$ data points), | ||
| /// (overfitting)(https://en.wikipedia.org/wiki/Overfitting) may happen so the trained model is good at describing training data but may fail to predict correct results in unseen events. |
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so [](start = 76, length = 2)
when #Resolved
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| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
| Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. | ||
| For high-dimention and sparse data set, if user carefully select the coefficient of L1-norm, it is possible to achieve a good prediction quality with a model with a few of non-zeros (e.g., 1% values) in $\textbf{w}_1,\dots,\textbf{w}_m$ without affecting its . |
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dimention [](start = 9, length = 9)
dimension
please use spell checker plugin #Resolved
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It doesn't show anything.. It was ok yesterday. Let me try Vim's.
In reply to: 278724688 [](ancestors = 278724688)
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you're right, the plugin is useless in markdown sections. not sure it there's option to make it look at those sections. You might be able to temporarily remove the CDATA tags, and see if typos show up like the way they do in
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
| Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. | ||
| For high-dimention and sparse data set, if user carefully select the coefficient of L1-norm, it is possible to achieve a good prediction quality with a model with a few of non-zeros (e.g., 1% values) in $\textbf{w}_1,\dots,\textbf{w}_m$ without affecting its . |
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with a few of non-zeros [](start = 158, length = 23)
that has only a few non-zero weights #Resolved
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
| Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. | ||
| For high-dimention and sparse data set, if user carefully select the coefficient of L1-norm, it is possible to achieve a good prediction quality with a model with a few of non-zeros (e.g., 1% values) in $\textbf{w}_1,\dots,\textbf{w}_m$ without affecting its . |
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values [](start = 192, length = 6)
1% of weights #Resolved
| (overfitting)(https://en.wikipedia.org/wiki/Overfitting) may happen so the trained model is good at describing training data but may fail to predict correct results in unseen events. | ||
| [Regularization](https://en.wikipedia.org/wiki/Regularization_(mathematics)) is a common technique to alleviate such a phenomenon by penalizing the magnitude (usually measureed by [norm function](https://en.wikipedia.org/wiki/Norm_(mathematics))) of model parameters. | ||
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. |
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add blank line for new paragraph. #Resolved
| [Regularization](https://en.wikipedia.org/wiki/Regularization_(mathematics)) is a common technique to alleviate such a phenomenon by penalizing the magnitude (usually measureed by [norm function](https://en.wikipedia.org/wiki/Norm_(mathematics))) of model parameters. | ||
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) which penalizing a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations. | ||
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
| Togehter with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the $\textbf{w}_1,\dots,\textbf{w}_m$. |
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$\textbf{w}_1,\dots,\textbf{w}_m$ [](start = 110, length = 33)
let's call this something like, model weights, or model parameters, and not repeat it below. #Resolved
| values to the error of the hypothesis. An accurate model with extreme | ||
| coefficient values would be penalized more, but a less accurate model with more | ||
| conservative values would be penalized less. | ||
| This class use [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) to formulate the optimized problem built upon collected data. |
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This class use empricial risk minimization to formulat [](start = 0, length = 115)
please limit the line width, so that it's easy to review without going left and right. it's also a good practice for viewing the file on github. #Resolved
| This class use [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) | ||
| to formulate the optimization problem built upon collected data. | ||
| If the training data does not contain enough data points | ||
| (for example, to train a linear model in $n$-dimensional space, we at least need $n$ data points), |
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at least need
$n$ [](start = 67, length = 17)
need at least
| [Regularization](https://en.wikipedia.org/wiki/Regularization_(mathematics)) is a common technique to alleviate | ||
| such a phenomenon by penalizing the magnitude (usually measured by | ||
| [norm function](https://en.wikipedia.org/wiki/Norm_(mathematics))) of model parameters. | ||
| This trainer supports [elastic net regularization](https://en.wikipedia.org/wiki/Elastic_net_regularization) |
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) [](start = 107, length = 1)
Add comma after ) (in general whenever the next word is "which") #Resolved
| Sometimes, using L2-norm leads to a better prediction quality, so users may still want to try it and fine tune the coefficients of L1-norm and L2-norm. | ||
| Note that conceptually, using L1-norm implies that the distribution of all model parameters is a | ||
| [Laplace distribution](https://en.wikipedia.org/wiki/Laplace_distribution) while | ||
| L2-norm means that a [Gaussian distribution](https://en.wikipedia.org/wiki/Normal_distribution) for them. |
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means that a [](start = 8, length = 12)
"assumes a Gaussian distribution" or "implies a Gaussian distribution" #Resolved
| L2-norm means that a [Gaussian distribution](https://en.wikipedia.org/wiki/Normal_distribution) for them. | ||
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| An aggressive regularization (that is, assigning large coefficients to L1-norm or L2-norm regularization terms) | ||
| can harm predictive capacity by excluding important variables out of the model. |
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out of the model [](start = 62, length = 16)
from the model #Resolved
| /// ### Training Algorithm Details | ||
| /// The optimization algorithm is an extension of (http://jmlr.org/papers/volume14/shalev-shwartz13a/shalev-shwartz13a.pdf) following a similar path proposed in an earlier [paper](https://www.csie.ntu.edu.tw/~cjlin/papers/maxent_dual.pdf). | ||
| /// It is usually much faster than [L-BFGS](https://en.wikipedia.org/wiki/Limited-memory_BFGS) and [truncated Newton methods](https://en.wikipedia.org/wiki/Truncated_Newton_method) for large-scale and sparse data set. | ||
| /// The optimization algorithm is an extension of (http://jmlr.org/papers/volume14/shalev-shwartz13a/shalev-shwartz13a.pdf) |
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(http://jmlr.org/papers/volume14/shalev-shwartz13a/shalev-shwartz13a.pdf) [](start = 54, length = 73)
Maybe give this link some display text [like this] ? #Resolved
| /// The optimization algorithm is an extension of (http://jmlr.org/papers/volume14/shalev-shwartz13a/shalev-shwartz13a.pdf) | ||
| /// following a similar path proposed in an earlier [paper](https://www.csie.ntu.edu.tw/~cjlin/papers/maxent_dual.pdf). | ||
| /// It is usually much faster than [L-BFGS](https://en.wikipedia.org/wiki/Limited-memory_BFGS) and | ||
| /// [truncated Newton methods](https://en.wikipedia.org/wiki/Truncated_Newton_method) for large-scale and sparse data set. |
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data set [](start = 117, length = 8)
data sets #Resolved
| to formulate the optimization problem built upon collected data. | ||
| If the training data does not contain enough data points | ||
| (for example, to train a linear model in $n$-dimensional space, we at least need $n$ data points), | ||
| (overfitting)(https://en.wikipedia.org/wiki/Overfitting) may happen so that |
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(overfitting) [](start = 0, length = 13)
square brackets [overfitting] #Resolved
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
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| Together with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the model weights, $\textbf{w}_1,\dots,\textbf{w}_m$. | ||
| For high-dimension and sparse data set, if users carefully select the coefficient of L1-norm, |
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high-dimension [](start = 4, length = 14)
"high-dimensional" perhaps? #Resolved
| L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects. | ||
|
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| Together with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the model weights, $\textbf{w}_1,\dots,\textbf{w}_m$. | ||
| For high-dimension and sparse data set, if users carefully select the coefficient of L1-norm, |
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data set [](start = 30, length = 8)
data sets #Resolved
| @@ -0,0 +1,27 @@ | |||
| This class use [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) | |||
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Why is this in the l1-norm and l2-norm regularization include? #Resolved
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We are optimize regularized ERM. ER is also known as loss function. #Resolved
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I will add Note that empricial risk is usually measured by applying a loss function on the model's predictions on collected data points.
In reply to: 278778511 [](ancestors = 278778511)
| /// | ||
| /// An aggressive regularization (that is, assigning large coefficients to L1-norm or L2-norm regularization terms) can harm predictive capacity by excluding important variables out of the model. | ||
| /// Therefore, choosing the right regularization coefficients is important in practice. | ||
| /// [!include[regularization](~/../docs/samples/docs/api-reference/regularization-l1-l2.md)] |
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Why are we including this in the base class definition for multiclass and in the derived classes for binary classification? #ByDesign
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It's a common behavior shared by all derived classes. #Resolved
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But it is inconsistent between the multiclass and binary ... #Resolved
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The reason is that they are written by different persons. Multiclass' XML doc is referenced in derived classes' XML docs, so there is no difference actually. I honestly don't have much time for writing style.
In reply to: 278777687 [](ancestors = 278777687)
Co-Authored-By: wschin <wschin@outlook.com>
Co-Authored-By: wschin <wschin@outlook.com>
Codecov Report
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Coverage ? 72.76%
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| /// | ||
| /// An aggressive regularization (that is, assigning large coefficients to L1-norm or L2-norm regularization terms) can harm predictive capacity by excluding important variables out of the model. | ||
| /// Therefore, choosing the right regularization coefficients is important in practice. | ||
| /// [!include[regularization](~/../docs/samples/docs/api-reference/regularization-l1-l2.md)] |
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But it is inconsistent between the multiclass and binary ... #Resolved
| @@ -0,0 +1,27 @@ | |||
| This class uses [empirical risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) | |||
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It's not clear how this connects with l1-norm and l2-norm regularization. Let me know if you want to discuss this offline. #ByDesign
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ERM ---> without enough data ---> overfit ---> use regularization to overcome overfit.
In reply to: 279009294 [](ancestors = 279009294)
| @@ -0,0 +1,27 @@ | |||
| This class uses [empricial risk minimization](https://en.wikipedia.org/wiki/Empirical_risk_minimization) (i.e., ERM) | |||
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This [](start = 0, length = 5)
Please add a header '### Regularization' so that the following text becomes a separate section. Also move it after all the algo details.
#ByDesign
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No. I don't only mean regularization. It is a brief introduction to the whole optimization problem.
In reply to: 284914354 [](ancestors = 284914354)
| Minimization.](http://www.jmlr.org/papers/volume14/shalev-shwartz13a/shalev-shwartz13a.pdf) | ||
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| Check the See Also section for links to examples of the usage. No newline at end of file |
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you don't need this file. you can just move it to the end of algo-details.sdca.md. It's ok that regularization details come after this, b/c it will be a separate section. #ByDesign
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It looks super strange... Regularization is the reason why SDCA can exist. As you may have known, SDCA solves "Dual form" of the original optimization problem. Without regularization, that dual form may not exist. Consequently, user should learn regularization before jumping into details of SDCA.
In reply to: 284914930 [](ancestors = 284914930)
ghost
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Fix #3356.