From 9460449a4cd8275183a43b57c583b12389a4f3bb Mon Sep 17 00:00:00 2001 From: Elijah Melton Date: Sun, 5 Jan 2025 00:00:04 -0800 Subject: [PATCH] sync changes: - A Untitled.java - A natural-language-processing/multinomial-logistic-regression.md - A site/Untitled.java --- Untitled.java | 142 +++++++++++++ .../multinomial-logistic-regression.md | 17 ++ site/Untitled.java | 142 +++++++++++++ site/categories/algorithm analysis.html | 2 +- site/categories/algorithms.html | 2 +- site/categories/computer science.html | 2 +- site/categories/database design.html | 2 +- site/categories/database systems.html | 2 +- site/categories/distributed systems.html | 2 +- site/categories/graph theory.html | 2 +- site/categories/index.html | 2 +- site/categories/mathematics.html | 2 +- site/categories/operations research.html | 2 +- site/categories/software engineering.html | 2 +- site/index.html | 14 +- .../multinomial-logistic-regression.html | 200 ++++++++++++++++++ site/tags/acyclic graphs.html | 2 +- site/tags/algorithm-analysis.html | 2 +- site/tags/algorithm.html | 2 +- site/tags/algorithms.html | 2 +- site/tags/approximation.html | 2 +- site/tags/asymptotic notation.html | 2 +- site/tags/batch processing.html | 2 +- site/tags/bipartite graphs.html | 2 +- site/tags/bipartite matching.html | 2 +- site/tags/breadth-first search.html | 2 +- site/tags/column-oriented storage.html | 2 +- site/tags/compatibility.html | 2 +- site/tags/complexity analysis.html | 2 +- site/tags/complexity-analysis.html | 2 +- site/tags/connected components.html | 2 +- site/tags/connected graphs.html | 2 +- site/tags/data analysis.html | 2 +- site/tags/data modeling.html | 2 +- site/tags/data replication.html | 2 +- site/tags/data serialization.html | 2 +- site/tags/data structures.html | 2 +- site/tags/data systems.html | 2 +- site/tags/depth first search.html | 2 +- site/tags/depth-first search.html | 2 +- site/tags/distributed filesystems.html | 2 +- site/tags/document databases.html | 2 +- site/tags/dynamic-programming.html | 2 +- site/tags/efficiency.html | 2 +- site/tags/encoding formats.html | 2 +- site/tags/etl.html | 2 +- site/tags/failover.html | 2 +- site/tags/ford-fulkerson algorithm.html | 2 +- site/tags/gale-shapley.html | 2 +- site/tags/graph coloring.html | 2 +- site/tags/graph databases.html | 2 +- site/tags/graph fundamentals.html | 2 +- site/tags/graph properties.html | 2 +- site/tags/graph representation.html | 2 +- site/tags/graph theory.html | 2 +- site/tags/graph traversal.html | 2 +- site/tags/graph-theory.html | 2 +- site/tags/graph-traversal.html | 2 +- site/tags/graph.html | 2 +- site/tags/greedy-algorithms.html | 2 +- site/tags/independent set.html | 2 +- site/tags/index.html | 2 +- site/tags/indexing.html | 2 +- site/tags/induction proofs.html | 2 +- site/tags/induction.html | 2 +- site/tags/interval.html | 2 +- site/tags/leader-follower model.html | 2 +- site/tags/linear programs.html | 2 +- site/tags/linear systems.html | 2 +- site/tags/maintainability.html | 2 +- site/tags/mapreduce.html | 2 +- site/tags/matching.html | 2 +- site/tags/max flow min cut.html | 2 +- site/tags/message passing.html | 2 +- site/tags/odd cycles.html | 2 +- site/tags/oltp vs olap.html | 2 +- site/tags/optimization.html | 2 +- site/tags/partitioning.html | 2 +- site/tags/performance.html | 2 +- site/tags/pigeonhole principle.html | 2 +- site/tags/problem-solving.html | 2 +- site/tags/proof techniques.html | 2 +- site/tags/query languages.html | 2 +- site/tags/relational databases.html | 2 +- site/tags/reliability.html | 2 +- site/tags/replication logs.html | 2 +- site/tags/scalability.html | 2 +- site/tags/scheduling.html | 2 +- site/tags/schema evolution.html | 2 +- site/tags/set cover.html | 2 +- site/tags/shortest-paths.html | 2 +- site/tags/spanning trees.html | 2 +- site/tags/stable matching.html | 2 +- site/tags/synchronous vs asynchronous.html | 2 +- site/tags/time complexity.html | 2 +- site/tags/trees.html | 2 +- site/tags/vertex cover.html | 2 +- 97 files changed, 600 insertions(+), 99 deletions(-) create mode 100644 Untitled.java create mode 100644 natural-language-processing/multinomial-logistic-regression.md create mode 100644 site/Untitled.java create mode 100644 site/natural-language-processing/multinomial-logistic-regression.html diff --git a/Untitled.java b/Untitled.java new file mode 100644 index 0000000..81c47ec --- /dev/null +++ b/Untitled.java @@ -0,0 +1,142 @@ +# Multinomial Logistic Regression (MLR) + +## Introduction + +Multinomial Logistic Regression (MLR) is a generalization of binary logistic regression that allows for classification into multiple classes. It's a powerful and interpretable model widely used in Natural Language Processing (NLP) and other fields. + +## Key Concepts + +### Features + +In MLR, features are functions that map input-output pairs to real numbers: + +$f_j: \mathcal{V}^* \times \mathcal{L} \rightarrow \mathbb{R}$ + +Where $\mathcal{V}^*$ is the set of all possible input sequences and $\mathcal{L}$ is the set of all possible labels. + +A common template for features is: + +$f_{\ell,\phi}(x, y) = \phi(x) \cdot \mathbb{1}\{y = \ell\}$ + +Where $\phi(x)$ is some function of the input and $\mathbb{1}\{y = \ell\}$ is an indicator function. + +```python +import numpy as np + +def feature_template(phi, l): + def feature(x, y): + return phi(x) * (y == l) + return feature + +# Example usage +def word_count(x, word): + return x.lower().count(word) + +f_sports_vodka = feature_template(lambda x: word_count(x, 'vodka'), 'sports') +``` + +### Model Definition + +An MLR model is defined by: + +1. A set of feature functions $f_1, ..., f_d$ +2. A weight vector $\theta \in \mathbb{R}^d$ + +The score for a given input-output pair is: + +$\text{score}_{\text{MLR}}(x, y; \theta) = \sum_{j=1}^d \theta_j f_j(x, y) = \theta^T f(x, y)$ + +The classification rule is: + +$\text{classify}_{\text{MLR}}(x) = \arg\max_{y \in \mathcal{L}} \text{score}_{\text{MLR}}(x, y; \theta)$ + +```python +def score_mlr(x, y, theta, features): + return sum(theta[j] * f(x, y) for j, f in enumerate(features)) + +def classify_mlr(x, theta, features, labels): + return max(labels, key=lambda y: score_mlr(x, y, theta, features)) +``` + +### Probabilistic Interpretation + +MLR defines a probability distribution over labels: + +$p_{\text{MLR}}(Y | X = x; \theta) = \text{softmax}(\langle\text{score}_{\text{MLR}}(x, \ell; \theta)\rangle_{\ell \in \mathcal{L}})$ + +Where softmax is defined as: + +$\text{softmax}(t_1, ..., t_k) = (\frac{e^{t_1}}{\sum_{j=1}^k e^{t_j}}, ..., \frac{e^{t_k}}{\sum_{j=1}^k e^{t_j}})$ + +```python +def softmax(scores): + exp_scores = np.exp(scores) + return exp_scores / np.sum(exp_scores) + +def p_mlr(x, theta, features, labels): + scores = [score_mlr(x, y, theta, features) for y in labels] + return softmax(scores) +``` + +## Learning + +The learning objective for MLR is to minimize the negative log-likelihood (also known as cross-entropy loss) plus a regularization term: + +$\theta^* = \arg\min_{\theta \in \mathbb{R}^d} \sum_{i=1}^n -\log p_{\text{MLR}}(Y = y_i | X = x_i; \theta) + \lambda \|\theta\|_p^p$ + +Where $\lambda > 0$ is a hyperparameter and $p = 1$ or $2$ for L1 or L2 regularization respectively. + +```python +def negative_log_likelihood(theta, X, y, features, labels, lambda_, p): + nll = sum(-np.log(p_mlr(x, theta, features, labels)[labels.index(y_i)]) + for x, y_i in zip(X, y)) + reg = lambda_ * np.linalg.norm(theta, ord=p)**p + return nll + reg + +from scipy.optimize import minimize + +def train_mlr(X, y, features, labels, lambda_=0.1, p=2): + d = len(features) + theta_init = np.zeros(d) + result = minimize(lambda theta: negative_log_likelihood(theta, X, y, features, labels, lambda_, p), + theta_init, method='L-BFGS-B') + return result.x +``` + +## Implementation Considerations + +1. Feature engineering is crucial for MLR performance. +2. L1 regularization can lead to sparse models, effectively performing feature selection. +3. MLR can be computationally expensive for large label sets due to the normalization factor in the softmax. +4. Stochastic gradient descent or its variants are often used for large-scale problems. + +## Evaluation + +Common evaluation metrics include accuracy, precision, recall, and F1 score. Cross-validation is often used to get more reliable estimates of model performance. + +```python +from sklearn.model_selection import cross_val_score +from sklearn.metrics import make_scorer, accuracy_score, precision_score, recall_score, f1_score + +def mlr_classifier(theta, features, labels): + return lambda X: [classify_mlr(x, theta, features, labels) for x in X] + +def evaluate_mlr(X, y, features, labels, lambda_=0.1, p=2, cv=5): + theta = train_mlr(X, y, features, labels, lambda_, p) + clf = mlr_classifier(theta, features, labels) + + scorers = { + 'accuracy': make_scorer(accuracy_score), + 'precision': make_scorer(precision_score, average='weighted'), + 'recall': make_scorer(recall_score, average='weighted'), + 'f1': make_scorer(f1_score, average='weighted') + } + + scores = {metric: cross_val_score(clf, X, y, cv=cv, scoring=scorer) + for metric, scorer in scorers.items()} + + return {metric: (np.mean(score), np.std(score)) for metric, score in scores.items()} +``` + +This implementation provides a comprehensive overview of Multinomial Logistic Regression, including its key concepts, model definition, learning process, and evaluation methods. The code examples demonstrate how to implement these concepts using NumPy and Python. + diff --git a/natural-language-processing/multinomial-logistic-regression.md b/natural-language-processing/multinomial-logistic-regression.md new file mode 100644 index 0000000..365da5b --- /dev/null +++ b/natural-language-processing/multinomial-logistic-regression.md @@ -0,0 +1,17 @@ +# Multinomial Logistic Regression + +## Classification + +Input can be anything (document, image, etc.) and output is a class label from the finite set $\mathcal{L}$. + +$$ +classify : \mathcal{V}* \rightarrow \mathcal{L} +$$ + +$\mathcal{V}$ is the set of words in our vocabulary. + +$X$ is a random variable representing the input, in a given instance taking values from $\mathcal{V}*$. + +$Y$ is a random variable representing the output, taking values from $\mathcal{L}$. + +$p(X, Y)$ is the "true" distribution of labeled texts. $p(Y)$ is the distribution of labels. We don't normally know this without looking at the data. diff --git a/site/Untitled.java b/site/Untitled.java new file mode 100644 index 0000000..81c47ec --- /dev/null +++ b/site/Untitled.java @@ -0,0 +1,142 @@ +# Multinomial Logistic Regression (MLR) + +## Introduction + +Multinomial Logistic Regression (MLR) is a generalization of binary logistic regression that allows for classification into multiple classes. It's a powerful and interpretable model widely used in Natural Language Processing (NLP) and other fields. + +## Key Concepts + +### Features + +In MLR, features are functions that map input-output pairs to real numbers: + +$f_j: \mathcal{V}^* \times \mathcal{L} \rightarrow \mathbb{R}$ + +Where $\mathcal{V}^*$ is the set of all possible input sequences and $\mathcal{L}$ is the set of all possible labels. + +A common template for features is: + +$f_{\ell,\phi}(x, y) = \phi(x) \cdot \mathbb{1}\{y = \ell\}$ + +Where $\phi(x)$ is some function of the input and $\mathbb{1}\{y = \ell\}$ is an indicator function. + +```python +import numpy as np + +def feature_template(phi, l): + def feature(x, y): + return phi(x) * (y == l) + return feature + +# Example usage +def word_count(x, word): + return x.lower().count(word) + +f_sports_vodka = feature_template(lambda x: word_count(x, 'vodka'), 'sports') +``` + +### Model Definition + +An MLR model is defined by: + +1. A set of feature functions $f_1, ..., f_d$ +2. A weight vector $\theta \in \mathbb{R}^d$ + +The score for a given input-output pair is: + +$\text{score}_{\text{MLR}}(x, y; \theta) = \sum_{j=1}^d \theta_j f_j(x, y) = \theta^T f(x, y)$ + +The classification rule is: + +$\text{classify}_{\text{MLR}}(x) = \arg\max_{y \in \mathcal{L}} \text{score}_{\text{MLR}}(x, y; \theta)$ + +```python +def score_mlr(x, y, theta, features): + return sum(theta[j] * f(x, y) for j, f in enumerate(features)) + +def classify_mlr(x, theta, features, labels): + return max(labels, key=lambda y: score_mlr(x, y, theta, features)) +``` + +### Probabilistic Interpretation + +MLR defines a probability distribution over labels: + +$p_{\text{MLR}}(Y | X = x; \theta) = \text{softmax}(\langle\text{score}_{\text{MLR}}(x, \ell; \theta)\rangle_{\ell \in \mathcal{L}})$ + +Where softmax is defined as: + +$\text{softmax}(t_1, ..., t_k) = (\frac{e^{t_1}}{\sum_{j=1}^k e^{t_j}}, ..., \frac{e^{t_k}}{\sum_{j=1}^k e^{t_j}})$ + +```python +def softmax(scores): + exp_scores = np.exp(scores) + return exp_scores / np.sum(exp_scores) + +def p_mlr(x, theta, features, labels): + scores = [score_mlr(x, y, theta, features) for y in labels] + return softmax(scores) +``` + +## Learning + +The learning objective for MLR is to minimize the negative log-likelihood (also known as cross-entropy loss) plus a regularization term: + +$\theta^* = \arg\min_{\theta \in \mathbb{R}^d} \sum_{i=1}^n -\log p_{\text{MLR}}(Y = y_i | X = x_i; \theta) + \lambda \|\theta\|_p^p$ + +Where $\lambda > 0$ is a hyperparameter and $p = 1$ or $2$ for L1 or L2 regularization respectively. + +```python +def negative_log_likelihood(theta, X, y, features, labels, lambda_, p): + nll = sum(-np.log(p_mlr(x, theta, features, labels)[labels.index(y_i)]) + for x, y_i in zip(X, y)) + reg = lambda_ * np.linalg.norm(theta, ord=p)**p + return nll + reg + +from scipy.optimize import minimize + +def train_mlr(X, y, features, labels, lambda_=0.1, p=2): + d = len(features) + theta_init = np.zeros(d) + result = minimize(lambda theta: negative_log_likelihood(theta, X, y, features, labels, lambda_, p), + theta_init, method='L-BFGS-B') + return result.x +``` + +## Implementation Considerations + +1. Feature engineering is crucial for MLR performance. +2. L1 regularization can lead to sparse models, effectively performing feature selection. +3. MLR can be computationally expensive for large label sets due to the normalization factor in the softmax. +4. Stochastic gradient descent or its variants are often used for large-scale problems. + +## Evaluation + +Common evaluation metrics include accuracy, precision, recall, and F1 score. Cross-validation is often used to get more reliable estimates of model performance. + +```python +from sklearn.model_selection import cross_val_score +from sklearn.metrics import make_scorer, accuracy_score, precision_score, recall_score, f1_score + +def mlr_classifier(theta, features, labels): + return lambda X: [classify_mlr(x, theta, features, labels) for x in X] + +def evaluate_mlr(X, y, features, labels, lambda_=0.1, p=2, cv=5): + theta = train_mlr(X, y, features, labels, lambda_, p) + clf = mlr_classifier(theta, features, labels) + + scorers = { + 'accuracy': make_scorer(accuracy_score), + 'precision': make_scorer(precision_score, average='weighted'), + 'recall': make_scorer(recall_score, average='weighted'), + 'f1': make_scorer(f1_score, average='weighted') + } + + scores = {metric: cross_val_score(clf, X, y, cv=cv, scoring=scorer) + for metric, scorer in scorers.items()} + + return {metric: (np.mean(score), np.std(score)) for metric, score in scores.items()} +``` + +This implementation provides a comprehensive overview of Multinomial Logistic Regression, including its key concepts, model definition, learning process, and evaluation methods. The code examples demonstrate how to implement these concepts using NumPy and Python. + diff --git a/site/categories/algorithm analysis.html b/site/categories/algorithm analysis.html index b8a376d..2f0adc2 100644 --- a/site/categories/algorithm analysis.html +++ b/site/categories/algorithm analysis.html @@ -179,7 +179,7 @@

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+
+ Home ยป Multinomial Logistic Regression +
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Multinomial Logistic Regression

+

Classification

+

Input can be anything (document, image, etc.) and output is a class label from the finite set $\mathcal{L}$.

+

$$ +classify : \mathcal{V}* \rightarrow \mathcal{L} +$$

+

$\mathcal{V}$ is the set of words in our vocabulary.

+

$X$ is a random variable representing the input, in a given instance taking values from $\mathcal{V}*$.

+

$Y$ is a random variable representing the output, taking values from $\mathcal{L}$.

+

$p(X, Y)$ is the "true" distribution of labeled texts. $p(Y)$ is the distribution of labels. We don't normally know this without looking at the data.

+
+ +
+ + \ No newline at end of file diff --git a/site/tags/acyclic graphs.html b/site/tags/acyclic graphs.html index 1d614f1..79f4a3c 100644 --- a/site/tags/acyclic graphs.html +++ b/site/tags/acyclic graphs.html @@ -179,7 +179,7 @@

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Tag: connected graphs

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Tag: data analysis

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Tag: data modeling

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Tag: data replication

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Tag: data serialization

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Tag: data structures

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Tag: data systems

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Tag: depth first search

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Tag: depth-first search

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Tag: distributed filesystems

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Tag: document databases

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Tag: dynamic-programming

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Tag: efficiency

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Tag: encoding formats

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Tag: etl

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Tag: failover

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Tag: ford-fulkerson algorithm

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Tag: gale-shapley

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Tag: graph coloring

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diff --git a/site/tags/graph databases.html b/site/tags/graph databases.html index 55eb055..8a213a2 100644 --- a/site/tags/graph databases.html +++ b/site/tags/graph databases.html @@ -179,7 +179,7 @@

Tag: graph databases

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Tag: graph fundamentals

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Tag: graph properties

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Tag: graph representation

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Tag: graph theory

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Tag: graph traversal

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Tag: graph-theory

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Tag: graph-traversal

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Tag: graph

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Tag: greedy-algorithms

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Tag: independent set

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Tags

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Tag: indexing

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Tag: induction proofs

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Tag: induction

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Tag: interval

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Tag: leader-follower model

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Tag: linear programs

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Tag: linear systems

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Tag: maintainability

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Tag: mapreduce

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Tag: matching

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Tag: max flow min cut

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Tag: message passing

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Tag: odd cycles

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Tag: oltp vs olap

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Tag: optimization

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Tag: partitioning

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Tag: performance

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Tag: pigeonhole principle

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Tag: problem-solving

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Tag: proof techniques

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Tag: query languages

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diff --git a/site/tags/relational databases.html b/site/tags/relational databases.html index e1af97e..a11e9e5 100644 --- a/site/tags/relational databases.html +++ b/site/tags/relational databases.html @@ -179,7 +179,7 @@

Tag: relational databases

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Tag: reliability

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Tag: replication logs

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Tag: scalability

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Tag: scheduling

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Tag: schema evolution

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Tag: set cover

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Tag: shortest-paths

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diff --git a/site/tags/spanning trees.html b/site/tags/spanning trees.html index 3fe73a1..50b95e4 100644 --- a/site/tags/spanning trees.html +++ b/site/tags/spanning trees.html @@ -179,7 +179,7 @@

Tag: spanning trees

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Tag: stable matching

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diff --git a/site/tags/synchronous vs asynchronous.html b/site/tags/synchronous vs asynchronous.html index edc600a..f21733f 100644 --- a/site/tags/synchronous vs asynchronous.html +++ b/site/tags/synchronous vs asynchronous.html @@ -179,7 +179,7 @@

Tag: synchronous vs asynchronous

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Tag: time complexity

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Tag: trees

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Tag: vertex cover

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