From b62d729c38cec4ca7db1561a3841d5bc914c223a Mon Sep 17 00:00:00 2001 From: Elijah Melton Date: Sun, 12 Jan 2025 12:57:17 -0800 Subject: [PATCH] sync changes: - D site/Untitled.java - M "site/categories/algorithm analysis.html" - M site/categories/algorithms.html --- 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 | 6 +- site/categories/mathematics.html | 2 +- site/categories/operations research.html | 2 +- site/categories/research.html | 195 +++++++++++++++++ site/categories/software engineering.html | 2 +- site/categories/system-design.html | 193 +++++++++++++++++ site/index.html | 56 ++--- .../hints-for-computer-system-design.html | 14 +- .../systems-research/how-to-read-a-paper.html | 14 +- .../paper-review-template.html | 10 +- site/systems-research/strong-inference.html | 10 +- 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/caching.html | 193 +++++++++++++++++ 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 | 10 +- 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/meta.html | 195 +++++++++++++++++ site/tags/odd cycles.html | 2 +- site/tags/oltp vs olap.html | 2 +- site/tags/optimization.html | 2 +- site/tags/paper.html | 196 ++++++++++++++++++ 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/research.html | 195 +++++++++++++++++ site/tags/review.html | 196 ++++++++++++++++++ site/tags/scalability.html | 2 +- site/tags/scaling.html | 193 +++++++++++++++++ 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/systems.html | 193 +++++++++++++++++ site/{test.html => tags/template.html} | 14 +- site/tags/time complexity.html | 2 +- site/tags/trees.html | 2 +- site/tags/vertex cover.html | 2 +- .../hints-for-computer-system-design.md | 8 + systems-research/how-to-read-a-paper.md | 8 + systems-research/paper-review-template.md | 7 + systems-research/strong-inference.md | 8 + test.md | 1 - 113 files changed, 1947 insertions(+), 290 deletions(-) delete mode 100644 site/Untitled.java create mode 100644 site/categories/research.html create mode 100644 site/categories/system-design.html create mode 100644 site/tags/caching.html create mode 100644 site/tags/meta.html create mode 100644 site/tags/paper.html create mode 100644 site/tags/research.html create mode 100644 site/tags/review.html create mode 100644 site/tags/scaling.html create mode 100644 site/tags/systems.html rename site/{test.html => tags/template.html} (92%) delete mode 100644 test.md diff --git a/site/Untitled.java b/site/Untitled.java deleted file mode 100644 index 81c47ec..0000000 --- a/site/Untitled.java +++ /dev/null @@ -1,142 +0,0 @@ -# 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. 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    Tags

  • bipartite graphs (1 pages)
  • bipartite matching (1 pages)
  • breadth-first search (2 pages)
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  • column-oriented storage (1 pages)
  • compatibility (1 pages)
  • complexity analysis (1 pages)
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  • matching (1 pages)
  • max flow min cut (1 pages)
  • message passing (1 pages)
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  • odd cycles (1 pages)
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  • optimization (3 pages)
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  • partitioning (1 pages)
  • performance (1 pages)
  • pigeonhole principle (1 pages)
    • @@ -253,7 +256,10 @@

    Tags

  • relational databases (1 pages)
  • reliability (1 pages)
  • replication logs (1 pages)
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  • scalability (1 pages)
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  • scheduling (1 pages)
  • schema evolution (1 pages)
  • set cover (1 pages)
    • @@ -261,6 +267,8 @@

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  • stable matching (1 pages)
  • synchronous vs asynchronous (1 pages)
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