# chernoff bound

Published articles for chernoff bound.

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## The Inequality

DevFeed: [The Inequality](<https://devfeed.tech/articles/the-inequality-40392.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/11/23/the-inequality/>)

Published: 2015-11-23T22:06:27Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>), [Computer science](<https://devfeed.tech/topics/computer-science.md>), [Machine Learning & Artificial Intelligence](<https://devfeed.tech/topics/machine-learning-artificial-intelligence.md>), [Algorithms](<https://devfeed.tech/topics/algorithms.md>)

Tags: [calculus](<https://devfeed.tech/tags/calculus.md>), [cauchy-schwarz](<https://devfeed.tech/tags/cauchy-schwarz.md>), [chernoff-bound](<https://devfeed.tech/tags/chernoff-bound.md>), [computer-science](<https://devfeed.tech/tags/computer-science.md>), [euler-s-number](<https://devfeed.tech/tags/euler-s-number.md>), [inequality](<https://devfeed.tech/tags/inequality.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [math](<https://devfeed.tech/tags/math.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [taylor-series](<https://devfeed.tech/tags/taylor-series.md>)

### AI overview

This tutorial explains how the inequality 1+x ≤ e^x is used to bound the probability of bad events in machine-learning algorithm analysis and introduces its use in proving the weighted arithmetic-geometric mean inequality.

### Source excerpt

Math and computer science are full of inequalities, but there is one that shows up more often in my work than any other. Of course, I'm talking about $$\displaystyle 1+x \leq e^{x}$$ This is The Inequality. I've been told on many occasions that the entire field of machine learning reduces to The Inequality combined with the Chernoff bound (which is proved using The Inequality). Why does it show up so often in machine learning?

## The Boosting Margin, or Why Boosting Doesn't Overfit

DevFeed: [The Boosting Margin, or Why Boosting Doesn't Overfit](<https://devfeed.tech/articles/the-boosting-margin-or-why-boosting-doesn-t-overfit-40388.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/09/21/the-boosting-margin-or-why-boosting-doesnt-overfit/>)

Published: 2015-09-21T11:33:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [machine learning overfitting](<https://devfeed.tech/topics/machine-learning-overfitting.md>), [Machine learning](<https://devfeed.tech/topics/machine-learning.md>), [generalization in machine learning](<https://devfeed.tech/topics/generalization-in-machine-learning.md>), [Occam's razor machine learning](<https://devfeed.tech/topics/occam-s-razor-machine-learning.md>), [Algorithm](<https://devfeed.tech/topics/algorithm.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [boosting](<https://devfeed.tech/tags/boosting.md>), [chernoff-bound](<https://devfeed.tech/tags/chernoff-bound.md>), [classficiation](<https://devfeed.tech/tags/classficiation.md>), [complexity](<https://devfeed.tech/tags/complexity.md>), [conditional-probability](<https://devfeed.tech/tags/conditional-probability.md>), [error](<https://devfeed.tech/tags/error.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [margins](<https://devfeed.tech/tags/margins.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [model](<https://devfeed.tech/tags/model.md>), [occam-s-razor](<https://devfeed.tech/tags/occam-s-razor.md>), [overfitting](<https://devfeed.tech/tags/overfitting.md>), [training-data](<https://devfeed.tech/tags/training-data.md>), [vc-dimension](<https://devfeed.tech/tags/vc-dimension.md>)

### AI overview

This article explains why boosting can continue improving generalization after reaching zero training error. It introduces the margin-based theoretical explanation for this behavior and defines the confidence and margin of AdaBoost classifiers.

### Source excerpt

There's a well-understood phenomenon in machine learning called overfitting. The idea is best shown by a graph: overfitting Let me explain. The vertical axis represents the error of a hypothesis. The horizontal axis represents the complexity of the hypothesis. The blue curve represents the error of a machine learning algorithm's output on its training data, and the red curve represents the generalization of that hypothesis to the real world. The overfitting phenomenon is marker in the middle of the graph, before which the training error and generalization error both go down, but after which the training error continues to fall while the generalization error rises.

## Occam's Razor and PAC-learning

DevFeed: [Occam's Razor and PAC-learning](<https://devfeed.tech/articles/occam-s-razor-and-pac-learning-40366.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2014/09/19/occams-razor-and-pac-learning/>)

Published: 2014-09-19T10:00:47Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Occam's razor machine learning](<https://devfeed.tech/topics/occam-s-razor-machine-learning.md>), [Learning](<https://devfeed.tech/topics/learning.md>), [data](<https://devfeed.tech/topics/data.md>)

Tags: [chernoff](<https://devfeed.tech/tags/chernoff.md>), [chernoff-bound](<https://devfeed.tech/tags/chernoff-bound.md>), [computational-learning-theory](<https://devfeed.tech/tags/computational-learning-theory.md>), [learning-theory](<https://devfeed.tech/tags/learning-theory.md>), [occam-s-razor](<https://devfeed.tech/tags/occam-s-razor.md>), [pac-learning](<https://devfeed.tech/tags/pac-learning.md>), [vc-dimension](<https://devfeed.tech/tags/vc-dimension.md>)

### AI overview

This tutorial develops a simple PAC-learning theorem for a finite hypothesis class. It explains that if a hypothesis consistent with observed data can always be found, the class can be learned efficiently, and it uses the Chernoff-Hoeffding bound to quantify the sample requirement for achieving low error.

### Source excerpt

So far our discussion of learning theory has been seeing the definition of PAC-learning, tinkering with it, and seeing simple examples of learnable concept classes. We've said that our real interest is in proving big theorems about what big classes of problems can and can't be learned. One major tool for doing this with PAC is the concept of VC-dimension, but to set the stage we're going to prove a simpler theorem that gives a nice picture of PAC-learning when your hypothesis class is small.

## Probabilistic Bounds -- A Primer

DevFeed: [Probabilistic Bounds -- A Primer](<https://devfeed.tech/articles/probabilistic-bounds-a-primer-40312.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/04/15/probabilistic-bounds-a-primer/>)

Published: 2013-04-15T11:14:32Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Algorithms, Complexity](<https://devfeed.tech/topics/algorithms-complexity.md>), [math](<https://devfeed.tech/topics/math.md>), [Learning](<https://devfeed.tech/topics/learning.md>), [Streaming](<https://devfeed.tech/topics/streaming.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [algorithms](<https://devfeed.tech/tags/algorithms.md>), [chebyshev](<https://devfeed.tech/tags/chebyshev.md>), [chernoff](<https://devfeed.tech/tags/chernoff.md>), [chernoff-bound](<https://devfeed.tech/tags/chernoff-bound.md>), [inequality](<https://devfeed.tech/tags/inequality.md>), [learning-theory](<https://devfeed.tech/tags/learning-theory.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [markov](<https://devfeed.tech/tags/markov.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [probabilistic](<https://devfeed.tech/tags/probabilistic.md>), [probabilistic-method](<https://devfeed.tech/tags/probabilistic-method.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>), [streaming](<https://devfeed.tech/tags/streaming.md>), [streaming-algorithms](<https://devfeed.tech/tags/streaming-algorithms.md>), [variance](<https://devfeed.tech/tags/variance.md>)

### AI overview

This tutorial introduces probabilistic bounds used in algorithm analysis, machine learning theory, randomized algorithms, and streaming algorithms. It focuses on the Chernoff bound and presents simpler bounds from Markov's and Chebyshev's inequalities, including short proofs.

### Source excerpt

Probabilistic arguments are a key tool for the analysis of algorithms in machine learning theory and probability theory. They also assume a prominent role in the analysis of randomized and streaming algorithms, where one imposes a restriction on the amount of storage space an algorithm is allowed to use for its computations (usually sublinear in the size of the input). While a whole host of probabilistic arguments are used, one theorem in particular (or family of theorems) is ubiquitous: the Chernoff bound.