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Related Concept Videos

Quadratic Models01:23

Quadratic Models

Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Prediction Intervals01:03

Prediction Intervals

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Slant Asymptotes01:27

Slant Asymptotes

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Related Experiment Videos

Worst-case quadratic loss bounds for prediction using linear functions and gradient descent.

N Cesa-Bianchi1, P M Long, M K Warmuth

  • 1Dipartimento di Sci. dell'Inf., Milan Univ.

IEEE Transactions on Neural Networks
|January 1, 1996
PubMed
Summary

This study analyzes gradient descent (GD) for online linear prediction, providing worst-case error bounds. The findings offer insights into prediction accuracy with limited prior information, crucial for machine learning applications.

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Area of Science:

  • Machine Learning
  • Optimization Algorithms
  • Statistical Learning Theory

Background:

  • Online linear prediction is a fundamental problem in machine learning.
  • Gradient descent (GD) is a widely used optimization algorithm.
  • Understanding GD's performance in online settings is crucial for developing robust prediction models.

Purpose of the Study:

  • To analyze the performance of gradient descent (GD) in online linear prediction within arbitrary inner product spaces.
  • To establish worst-case bounds on prediction errors under varying levels of prior information.
  • To explore the applicability of these findings to predicting smooth functions.

Main Methods:

  • Utilized variants and extensions of online gradient descent algorithms.
  • Derived theoretical bounds on the sum of squared prediction errors.
  • Expressed bounds as a function of the best fixed linear predictor's performance.
  • Investigated the tightness of derived upper bounds with matching lower bounds where applicable.

Main Results:

  • Established worst-case performance bounds for online gradient descent in linear prediction.
  • Demonstrated that prediction error bounds are often related to the performance of the best bounded-norm linear predictor.
  • Showcased the applicability of the developed methods to online prediction of smooth functions.
  • Proved that derived upper bounds are tight to within constants, with matching lower bounds in certain scenarios.

Conclusions:

  • The study provides a theoretical framework for understanding gradient descent's efficacy in online linear prediction.
  • The derived bounds offer valuable insights into the trade-offs between prediction accuracy and available prior information.
  • The research extends the application of gradient descent-based methods to complex function prediction tasks.