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Related Concept Videos

Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
Lagrange Multipliers: Problem Solving01:30

Lagrange Multipliers: Problem Solving

A silo with a cylindrical base, flat bottom, and hemispherical roof is a common design in agricultural and industrial storage due to its structural efficiency and ease of construction. Optimizing its dimensions to maximize storage capacity for a given amount of material—i.e., a fixed surface area—is a classic problem in applied calculus and engineering design. The key parameters are the radius r of the base and the height h of the cylindrical section.The total volume of the silo is obtained by...
Maximizing the Directional Derivative01:25

Maximizing the Directional Derivative

The directional derivative is a central concept in multivariable calculus that describes how a function changes at a given point when moving in a specified direction. This direction is represented by a unit vector, ensuring that only the orientation influences the rate of change. By varying the direction, different rates of change can be observed, demonstrating that the directional derivative depends strongly on the chosen direction.The directional derivative is computed using the gradient...
Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
Second Order systems I01:20

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...

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Related Experiment Videos

From Stochastic Conjugate Gradient to Stochastic Second-Order Optimization, Driven by Conjugate Coefficient With

Zhuang Yang

    IEEE Transactions on Neural Networks and Learning Systems
    |June 11, 2026
    PubMed
    Summary

    We introduce novel stochastic second-order conjugate gradient (S2CG) algorithms that effectively combine conjugate gradient and second-order information for machine learning. These S2CG methods demonstrate superior performance over existing algorithms in various tasks.

    Related Experiment Videos

    Area of Science:

    • Optimization Algorithms
    • Machine Learning Theory

    Background:

    • Stochastic first-order (SFO) algorithms are crucial for machine learning.
    • Conjugate gradient (CG) and second-order information (SOI) methods enhance SFO algorithms separately.
    • Existing research often focuses on CG or SOI in isolation, limiting potential improvements.

    Purpose of the Study:

    • To investigate the combined effect of CG and SOI on SFO algorithms.
    • To develop efficient and effective stochastic second-order conjugate gradient (S2CG) algorithms.
    • To bridge the gap between theoretical and empirical studies of CG and SOI in machine learning.

    Main Methods:

    • Developed novel S2CG algorithms integrating CG and SOI.
    • Analyzed the transition from stochastic CG (SCG) to stochastic second-order (SSO) driven by conjugate coefficients with SOI.
    • Provided theoretical convergence guarantees for S2CG in strongly convex, general convex, and Polyak-Łojasiewicz (PL) settings.

    Main Results:

    • Demonstrated that S2CG algorithms are fast and low-cost.
    • Established theoretical guarantees for S2CG across different convexity assumptions.
    • Empirically validated S2CG superiority on Support Vector Machines (SVM), Logistic Regression (LR), Federated Learning (FL), and Neural Networks (NNs).

    Conclusions:

    • S2CG algorithms represent a significant advancement over existing SFO, SCG, and SSO methods.
    • The integration of CG and SOI offers substantial benefits for machine learning optimization.
    • S2CG algorithms show robustness and superior performance across diverse machine learning applications.