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2011.04.29.( Fri) Computational Modeling of Intelligence Joon Shik Kim

A Global Minimization Algorithm Based on a Geodesic of a Lagrangian Formulation of Newtonian Dynamics . 2011.04.29.( Fri) Computational Modeling of Intelligence Joon Shik Kim . Introduction (1/2). A risk function or cost function can be described

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2011.04.29.( Fri) Computational Modeling of Intelligence Joon Shik Kim

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  1. A Global Minimization Algorithm Based on a Geodesic of a Lagrangian Formulation of Newtonian Dynamics 2011.04.29.(Fri) Computational Modeling of Intelligence JoonShik Kim

  2. Introduction (1/2) • A risk function or cost function can be described as a multi-minima error potential. • Attouch et al’s (2000) heavy ball with friction (HBF) depends on the damped oscillator equation. • Qian’s(1999) steepest descent algorithm depends on a critically damped oscillator equation.

  3. Introduction (2/2) • Above two algorithms use a constant damping coefficient and we suggest an adaptive damping coefficient. • We calculated the geodesic of classical dynamics Lagrangian and this geodesic produces an adaptively damped oscillator equation. • We discretized it andobtained first-order adaptive steepest descent. Then, this first-order rule was appliedto Rosenbrock and Griewank potentials

  4. Theory (1/4) • Classical dynamics LagrangianLcis defined as follows: , (1) m is aparticle mass, wiis a weight coordinates and V is a potential

  5. Theory (2/4) • We introduce a affine parameter σ in which • newly defined LagrangianLN is a constant 1. (2) • Then we apply Euler-Lagrange equation to • LN as follows; (3)

  6. Theory (3/4) • We obtain following second-order equation in an affine parameter σ: (4) • If we change independent variable from σ • to ordinary time t, we can derive following • second-order adaptively damped oscillator • equation; (5)

  7. Theory (4/4) • We discretizedabove equation to obtain first-order adaptive steepest descent; (6)

  8. Global minimum search examples (1/3) • Rosenbrock potential Two initial points are (-1.2, 1) and global minimum is located in (1, 1). m=10000 and Δt=1.

  9. Global minimum search examples (2/3) • Griewank potential(1) Initial two points are (10, 10) and global minimum is located at (0, 0). m=1000, Δt=1.

  10. Global minimum search examples (3/3) • Griewank potential(2) In Attouch et al’s case, the global search consists of several new shooting from the point in the path connecting each local minimum. In contrast, our method makes the ball autonomously arrive at the global minimum. Initial two points are (7, 7) and global minimum is located at (0, 0). m=1000, Δt=1.

  11. Discussion • We derived a novel adaptive steepest descent which shows good performance for global minimum search problems. • We used the calculus of variation method for the global minimum search problem. • Our learning rule is valid under the constraint that the global minimum value is zero.

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