1. 2012.05.17.(Thu) Computational Models of Intelligence Joon Shik Kim
A Global Minimization Algorithm Based on a Geodesic of a Lagrangian Formulation of Newtonian Dynamics
(Thu)
Computational Models of Intelligence
Joon Shik 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 and obtained first-order adaptive steepest descent. Then, this first-order rule was applied to Rosenbrock and Griewank potentials

4. Theory (1/4) Classical dynamics Lagrangian Lc is defined as follows:
, (1)
m is a particle mass, wi is a weight
coordinates and V is a potential

5. Theory (2/4) We introduce a affine parameter σ in which
newly defined Lagrangian LN 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 discretized above 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.