Gradient_descent.gif
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Summary
| Description |
English:
Gradient descent is a simple method to find the minimum of a function, where at each iteration a small step is made in the direction of the steepest descent. It tends to get stuck in a local minimum, so it is often run with several initial conditions.
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| Date | |
| Source | https://twitter.com/j_bertolotti/status/1121054414066810881 |
| Author | Jacopo Bertolotti |
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Permission
( Reusing this file ) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 11.0 code
f = Evaluate[x^2 + y^2 + Total@Table[RandomReal[{0,2}] E^(-(((x - RandomReal[{-1, 1}])^2 + (y + RandomReal[{-1, 1}])^2)/(2 RandomReal[{0.1, 0.4}]^2))), {10}]];
step = 0.02;
nstep = 100;
coord = {-1, 1};
pos = {coord[[1]], coord[[2]], f /. {x -> coord[[1]], y -> coord[[2]]}};
evo = Reap[Do[
subst = MapThread[Rule, {{x, y, z}, pos}];
dfx = (D[{x, y, f}, x] /. subst);
dfy = (D[{x, y, f}, y] /. subst);
tmp = {D[f, x] /. subst, D[f, y] /. subst};
pos = pos - step {tmp[[1]], tmp[[2]], 0};
pos[[3]] = Evaluate[f /. {x -> pos[[1]], y -> pos[[2]]}];
Sow[pos];
, nstep];][[2, 1]];
coord = {-1, -1};
pos = {coord[[1]], coord[[2]], f /. {x -> coord[[1]], y -> coord[[2]]}};
evo1 = Reap[Do[
subst = MapThread[Rule, {{x, y, z}, pos}];
dfx = (D[{x, y, f}, x] /. subst);
dfy = (D[{x, y, f}, y] /. subst);
tmp = {D[f, x] /. subst, D[f, y] /. subst};
pos = pos - step {tmp[[1]], tmp[[2]], 0};
pos[[3]] = Evaluate[f /. {x -> pos[[1]], y -> pos[[2]]}];
Sow[pos];
, nstep];][[2, 1]];
coord = {1, 1};
pos = {coord[[1]], coord[[2]], f /. {x -> coord[[1]], y -> coord[[2]]}};
evo2 = Reap[Do[
subst = MapThread[Rule, {{x, y, z}, pos}];
dfx = (D[{x, y, f}, x] /. subst);
dfy = (D[{x, y, f}, y] /. subst);
tmp = {D[f, x] /. subst, D[f, y] /. subst};
pos = pos - step {tmp[[1]], tmp[[2]], 0};
pos[[3]] = Evaluate[f /. {x -> pos[[1]], y -> pos[[2]]}];
Sow[pos];
, nstep];][[2, 1]];
p1 = Table[
Show[
Plot3D[f, {x, -1.35, 1.35}, {y, -1.35, 1.35}, Boxed -> False, Axes -> False(*,PlotStyle\[Rule]{Opacity[0.3]}*)],
Graphics3D[{PointSize[0.03], Point[evo[[j]] ], Thick, Line[evo[[1 ;; j]] ], Point[evo1[[j]] ], Line[evo1[[1 ;; j]]] , Point[evo2[[j]] ], Line[evo2[[1 ;; j]]] }]
]
, {j, 1, nstep, 2}];
ListAnimate[p1]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication . |
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The person who associated a work with this deed has dedicated the work to the
public domain
by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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