Simple Stochastic Gradient Descent (SGD) Example¶
In [1]:
#### initialization
import numpy as np
from matplotlib import pyplot as plt
%matplotlib inline
np.random.seed(123)
In [2]:
#### generate data
X = np.concatenate([ np.random.multivariate_normal([-3, -3], [[1, 0], [0, 1]], 10),
np.random.multivariate_normal([ 3, 3], [[1, 0], [0, 1]], 10) ])
y = np.concatenate([[0] * 10, [1] * 10])
X = np.insert(X, 2, 1, axis = 1)
plt.scatter(X[:,0], X[:,1], c = y, marker = 'x')
plt.show()
In [3]:
#### initialize weights to small random values
w = np.array([-0.02, -0.01, 0.0]) # wrong direction!!!
print(w)
[-0.02 -0.01 0. ]
In [4]:
#### make our first prediction
def PredictProbability(X, w):
return 1.0 / (1.0 + np.exp(- np.dot(X, w)))
p = PredictProbability(X, w)
print(p)
[ 0.52541287 0.5248304 0.5212516 0.53564511 0.51832895 0.52610744 0.51663417 0.5257829 0.5060031 0.51650826 0.47012197 0.47925152 0.48536796 0.47655338 0.48036483 0.48577841 0.4881096 0.47332214 0.47578417 0.48119748]
In [5]:
#### estimate loss of our current predictor
def ComputeLoss(y, p):
return - np.mean(np.log(y * p + (1 - y) * (1 - p)))
print(ComputeLoss(y, p)) # cross entropy
print(np.exp(- ComputeLoss(y, p))) # the associated probability for the expected log loss
0.736217912002 0.478921819005
In [6]:
#### weight update example: the predicted probability improves ... we've learned!
def ComputeGradient(X, w, y):
return ((1.0 / (1.0 + np.exp(- np.dot(X, w))) - y) * X)
j = 1
print(X[j,:])
print(y[j])
print(w)
print(PredictProbability(X[j,:], w))
print(ComputeGradient(X[j,:], w, y[j]))
print(w - ComputeGradient(X[j,:], w, y[j]))
print(PredictProbability(X[j,:], w - ComputeGradient(X[j,:], w, y[j])))
[-2.7170215 -4.50629471 1. ] 0 [-0.02 -0.01 0. ] 0.524830401829 [-1.42597549 -2.36504047 0.5248304 ] [ 1.40597549 2.35504047 -0.5248304 ] 3.19214434967e-07
In [7]:
#### train the model (without regularization; weights continue to grow)
learningRate = 0.5
epochs = 5000
for epoch in range(epochs):
index = list(range(X.shape[0]))
np.random.shuffle(index)
for j in index:
w = w - learningRate * ComputeGradient(X[j,:], w, y[j])
p = PredictProbability(X, w)
print([epoch, ComputeLoss(y, p), w])
[0, 0.016564383455020142, array([ 0.92674277, 1.14325754, 0.18081833])] [1, 0.011063084660567988, array([ 1.11028213, 1.28235886, 0.11731881])] [2, 0.0085521535785665279, array([ 1.23092981, 1.37481961, 0.06539341])] [3, 0.0070020567923956348, array([ 1.32601845, 1.44970767, 0.02397545])] [4, 0.0059706231120360258, array([ 1.40372911, 1.50810422, -0.0115258 ])] [5, 0.0052176248607362259, array([ 1.46935177, 1.55776005, -0.04375374])] [6, 0.0046438802974535631, array([ 1.52538006, 1.60127413, -0.07331536])] [7, 0.0041875272505190863, array([ 1.57505442, 1.64005569, -0.10027255])] [8, 0.0038161116761212068, array([ 1.61969767, 1.67478824, -0.12503419])] [9, 0.0035062194510308833, array([ 1.66072596, 1.70650176, -0.14732325])] [10, 0.0032443066522490301, array([ 1.69828638, 1.73555861, -0.16812741])] [11, 0.0030198003450907407, array([ 1.73291821, 1.76235893, -0.1876961 ])] [12, 0.0028255119964473176, array([ 1.76469508, 1.78728342, -0.20647275])] [13, 0.0026550513451190231, array([ 1.79441159, 1.81059959, -0.22417735])] [14, 0.0025045409056646669, array([ 1.82244994, 1.83232971, -0.24075338])] [15, 0.0023705096467526564, array([ 1.84867791, 1.85289624, -0.25665482])] [16, 0.0022500702758181394, array([ 1.87367687, 1.87238917, -0.27156259])] [17, 0.0021416073740656424, array([ 1.89719651, 1.89089588, -0.28595536])] [18, 0.0020431908652720622, array([ 1.91959261, 1.90847928, -0.29974158])] [19, 0.0019535207711806005, array([ 1.94104793, 1.92523175, -0.31280147])] [20, 0.0018715280384540821, array([ 1.96150278, 1.9412507 , -0.32535597])] [21, 0.0017962445915375765, array([ 1.98099072, 1.95660561, -0.33753467])] [22, 0.0017268249285770081, array([ 1.99969099, 1.97134859, -0.34925026])] [23, 0.0016626518505203785, array([ 2.01764272, 1.98550598, -0.36055711])] [24, 0.0016031284819542206, array([ 2.03491193, 1.99912396, -0.37148639])] [25, 0.0015477342498061391, array([ 2.05163463, 2.01222273, -0.38197983])] [26, 0.0014960830297231194, array([ 2.06775148, 2.02487661, -0.39214201])] [27, 0.0014477945775444681, array([ 2.0833282 , 2.03709349, -0.4019945 ])] [28, 0.0014025941738515704, array([ 2.09831561, 2.04890247, -0.41164006])] [29, 0.00136011996282643, array([ 2.11289215, 2.06035104, -0.42092481])] [30, 0.0013201917648151622, array([ 2.12695786, 2.07143729, -0.43002589])] [31, 0.0012825393988866753, array([ 2.1406566 , 2.08219855, -0.43881251])] [32, 0.0012470121720729046, array([ 2.15390784, 2.09264225, -0.44742664])] [33, 0.0012134050954586758, array([ 2.16678842, 2.10279985, -0.45581671])] [34, 0.001181578674968264, array([ 2.17931736, 2.11268837, -0.46397825])] [35, 0.0011513742271062239, array([ 2.19155307, 2.12231339, -0.47189848])] [36, 0.0011226964158220641, array([ 2.20343151, 2.13168421, -0.47968185])] [37, 0.0010954168011410334, array([ 2.21504495, 2.14081993, -0.48723366])] [38, 0.0010694489666079545, array([ 2.22633521, 2.14972675, -0.49466153])] [39, 0.0010446796776196522, array([ 2.2373926 , 2.1584227 , -0.50187715])] [40, 0.0010210476181917716, array([ 2.24815302, 2.16691394, -0.50897996])] [41, 0.00099846537520009173, array([ 2.25869575, 2.17519746, -0.51590524])] [42, 0.00097686597876155872, array([ 2.26899984, 2.18330718, -0.52267533])] [43, 0.00095618886825551768, array([ 2.27905473, 2.19123722, -0.52934104])] [44, 0.00093637224169348303, array([ 2.28889503, 2.1989979 , -0.53587298])] [45, 0.00091735985573290226, array([ 2.29853076, 2.20661096, -0.54226296])] [46, 0.00089910667267898705, array([ 2.30798821, 2.21406137, -0.54850956])] [47, 0.00088157147371096547, array([ 2.31725505, 2.22135097, -0.5546476 ])] [48, 0.00086471100516056967, array([ 2.32633777, 2.22849706, -0.56067237])] [49, 0.00084848704794993882, array([ 2.33522404, 2.23551866, -0.56660124])] [50, 0.00083286448278531585, array([ 2.34394253, 2.24239975, -0.57242965])] [51, 0.00081780670203235838, array([ 2.35251724, 2.24915376, -0.5781314 ])] [52, 0.00080328699936707516, array([ 2.36093327, 2.25577928, -0.58374209])] [53, 0.00078927837779438001, array([ 2.3691789 , 2.26229111, -0.58927628])] [54, 0.00077575212609962813, array([ 2.3772804 , 2.26868603, -0.59471873])] [55, 0.00076268305650780397, array([ 2.38525623, 2.27496514, -0.60005703])] [56, 0.00075004692132625767, array([ 2.39309825, 2.28114426, -0.6053019 ])] [57, 0.00073782581469971796, array([ 2.40079271, 2.28722447, -0.61047965])] [58, 0.00072599698839699936, array([ 2.40837481, 2.29319643, -0.61556715])] [59, 0.0007145428598705878, array([ 2.41583334, 2.29907243, -0.62057684])] [60, 0.0007034467252866615, array([ 2.42315944, 2.30486259, -0.62552012])] [61, 0.0006926900326842798, array([ 2.43038473, 2.31055775, -0.63037616])] [62, 0.00068226026653853957, array([ 2.43749543, 2.31615942, -0.6351683 ])] [63, 0.00067214144339136617, array([ 2.44448634, 2.32168012, -0.6399027 ])] [64, 0.00066231800666051281, array([ 2.45138444, 2.32711645, -0.64455711])] [65, 0.00065277986411110876, array([ 2.45816785, 2.33247568, -0.64915875])] [66, 0.00064351361550773621, array([ 2.46485367, 2.33775538, -0.65369668])] [67, 0.00063450629300971118, array([ 2.47145436, 2.34295841, -0.65816111])] [68, 0.00062574876706961819, array([ 2.47796188, 2.34808938, -0.6625629 ])] [69, 0.00061723109742114405, array([ 2.48436676, 2.35315074, -0.66692031])] [70, 0.00060894256387768035, array([ 2.490694 , 2.3581382 , -0.67121129])] [71, 0.00060087419493718677, array([ 2.49693489, 2.36306226, -0.67544324])] [72, 0.00059301714040190705, array([ 2.50309291, 2.36791802, -0.67962406])] [73, 0.00058536446784764463, array([ 2.50915958, 2.37271209, -0.68376203])] [74, 0.00057790738523916967, array([ 2.5151472 , 2.3774437 , -0.68784862])] [75, 0.00057063862141625444, array([ 2.52105857, 2.38211326, -0.69188501])] [76, 0.00056355086568516842, array([ 2.52689493, 2.38672376, -0.69587244])] [77, 0.00055663741312223045, array([ 2.53265708, 2.391278 , -0.69981219])] [78, 0.00054989185831355591, array([ 2.53835626, 2.39577221, -0.7036977 ])] [79, 0.00054330842845133067, array([ 2.54397742, 2.40021517, -0.70754538])] [80, 0.00053688047075149106, array([ 2.54953998, 2.40460409, -0.71133738])] [81, 0.00053060383025860486, array([ 2.55503462, 2.40893765, -0.71509019])] [82, 0.00052447266165628489, array([ 2.56045734, 2.41322557, -0.71880473])] [83, 0.00051848151460667617, array([ 2.56582393, 2.417462 , -0.72247037])] [84, 0.0005126261542060304, array([ 2.57112883, 2.42164677, -0.72609839])] [85, 0.00050690199300561429, array([ 2.57636653, 2.42578887, -0.72969073])] [86, 0.00050130450247102556, array([ 2.58155146, 2.42987975, -0.73324074])] [87, 0.00049582942792833202, array([ 2.58667844, 2.43392788, -0.73675038])] [88, 0.00049047292428565175, array([ 2.59174885, 2.43793153, -0.7402231 ])] [89, 0.00048523159299396505, array([ 2.59676378, 2.44188804, -0.7436628 ])] [90, 0.0004801014152600312, array([ 2.60171879, 2.44580627, -0.7470704 ])] [91, 0.00047507887237864153, array([ 2.60662656, 2.44967843, -0.75043983])] [92, 0.00047016069295215135, array([ 2.61147632, 2.45351242, -0.75378084])] [93, 0.00046534342326604028, array([ 2.61627578, 2.4573069 , -0.75708769])] [94, 0.0004606240513883251, array([ 2.62102583, 2.46106338, -0.76036023])] [95, 0.00045599959661549682, array([ 2.62573263, 2.46477802, -0.76359706])] [96, 0.00045146733242827956, array([ 2.63038692, 2.46845673, -0.76680744])] [97, 0.00044702427911327776, array([ 2.63499995, 2.47209841, -0.76997995])] [98, 0.00044266805924549448, array([ 2.63956706, 2.47570395, -0.77312231])] [99, 0.00043839622691925625, array([ 2.64408843, 2.47927428, -0.77623566])] [100, 0.00043420621803802299, array([ 2.64856585, 2.48281024, -0.77931951])] [101, 0.00043009580505774369, array([ 2.65299526, 2.48631241, -0.78238126])] [102, 0.00042606278609377764, array([ 2.65738203, 2.48978073, -0.7854155 ])] [103, 0.00042210489693798029, array([ 2.6617273 , 2.49321622, -0.78842215])] [104, 0.00041821981499777722, array([ 2.66603254, 2.4966213 , -0.79139944])] [105, 0.00041440575136136349, array([ 2.67030204, 2.49999165, -0.79434768])] [106, 0.00041066088941758545, array([ 2.67453206, 2.50333048, -0.79727036])] [107, 0.00040698322339763712, array([ 2.67871938, 2.50664213, -0.80016987])] [108, 0.00040337089261950154, array([ 2.68286913, 2.50992424, -0.80304373])] [109, 0.00039982213396778325, array([ 2.68698567, 2.51317589, -0.80588888])] [110, 0.0003963353971840768, array([ 2.69106567, 2.51639872, -0.80870979])] [111, 0.00039290924678490446, array([ 2.6951058 , 2.51959401, -0.8115111 ])] [112, 0.00038954195060398206, array([ 2.69911452, 2.52275884, -0.81428639])] [113, 0.00038623187797265769, array([ 2.70308911, 2.52589746, -0.81703704])] [114, 0.00038297771935086506, array([ 2.70702956, 2.5290092 , -0.81976505])] [115, 0.00037977816992977532, array([ 2.71093598, 2.53209316, -0.82247266])] [116, 0.00037663179413517677, array([ 2.7148094 , 2.53515105, -0.82515837])] [117, 0.00037353724214651043, array([ 2.71864671, 2.53818513, -0.82782591])] [118, 0.00037049300862096907, array([ 2.7224561 , 2.54119445, -0.830467 ])] [119, 0.00036749820891505952, array([ 2.72623399, 2.54417856, -0.83308761])] [120, 0.00036455162998463554, array([ 2.7299775 , 2.5471385 , -0.83569227])] [121, 0.00036165201747398186, array([ 2.73369112, 2.5500741 , -0.83827661])] [122, 0.00035879817475464941, array([ 2.73737484, 2.55298691, -0.84084063])] [123, 0.00035598899442270879, array([ 2.74103261, 2.55587659, -0.84338058])] [124, 0.00035322366487819645, array([ 2.74465819, 2.55874342, -0.84590556])] [125, 0.00035050091307628538, array([ 2.74825868, 2.56158791, -0.84840717])] [126, 0.00034782009180553891, array([ 2.75182783, 2.56440984, -0.85089459])] [127, 0.00034517990316956847, array([ 2.75537268, 2.56721014, -0.85335935])] [128, 0.00034257954955785379, array([ 2.75889051, 2.56998922, -0.85580563])] [129, 0.00034001823883823438, array([ 2.76237869, 2.57274782, -0.85823716])] [130, 0.00033749501694907056, array([ 2.76584058, 2.57548563, -0.86065103])] [131, 0.00033500897602734247, array([ 2.7692767 , 2.5782033 , -0.86304717])] [132, 0.00033255935751081908, array([ 2.77269 , 2.58089882, -0.86542454])] [133, 0.000330145329571929, array([ 2.77607846, 2.5835752 , -0.8677839 ])] [134, 0.0003277662745912645, array([ 2.77943909, 2.58623187, -0.87013062])] [135, 0.00032542132220150647, array([ 2.7827754 , 2.58886919, -0.87246083])] [136, 0.00032310966936808193, array([ 2.78608754, 2.5914882 , -0.87477452])] [137, 0.00032083075010539824, array([ 2.78937612, 2.59408778, -0.87707249])] [138, 0.00031858370360428453, array([ 2.79264412, 2.59666911, -0.8793509 ])] [139, 0.00031636797526338759, array([ 2.79588637, 2.59923342, -0.88161625])] [140, 0.00031418288900251774, array([ 2.7991083 , 2.60177893, -0.88386384])] [141, 0.00031202788888537664, array([ 2.80230531, 2.60430732, -0.88609896])] [142, 0.00030990234690598726, array([ 2.80548004, 2.60681744, -0.88832013])] [143, 0.00030780550137621704, array([ 2.80863557, 2.60931065, -0.89052302])] [144, 0.00030573690186436224, array([ 2.81176962, 2.61178676, -0.89271141])] [145, 0.00030369599314715256, array([ 2.81488018, 2.61424686, -0.89488774])] [146, 0.00030168226244447723, array([ 2.8179695 , 2.61668954, -0.89705107])] [147, 0.00029969505081197485, array([ 2.82104058, 2.61911584, -0.89919733])] [148, 0.00029773385343190604, array([ 2.82409141, 2.62152624, -0.90132958])] [149, 0.00029579818851177417, array([ 2.82712209, 2.62392084, -0.90344824])] [150, 0.00029388766992394287, array([ 2.83013056, 2.62629948, -0.9055567 ])] [151, 0.00029200170081354957, array([ 2.83311945, 2.62866274, -0.90765185])] [152, 0.00029013983401866871, array([ 2.83608926, 2.63101037, -0.90973389])] [153, 0.00028830159297221585, array([ 2.83903976, 2.63334325, -0.91180297])] [154, 0.00028648645447874248, array([ 2.84197395, 2.6356611 , -0.91385624])] [155, 0.00028469408829877828, array([ 2.84488935, 2.63796367, -0.91589771])] [156, 0.00028292403720381077, array([ 2.84778626, 2.64025245, -0.91792613])] [157, 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2.3402772018723957e-05, array([ 3.99892238, 3.54833458, -1.7462842 ])] [1946, 2.3390788709015986e-05, array([ 3.99915828, 3.54852008, -1.74645682])] [1947, 2.3378817676103298e-05, array([ 3.99939405, 3.54870548, -1.74662937])] [1948, 2.3366858888932547e-05, array([ 3.99962972, 3.54889079, -1.74680182])] [1949, 2.3354912333927803e-05, array([ 3.99986526, 3.549076 , -1.74697418])] [1950, 2.3342978001567143e-05, array([ 4.00010067, 3.54926112, -1.74714646])] [1951, 2.3331055847216158e-05, array([ 4.00033597, 3.54944615, -1.74731864])] [1952, 2.3319145870700804e-05, array([ 4.00057113, 3.54963108, -1.74749075])] [1953, 2.3307248054726444e-05, array([ 4.00080619, 3.54981592, -1.74766275])] [1954, 2.3295362376380577e-05, array([ 4.00104113, 3.55000066, -1.74783467])] [1955, 2.3283488828086079e-05, array([ 4.00127594, 3.55018531, -1.74800652])] [1956, 2.3271627382174601e-05, array([ 4.00151063, 3.55036986, -1.74817828])] [1957, 2.3259778018152286e-05, array([ 4.00174521, 3.55055432, 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3.5529437 , -1.75057379])] [1971, 2.3095145487232567e-05, array([ 4.0050167 , 3.55312685, -1.75074425])] [1972, 2.3083475186122011e-05, array([ 4.00524948, 3.55330991, -1.75091463])] [1973, 2.3071816679382503e-05, array([ 4.00548216, 3.55349287, -1.75108492])] [1974, 2.3060169953892261e-05, array([ 4.00571471, 3.55367573, -1.75125513])] [1975, 2.3048534977706862e-05, array([ 4.00594715, 3.55385851, -1.75142525])] [1976, 2.3036911746915475e-05, array([ 4.00617946, 3.55404119, -1.75159529])] [1977, 2.3025300225748353e-05, array([ 4.00641166, 3.55422378, -1.75176524])] [1978, 2.3013700411856753e-05, array([ 4.00664375, 3.55440628, -1.7519351 ])] [1979, 2.3002112281105191e-05, array([ 4.00687572, 3.55458869, -1.75210488])] [1980, 2.2990535834238625e-05, array([ 4.00710758, 3.55477099, -1.75227456])] [1981, 2.2978971023291426e-05, array([ 4.00733932, 3.55495321, -1.75244416])] [1982, 2.2967417850431263e-05, array([ 4.00757093, 3.55513535, -1.75261369])] [1983, 2.2955876297687133e-05, array([ 4.00780243, 3.55531738, -1.75278314])] [1984, 2.2944346349396438e-05, array([ 4.00803382, 3.55549932, -1.75295249])] [1985, 2.2932827966310228e-05, array([ 4.0082651 , 3.55568117, -1.75312175])] [1986, 2.2921321163039054e-05, array([ 4.00849625, 3.55586293, -1.75329094])] [1987, 2.2909825892502574e-05, array([ 4.00872729, 3.5560446 , -1.75346003])] [1988, 2.2898342150002951e-05, array([ 4.00895822, 3.55622617, -1.75362904])] [1989, 2.2886869926728964e-05, array([ 4.00918902, 3.55640766, -1.75379798])] [1990, 2.2875409197258269e-05, array([ 4.0094197 , 3.55658905, -1.75396683])] [1991, 2.2863959947615882e-05, array([ 4.00965027, 3.55677036, -1.75413561])] [1992, 2.2852522148272073e-05, array([ 4.00988074, 3.55695157, -1.75430428])] [1993, 2.2841095795644489e-05, array([ 4.01011109, 3.55713269, -1.75447287])] [1994, 2.2829680862164335e-05, array([ 4.01034132, 3.55731372, -1.75464137])] [1995, 2.2818277344153124e-05, array([ 4.01057143, 3.55749465, -1.75480981])] [1996, 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4.43191321, 3.8882977 , -2.06422589])] [4994, 9.1365836936606729e-06, array([ 4.43200524, 3.88836984, -2.06429366])] [4995, 9.1347578155273095e-06, array([ 4.43209725, 3.88844197, -2.06436142])] [4996, 9.1329326674135285e-06, array([ 4.43218925, 3.88851408, -2.06442917])] [4997, 9.1311082490913867e-06, array([ 4.43228123, 3.88858618, -2.0644969 ])] [4998, 9.1292845590888931e-06, array([ 4.43237318, 3.88865827, -2.06456463])] [4999, 9.1274615983949097e-06, array([ 4.43246512, 3.88873034, -2.06463234])]
In [8]:
#### add regularization to stochastic gradient descent (to keep the weights small)
def ComputeRegularizedLoss(y, p, w, weightDecayRate):
return (- np.mean(np.log(y * p + (1 - y) * (1 - p))) + 0.5 * weightDecayRate * ((w[0] ** 2) + (w[1] ** 2)))
def ComputeRegularizedGradient(X, w, y, weightDecayRate):
return ((1.0 / (1.0 + np.exp(- np.dot(X, w))) - y) * X + weightDecayRate * np.array([ w[0], w[1], 0 ]))
w = np.array([-0.02, -0.01, 0.0])
learningRate = 0.1
weightDecayRate = 0.01
for epoch in range(epochs):
index = list(range(X.shape[0]))
np.random.shuffle(index)
for j in index:
w = w - learningRate * ComputeRegularizedGradient(X[j,:], w, y[j], weightDecayRate)
print([epoch, ComputeRegularizedLoss(y, p, w, weightDecayRate), w])
[0, 0.0041163690599516148, array([ 0.67476091, 0.60510001, -0.02747105])] [1, 0.0057742758798203387, array([ 0.79973105, 0.71656119, -0.03335005])] [2, 0.006981802450420886, array([ 0.87990645, 0.78759104, -0.03879932])] [3, 0.0079207631769337254, array([ 0.93822783, 0.83788763, -0.04545204])] [4, 0.0086756716062421201, array([ 0.98242537, 0.87644123, -0.0522819 ])] [5, 0.0092879324284771657, array([ 1.01713134, 0.90620353, -0.05944661])] [6, 0.009820728669638722, array([ 1.0464855 , 0.93122948, -0.06658307])] [7, 0.010283357102127886, array([ 1.07131904, 0.95242924, -0.07389927])] [8, 0.010700140154966946, array([ 1.09346969, 0.97083808, -0.08089473])] [9, 0.011059442114054036, array([ 1.11229785, 0.98633484, -0.0878133 ])] [10, 0.01136129831822274, array([ 1.12793388, 0.99909927, -0.09463135])] [11, 0.011642044657172128, array([ 1.1423569 , 1.01074436, -0.10142019])] [12, 0.011878160505055833, array([ 1.15446358, 1.0203041 , -0.10811677])] [13, 0.012082939358363654, array([ 1.16499449, 1.02837261, -0.11472431])] [14, 0.012267238402772943, array([ 1.1744199 , 1.03554821, -0.1211999 ])] [15, 0.012448223796612631, array([ 1.18359622, 1.0425542 , -0.12759064])] [16, 0.012611188085488017, array([ 1.19187655, 1.04873372, -0.13383492])] [17, 0.012748348637221178, array([ 1.19891984, 1.05377201, -0.13992797])] [18, 0.012871855296347545, array([ 1.20522114, 1.05829466, -0.14593968])] [19, 0.012983710665076439, array([ 1.21097375, 1.06228961, -0.15183457])] [20, 0.013084362889308837, array([ 1.21626395, 1.0657153 , -0.15764017])] [21, 0.013177633658381411, array([ 1.22110136, 1.06893064, -0.16342625])] [22, 0.013276288311394569, array([ 1.22616288, 1.07236037, -0.16912316])] [23, 0.013350135527166096, array([ 1.23009394, 1.07474207, -0.17470158])] [24, 0.013425748761059519, array([ 1.23408632, 1.07719785, -0.18014564])] [25, 0.013488676994445253, array([ 1.23745051, 1.07917845, -0.185506 ])] [26, 0.013538253060329641, array([ 1.24019927, 1.08061597, -0.19081175])] [27, 0.013593834418136439, array([ 1.24324911, 1.08225368, -0.19595851])] [28, 0.013640062778043636, array([ 1.24589986, 1.08347616, -0.20105921])] [29, 0.01369812673125608, array([ 1.24899292, 1.08527257, -0.20602312])] [30, 0.013734591187093963, array([ 1.2511116 , 1.08619174, -0.21101183])] [31, 0.013772263264723203, array([ 1.2531928 , 1.0872603 , -0.21599605])] [32, 0.013810499738626152, array([ 1.25534921, 1.08828894, -0.22079368])] [33, 0.013843757798887527, array([ 1.25725218, 1.08914785, -0.22563951])] [34, 0.013874393611428386, array([ 1.2591369 , 1.08978323, -0.23026237])] [35, 0.013892741128017414, array([ 1.2603613 , 1.09005143, -0.23491385])] [36, 0.013914673463440112, array([ 1.26177954, 1.09042267, -0.23943606])] [37, 0.013945504376392382, array([ 1.26360679, 1.09113394, -0.2438265 ])] [38, 0.013966124715170305, array([ 1.26494108, 1.09147767, -0.24818244])] [39, 0.013984478248984449, array([ 1.26618742, 1.09171405, -0.25253683])] [40, 0.014000725333221565, array([ 1.26743729, 1.09175193, -0.25669512])] [41, 0.01400398727535406, array([ 1.26799654, 1.09140127, -0.26096209])] [42, 0.014025216349508611, array([ 1.2693831 , 1.09173455, -0.26510126])] [43, 0.014039798531181567, array([ 1.27045667, 1.09182144, -0.26914278])] [44, 0.014054283525041131, array([ 1.27145561, 1.09198528, -0.27320627])] [45, 0.014066506857825296, array([ 1.27244364, 1.09195378, -0.27705147])] [46, 0.014081065400473367, array([ 1.2733862 , 1.09218825, -0.28090834])] [47, 0.014098161083767666, array([ 1.27456433, 1.09237929, -0.2847168 ])] [48, 0.014106849958054786, array([ 1.27533055, 1.09228041, -0.28842842])] [49, 0.014127627781517587, array([ 1.27665185, 1.09263906, -0.29206348])] [50, 0.014130839349306375, array([ 1.27714397, 1.09235784, -0.29570558])] [51, 0.014136764587961585, array([ 1.27779474, 1.0921392 , -0.29928603])] [52, 0.014150427740756434, array([ 1.27879085, 1.09222435, -0.30276679])] [53, 0.014151750795207524, array([ 1.27918224, 1.09188711, -0.30621772])] [54, 0.01416511579761057, array([ 1.28003134, 1.09211604, -0.30966662])] [55, 0.014172399673656895, array([ 1.28067866, 1.09202409, -0.31305921])] [56, 0.014179871857095135, array([ 1.28131791, 1.09195847, -0.31633196])] [57, 0.01418071133363148, array([ 1.28175643, 1.0915206 , -0.3195097 ])] [58, 0.014190296786280865, array([ 1.28245879, 1.09157378, -0.32271185])] [59, 0.014196799963812476, array([ 1.28299023, 1.09154504, -0.32589814])] [60, 0.014200036410958209, array([ 1.28348176, 1.09126366, -0.32890966])] [61, 0.014196868253943864, array([ 1.28365947, 1.09076419, -0.33199629])] [62, 0.014196768206360956, array([ 1.28391181, 1.09045799, -0.33497539])] [63, 0.01420535040644183, array([ 1.2846371 , 1.0903908 , -0.33797841])] [64, 0.014214551064893746, array([ 1.28537831, 1.0903611 , -0.34087804])] [65, 0.014212784892575115, array([ 1.28559743, 1.0899407 , -0.34371588])] [66, 0.014220665539544788, array([ 1.2861597 , 1.09000039, -0.34660962])] [67, 0.014222782396796306, array([ 1.28655151, 1.08973217, -0.34937584])] [68, 0.014214233208720548, array([ 1.2864763 , 1.08903621, -0.35212311])] [69, 0.014216802024929644, array([ 1.28693356, 1.0887318 , -0.35475852])] [70, 0.014216372050328396, array([ 1.28717291, 1.08840931, -0.35749663])] [71, 0.014226084671942744, array([ 1.28787695, 1.08846884, -0.36010491])] [72, 0.014227483535621817, array([ 1.28821134, 1.08820162, -0.36265873])] [73, 0.014244464199797047, array([ 1.28927948, 1.08849702, -0.36512277])] [74, 0.014248254448274425, array([ 1.28969027, 1.08835858, -0.36767885])] [75, 0.014257957637386348, array([ 1.29033579, 1.088485 , -0.37009941])] [76, 0.014265867124573457, array([ 1.29087127, 1.08857673, -0.3724827 ])] [77, 0.014267060575832231, array([ 1.29127988, 1.08820168, -0.374761 ])] [78, 0.014271972585690979, array([ 1.29182192, 1.08800972, -0.37699334])] [79, 0.01427992843123875, array([ 1.29247751, 1.08796235, -0.37927172])] [80, 0.014292854856070989, array([ 1.29334455, 1.0881201 , -0.38150631])] [81, 0.014299882738941404, array([ 1.29394939, 1.08804688, -0.38366556])] [82, 0.014311970839784898, array([ 1.29472287, 1.08823774, -0.38577564])] [83, 0.014306679902463626, array([ 1.29468802, 1.08779291, -0.38788592])] [84, 0.014310084313193758, array([ 1.29507151, 1.08764937, -0.3899307 ])] [85, 0.014308264387839189, array([ 1.29519838, 1.08733092, -0.3920527 ])] [86, 0.014312890362979449, array([ 1.2956989 , 1.08716003, -0.39403135])] [87, 0.014313407914151274, array([ 1.29596287, 1.08689297, -0.39605117])] [88, 0.01431757985336264, array([ 1.29637525, 1.08678503, -0.39798723])] [89, 0.014317069845794001, array([ 1.29650895, 1.08657859, -0.39989942])] [90, 0.014311794290415489, array([ 1.29643467, 1.08618162, -0.40186415])] [91, 0.014313519512833703, array([ 1.29665626, 1.08607594, -0.40387047])] [92, 0.01430488735749175, array([ 1.29647292, 1.08549986, -0.40580595])] [93, 0.014315835991484904, array([ 1.29714231, 1.08570877, -0.40762397])] [94, 0.014322551113574439, array([ 1.29765571, 1.08571377, -0.40939103])] [95, 0.01432753152540547, array([ 1.29817466, 1.08555211, -0.41110591])] [96, 0.014325167824542203, array([ 1.29821011, 1.08529193, -0.41279628])] [97, 0.014329624128913393, array([ 1.29869662, 1.08512046, -0.41440392])] [98, 0.014338191079002407, array([ 1.29931272, 1.08517241, -0.4160361 ])] [99, 0.014345184179876394, array([ 1.29986162, 1.08515949, -0.41767992])] [100, 0.014350206694044198, array([ 1.30032438, 1.08506791, -0.41922145])] [101, 0.014359940539782385, array([ 1.30092395, 1.08524628, -0.42080985])] [102, 0.014361854233968267, array([ 1.30123386, 1.08505105, -0.42241225])] [103, 0.014363486442015258, array([ 1.30153617, 1.08483888, -0.42384417])] [104, 0.014363671186286512, array([ 1.30168535, 1.08467691, -0.42536569])] [105, 0.014360582958664134, array([ 1.30185273, 1.08419121, -0.42680124])] [106, 0.014372960284218889, array([ 1.30260091, 1.08443416, -0.42823403])] [107, 0.014360930348458242, array([ 1.3022423 , 1.08375531, -0.4297916 ])] [108, 0.014360064758740104, array([ 1.30236428, 1.08352884, -0.43123801])] [109, 0.014361452616393239, array([ 1.30263361, 1.08333315, -0.43252903])] [110, 0.0143664841907958, array([ 1.3030267 , 1.08332486, -0.43393822])] [111, 0.014360833047027259, array([ 1.30283326, 1.08303583, -0.43528151])] [112, 0.014353359526923926, array([ 1.3026174 , 1.08260534, -0.43666695])] [113, 0.014362855873168628, array([ 1.30321101, 1.0827681 , -0.43799971])] [114, 0.014357042991949944, array([ 1.30307129, 1.08239933, -0.43937659])] [115, 0.014356599394595153, array([ 1.30325753, 1.0821341 , -0.44064065])] [116, 0.014349644525957885, array([ 1.30313597, 1.08163768, -0.44193192])] [117, 0.014351469928489475, array([ 1.30324287, 1.08167764, -0.44314642])] [118, 0.014354057808805144, array([ 1.30352243, 1.08158002, -0.44433238])] [119, 0.014361280913375312, array([ 1.30407908, 1.08157683, -0.44545742])] [120, 0.014365627921030818, array([ 1.30442954, 1.08155613, -0.44667688])] [121, 0.014369166659131365, array([ 1.30480483, 1.08143063, -0.44778542])] [122, 0.014372216468560907, array([ 1.30503963, 1.08142932, -0.44891882])] [123, 0.014373755329531437, array([ 1.30525535, 1.08131126, -0.4500863 ])] [124, 0.01437974454847301, array([ 1.3056879 , 1.08134293, -0.45113392])] [125, 0.014391836203386918, array([ 1.30635194, 1.08165907, -0.45219922])] [126, 0.014388160661328649, array([ 1.30630308, 1.08137824, -0.45328873])] [127, 0.014389753277109587, array([ 1.30651616, 1.08126809, -0.45431452])] [128, 0.014386458473546757, array([ 1.30644219, 1.08105273, -0.45539837])] [129, 0.014394705402532588, array([ 1.30694085, 1.08121284, -0.45639193])] [130, 0.01439672706678358, array([ 1.30710479, 1.08120165, -0.45742161])] [131, 0.014400479206318912, array([ 1.30745224, 1.08112857, -0.45840745])] [132, 0.014405905806983858, array([ 1.30787731, 1.08111638, -0.4594351 ])] [133, 0.014424314682985877, array([ 1.30881738, 1.08168133, -0.46026628])] [134, 0.014422497773604602, array([ 1.30891166, 1.08139925, -0.46122581])] [135, 0.014420296511429002, array([ 1.30885762, 1.08126109, -0.4622223 ])] [136, 0.014420542458698567, array([ 1.30897628, 1.08114019, -0.46319009])] [137, 0.014418300611238746, array([ 1.30900303, 1.08090041, -0.46407943])] [138, 0.014422668632663622, array([ 1.30930152, 1.080943 , -0.46490319])] [139, 0.014434309191298567, array([ 1.30991649, 1.08127478, -0.46576733])] [140, 0.014434043005937352, array([ 1.30999534, 1.08115462, -0.46658947])] [141, 0.014439940549955271, array([ 1.3104013 , 1.08120815, -0.4674823 ])] [142, 0.014442345380819963, array([ 1.3105963 , 1.08119421, -0.46833823])] [143, 0.014442327504214508, array([ 1.31072637, 1.08103487, -0.46922585])] [144, 0.014444600128092275, array([ 1.31096474, 1.08095605, -0.46999663])] [145, 0.014441770368344724, array([ 1.31091193, 1.08075829, -0.47079293])] [146, 0.014434269718121756, array([ 1.31057931, 1.08046764, -0.47163944])] [147, 0.014441782902077357, array([ 1.31092169, 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