Vectorized Multivariable Linear Regression from Scratch

Gradient Descent for multiple features

The following describes in detail the vectorized imiplementation of the algorithm of gradient descent for mltiple features represented by the vector x.

Cost Function

def cost_func(x,y,w,b):
  cost = 0
  n_points = y.shape[0]
  for i in range(n_points):
    f = np.dot(w,x[i])+b
    cost += (f -y[i])**2
  cost = cost/(2*n_points)
  return cost

Derivative Function

def derivative_func(x,y,w,b):
  n_points = y.shape[0]
  n_features = x[0].shape[0]
  dw = np.array([0]*n_features)
  db = 0
  for i in range(n_points):
    f = np.dot(w,x[i]) + b
    for j in range(n_features):
      dw[j] += (f - y[i])*x[i][j]/n_points
    db += (f - y[i])/n_points
  return dw, db

Gradient Descent Function

def grad_desc(x,y,w,b,a,n,derivative_func, cost_func):
  cost_arr = []
  for i in range(n):
    dw,db = derivative_func(x,y,w,b)
    w = w - a * dw
    b = b - a * db
    cost = cost_func(x,y,w,b)
    cost_arr.append(cost)
    if(i%100==0 or i==n-1):
      print("iteration:",i+1,"w:", w,"b:", b, "cost:", cost)
  return w, b, cost, cost_arr

Implementation

import numpy as np
x_train = np.array([[2104, 5, 1, 45], [1416, 3, 2, 40], [852, 2, 1, 35]])
y_train = np.array([460, 232, 178])
w = np.array([0,0,0,0])
b = 0
a = 0.0000005
n = 100000

w,b,cost,cost_arr = grad_desc(x_train, y_train, w, b, a, n, derivative_func, cost_func)

iteration: 1 w: [2.413345e-01 5.580000e-04 1.830000e-04 6.034500e-03] b: 0.000145 cost: 2529.445907451492
iteration: 101 w: [ 0.2023515  0.0007205 -0.000998  -0.0021815] b: -0.00011984671618528004 cost: 695.9930856678557
iteration: 201 w: [ 0.2025335  0.0009705 -0.002148  -0.009355 ] b: -0.00035963602268744473 cost: 694.9289935685856
iteration: 301 w: [ 0.202715   0.0012205 -0.003298  -0.016518 ] b: -0.0005982983702994033 cost: 693.8710622039179
iteration: 401 w: [ 0.202896  0.001445 -0.004448 -0.023628] b: -0.0008358359588454846 cost: 692.8254981266464
iteration: 501 w: [ 0.2030755  0.001675  -0.005568  -0.0307165] b: -0.0010722555533169366 cost: 691.7882133976278
iteration: 601 w: [ 0.2032545  0.001925  -0.006668  -0.037777 ] b: -0.001307562678074441 cost: 690.7597559872951
iteration: 701 w: [ 0.203433   0.002175  -0.007768  -0.0447935] b: -0.001541761106294673 cost: 689.7419977161079
iteration: 801 w: [ 0.203611   0.002425  -0.008868  -0.0517985] b: -0.0017748583098112362 cost: 688.7303041602801
iteration: 901 w: [ 0.203787  0.002675 -0.009968 -0.05875 ] b: -0.002006856060060288 cost: 687.7305146449748
iteration: 1001 w: [ 0.203963   0.002925  -0.011068  -0.0656845] b: -0.0022377620291096846 cost: 686.7374556983464
iteration: 1101 w: [ 0.204138  0.003175 -0.012168 -0.072588] b: -0.0024675777448512647 cost: 685.7530337759337
iteration: 1201 w: [ 0.204312   0.003425  -0.013268  -0.0794485] b: -0.0026963111772917827 cost: 684.7788423460347
iteration: 1301 w: [ 0.204486  0.003675 -0.014368 -0.086298] b: -0.0029239663570966323 cost: 683.810415310279
iteration: 1401 w: [ 0.2046585  0.0039135 -0.015468  -0.0930935] b: -0.0031505485097711224 cost: 682.8536764534834
iteration: 1501 w: [ 0.20483    0.0041135 -0.016568  -0.0998695] b: -0.0033760631581245247 cost: 681.9040178588283
iteration: 1601 w: [ 0.205002   0.004328  -0.0176535 -0.106625 ] b: -0.0036005085376852666 cost: 680.9615129263806
iteration: 1701 w: [ 0.205172   0.004578  -0.0187035 -0.1133365] b: -0.003823897765948446 cost: 680.0296293812801
iteration: 1801 w: [ 0.2053415  0.004828  -0.0197535 -0.120036 ] b: -0.004046235485751484 cost: 679.1034286909584
iteration: 1901 w: [ 0.205511   0.005078  -0.0208035 -0.1266875] b: -0.004267518384910261 cost: 678.187686912896
iteration: 2001 w: [ 0.205679   0.005328  -0.0218535 -0.133312 ] b: -0.004487759825311172 cost: 677.2795368977503
iteration: 2101 w: [ 0.2058465  0.005596  -0.0229035 -0.1399215] b: -0.004706960723743415 cost: 676.3772234638445
iteration: 2201 w: [ 0.206013   0.005896  -0.0239535 -0.1464775] b: -0.004925125624179499 cost: 675.4856970809177
iteration: 2301 w: [ 0.206179   0.006196  -0.0250035 -0.153025 ] b: -0.005142256578274909 cost: 674.5991745576366
iteration: 2401 w: [ 0.206344   0.006496  -0.0260535 -0.15954  ] b: -0.005358360241940179 cost: 673.7207726269572
iteration: 2501 w: [ 0.2065085  0.0067745 -0.027125  -0.1660135] b: -0.0055734435777244845 cost: 672.8512633960598
iteration: 2601 w: [ 0.206672   0.0070245 -0.028225  -0.1724735] b: -0.0057875101964774515 cost: 671.9868482788764
iteration: 2701 w: [ 0.206835   0.0072745 -0.029325  -0.1788865] b: -0.006000561348086028 cost: 671.1322623266645
iteration: 2801 w: [ 0.2069975  0.0075245 -0.030425  -0.185284 ] b: -0.006212606472340106 cost: 670.2833886731445
iteration: 2901 w: [ 0.207159   0.0077745 -0.031525  -0.191667 ] b: -0.00642364451026305 cost: 669.4400745262191
iteration: 3001 w: [ 0.2073195  0.0080245 -0.032625  -0.197989 ] b: -0.0066336814964713135 cost: 668.6081996289139
iteration: 3101 w: [ 0.207479   0.0082745 -0.033725  -0.2042985] b: -0.006842726206046504 cost: 667.7815296096983
iteration: 3201 w: [ 0.207639   0.0085245 -0.034825  -0.2105845] b: -0.007050775211079575 cost: 666.9614207743449
iteration: 3301 w: [ 0.2077975  0.0087745 -0.035925  -0.2168315] b: -0.007257836447118926 cost: 666.1497838167132
iteration: 3401 w: [ 0.207956   0.0090245 -0.037025  -0.223065 ] b: -0.007463916293745738 cost: 665.3433667135956
iteration: 3501 w: [ 0.2081135  0.0092745 -0.038125  -0.229261 ] b: -0.0076690160776340536 cost: 664.5451307273105
iteration: 3601 w: [ 0.2082695  0.0095245 -0.039225  -0.2354185] b: -0.007873146832570208 cost: 663.7551382261652
iteration: 3701 w: [ 0.2084255  0.0097745 -0.040325  -0.2415725] b: -0.008076305656634936 cost: 662.9690130199062
iteration: 3801 w: [ 0.208581   0.0100245 -0.041425  -0.2476855] b: -0.008278497331775565 cost: 662.1913455782793
iteration: 3901 w: [ 0.208735   0.010317  -0.0424825 -0.2537675] b: -0.008479729273751815 cost: 661.4215053114478
iteration: 4001 w: [ 0.2088885  0.010617  -0.0435325 -0.259838 ] b: -0.008680002444220802 cost: 660.6565300223458
iteration: 4101 w: [ 0.209042   0.010917  -0.0445825 -0.265858 ] b: -0.008879322789329899 cost: 659.9009983958877
iteration: 4201 w: [ 0.209194   0.011217  -0.0456325 -0.271867 ] b: -0.009077697385785134 cost: 659.1500791236687
iteration: 4301 w: [ 0.209346   0.011517  -0.0466825 -0.2778635] b: -0.009275126539940259 cost: 658.4039296265429
iteration: 4401 w: [ 0.209496   0.011817  -0.0477325 -0.283797 ] b: -0.00947161669159306 cost: 657.6685605755812
iteration: 4501 w: [ 0.2096465  0.012117  -0.0487825 -0.289721 ] b: -0.00966717154421513 cost: 656.9375086290565
iteration: 4601 w: [ 0.2097965  0.012417  -0.0498325 -0.2956425] b: -0.009861794300524621 cost: 656.2099347808465
iteration: 4701 w: [ 0.209945   0.012717  -0.0508825 -0.3015015] b: -0.010055491971623896 cost: 655.4929004964749
iteration: 4801 w: [ 0.2100935  0.013008  -0.0519325 -0.3073465] b: -0.010248264069244464 cost: 654.7806761471173
iteration: 4901 w: [ 0.2102415  0.013258  -0.0529825 -0.3131835] b: -0.010440117224497688 cost: 654.0727457961598
iteration: 5001 w: [ 0.210388   0.013508  -0.0540325 -0.3189695] b: -0.010631055309118353 cost: 653.373832197853
iteration: 5101 w: [ 0.2105345  0.013758  -0.0550825 -0.3247415] b: -0.010821084725986743 cost: 652.6795723436072
iteration: 5201 w: [ 0.21068    0.014008  -0.0561325 -0.3304895] b: -0.011010203536947682 cost: 651.9910995793263
iteration: 5301 w: [ 0.210825   0.0142615 -0.0571825 -0.336195 ] b: -0.011198420491851319 cost: 651.3104716657031
iteration: 5401 w: [ 0.21097    0.0145615 -0.0582325 -0.341889 ] b: -0.011385740109903942 cost: 650.6338118623238
iteration: 5501 w: [ 0.2111135  0.0148615 -0.0592825 -0.3475665] b: -0.011572162149229777 cost: 649.9619720741375
iteration: 5601 w: [ 0.2112565  0.0151615 -0.0603325 -0.353189 ] b: -0.011757693805196872 cost: 649.2992767667502
iteration: 5701 w: [ 0.211399   0.0154615 -0.0613825 -0.3588115] b: -0.011942339549135714 cost: 648.6394511780375
iteration: 5801 w: [ 0.211541   0.015804  -0.0624325 -0.36442  ] b: -0.012126100113712401 cost: 647.9837941514609
iteration: 5901 w: [ 0.211682   0.016154  -0.0634825 -0.3699655] b: -0.012308980410778101 cost: 647.3379762581632
iteration: 6001 w: [ 0.211822   0.016504  -0.0645325 -0.3755155] b: -0.012490987986764828 cost: 646.6944523563071
iteration: 6101 w: [ 0.2119625  0.016825  -0.0655825 -0.3810445] b: -0.012672123409456514 cost: 646.0562288339962
iteration: 6201 w: [ 0.2121005  0.017125  -0.0666325 -0.3865165] b: -0.012852392093039107 cost: 645.4271963544116
iteration: 6301 w: [ 0.212239   0.017425  -0.0676825 -0.391982 ] b: -0.013031797225555575 cost: 644.8015924971579
iteration: 6401 w: [ 0.212378   0.017725  -0.0687325 -0.39744  ] b: -0.013210340216809507 cost: 644.1795163791699
iteration: 6501 w: [ 0.212515   0.018025  -0.0697825 -0.402849 ] b: -0.013388026728848293 cost: 643.5654740729954
iteration: 6601 w: [ 0.212652   0.018325  -0.0708325 -0.4082395] b: -0.013564858977700602 cost: 642.9560887832505
iteration: 6701 w: [ 0.2127885  0.018625  -0.0718825 -0.4136275] b: -0.01374084259258523 cost: 642.3496126278868
iteration: 6801 w: [ 0.212924   0.018925  -0.0729325 -0.418967 ] b: -0.01391598386789509 cost: 641.750973972813
iteration: 6901 w: [ 0.2130585  0.019225  -0.0739825 -0.4242755] b: -0.014090284070279208 cost: 641.1582356076469
iteration: 7001 w: [ 0.213193   0.019525  -0.0750325 -0.4295835] b: -0.014263746409743042 cost: 640.568109942329
iteration: 7101 w: [ 0.213327   0.019825  -0.0760825 -0.4348655] b: -0.014436375252410026 cost: 639.9832945934037
iteration: 7201 w: [ 0.2134595  0.020125  -0.0771325 -0.4401045] b: -0.014608172944214843 cost: 639.4055458161864
iteration: 7301 w: [ 0.2135925  0.020425  -0.0781825 -0.445343 ] b: -0.01477914443959268 cost: 638.8303412585573
iteration: 7401 w: [ 0.213725   0.020712  -0.0792325 -0.450556 ] b: -0.014949292091960484 cost: 638.2603743385561
iteration: 7501 w: [ 0.213856   0.020962  -0.0802345 -0.455718 ] b: -0.015118622477034573 cost: 637.6994142844079
iteration: 7601 w: [ 0.213987   0.021212  -0.0812345 -0.460876 ] b: -0.015287140198373884 cost: 637.141328724567
iteration: 7701 w: [ 0.2141175  0.021462  -0.0822345 -0.4660195] b: -0.015454844998592595 cost: 636.5871528919143
iteration: 7801 w: [ 0.214247   0.021717  -0.0832295 -0.4711345] b: -0.015621740978801963 cost: 636.0383759096808
iteration: 7901 w: [ 0.214376   0.022017  -0.0841795 -0.4762155] b: -0.01578783423794423 cost: 635.4960978953343
iteration: 8001 w: [ 0.214504   0.022317  -0.0851295 -0.481276 ] b: -0.015953127487535804 cost: 634.9582480210778
iteration: 8101 w: [ 0.2146325  0.022617  -0.0860795 -0.4863355] b: -0.016117622136583992 cost: 634.4228215205202
iteration: 8201 w: [ 0.214759   0.022917  -0.0870295 -0.4913435] b: -0.01628132562817647 cost: 633.8949096425367
iteration: 8301 w: [ 0.214886   0.023217  -0.0879795 -0.4963405] b: -0.016444239940155186 cost: 633.3703751730842
iteration: 8401 w: [ 0.2150125  0.023517  -0.0889295 -0.501336 ] b: -0.016606366647562168 cost: 632.8482581924086
iteration: 8501 w: [ 0.2151375  0.023817  -0.0898795 -0.506274 ] b: -0.016767709530766145 cost: 632.3341237830863
iteration: 8601 w: [ 0.215262   0.024117  -0.0908295 -0.5111985] b: -0.016928273313438692 cost: 631.8235400798213
iteration: 8701 w: [ 0.215387   0.024417  -0.0917795 -0.516123 ] b: -0.01708806319070792 cost: 631.3151561998055
iteration: 8801 w: [ 0.215511   0.024713  -0.0927335 -0.521015 ] b: -0.01724707952602148 cost: 630.8121060432607
iteration: 8901 w: [ 0.2156345  0.024963  -0.0937335 -0.525869 ] b: -0.017405325612922583 cost: 630.3142816967135
iteration: 9001 w: [ 0.2157575  0.025213  -0.0947335 -0.5307105] b: -0.017562809157736888 cost: 629.8198151695871
iteration: 9101 w: [ 0.21588    0.025463  -0.0957335 -0.5355455] b: -0.017719526577670713 cost: 629.3281094657372
iteration: 9201 w: [ 0.216002   0.025713  -0.0967335 -0.5403505] b: -0.017875484390204293 cost: 628.8414195222083
iteration: 9301 w: [ 0.2161225  0.025963  -0.0977335 -0.545109 ] b: -0.018030688326887326 cost: 628.3612912918138
iteration: 9401 w: [ 0.2162435  0.026213  -0.0987335 -0.5498765] b: -0.01818514099529271 cost: 627.8823616070082
iteration: 9501 w: [ 0.2163645  0.026463  -0.0997335 -0.55463  ] b: -0.01833884113036277 cost: 627.406829126374
iteration: 9601 w: [ 0.2164835  0.026713  -0.1007335 -0.55933  ] b: -0.018491796348226376 cost: 626.9384182136115
iteration: 9701 w: [ 0.2166025  0.026963  -0.1017335 -0.5640205] b: -0.0186440142509367 cost: 626.4729114274149
iteration: 9801 w: [ 0.2167215  0.0272455 -0.1027335 -0.5686955] b: -0.018795488411498525 cost: 626.0106572418074
iteration: 9901 w: [ 0.2168395  0.0275455 -0.1037335 -0.5733615] b: -0.018946223421659778 cost: 625.5511251713053
iteration: 10001 w: [ 0.2169565  0.0278455 -0.1047335 -0.5779825] b: -0.019096231305416184 cost: 625.0977588586564
iteration: 10101 w: [ 0.217073   0.0281555 -0.1057335 -0.582588 ] b: -0.019245509266673105 cost: 624.6477093062289
iteration: 10201 w: [ 0.2171905  0.0285055 -0.1067335 -0.5871965] b: -0.019394061233485523 cost: 624.1990628401919
iteration: 10301 w: [ 0.2173055  0.0288555 -0.1077335 -0.591755 ] b: -0.019541889810656344 cost: 623.756940828035
iteration: 10401 w: [ 0.217421   0.0291565 -0.1087335 -0.596304 ] b: -0.019688998492371303 cost: 623.3178810248277
iteration: 10501 w: [ 0.2175355  0.0294565 -0.1097335 -0.600827 ] b: -0.01983539125795989 cost: 622.883076942993
iteration: 10601 w: [ 0.2176505  0.0297565 -0.1107335 -0.60536  ] b: -0.019981068756584914 cost: 622.4492294594884
iteration: 10701 w: [ 0.217764   0.0300565 -0.1117335 -0.609837 ] b: -0.020126033527300156 cost: 622.0223009308603
iteration: 10801 w: [ 0.217877   0.0303565 -0.1127335 -0.614291 ] b: -0.020270294817821766 cost: 621.5992617284829
iteration: 10901 w: [ 0.21799    0.0306565 -0.1137335 -0.618751 ] b: -0.02041385061047308 cost: 621.1774937822787
iteration: 11001 w: [ 0.2181025  0.0309565 -0.1147335 -0.623193 ] b: -0.020556707946261358 cost: 620.7591386769445
iteration: 11101 w: [ 0.218214   0.0312565 -0.1157335 -0.627589 ] b: -0.02069886562388622 cost: 620.346653647179
iteration: 11201 w: [ 0.2183255  0.0315565 -0.1167335 -0.631963 ] b: -0.02084033257743231 cost: 619.9378677314106
iteration: 11301 w: [ 0.218436   0.0318565 -0.1177335 -0.6363535] b: -0.020981104941769094 cost: 619.5293800742135
iteration: 11401 w: [ 0.2185465  0.0321565 -0.1187335 -0.6407095] b: -0.02112118541533878 cost: 619.1256610414683
iteration: 11501 w: [ 0.218656   0.0324565 -0.1197335 -0.645023 ] b: -0.021260582653791514 cost: 618.7273677374055
iteration: 11601 w: [ 0.218765   0.0327565 -0.1207335 -0.6493335] b: -0.021399300515166655 cost: 618.3310315353523
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iteration: 12001 w: [ 0.2191975  0.0339175 -0.1247335 -0.6663905] b: -0.021947397814013597 cost: 616.7785953545737
iteration: 12101 w: [ 0.2193045  0.0341675 -0.1257335 -0.6706005] b: -0.022082747048806018 cost: 616.3994721755062
iteration: 12201 w: [ 0.219412   0.0344175 -0.1267335 -0.6748215] b: -0.02221742695768332 cost: 616.0210374030172
iteration: 12301 w: [ 0.2195175  0.0346675 -0.1277335 -0.6789725] b: -0.02235145143549944 cost: 615.6501030944033
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iteration: 12501 w: [ 0.219728   0.0351675 -0.1297335 -0.6872765] b: -0.022617527643841392 cost: 614.9127850177091
iteration: 12601 w: [ 0.219833   0.0354175 -0.1307335 -0.6914245] b: -0.0227495818960631 cost: 614.5468166786753
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iteration: 12801 w: [ 0.2200405  0.0360035 -0.1326475 -0.69959  ] b: -0.023011749972109922 cost: 613.8314674589074
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iteration: 13001 w: [ 0.220247   0.0366035 -0.1345475 -0.707723 ] b: -0.02327133873010081 cost: 613.1250323657525
iteration: 13101 w: [ 0.2203485  0.0369035 -0.1354975 -0.711743 ] b: -0.02340017468047057 cost: 612.7778432097031
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iteration: 99101 w: [ 0.2421585  0.2527055 -0.827416  -1.570597 ] b: -0.04253521993207394 cost: 563.8862294425503
iteration: 99201 w: [ 0.2421595  0.2529555 -0.828166  -1.570647 ] b: -0.04252413312891034 cost: 563.873154925134
iteration: 99301 w: [ 0.242161   0.2532055 -0.828916  -1.570697 ] b: -0.04251303764582283 cost: 563.8600807174487
iteration: 99401 w: [ 0.2421625  0.2534555 -0.829666  -1.570747 ] b: -0.04250194477709257 cost: 563.8470074742296
iteration: 99501 w: [ 0.2421635  0.2537055 -0.830416  -1.570797 ] b: -0.04249081189654357 cost: 563.8339354312387
iteration: 99601 w: [ 0.242165   0.2539555 -0.831166  -1.570847 ] b: -0.04247967325288792 cost: 563.8208638547225
iteration: 99701 w: [ 0.2421665  0.2542055 -0.831916  -1.570897 ] b: -0.04246853139665554 cost: 563.8077932413789
iteration: 99801 w: [ 0.242168   0.2544555 -0.832666  -1.570947 ] b: -0.042457368840363024 cost: 563.7947235873087
iteration: 99901 w: [ 0.2421685  0.2547055 -0.833416  -1.570997 ] b: -0.042446192506948185 cost: 563.7816551667245
iteration: 100000 w: [ 0.24217    0.254953  -0.8341585 -1.5710465] b: -0.04243509994948715 cost: 563.768717592519

Result of the model

print("After 100000 iterations")
print("cost:", cost)

After 100000 iterations
cost: 563.768717592519

Visualizing the cost

import matplotlib.pyplot as plt
plt.plot(np.arange(1,100001,1), cost_arr)
plt.title("Cost vs Number of iterations")
plt.xlabel("Number of iterations")
plt.ylabel("Cost")
plt.show()