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gradient_descent [2025/08/01 17:02] – hkimscilgradient_descent [2025/08/01 18:57] (current) – [R output] hkimscil
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 ====== Gradient Descent ====== ====== Gradient Descent ======
 +====== explanation ======
 점차하강 = 조금씩 깍아서 원하는 기울기 (미분값) 찾기 점차하강 = 조금씩 깍아서 원하는 기울기 (미분값) 찾기
 prerequisite:  prerequisite: 
Line 22: Line 23:
 & = & \sum{2 \frac{1}{N} (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\ & = & \sum{2 \frac{1}{N} (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
 & \because & \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\ & \because & \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
-& = & -2 \frac{1}{N} \sum{(Y_i - (a + bX_i))} \\ +& = & -2 \frac{\sum{(Y_i - (a + bX_i))}}{N} \\ 
 +& = & -2 * \text{mean of residuals} \\ 
 \end{eqnarray*}  \end{eqnarray*} 
 +아래 R code에서 gradient function을 참조.
  
 </WRAP> </WRAP>
Line 40: Line 43:
 (미분을 이해한다는 것을 전제로) 위의 식은 b값이 변할 때 msr (mean square residual) 값이 어떻게 변하는가를 알려주는 것이다. 그리고 그것은 b값에 대한 residual의 총합에 (-2/N)*X값을 곱한 값이다.  (미분을 이해한다는 것을 전제로) 위의 식은 b값이 변할 때 msr (mean square residual) 값이 어떻게 변하는가를 알려주는 것이다. 그리고 그것은 b값에 대한 residual의 총합에 (-2/N)*X값을 곱한 값이다. 
 </WRAP> </WRAP>
 +====== R code ======
 <code> <code>
-library(tidyverse) +# d statquest explanation 
-# a simple example +# x <- c(0.5, 2.3, 2.9) 
-# statquest explanation +# y <- c(1.4, 1.9, 3.2)
-x <- c(0.5, 2.3, 2.9) +
-y <- c(1.4, 1.9, 3.2)+
  
 rm(list=ls()) rm(list=ls())
 # set.seed(191) # set.seed(191)
-n <- 500+n <- 300
 x <- rnorm(n, 5, 1.2) x <- rnorm(n, 5, 1.2)
 y <- 2.14 * x + rnorm(n, 0, 4) y <- 2.14 * x + rnorm(n, 0, 4)
Line 56: Line 57:
 # data <- data.frame(x, y) # data <- data.frame(x, y)
 data <- tibble(x = x, y = y) data <- tibble(x = x, y = y)
-data 
  
 mo <- lm(y~x) mo <- lm(y~x)
Line 65: Line 65:
 b1 = rnorm(1) b1 = rnorm(1)
 b0 = rnorm(1) b0 = rnorm(1)
 +
 +b1.init <- b1
 +b0.init <- b0
  
 # Predict function: # Predict function:
Line 85: Line 88:
 loss = loss_mse(predictions, y) loss = loss_mse(predictions, y)
  
-temp.sum <- data.frame(x, y, b0, b1,predictions, residuals) +data <- tibble(data.frame(x, y, predictions, residuals))
-temp.sum+
  
 print(paste0("Loss is: ", round(loss))) print(paste0("Loss is: ", round(loss)))
Line 107: Line 109:
  
 # Record Loss for each epoch: # Record Loss for each epoch:
-logs = list() +# logs = list() 
-bs=list()+# bs=list()
 b0s = c() b0s = c()
 b1s = c() b1s = c()
Line 119: Line 121:
   loss = loss_mse(predictions, y)   loss = loss_mse(predictions, y)
   mse = append(mse, loss)   mse = append(mse, loss)
-   +  # logs = append(logs, loss)
-  logs = append(logs, loss)+
      
   if (epoch %% 10 == 0){   if (epoch %% 10 == 0){
Line 135: Line 136:
   b1s <- append(b1s, b1)   b1s <- append(b1s, b1)
 } }
-# I must unscale coefficients to make them comprehensible+ 
 +# unscale coefficients to make them comprehensible
 b0 =  b0 - (mean(x) / sd(x)) * b1 b0 =  b0 - (mean(x) / sd(x)) * b1
 b1 = b1 / sd(x) b1 = b1 / sd(x)
  
 +# changes of estimators
 b0s <- b0s - (mean(x) /sd(x)) * b1s b0s <- b0s - (mean(x) /sd(x)) * b1s
 b1s <- b1s / sd(x) b1s <- b1s / sd(x)
Line 144: Line 147:
 parameters <- tibble(data.frame(b0s, b1s, mse)) parameters <- tibble(data.frame(b0s, b1s, mse))
  
-cat(paste0("Inclination: ", b1, ", \n", "Intercept: ", b0, "\n"))+cat(paste0("Slope: ", b1, ", \n", "Intercept: ", b0, "\n"))
 summary(lm(y~x))$coefficients summary(lm(y~x))$coefficients
  
Line 150: Line 153:
   geom_point(size = 2) +    geom_point(size = 2) + 
   geom_abline(aes(intercept = b0s, slope = b1s),   geom_abline(aes(intercept = b0s, slope = b1s),
-              data = parameters, linewidth = 0.5, color = 'red') + +              data = parameters, linewidth = 0.5,  
 +              color = 'green') + 
   theme_classic() +   theme_classic() +
   geom_abline(aes(intercept = b0s, slope = b1s),    geom_abline(aes(intercept = b0s, slope = b1s), 
               data = parameters %>% slice_head(),                data = parameters %>% slice_head(), 
-              linewidth = 0.5, color = 'blue') + +              linewidth = 1, color = 'blue') + 
   geom_abline(aes(intercept = b0s, slope = b1s),    geom_abline(aes(intercept = b0s, slope = b1s), 
               data = parameters %>% slice_tail(),                data = parameters %>% slice_tail(), 
-              linewidth = 1, color = 'green') + +              linewidth = 1, color = 'red') + 
-  labs(title = 'Gradient descent: blue: start, green: end')+  labs(title = 'Gradient descent. blue: start, red: end, green: gradients') 
 + 
 +b0.init 
 +b1.init 
 data data
 parameters parameters
  
-</code> 
  
 +</code>
 +====== R output =====
 <code> <code>
 > rm(list=ls()) > rm(list=ls())
 > # set.seed(191) > # set.seed(191)
-> n <- 500+> n <- 300
 > x <- rnorm(n, 5, 1.2) > x <- rnorm(n, 5, 1.2)
 > y <- 2.14 * x + rnorm(n, 0, 4) > y <- 2.14 * x + rnorm(n, 0, 4)
Line 173: Line 182:
 > # data <- data.frame(x, y) > # data <- data.frame(x, y)
 > data <- tibble(x = x, y = y) > data <- tibble(x = x, y = y)
-> data 
-# A tibble: 500 × 2 
-       x     y 
-   <dbl> <dbl> 
- 1  4.48 11.1  
- 2  6.45 10.2  
- 3  6.41 11.7  
- 4  5.35 15.8  
- 5  5.17  8.84 
- 6  3.64  1.36 
- 7  6.35 10.9  
- 8  3.30 10.7  
- 9  6.30  6.98 
-10  3.81  5.22 
-# ℹ 490 more rows 
-# ℹ Use `print(n = ...)` to see more rows 
 >  > 
 > mo <- lm(y~x) > mo <- lm(y~x)
Line 197: Line 190:
  
 Residuals: Residuals:
-     Min       1Q   Median       3Q      Max  +   Min     1Q Median     3Q    Max  
--10.2534  -2.6615   0.0087   2.7559   9.7626 +-9.754 -2.729 -0.135  2.415 10.750 
  
 Coefficients: Coefficients:
             Estimate Std. Error t value Pr(>|t|)                 Estimate Std. Error t value Pr(>|t|)    
-(Intercept)   0.1281     0.7108    0.18    0.857     +(Intercept)  -0.7794     0.9258  -0.842    0.401     
-x             2.1606     0.1388   15.57   <2e-16 ***+x             2.2692     0.1793  12.658   <2e-16 ***
 --- ---
 Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  
-Residual standard error: 3.822 on 498 degrees of freedom +Residual standard error: 3.951 on 298 degrees of freedom 
-Multiple R-squared:  0.3273, Adjusted R-squared:  0.326  +Multiple R-squared:  0.3497, Adjusted R-squared:  0.3475  
-F-statistic: 242.3 on 1 and 498 DF,  p-value: < 2.2e-16+F-statistic: 160.2 on 1 and 298 DF,  p-value: < 2.2e-16
  
 >  > 
Line 216: Line 209:
 > b1 = rnorm(1) > b1 = rnorm(1)
 > b0 = rnorm(1) > b0 = rnorm(1)
 +> 
 +> b1.init <- b1
 +> b0.init <- b0
 >  > 
 > # Predict function: > # Predict function:
Line 236: Line 232:
 > loss = loss_mse(predictions, y) > loss = loss_mse(predictions, y)
 >  > 
-> temp.sum <- data.frame(x, y, b0, b1,predictions, residuals) +> data <- tibble(data.frame(x, y, predictions, residuals))
-> temp.sum +
-           x          y        b0        b1 predictions   residuals +
-1   4.479742 11.1333005 0.5808843 0.9861742    4.998691  6.13460981 +
-2   6.452609 10.1579559 0.5808843 0.9861742    6.944280  3.21367543 +
-3   6.413290 11.6979692 0.5808843 0.9861742    6.905505  4.79246407 +
-4   5.345206 15.7547560 0.5808843 0.9861742    5.852188  9.90256769 +
-5   5.173454  8.8357907 0.5808843 0.9861742    5.682811  3.15297971 +
-6   3.636604  1.3575950 0.5808843 0.9861742    4.167209 -2.80961437 +
-7   6.348325 10.8649166 0.5808843 0.9861742    6.841439  4.02347773 +
-8   3.299825 10.7060946 0.5808843 0.9861742    3.835087  6.87100778 +
-9   6.300136  6.9805495 0.5808843 0.9861742    6.793915  0.18663406 +
-10  3.812044  5.2219952 0.5808843 0.9861742    4.340224  0.88177101 +
-11  3.436925  9.7518360 0.5808843 0.9861742    3.970291  5.78154462 +
-12  5.883357 14.4497406 0.5808843 0.9861742    6.382899  8.06684128 +
-13  4.328653 14.5001264 0.5808843 0.9861742    4.849690  9.65043604 +
-14  4.130057 10.0931558 0.5808843 0.9861742    4.653840  5.43931617 +
-15  5.322393  9.1121695 0.5808843 0.9861742    5.829691  3.28247861 +
-16  4.526528  8.1055219 0.5808843 0.9861742    5.044829  3.06069276 +
-17  3.817400  4.4323299 0.5808843 0.9861742    4.345505  0.08682455 +
-18  3.387983 -0.2180968 0.5808843 0.9861742    3.922026 -4.14012281 +
-19  4.270354  8.2876796 0.5808843 0.9861742    4.792198  3.49548203 +
-20  5.822266 10.5076073 0.5808843 0.9861742    6.322653  4.18495463 +
-21  6.009412  9.8284624 0.5808843 0.9861742    6.507211  3.32125112 +
-22  5.785644 12.2267578 0.5808843 0.9861742    6.286537  5.94022063 +
-23  5.103190 10.6399113 0.5808843 0.9861742    5.613518  5.02639300 +
-24  5.381166 18.1917469 0.5808843 0.9861742    5.887652 12.30409506 +
-25  2.812116 11.6562811 0.5808843 0.9861742    3.354121  8.30216030 +
-26  3.146225  5.1230054 0.5808843 0.9861742    3.683610  1.43939522 +
-27  4.883188 10.6312680 0.5808843 0.9861742    5.396558  5.23470953 +
-28  4.955458 11.8617949 0.5808843 0.9861742    5.467830  6.39396521 +
-29  3.952036  8.4087869 0.5808843 0.9861742    4.478281  3.93050617 +
-30  6.739458 18.9309648 0.5808843 0.9861742    7.227164 11.70380055 +
-31  4.882959  9.7563509 0.5808843 0.9861742    5.396333  4.36001803 +
-32  5.070600 17.4532502 0.5808843 0.9861742    5.581379 11.87187072 +
-33  5.257397 10.4619521 0.5808843 0.9861742    5.765594  4.69635807 +
-34  3.921518  0.6379410 0.5808843 0.9861742    4.448184 -3.81024292 +
-35  5.112554  7.8998652 0.5808843 0.9861742    5.622753  2.27711243 +
-36  5.783692 17.7148068 0.5808843 0.9861742    6.284612 11.43019492 +
-37  5.756150 10.1595169 0.5808843 0.9861742    6.257451  3.90206554 +
-38  6.010004 15.6295228 0.5808843 0.9861742    6.507795  9.12172776 +
-39  7.527341 10.8972192 0.5808843 0.9861742    8.004154  2.89306557 +
-40  3.718376 11.4857450 0.5808843 0.9861742    4.247851  7.23789408 +
-41  3.816369  2.7557710 0.5808843 0.9861742    4.344489 -1.58871815 +
-42  3.904699 11.9299211 0.5808843 0.9861742    4.431598  7.49832314 +
-43  4.889957 17.8620975 0.5808843 0.9861742    5.403234 12.45886371 +
-44  3.456463  7.4961713 0.5808843 0.9861742    3.989559  3.50661222 +
-45  5.274541 15.1190395 0.5808843 0.9861742    5.782501  9.33653899 +
-46  5.064607 13.5273619 0.5808843 0.9861742    5.575469  7.95189250 +
-47  3.175056  3.5614626 0.5808843 0.9861742    3.712043 -0.15058044 +
-48  7.179882 10.5044662 0.5808843 0.9861742    7.661499  2.84296697 +
-49  4.098562 16.6057557 0.5808843 0.9861742    4.622780 11.98297543 +
-50  4.532198  8.3301716 0.5808843 0.9861742    5.050421  3.27975067 +
-51  7.248676 12.4619092 0.5808843 0.9861742    7.729341  4.73256789 +
-52  4.440335 14.0170577 0.5808843 0.9861742    4.959828  9.05722939 +
-53  6.565150 15.0841504 0.5808843 0.9861742    7.055266  8.02888421 +
-54  5.886684  8.9326774 0.5808843 0.9861742    6.386180  2.54649717 +
-55  3.417331  3.3471923 0.5808843 0.9861742    3.950968 -0.60377576 +
-56  5.917124  6.0906638 0.5808843 0.9861742    6.416200 -0.32553581 +
-57  3.453444 11.2981952 0.5808843 0.9861742    3.986582  7.31161320 +
-58  4.825523  4.3895944 0.5808843 0.9861742    5.339690 -0.95009592 +
-59  4.649551  4.5605470 0.5808843 0.9861742    5.166152 -0.60560470 +
-60  5.065955 15.8326852 0.5808843 0.9861742    5.576798 10.25588679 +
-61  5.189293 11.3101911 0.5808843 0.9861742    5.698431  5.61175986 +
-62  5.769762  9.5544913 0.5808843 0.9861742    6.270875  3.28361629 +
-63  6.136073 14.2736277 0.5808843 0.9861742    6.632121  7.64150631 +
-64  5.079882  9.4934102 0.5808843 0.9861742    5.590533  3.90287732 +
-65  2.407474  1.5588896 0.5808843 0.9861742    2.955073 -1.39618346 +
-66  5.754148 13.7113431 0.5808843 0.9861742    6.255477  7.45586639 +
-67  4.674475 15.8016174 0.5808843 0.9861742    5.190731 10.61088686 +
-68  5.690545 19.2021795 0.5808843 0.9861742    6.192753 13.00942611 +
-69  5.204651  8.6483243 0.5808843 0.9861742    5.713577  2.93474768 +
-70  4.331535 11.2805649 0.5808843 0.9861742    4.852533  6.42803216 +
-71  3.605775  5.2645847 0.5808843 0.9861742    4.136806  1.12777848 +
-72  6.863329 10.9046518 0.5808843 0.9861742    7.349322  3.55532960 +
-73  6.296937  6.9897624 0.5808843 0.9861742    6.790761  0.19900104 +
-74  5.377210 13.8655597 0.5808843 0.9861742    5.883750  7.98180970 +
-75  5.403542  9.0932758 0.5808843 0.9861742    5.909718  3.18355804 +
-76  4.097157  7.8821791 0.5808843 0.9861742    4.621395  3.26078434 +
-77  3.994292  3.7220445 0.5808843 0.9861742    4.519952 -0.79790775 +
-78  3.898466  8.3450728 0.5808843 0.9861742    4.425451  3.91962211 +
-79  6.201434 19.8080454 0.5808843 0.9861742    6.696578 13.11146709 +
-80  6.972587 15.4954552 0.5808843 0.9861742    7.457070  8.03838546 +
-81  5.512087 10.1381343 0.5808843 0.9861742    6.016763  4.12137159 +
-82  5.463011 12.2079314 0.5808843 0.9861742    5.968365  6.23956610 +
-83  5.840064 15.6354616 0.5808843 0.9861742    6.340205  9.29525659 +
-84  4.628974  8.7689346 0.5808843 0.9861742    5.145859  3.62307535 +
-85  3.775477  7.9325251 0.5808843 0.9861742    4.304162  3.62836296 +
-86  4.789949  5.2633379 0.5808843 0.9861742    5.304608 -0.04127026 +
-87  5.878382 22.5915677 0.5808843 0.9861742    6.377993 16.21357470 +
-88  4.992651 14.0194907 0.5808843 0.9861742    5.504508  8.51498287 +
-89  3.181054 10.1407775 0.5808843 0.9861742    3.717958  6.42281930 +
-90  5.530133 11.2858888 0.5808843 0.9861742    6.034559  5.25133021 +
-91  5.141758  8.7832124 0.5808843 0.9861742    5.651554  3.13165855 +
-92  4.911979 13.4673585 0.5808843 0.9861742    5.424951  8.04240727 +
-93  6.363601  9.6015990 0.5808843 0.9861742    6.856503  2.74509561 +
-94  4.590408 13.5678458 0.5808843 0.9861742    5.107827  8.46001920 +
-95  3.394860 11.9961020 0.5808843 0.9861742    3.928808  8.06729420 +
-96  5.054608 10.9710834 0.5808843 0.9861742    5.565608  5.40547500 +
-97  5.631312 15.0984710 0.5808843 0.9861742    6.134339  8.96413204 +
-98  4.528634 13.5841385 0.5808843 0.9861742    5.046906  8.53723213 +
-99  6.368627 14.4842546 0.5808843 0.9861742    6.861460  7.62279469 +
-100 4.502220  6.1382455 0.5808843 0.9861742    5.020858  1.11738800 +
-101 3.363460  5.3607519 0.5808843 0.9861742    3.897841  1.46291038 +
-102 5.489312  9.6561866 0.5808843 0.9861742    5.994303  3.66188397 +
-103 2.208872 -0.2618084 0.5808843 0.9861742    2.759217 -3.02102560 +
-104 4.703816 15.7759129 0.5808843 0.9861742    5.219667 10.55624606 +
-105 2.594336 10.8674426 0.5808843 0.9861742    3.139352  7.72809101 +
-106 4.360380 10.0070566 0.5808843 0.9861742    4.880979  5.12607762 +
-107 4.255169  2.7966159 0.5808843 0.9861742    4.777223 -1.98060674 +
-108 6.229481 11.1195615 0.5808843 0.9861742    6.724238  4.39532397 +
-109 3.429806  6.8987700 0.5808843 0.9861742    3.963271  2.93549927 +
-110 8.152707 22.5016746 0.5808843 0.9861742    8.620874 13.88080087 +
-111 2.640736 -3.1625558 0.5808843 0.9861742    3.185110 -6.34766542 +
-112 4.741218 18.5919552 0.5808843 0.9861742    5.256551 13.33540377 +
-113 5.488745 12.9386181 0.5808843 0.9861742    5.993743  6.94487516 +
-114 3.227389  5.3641700 0.5808843 0.9861742    3.763652  1.60051759 +
-115 4.443698 13.9122126 0.5808843 0.9861742    4.963145  8.94906779 +
-116 5.338956  8.8612430 0.5808843 0.9861742    5.846025  3.01521767 +
-117 6.797698 15.9969094 0.5808843 0.9861742    7.284598  8.71231099 +
-118 7.022180 13.5295313 0.5808843 0.9861742    7.505977  6.02355391 +
-119 5.473466 11.9334015 0.5808843 0.9861742    5.978675  5.95472630 +
-120 6.024003 13.1870525 0.5808843 0.9861742    6.521600  6.66545200 +
-121 5.091827  3.2878229 0.5808843 0.9861742    5.602313 -2.31449027 +
-122 4.493815 12.2146866 0.5808843 0.9861742    5.012568  7.20211816 +
-123 7.112794 19.9524515 0.5808843 0.9861742    7.595338 12.35711319 +
-124 5.225292 16.7979964 0.5808843 0.9861742    5.733932 11.06406408 +
-125 5.064472 15.5614482 0.5808843 0.9861742    5.575336  9.98611181 +
-126 5.552849  9.4032924 0.5808843 0.9861742    6.056961  3.34633146 +
-127 3.902294  8.0228812 0.5808843 0.9861742    4.429226  3.59365505 +
-128 6.951468 17.2616673 0.5808843 0.9861742    7.436242  9.82542486 +
-129 5.217489 14.3563415 0.5808843 0.9861742    5.726237  8.63010443 +
-130 1.832789  6.8450024 0.5808843 0.9861742    2.388333  4.45666924 +
-131 5.170683  9.6343272 0.5808843 0.9861742    5.680078  3.95424866 +
-132 6.104459 11.7304563 0.5808843 0.9861742    6.600944  5.12951245 +
-133 4.584068 14.0684806 0.5808843 0.9861742    5.101574  8.96690638 +
-134 6.594802 11.5921896 0.5808843 0.9861742    7.084508  4.50768117 +
-135 4.492839 11.3445727 0.5808843 0.9861742    5.011606  6.33296667 +
-136 4.835051 10.0457136 0.5808843 0.9861742    5.349087  4.69662702 +
-137 4.495956 12.9092820 0.5808843 0.9861742    5.014680  7.89460166 +
-138 4.704205 11.3171541 0.5808843 0.9861742    5.220049  6.09710462 +
-139 6.495443  8.1764858 0.5808843 0.9861742    6.986523  1.18996263 +
-140 4.475086  5.9729101 0.5808843 0.9861742    4.994099  0.97881090 +
-141 3.089286  2.3112953 0.5808843 0.9861742    3.627459 -1.31616349 +
-142 5.959849 12.4859299 0.5808843 0.9861742    6.458334  6.02759586 +
-143 6.474426 13.7818185 0.5808843 0.9861742    6.965796  6.81602242 +
-144 5.567374 14.4706101 0.5808843 0.9861742    6.071285  8.39932527 +
-145 4.052363  7.9027988 0.5808843 0.9861742    4.577220  3.32557871 +
-146 5.074791 10.0325838 0.5808843 0.9861742    5.585512  4.44707188 +
-147 6.831115  9.3577103 0.5808843 0.9861742    7.317553  2.04015701 +
-148 3.607346 10.5973814 0.5808843 0.9861742    4.138356  6.45902511 +
-149 6.896116 21.4919066 0.5808843 0.9861742    7.381656 14.11025058 +
-150 6.317008 15.9406225 0.5808843 0.9861742    6.810555  9.13006776 +
-151 5.168403 15.8355851 0.5808843 0.9861742    5.677830 10.15775499 +
-152 4.434618  7.1342370 0.5808843 0.9861742    4.954190  2.18004689 +
-153 5.891072  8.7559727 0.5808843 0.9861742    6.390507  2.36546522 +
-154 3.512759  8.4225211 0.5808843 0.9861742    4.045077  4.37744430 +
-155 3.946038 11.5013653 0.5808843 0.9861742    4.472365  7.02900004 +
-156 6.756402 13.1194434 0.5808843 0.9861742    7.243873  5.87557008 +
-157 4.691839 13.2412345 0.5808843 0.9861742    5.207855  8.03337960 +
-158 4.691552 11.1985804 0.5808843 0.9861742    5.207571  5.99100894 +
-159 4.025405 13.5279553 0.5808843 0.9861742    4.550635  8.97732006 +
-160 6.330436 12.8980948 0.5808843 0.9861742    6.823797  6.07429736 +
-161 5.213079 15.7716904 0.5808843 0.9861742    5.721888 10.04980247 +
-162 6.378086 11.6205214 0.5808843 0.9861742    6.870788  4.74973340 +
-163 7.228954 11.5549520 0.5808843 0.9861742    7.709892  3.84505980 +
-164 3.621615 14.6260406 0.5808843 0.9861742    4.152427 10.47361310 +
-165 5.235016  8.3091022 0.5808843 0.9861742    5.743522  2.56558049 +
-166 3.767853 11.1148677 0.5808843 0.9861742    4.296644  6.81822388 +
- [ reached 'max' / getOption("max.print") -- omitted 334 rows ]+
 >  > 
 > print(paste0("Loss is: ", round(loss))) > print(paste0("Loss is: ", round(loss)))
-[1] "Loss is: 46"+[1] "Loss is: 393"
 >  > 
 > gradient <- function(x, y, predictions){ > gradient <- function(x, y, predictions){
Line 421: Line 248:
 > print(gradients) > print(gradients)
 $db1 $db1
-[1] -57.11316+[1] -200.6834
  
 $db0 $db0
-[1] -10.77174+[1] -37.76994
  
 >  > 
Line 433: Line 260:
 >  > 
 > # Record Loss for each epoch: > # Record Loss for each epoch:
-> logs = list() +> # logs = list() 
-> bs=list()+> # bs=list()
 > b0s = c() > b0s = c()
 > b1s = c() > b1s = c()
Line 445: Line 272:
 +   loss = loss_mse(predictions, y) +   loss = loss_mse(predictions, y)
 +   mse = append(mse, loss) +   mse = append(mse, loss)
-+    ++   # logs = append(logs, loss)
-+   logs = append(logs, loss)+
 +    +   
 +   if (epoch %% 10 == 0){ +   if (epoch %% 10 == 0){
Line 461: Line 287:
 +   b1s <- append(b1s, b1) +   b1s <- append(b1s, b1)
 + } + }
-[1] "Epoch: 10, Loss: 16.50445" +[1] "Epoch: 10, Loss: 18.5393" 
-[1] "Epoch: 20, Loss: 14.56909" +[1] "Epoch: 20, Loss: 15.54339" 
-[1] "Epoch: 30, Loss: 14.54677" +[1] "Epoch: 30, Loss: 15.50879" 
-[1] "Epoch: 40, Loss: 14.54651" +[1] "Epoch: 40, Loss: 15.50839" 
-[1] "Epoch: 50, Loss: 14.54651" +[1] "Epoch: 50, Loss: 15.50839" 
-[1] "Epoch: 60, Loss: 14.54651" +[1] "Epoch: 60, Loss: 15.50839" 
-[1] "Epoch: 70, Loss: 14.54651" +[1] "Epoch: 70, Loss: 15.50839" 
-[1] "Epoch: 80, Loss: 14.54651" +[1] "Epoch: 80, Loss: 15.50839" 
-> # I must unscale coefficients to make them comprehensible+>  
 +> # unscale coefficients to make them comprehensible
 > b0 =  b0 - (mean(x) / sd(x)) * b1 > b0 =  b0 - (mean(x) / sd(x)) * b1
 > b1 = b1 / sd(x) > b1 = b1 / sd(x)
 >  > 
 +> # changes of estimators
 > b0s <- b0s - (mean(x) /sd(x)) * b1s > b0s <- b0s - (mean(x) /sd(x)) * b1s
 > b1s <- b1s / sd(x) > b1s <- b1s / sd(x)
Line 478: Line 306:
 > parameters <- tibble(data.frame(b0s, b1s, mse)) > parameters <- tibble(data.frame(b0s, b1s, mse))
 >  > 
-> cat(paste0("Inclination: ", b1, ", \n", "Intercept: ", b0, "\n")) +> cat(paste0("Slope: ", b1, ", \n", "Intercept: ", b0, "\n")) 
-Inclination: 2.16059976407543,  +Slope: 2.26922511738252,  
-Intercept: 0.128130381671001+Intercept: -0.779435058320381
 > summary(lm(y~x))$coefficients > summary(lm(y~x))$coefficients
-             Estimate Std. Error    t value     Pr(>|t|) +              Estimate Std. Error    t value     Pr(>|t|) 
-(Intercept) 0.1281304  0.7108462  0.1802506 8.570292e-01 +(Intercept) -0.7794352  0.9258064 -0.8418986 4.005198e-01 
-x           2.1605998  0.1387908 15.5673144 8.229814e-45+x            2.2692252  0.1792660 12.6584242 1.111614e-29
 >  > 
 > ggplot(data, aes(x = x, y = y)) +  > ggplot(data, aes(x = x, y = y)) + 
 +   geom_point(size = 2) +  +   geom_point(size = 2) + 
 +   geom_abline(aes(intercept = b0s, slope = b1s), +   geom_abline(aes(intercept = b0s, slope = b1s),
-+               data = parameters, linewidth = 0.5, color = 'red') + ++               data = parameters, linewidth = 0.5,  
 ++               color = 'green') + 
 +   theme_classic() + +   theme_classic() +
 +   geom_abline(aes(intercept = b0s, slope = b1s),  +   geom_abline(aes(intercept = b0s, slope = b1s), 
 +               data = parameters %>% slice_head(),  +               data = parameters %>% slice_head(), 
-+               linewidth = 0.5, color = 'blue') + ++               linewidth = 1, color = 'blue') + 
 +   geom_abline(aes(intercept = b0s, slope = b1s),  +   geom_abline(aes(intercept = b0s, slope = b1s), 
 +               data = parameters %>% slice_tail(),  +               data = parameters %>% slice_tail(), 
-+               linewidth = 1, color = 'green') + ++               linewidth = 1, color = 'red') + 
-+   labs(title = 'Gradient descent: blue: start, green: end')++   labs(title = 'Gradient descent. blue: start, red: end, green: gradients') 
 +>  
 +> b0.init 
 +[1] -1.67967 
 +> b1.init 
 +[1] -1.323992 
 +> 
 > data > data
-# A tibble: 500 × 2 +# A tibble: 300 × 4 
-       x     y +       x     y predictions residuals 
-   <dbl> <dbl> +   <dbl> <dbl>       <dbl>     <dbl> 
- 1  4.48 11.1  + 1  4.13  6.74       -7.14     13.9  
- 2  6.45 10.2  + 2  7.25 14.0       -11.3      25.3  
- 3  6.41 11.7  + 3  6.09 13.5        -9.74     23.3  
- 4  5.35 15.8  + 4  6.29 15.1       -10.0      25.1  
- 5  5.17  8.84 + 5  4.40  3.81       -7.51     11.3  
- 6  3.64  1.36 + 6  6.03 13.9        -9.67     23.5  
- 7  6.35 10.9  + 7  6.97 12.1       -10.9      23.0  
- 8  3.30 10.7  + 8  4.84 12.8        -8.09     20.9  
- 9  6.30  6.98 + 9  6.85 17.2       -10.7      28.0  
-10  3.81  5.22 +10  3.33  3.80       -6.08      9.88 
-# ℹ 490 more rows+# ℹ 290 more rows
 # ℹ Use `print(n = ...)` to see more rows # ℹ Use `print(n = ...)` to see more rows
 > parameters > parameters
 # A tibble: 80 × 3 # A tibble: 80 × 3
-      b0s   b1s   mse +       b0s    b1s   mse 
-    <dbl> <dbl> <dbl> +     <dbl>  <dbl> <dbl> 
- 1 -2.69   1.07 123.  + 1  2.67   -0.379 183.  
- 2 -2.12   1.29  84.1 + 2  1.99    0.149 123.  
- 3 -1.67   1.46  59.1 + 3  1.44    0.571  84.3 
- 4 -1.31   1.60  43.0 + 4  1.00    0.910  59.6 
- 5 -1.02   1.71  32.8 + 5  0.652   1.18   43.7 
- 6 -0.791  1.80  26.2 + 6  0.369   1.40   33.6 
- 7 -0.606  1.87  22.0 + 7  0.142   1.57   27.1 
- 8 -0.459  1.93  19.3 + 8 -0.0397  1.71   22.9 
- 9 -0.341  1.98  17.6 + 9 -0.186   1.82   20.2 
-10 -0.247  2.01  16.5+10 -0.303   1.91   18.5
 # ℹ 70 more rows # ℹ 70 more rows
-# ℹ Use `print(n = ...)` to see more rows +# 
-> +
 </code> </code>
-{{:pasted:20250801-135321.png}}+ 
 +{{:pasted:20250801-185727.png}}
  
gradient_descent.1754035324.txt.gz · Last modified: 2025/08/01 17:02 by hkimscil
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