设$f(x)=x^2$，其导数为$f'(x)=2x^{2-1}=2x$。
# 初始化参数
x <- 5
learning_rate <- 0.1
num_iterations <- 100
# 梯度下降算法
for (i in 1:num_iterations) {
gradient <- 2 * x # 计算梯度
x <- x - learning_rate * gradient # 更新参数
cat("Iteration", i, ": x =", x, ", f(x) =", x^2, "\n")
}
cat("Minimum value of f(x) is approximately at x =", x, "\n")
梯度下降法是一种迭代优化算法，用于寻找函数的最小值。它在机器学习和深度学习中被广泛用于优化模型参数，以最小化损失函数。
展开/折叠结果
Iteration 1 : x = 4 , f(x) = 16
Iteration 2 : x = 3.2 , f(x) = 10.24
Iteration 3 : x = 2.56 , f(x) = 6.5536
Iteration 4 : x = 2.048 , f(x) = 4.194304
Iteration 5 : x = 1.6384 , f(x) = 2.684355
Iteration 6 : x = 1.31072 , f(x) = 1.717987
Iteration 7 : x = 1.048576 , f(x) = 1.099512
Iteration 8 : x = 0.8388608 , f(x) = 0.7036874
Iteration 9 : x = 0.6710886 , f(x) = 0.45036
Iteration 10 : x = 0.5368709 , f(x) = 0.2882304
Iteration 11 : x = 0.4294967 , f(x) = 0.1844674
Iteration 12 : x = 0.3435974 , f(x) = 0.1180592
Iteration 13 : x = 0.2748779 , f(x) = 0.07555786
Iteration 14 : x = 0.2199023 , f(x) = 0.04835703
Iteration 15 : x = 0.1759219 , f(x) = 0.0309485
Iteration 16 : x = 0.1407375 , f(x) = 0.01980704
Iteration 17 : x = 0.11259 , f(x) = 0.01267651
Iteration 18 : x = 0.09007199 , f(x) = 0.008112964
Iteration 19 : x = 0.07205759 , f(x) = 0.005192297
Iteration 20 : x = 0.05764608 , f(x) = 0.00332307
Iteration 21 : x = 0.04611686 , f(x) = 0.002126765
Iteration 22 : x = 0.03689349 , f(x) = 0.001361129
Iteration 23 : x = 0.02951479 , f(x) = 0.0008711229
Iteration 24 : x = 0.02361183 , f(x) = 0.0005575186
Iteration 25 : x = 0.01888947 , f(x) = 0.0003568119
Iteration 26 : x = 0.01511157 , f(x) = 0.0002283596
Iteration 27 : x = 0.01208926 , f(x) = 0.0001461502
Iteration 28 : x = 0.009671407 , f(x) = 9.35361e-05
Iteration 29 : x = 0.007737125 , f(x) = 5.986311e-05
Iteration 30 : x = 0.0061897 , f(x) = 3.831239e-05
Iteration 31 : x = 0.00495176 , f(x) = 2.451993e-05
Iteration 32 : x = 0.003961408 , f(x) = 1.569275e-05
Iteration 33 : x = 0.003169127 , f(x) = 1.004336e-05
Iteration 34 : x = 0.002535301 , f(x) = 6.427752e-06
Iteration 35 : x = 0.002028241 , f(x) = 4.113761e-06
Iteration 36 : x = 0.001622593 , f(x) = 2.632807e-06
Iteration 37 : x = 0.001298074 , f(x) = 1.684997e-06
Iteration 38 : x = 0.001038459 , f(x) = 1.078398e-06
Iteration 39 : x = 0.0008307675 , f(x) = 6.901746e-07
Iteration 40 : x = 0.000664614 , f(x) = 4.417118e-07
Iteration 41 : x = 0.0005316912 , f(x) = 2.826955e-07
Iteration 42 : x = 0.000425353 , f(x) = 1.809251e-07
Iteration 43 : x = 0.0003402824 , f(x) = 1.157921e-07
Iteration 44 : x = 0.0002722259 , f(x) = 7.410694e-08
Iteration 45 : x = 0.0002177807 , f(x) = 4.742844e-08
Iteration 46 : x = 0.0001742246 , f(x) = 3.03542e-08
Iteration 47 : x = 0.0001393797 , f(x) = 1.942669e-08
Iteration 48 : x = 0.0001115037 , f(x) = 1.243308e-08
Iteration 49 : x = 8.920298e-05 , f(x) = 7.957172e-09
Iteration 50 : x = 7.136238e-05 , f(x) = 5.09259e-09
Iteration 51 : x = 5.708991e-05 , f(x) = 3.259258e-09
Iteration 52 : x = 4.567193e-05 , f(x) = 2.085925e-09
Iteration 53 : x = 3.653754e-05 , f(x) = 1.334992e-09
Iteration 54 : x = 2.923003e-05 , f(x) = 8.543948e-10
Iteration 55 : x = 2.338403e-05 , f(x) = 5.468127e-10
Iteration 56 : x = 1.870722e-05 , f(x) = 3.499601e-10
Iteration 57 : x = 1.496578e-05 , f(x) = 2.239745e-10
Iteration 58 : x = 1.197262e-05 , f(x) = 1.433437e-10
Iteration 59 : x = 9.578097e-06 , f(x) = 9.173994e-11
Iteration 60 : x = 7.662478e-06 , f(x) = 5.871356e-11
Iteration 61 : x = 6.129982e-06 , f(x) = 3.757668e-11
Iteration 62 : x = 4.903986e-06 , f(x) = 2.404908e-11
Iteration 63 : x = 3.923189e-06 , f(x) = 1.539141e-11
Iteration 64 : x = 3.138551e-06 , f(x) = 9.850502e-12
Iteration 65 : x = 2.510841e-06 , f(x) = 6.304321e-12
Iteration 66 : x = 2.008673e-06 , f(x) = 4.034765e-12
Iteration 67 : x = 1.606938e-06 , f(x) = 2.58225e-12
Iteration 68 : x = 1.28555e-06 , f(x) = 1.65264e-12
Iteration 69 : x = 1.02844e-06 , f(x) = 1.05769e-12
Iteration 70 : x = 8.227523e-07 , f(x) = 6.769213e-13
Iteration 71 : x = 6.582018e-07 , f(x) = 4.332296e-13
Iteration 72 : x = 5.265615e-07 , f(x) = 2.77267e-13
Iteration 73 : x = 4.212492e-07 , f(x) = 1.774509e-13
Iteration 74 : x = 3.369993e-07 , f(x) = 1.135686e-13
Iteration 75 : x = 2.695995e-07 , f(x) = 7.268387e-14
Iteration 76 : x = 2.156796e-07 , f(x) = 4.651768e-14
Iteration 77 : x = 1.725437e-07 , f(x) = 2.977131e-14
Iteration 78 : x = 1.380349e-07 , f(x) = 1.905364e-14
Iteration 79 : x = 1.104279e-07 , f(x) = 1.219433e-14
Iteration 80 : x = 8.834235e-08 , f(x) = 7.804371e-15
Iteration 81 : x = 7.067388e-08 , f(x) = 4.994798e-15
Iteration 82 : x = 5.653911e-08 , f(x) = 3.196671e-15
Iteration 83 : x = 4.523128e-08 , f(x) = 2.045869e-15
Iteration 84 : x = 3.618503e-08 , f(x) = 1.309356e-15
Iteration 85 : x = 2.894802e-08 , f(x) = 8.37988e-16
Iteration 86 : x = 2.315842e-08 , f(x) = 5.363123e-16
Iteration 87 : x = 1.852673e-08 , f(x) = 3.432399e-16
Iteration 88 : x = 1.482139e-08 , f(x) = 2.196735e-16
Iteration 89 : x = 1.185711e-08 , f(x) = 1.405911e-16
Iteration 90 : x = 9.485688e-09 , f(x) = 8.997828e-17
Iteration 91 : x = 7.58855e-09 , f(x) = 5.75861e-17
Iteration 92 : x = 6.07084e-09 , f(x) = 3.68551e-17
Iteration 93 : x = 4.856672e-09 , f(x) = 2.358727e-17
Iteration 94 : x = 3.885338e-09 , f(x) = 1.509585e-17
Iteration 95 : x = 3.10827e-09 , f(x) = 9.661344e-18
Iteration 96 : x = 2.486616e-09 , f(x) = 6.18326e-18
Iteration 97 : x = 1.989293e-09 , f(x) = 3.957286e-18
Iteration 98 : x = 1.591434e-09 , f(x) = 2.532663e-18
Iteration 99 : x = 1.273147e-09 , f(x) = 1.620905e-18
Iteration 100 : x = 1.018518e-09 , f(x) = 1.037379e-18
> cat("Minimum value of f(x) is approximately at x =", x, "\n")
Minimum value of f(x) is approximately at x = 1.018518e-09
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