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# Minimizing Loss with Calculus

Unsolved###### Calculus

###### Supervised

Educational Resource: http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture3.pdf

##### Problem reported in interviews at

Here, we will apply the Calculus to

**take the derivative of the loss function with respect to to the weights of the model and equate the derivative to zero**to get optimum weight vector for which the loss would be minimized. In this case, we are taking

**mean squared loss**as our loss function (defined below).

Let's try out this method and calculate the optimum weights at which loss is minimized for a given set of training examples. The model we are trying to learn is:

f(x) = wX + w_{o}

The corresponding loss function is:

Loss = \(\tfrac{n}{2}\sum_{i=1}^{i=n}(y_{i}-f(x_{i}))^{2}\)

n = number of training examples

y = actual labels of training examples

**You have to minimize loss with respect to w and equate the loss function's derivative to zero to get optimal w.**

You are given as input:

X: X is a matrix that has n training examples

y: output labels for X

**Output:**

w: a weight vector that results in the minimum loss

Hint:

https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse

First element of W should be W_{o} i.e., if Y = wX + w_{o} then W = [w_{o ,} elements of w]

To get the above W stack a column of ones to X at the beginning of X

##### Sample Input:

`X: [[0.02019019, 0.2709028, 0.58130202, 0.96772135, 0.40203406, 0.68584788, 0.39907061, 0.8020824, 0.59922299, 0.89880733, 0.52469613, 0.2320425, 0.95274654, 0.57433355, 0.60232225, 0.12846666, 0.62600488, 0.75885062, 0.09451081, 0.5949559], [0.30851973, 0.92397012, 0.45213047, 0.31350788, 0.47674255, 0.99656559, 0.31500455, 0.67267856, 0.96399447, 0.03283226, 0.36633299, 0.17880501, 0.1446183, 0.66984436, 0.46706976, 0.03951692, 0.22784513, 0.82142612, 0.07370921, 0.33906808], [0.73137758, 0.66755197, 0.03242705, 0.29475688, 0.15821178, 0.26910982, 0.91078273, 0.70083964, 0.33650157, 0.26706248, 0.82314416, 0.17939914, 0.77290582, 0.42405843, 0.76003435, 0.30173729, 0.4576326, 0.57653883, 0.30827412, 0.85723128], [0.39220308, 0.17726624, 0.99721065, 0.76477702, 0.3495421, 0.44473093, 0.71881405, 0.3056014, 0.68752376, 0.38752906, 0.12147931, 0.13051651, 0.37855929, 0.180905, 0.59225714, 0.80858205, 0.92062923, 0.03778071, 0.81331967, 0.53359333], [0.07207221, 0.17216235, 0.10151733, 0.77153031, 0.95657225, 0.47837661, 0.30229184, 0.93057551, 0.04504919, 0.82969811, 0.16200046, 0.34971251, 0.04055285, 0.57460145, 0.58058546, 0.40074296, 0.55627212, 0.50018151, 0.10975096, 0.41768908], [0.64015041, 0.12854868, 0.09152847, 0.81948493, 0.64439314, 0.1571958, 0.86136143, 0.96147962, 0.25692185, 0.91043344, 0.0031553, 0.50970494, 0.09239349, 0.32355557, 0.02523059, 0.78752319, 0.42300025, 0.83668593, 0.45169828, 0.23836782], [0.15067413, 0.17847288, 0.34651435, 0.88828655, 0.70997458, 0.3427081, 0.22473914, 0.47540793, 0.12765958, 0.39060752, 0.60724015, 0.72244086, 0.51583481, 0.49931979, 0.2823342, 0.80178407, 0.62729361, 0.28726125, 0.34917165, 0.60606349], [0.79312299, 0.37268688, 0.58758082, 0.55694459, 0.35795187, 0.08427892, 0.80678922, 0.32206049, 0.55948097, 0.06842917, 0.46172133, 0.16606839, 0.150101, 0.84987818, 0.2154061, 0.17590738, 0.16686814, 0.81569503, 0.1565211, 0.04867117], [0.0049085, 0.37890289, 0.89008346, 0.9281833, 0.33230683, 0.91271647, 0.02732307, 0.54477629, 0.01319123, 0.30752729, 0.79310271, 0.39230414, 0.49702199, 0.95427926, 0.48890644, 0.61628695, 0.50168302, 0.36486363, 0.01473803, 0.77408549], [0.903083, 0.67791114, 0.61581447, 0.05392819, 0.91796546, 0.92341286, 0.40200506, 0.92874617, 0.35434179, 0.03242402, 0.04678233, 0.88053641, 0.79735547, 0.9512864, 0.73243777, 0.24907558, 0.26310536, 0.80493796, 0.68572655, 0.45102886]]`

<class 'list'>

`y: [[1], [0], [1], [0], [1], [1], [1], [0], [1], [1]]`

##### Expected Output:

```
[[-0.07140561]
[ 0.0649857 ]
[-0.07455911]
[-0.25456612]
[ 0.02489909]
[-0.05027607]
[ 0.09235117]
[-0.0528404 ]
[ 0.43180331]
[-0.6064806 ]
[ 0.34821837]
[ 0.11929702]
[ 0.31914999]
[ 0.43665925]
[ 0.09250638]
[ 0.00743292]
[ 0.16578263]
[-0.0770846 ]
[ 0.16158202]
[-0.03285493]
[ 0.3250566 ]]
```

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Input Test Case

Please enter only one test case at a timenumpy has been already imported as np (import numpy as np)