# Tutor profile: Amadej Kristjan K.

## Questions

### Subject: Machine Learning

Suppose we are trying to use a machine learning algorithm to perform a classification task, which humans can do with $$98$$% accuracy. Our machine learning algorithm has $$95$$% accuracy on the training set and $$70$$% accuracy on the test set. What is our machine learning algorithm likely suffering from and how can we surmount this issue?

The algorithm is likely suffering from overfitting. This means that the algorithm 'memorises' a lot of irrelevant information that is specific to the set it's training on, but fails to generalise the patterns to the general data distribution. When faced with a yet unseen test set, it performs much worse. In such a scenario, what we can do several of the following: - we can increase the amount of training data (this way the algorithm is faced with a slightly more representative representation of the general data distribution), - we can reduce the representational capacity of the model (making the model simpler, to force the algorithm to only encode the most crucial, generalizable information into the model), - we can incorporate regularisation techniques to reduce overfitting (add a penalising term to the loss function as done in Ridge and Lasso regression or in case of using neural networks we can add dropout regularisation).

### Subject: Discrete Math

Prove that there are infinitely many prime numbers.

Suppose for the sake of contradiction that there are only finitely many prime numbers. Then we can list them as follows: $$ p_1,p_2,...,p_n$$. Now consider the number $$P = p_1p_2...p_n + 1$$. It is not divisible by either of the prime numbers listed, as it gives a remainder of $$1$$ when divided by any of them. Therefore it is only divisible by $$1$$ and itself, having only $$2$$ positive integer divisors, which makes $$P$$ a prime number. But we assumed the above list of primes contained all prime numbers. Since we found a prime outside of the list with supposedly all primes, we arrived at a contradiction. Hence, there are infinitely many prime numbers.

### Subject: Calculus

Let $$R$$ be a rectangle with side lengths $$a$$ and $$b$$. If we keep its perimeter $$P$$ fixed, at what $$a:b$$ ratio will the area $$A$$ of the rectangle be maximised?

The (fixed) perimeter $P$ of the rectangle is: $$P = 2(a+b)$$ The area is: $$ A = ab = a(\frac{P}{2}-a) = \frac{Pa}{2}-a^2$$ Thus, we can view the area of $$R$$ as a function of the variable $$a$$ and see at which value of $$a$$ the area is maximised subject to our given perimeter $$P$$. We can obtain the maximum by figuring out where the derivative of this function is $$0$$. $$ \frac{dA}{da} = \frac{P}{2}-2a$$ so we need to solve $$0 = \frac{P}{2}-2a$$, giving $$a = \frac{P}{4}$$. Now substituting this into $$P = 2(a+b)$$ gives $$b=\frac{P}{4}$$, meaning that for maximising the area we need $$a = b$$.

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