Неделя 1
Вступление
Описание современного машинного обучения.
"A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E."
Example: playing checkers.
E = the experience of playing many games of checkers
T = the task of playing checkers.
P = the probability that the program will win the next game.
В основном машинное обучение разделяется на обучение с учителем и обучение без учителя.
Обучение с учителем
Проблема классификации. Возможны выборы в системе и на основе входных данных слудет выбрать правильный выходные.
Мы рассматриваем классическую задачу реграссии. Любоая регресионная задача стоит в том, чтоб предсказать выходное значение.
Классификационная проблема состоит в том чтоб определить к какому классу относяться входные данные.
Обучение без учителя
Задача научить алгоритм научить самому классифицировать данные на основе выборки на неразмеченных данных.
Примеры
Кластеризация. Дана коллекция 1000000 разных генов. Найти пути автоматической группировки этих шенов в группы которые каким-то образом схожи по некоторым значениям таких как: время жизни, локация, роль и т.д.
Некластеризация. Алгоритм "Коктейльной вечеринки" который позволяет находить структуру в хаотическом окружающем пространстве(определять голоса и музыку с разных микрофонов и разделять их на разные дорожки без шумов).
Модель и оценка функции
Представление модели
Модель можеть задаваться различными видами, но чтоб в дальнейшем её обрабатывать мы должны формализировать её. Входные данные у нас будут представленны в виде Xi, а выхожные Yi
Саму модель мы представляем гипотезой h в которую входят данные, а выходят ответы.
Оценка функции
Мы можем измарить точность гипотезы используя оценочную функцию. Это среднее значение разницы результатов гипотезы с входящими х и у
If we try to think of it in visual terms, our training data set is scattered on the x-y plane. We are trying to make a straight line (defined by hθ(x)) which passes through these scattered data points.
Our objective is to get the best possible line. The best possible line will be such so that the average squared vertical distances of the scattered points from the line will be the least. Ideally, the line should pass through all the points of our training data set. In such a case, the value of J(θ0,θ1)
will be 0. The following example shows the ideal situation where we have a cost function of 0.
When θ1=1, we get a slope of 1 which goes through every single data point in our model. Conversely, when θ1=0.5, we see the vertical distance from our fit to the data points increase.
This increases our cost function to 0.58. Plotting several other points yields to the following graph:
Thus as a goal, we should try to minimize the cost function. In this case, θ1=1 is our global minimum.
A contour plot is a graph that contains many contour lines. A contour line of a two variable function has a constant value at all points of the same line. An example of such a graph is the one to the right below.
Taking any color and going along the 'circle', one would expect to get the same value of the cost function. For example, the three green points found on the green line above have the same value for J(θ0,θ1) and as a result, they are found along the same line. The circled x displays the value of the cost function for the graph on the left when θ0 = 800 and θ1= -0.15. Taking another h(x) and plotting its contour plot, one gets the following graphs:
When θ0 = 360 and θ1 = 0, the value of J(θ0,θ1) in the contour plot gets closer to the center thus reducing the cost function error. Now giving our hypothesis function a slightly positive slope results in a better fit of the data.
The graph above minimizes the cost function as much as possible and consequently, the result of θ1 and θ0 tend to be around 0.12 and 250 respectively. Plotting those values on our graph to the right seems to put our point in the center of the innermost 'circle'.
Изучение параметров
So we have our hypothesis function and we have a way of measuring how well it fits into the data. Now we need to estimate the parameters in the hypothesis function. That's where gradient descent comes in.
Imagine that we graph our hypothesis function based on its fields θ0 and θ1 (actually we are graphing the cost function as a function of the parameter estimates). We are not graphing x and y itself, but the parameter range of our hypothesis function and the cost resulting from selecting a particular set of parameters. We put θ0 on the x axis and θ1 on the y-axis, with the cost function on the vertical z-axis. The points on our graph will be the result of the cost function using our hypothesis with those specific theta parameters. The graph below depicts such a setup.
We will know that we have succeeded when our cost function is at the very bottom of the pits in our graph, i.e. when its value is the minimum. The red arrows show the minimum points in the graph.
The way we do this is by taking the derivative (the tangential line to a function) of our cost function. The slope of the tangent is the derivative at that point and it will give us a direction to move towards. We make steps down the cost function in the direction with the steepest descent. The size of each step is determined by the parameter α, which is called the learning rate.
For example, the distance between each 'star' in the graph above represents a step determined by our parameter α. A smaller α would result in a smaller step and a larger α results in a larger step. The direction in which the step is taken is determined by the partial derivative of J(θ0,θ1). Depending on where one starts on the graph, one could end up at different points. The image above shows us two different starting points that end up in two different places.
The gradient descent algorithm is:
repeat until convergence:
θj:=θj−α∂/∂θj*J(θ0,θ1)
where j=0,1 represents the feature index number.
At each iteration j, one should simultaneously update the parameters θ1,θ2,...,θn.Updating a specific parameter prior to calculating another one on the j(th) iteration would yield to a wrong implementation.
In this video we explored the scenario where we used one parameter θ1 and plotted its cost function to implement a gradient descent. Our formula for a single parameter was :
Repeat until convergence:
Regardless of the slope's sign for ddθ1*J(θ1), θ1 eventually converges to its minimum value. The following graph shows that when the slope is negative, the value of θ1 increases and when it is positive, the value of θ1 decreases.
On a side note, we should adjust our parameter α to ensure that the gradient descent algorithm converges in a reasonable time. Failure to converge or too much time to obtain the minimum value imply that our step size is wrong.
How does gradient descent converge with a fixed step size α? The intuition behind the convergence is that d/dθ1*J(θ1) approaches 0 as we approach the bottom of our convex function. At the minimum, the derivative will always be 0 and thus we get:
Note: [At 6:15 "h(x) = -900 - 0.1x" should be "h(x) = 900 - 0.1x"]
When specifically applied to the case of linear regression, a new form of the gradient descent equation can be derived. We can substitute our actual cost function and our actual hypothesis function and modify the equation to :
repeat until convergence: {
where m is the size of the training set, θ0 a constant that will be changing simultaneously with θ1 and xi,yiare values of the given training set (data).
Note that we have separated out the two cases for θj into separate equations for θ0 and θ1; and that for θ1 we are multiplying xi at the end due to the derivative. The following is a derivation of ∂/∂θjJ(θ) for a single example :
The point of all this is that if we start with a guess for our hypothesis and then repeatedly apply these gradient descent equations, our hypothesis will become more and more accurate.
So, this is simply gradient descent on the original cost function J. This method looks at every example in the entire training set on every step, and is called batch gradient descent. Note that, while gradient descent can be susceptible to local minima in general, the optimization problem we have posed here for linear regression has only one global, and no other local, optima; thus gradient descent always converges (assuming the learning rate α is not too large) to the global minimum. Indeed, J is a convex quadratic function. Here is an example of gradient descent as it is run to minimize a quadratic function.
The ellipses shown above are the contours of a quadratic function. Also shown is the trajectory taken by gradient descent, which was initialized at (48,30). The x’s in the figure (joined by straight lines) mark the successive values of θ that gradient descent went through as it converged to its minimum.