So far what we’ve been doing is point estimation using OLS. Another method of estimation with more mathematical rigor is max likelihood estimation. In this post we will look at likelihood, likelihood function, max likelihood, and how all this ties together in estimating the parameters of our bivariate linear regression model.

Probability

When we think of probability, we are thinking about the chance of something happening in the future. For example, if a coin is tossed, what is the chance of getting a head? It’s .

But how did we arrive at the conclusion that the probability of getting a head is ? This leads us to two varying schools of thoughts on probability - frequentism and bayesianism.

If the coin were tossed a 1000 times, about half the times the outcome would be a head and half the times it would be a tail. The more frequently an event happens, the more probable / certain it is. This is the frequentist definition of probability. Under this definition, probability is the ratio of the number of times we make the desired observation to the total number of observations.

In the Bayesian approach, probability is a measure of our certainty about something. In a coin toss, there’s a 50% chance of getting a head and a 50% chance of getting a tail because there’s only two possible outcomes.

This subtle difference in the definition of probability leads to vastly different statistical methods.

Under frequentism, it is all about the frequency of observations. The event that occurs more frequently is the more probable event. There is no concept of an underlying probability distribution for the event we are trying to observe.

Under Bayesianism, in contrast, each event that occurs follows an observed probability distribution which is based on some underlying true distribution. Calculating the probabilities involves estimating the parameters of the true distribution based on the observed distribution.

Likelihood

Likelihood is the hypothetical probability that an event that has already occurred would yield a specific outcome^[1].

For example, if a coin were tossed 10 times and out of those 10 times a person guessed the outcome correctly 8 times then what’s the probability that the person would guess equally correctly the next time a coin is tossed 10 times? This is the likelihood. Unlike probability, likelihood is about past events.

The likelihood of observing the data is represented by the likelihood function. The likelihood function is the probability density function of observing the given data. Suppose that we have observations - - of data which are realizations of a random variable . Let the function represent the probability density function of the random variable with parameters . The the likelihood of observing is given by the likelihood function as:

Max Likelihood Estimation

If we were to maximize the likelihood function, we’d end up with the highest probability of observing the data that we have observed i.e. the max likelihood. The data that we have observed follows some distribution we’d like to estimate. This would require us to estimate the parameters such that they maximize the likelihood function i.e. . The estimator is therefore the max likelihood estimator of the parameters such that it maximizes the likelihood function.

The procedure followed in maximizing the likelihood function to compute is called max likelihood estimation (MLE). MLE answers the following question: given that the we’ve observed data which is drawn from some distribution with parameters , to what value should we set these parameters in the sampling distribution so that the probability of observing this data is the maximum?

The mechanics of finding is an optimizaition problem involving a little bit of differential calculus. It involves calculating the partial derivative of the likelihood function with respect to each of the parameters in and setting them to zero. This would yield a system of equations which can then be solved to obtain estimates of the parameters which would maximize the function.

Calculating the partial derivatives like this becomes unwieldy very quickly since the likelihood function involves multiplication of lot of terms. We can instead calculate the maximum of log-likelihood since it will convert the product term to a summation term. It is defined as .

This provides us with enough background to apply MLE to find our regression parameters.

MLE for Regression

In regression, we have our data which we assume is normally distributed with mean and variance . The likelihood function for this can be written as:

In the equations above, because we assume to be normally distributed. As mentioned previously, computing the log-likelihood is easier. The log-likelihood is given as:

To find the ML estimators, we need to differentiate the log-likelihood function partially with respect to , , and and set the resulting equations to zero. This gives us:

Setting the above equations to zero and letting , , and represent the ML estimators, we get:

Simplifying the first two equation above, we get:

The equations above are the normal equations for OLS estimators. What this means is that the OLS estimators and are the same as the ML estimators and when we assume they’re normally distributed. The estimator for homoscedastic variance under ML is given by . This estimator underestimates the true i.e. it is biased downwards. However, in large samples, converges to the true value. This means that it is asymptotically unbiased.

Summary

In summary, MLE is an alternative to OLS. The method, however, requires that we make an explicit assumption about the probability distribution of the data. Under the normality assumption, the estimators for and are the same in both MLE and OLS. The ML estimator for is biased downwards but converges to the true value when the sample size increases.

The posts so far have covered estimators. An estimator looks at the sample data and comes up with an estimate for some true population parameter. However, how do we know how close, say , is to the true ? This is a question of statistical inference. To be able to make an inference, would need to follow some distribution. CLRM makes no assumptions about the distribution of data. CNLRM (Classic Normal Linear Regression Model), however, adds the assumption of normality i.e. the data and parameters are normally distributed.

In this post we will cover the normal distribution, CNLRM, and how the assumption of normality helps us.

Normal Distribution

A probability distribution is a function which provides the probabilities of the outcomes of a random phenomenon. For example, a random variable could describe the outcomes of rolling a fair dice. The probability distribution would then be a function that would return for each of the outcomes of .

The normal distribution is the most commonly encoutered probability distribution. It states that the averages (or any other statistic like sum or count) calculated by drawing samples of observations from a random variable tend to converge to the normal. The distribution is defined by two parameters - the mean (), and the variance (). If a random variable is normally distributed, it is written notationally as . The probability density function of normal distribution is given by:

Distribution of , , and

For us to be able to draw any inference about the estimators and , we will have to make some assumption about the probability distribution of . In one of the previous posts we saw that . This can alternatively be written as where . So we have

The , betas, and are fixed and therefore is a linear function of . Under CNLRM, we assume to be to be normally distributed. A property of normally distributed random variables is that their linear functions are also normally distributed. What this means is that since and are linear functions of , they are also normally distributed.
As mentioned previously, a normal distribution is described by its mean and variance. This means can also be described by its mean and variance as . What this shows us is that we assume, on average, for to be zero and there to be constant variance (homoscedasticity).

A further assumption we make is that where . This means that the and are uncorrelated and independently distributed because zero correlation between two normally distributed variables implies statstical independence.

Also note that since is normally distributed, so is . We can state this as

Why do we assume normality?

We assume normality for the following reasons:

First, the subsume the effect of a large number of factors that affect the dependent variable but are not included in the regression model. By using the central limit theorem we can state that the distribution of the sum of these variables will be normal.

Second, because we assume to be normal, both and are normal since they are linear functions of .

Third, the normal distribution is fairly easy to understand as it involves just two parameters, and is extensively studied.

Properties of estimators under CNLRM

1. They are unbiased.
2. They have minimum variance.

Combinining 1. and 2. means that the estimators are efficient - they are minimum-variance unbiased.

1. They are consistent. This means that as the size of the sample increases, the values of the estimators tend to converge to the true population value.

2. The estimators have the least variance in the class of all estimators - linear or otherwise. This makes the estimators best unbiased estimators (BUE) as opposed to best linear unbiased estimators (BLUE).

Misc

Earlier in the post I mentioned that . This section gives the derivation of this expression. We will look at the numerator for ease of calculation.

Finito.

In one of the previous posts we looked at the assumptions underlying the classic linear regression model. In this post we’ll take a look at the Gauss-Markov theorem which uses these assumptions to state that the OLS estimators are the best linear unbiased estimators, in the class of linear unbiased estimators.

Gauss-Markov Theorem

To recap briefly, some of the assumptions we made were that:

1. On average, the stochastic error term is zero i.e. .
2. The data is homoscedastic i.e. the variance / standard deviation of the stochastic error term is a finite value regardless of .
3. The model is linear in parameters i.e. and have power 1.
4. There is no autocorrelation between the disturbance terms.

The Gauss-Markov theorem states the following:

Given the assumptions underlying the CLRM, the OLS estimators, in the class of linear unbiased estimators, have minimum variance. That is, they are BLUE (Best Linear Unbiased Estimators).

We’ll go over what it means for an estimator to be best, linear, and unbiased.

Linear

The dependent variable is a linear function of the variables in the model. The model must be linear in parameters. Under this assumption and are both valid models since the estimators are linear in both of them.

Unbiased

This means that on average, the value of the estimator is the same as the parameter it is estimating i.e. .

Best

This means that among the class of linear estimators, the OLS estimators will have the least variance. Consider the two normal distributions shown in the graph above for two linear, unbiased estimators (represented by orange) and (represented by blue). Also let’s assume that the true value of is zero. This means that both of these are unbiased since, on average, their value is the same as the true value of the parameter. However, the OLS estimator is better since it has lesser variance compared to the other estimator.

Finito.

The calculation of the estimators and is based on sample data. As the sample drawn changes, the value of these estimators also changes. This leaves us with the question of how reliable these estimates are i.e. we’d like to determine the precision of these estimators. This can be determined by calculating the standard error or the coefficient of determination.

Standard Error

The standard error of a statistic^[1] is the standard deviation of the sampling distribution of that statistic.

Suppose we have a dataset which contains incomes of people. From this dataset we start drawing samples of size n and calculating the mean. Now if we plot the distribution (the sampling distribution) of the means we calculated, we’ll get a normal distribution centered on the population mean. The standard deviation of this sampling distribution is the standard error of the statistic which in our case is mean.

Standard error is given by the formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Where
is the standard deviation of the population.
is the sample size.

Calculating the standard error shows the sampling fluctuation. Sampling fluctuation shows the extent to which a statistic takes on different values.

Notice that the standard error has an inverse relation with the sample size n. This means that the larger the sample we draw from the population, the lesser the standard error will be. This will also result in a tighter normal distribution since the standard deviation will be less.

Standard Error of OLS Estimates

The standard error of the OLS estimators and is given by:

where is the square root of true but unknown constant of homoscedastic variance .

All of the terms in the equations above except can be calculated from the sample drawn. Therefore, we will need an unbiased estimator . The denominator represents the degrees of freedom. is the residual sum of squares.

Although , its summation can be computed with an alternative formula as $\sum{\hat{u}_i^2} = \sum{y_i^2} - \frac{(\sum{x_iy_i})^2}{\sum{x_i^2}}$.

All of the terms in the above formula would have already been computed as a part of computing the estimators.

How it all ties together

As you calculate the estimators and draw the sample regression curve that passes through your data, you need some numeric measure of how good the curve fits the data i.e. a measure of “goodness of fit”. This can be given as:

This is the positive square root of the estimator of homoscedastic variance. This is the standard deviation of the values about the regerssion curve.

The standard errors of the estimators and will show you how much they fluctuate as you draw the samples. The lesser their standard error, the better.

The square of the standard error is called the mean squared error.

Let’s see some code

To start off, let’s load the Boston housing dataset.

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import pandas as pd
from sklearn import datasets
boston = datasets.load_boston()
data, target = boston.data, boston.target
df = pd.DataFrame(data=data, columns=boston.feature_names)
df = df[['RM']]
df['RM'] = df['RM'].apply(round)
df['price'] = target 

I’m choosing to use the average number of rooms as the variable to see how it affects price. I’ve rounded it off to make the calculations easier. The scatter plot of the data looks the following:

This, ofcourse, is the entire population data so let’s draw a sample.

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sample = df.sample(n=100)

Now let’s bring back our function that calculates the OLS estimates:

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def ols(sample):
    Xbar = sample['X'].mean()
    Ybar = sample['Y'].mean()

    sample['x'] = sample['X'] - Xbar
    sample['x_sq'] = sample['x'] ** 2
    sample['y'] = sample['Y'] - Ybar
    sample['xy'] = sample['x'] * sample['y']

    beta2cap = sample['xy'].sum() / sample['x_sq'].sum()
    beta1cap = Ybar - (beta2cap * Xbar)

    sample['Ycap'] = beta1cap + beta2cap * sample['X']
    sample['ucap'] = sample['Y'] - sample['Ycap']

    return beta1cap, beta2cap

and now let’s see coefficients

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sample.rename(columns={'RM': 'X', 'price': 'Y'}, inplace=True)
intercept, slope = ols(sample)

The plot of the regression curve on the sample data looks the following:

The sample DataFrame has our intermediate calculation and we can use that to calculate the standard error. Let’s write a function which does that.

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from math import sqrt
def standard_error(sample):
    return sqrt((sample['ucap'] ** 2).sum() / (len(sample) - 2))

Finally, let’s see how much standard deviation we have around the regression line.

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standard_error(sample)
7.401174774558201

Coefficient of Determination

The coefficient of determination denoted by (for bivariate regression) or (for multivariate regression) is a ratio of the explained variance to the total variance of the data. The higher the coefficient of determination, the more accurately the regressors (average number of rooms in the house) explain the regressand (the price of the house) i.e. the better your regression model is.

For bivariate linear regression, value is given by:

can take on a value between 0 and 1.

Turning this into code, we have:

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def coeff_of_determination(sample):
    Ybar = sample['Y'].mean()
    return ((sample['Ycap'] - Ybar) ** 2).sum() / ((sample['Y'] - Ybar) ** 2).sum()

For the sample we’ve drawn, the value comes out to be 0.327. This means that only 32.7% of the variance in the total data is explained by the regression model.

Finito.

[1] A “statistic” is a numerical quantity such as mean, or median.

In the previous post we calculated the values of and from a given sample and in closing I mentioned that we’ll take a look at finding out whether the sample regression line we obtained is a good fit. In other words, we need to assess how close and are to and or how close is to . Recall that the population regression function (PRF) is given by . This shows that the value of depends on and . To be able to make any statistical inference about (and and ) we need to make assumptions about how the values of and were generated. We’ll discuss these assumptions in this post as they’re important for the valid interpretation of the regression estimates.

Classic Linear Regression Model (CLRM)

There are 7 assumptions that Gaussian or Classic linear regression model (CLRM) makes. These are:

1. The regression model is linear in parameters

Recall that the population regression function is given by . This is linear in parameters since and have power 1. The explanatory variables, however, do not have to be linear.

2. Values of X are fixed in repeated sampling

In this series of posts, we’ll be looking at fixed regressor models^[1] where the values of X are considered to be fixed when drawing a random sample. For example, in the previous post the values of X were fixed between 80 and 260. If another sample were to be drawn, the sample values of X would be used.

3. Zero mean value of the error term

This assumption states that for every given , the error term is zero, on average. This should be intuitive. The regression line passes through the conditional mean for every . The error term are distances of the individual points from the conditional mean. Some of them lie above the regression line and some of them lie below. However, they cancel each other out; the positive cancel out the negative and therefore is zero, on average. Notationally, .

What this assumption means that all those variables that were not considered to be a part of the regression model do not systematically affect i.e. their effect on average is zero on .

This assumption also means is that there is no specification error or specification bias. This happens when we chose the wrong functional form for the regression model, exclude necessary explanatory variables or include unnecessary ones.

4. Constant variance of (homoscedasticity)

This assumption states that the variation of is the same regardless of .

With this assumption, we assume that the variance of for a given (i.e. conditional variance) is a positive constant value given by . This means that the variance for the various populations is the same. What this also means is that the variance around the regression line is the same and it neither increases nor decreases with change in values of .

5. No autocorrelation between disturbances

Given any two values, and (), the correlation between any two and is zero. In other words, the observations are independently sampled. Notationally, . This assumption states that there is no serial correlation or autocorrelation i.e. and are uncorrelated.

To build an intuition, let’s consider the population regression function

Now suppose that and are positively correlated. This means that depends not only on but also on because that affects . Autocorrelation occurs in timeseries data like stock market trends where the the observation of one day depends on the observation of the previous day. What we assume is that each observation is independent as it is in case of the income-expense example we saw.

6. The sample size must be greater than the number of parameters to estimate

This is fairly simple to understand. To be able to estimate and , we need atleast 2 pairs of observations.

7. Nature of X variables

There are a few things we assume about the value of variables. One, for a given sample they are not all the same i.e. is a positive number. Second, there are no outliers among the values.

If we have all the values of the same in a given sample, the regression line would be a horizontal line and therefore it will be impossible to estimate and thus . This will happen because in the equation , the denominator will become zero since and will be the same.

Furthermore, we assume that there are no outliers in the values. Suppose there are a few values which are far apart from the mean value, the regression line which we get will be vastly different depending on whether the sample drawn contains these outliers or not.

These are the 7 assumptions underlying OLS. In the next section, I’d like to elucidate on assumption 3 and 5 a little more.

More on Assumption 3. and 5.

In assumption 3. we state that i.e. on average the disturbance term is zero. For the expected value of a variable given another variable be zero means the two variables are uncorrelated. We assume these two variables to be uncorrelated so that we can assess the impact each has on the predicted value of . If the disturbance and the explanatory variable were correlated positively or negatively then it means that contains some factor that affects the predicted value of i.e. we’ve skipped an important explanatory variable that should’ve been included in the regression model. This means we have functionally misspecified our regression model and introduced specification bias.

Functionally misspecifying our regression model will also introduce autocorrelation in our model. Consider the two plots shown below.

In the first plot the line fits the sample points properly. In the second one, it doesn’t fit properly. The sample points seem to form a curve of some sort while we try to fit a straight line through it. What we get is runs of negative residuals followed by positive residuals which indicates autocorrelation.

Autocorrelation can also be introduced not just by functionally misspecifying the regression model but also by the nature of data itself. For example, consider timeseries data representing stock market movement. The value of depends on i.e. the two are correlated.

When there is autocorrelation, the OLS estimators are no longer the best estimators for predicting the value of .

Conclusion

In this post we looked at the assumptions underlying OLS. In the coming post we’ll look at the Gauss-Markov theorem and the BLUE property of OLS estimators.

[1] There are also stochastic regression models where the values of X are not fixed.

In the previous post we built the intuition behind linear regression. In this post we’ll dig deeper into the simplest form of linear regression which involves one dependant variable and one explanatory variable. Since we only have two variables involved, it’s called bivariate linear regression.

Ordinary Least Squares

We defined sample regression function as:

The term is called the residual. If we re-write the sample regression function, we get . The residual then represents how far off our estimated value is from the actual value .^[1] A reasonable assumption to make is that we should pick and such that the sum of is the smallest. This would mean that our estimates are close to the actual values.

This assumption, however, does not work. Let’s assume that our values for are -10, 3, -3, 10. Here the sum is zero but we can see that the first and last predictions are far apart from the actual values. To mitigate this, we can add the squares of the residuals. We need to pick and such that the sum of the square of the residuals is the minimum i.e. is the least.

Again, assuming the values of to be -10, 3, -3, 10, the squares are 100, 9, 9, 100. This sums to 218. This shows us that the predicted values are far from the actual values. is called the residual sum of squares (RSS) or the squared error term.

The method of OLS provides us with estimators and such that, for a given sample set, is the least. These estimators are given by the formula:

where = , . These are the differences of individual values from their corresponding sample mean or . The estimators thus obtained are called least-squares estimators.

Now that we’ve covered significant ground, let’s solve a sum by hand.

Example

Let’s start by drawing a random sample from the original dataset. The sample I’ve drawn looks like this:

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income (X)  expenditure (Y)
        80               65
       100               70
       120               84
       140               80
       160              118
       180              130
       200              144
       220              137
       240              191
       260              150

To see how this contrasts with the original dataset, here’s a plot of the original dataset and the random sample. The faint green points represent the original dataset whereas the red ones are the sample we’ve drawn. As always, our task is to use the sample dataset to come up with a regression line that’s as close as possible to the population dataset i.e. use the red dots to come up with a line that’d be similar to the line we’d get if we had all the faint green dots.

I’ll write some Python + Pandas code to come up with the intermediate calculations and the final and values. I highly recommend that you solve this by hand to get the feel for it.

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def ols(sample):
    Xbar = sample['X'].mean()
    Ybar = sample['Y'].mean()

    sample['x'] = sample['X'].apply(lambda Xi: Xi - Xbar)
    sample['x_sq'] = sample['x'].apply(lambda x: x ** 2)
    sample['y'] = sample['Y'].apply(lambda Yi: Yi - Ybar)
    sample['xy'] = sample[['x', 'y']].apply(lambda row: row['x'] * row['y'], axis='columns')

    beta2cap = sample['xy'].sum() / sample['x_sq'].sum()
    beta1cap = Ybar - (beta2cap * Xbar)

    sample['Ycap'] = sample['X'].apply(lambda X: beta1cap + (beta2cap * X))

    sample['ucap'] = sample[['Y', 'Ycap']].apply(lambda row: row['Y'] - row['Ycap'], axis='columns')

    return beta1cap, beta2cap

On calling the function, these are the results I get:

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>>> ols(sample)
(9.696969696969703, 0.6306060606060606)

and the intermediate calculations are the following:

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     X    Y     x    x_sq     y      xy        Ycap       ucap
0   80   65 -90.0  8100.0 -51.9  4671.0   60.145455   4.854545
1  100   70 -70.0  4900.0 -46.9  3283.0   72.757576  -2.757576
2  120   84 -50.0  2500.0 -32.9  1645.0   85.369697  -1.369697
3  140   80 -30.0   900.0 -36.9  1107.0   97.981818 -17.981818
4  160  118 -10.0   100.0   1.1   -11.0  110.593939   7.406061
5  180  130  10.0   100.0  13.1   131.0  123.206061   6.793939
6  200  144  30.0   900.0  27.1   813.0  135.818182   8.181818
7  220  137  50.0  2500.0  20.1  1005.0  148.430303 -11.430303
8  240  191  70.0  4900.0  74.1  5187.0  161.042424  29.957576
9  260  150  90.0  8100.0  33.1  2979.0  173.654545 -23.654545

Let’s plot the line we obtained as a result of this.

Interpreting the Results

Our calculations gave us the results that and . Having a slope () of means that for every unit increase in income, there’s an increase of in expenditure. The intercept is where the regression line meets the Y axis. This means that even without any income, a persion would have an expenditure of . In a lot of cases, however, the intercept term doesn’t really matter as much as the slope and how to interpret the intercept depends upon what the Y axis represents.

Conclusion

That’s it for this post on bivariate linear regression. We saw how to calculate and and worked on a sample problem. We also saw how to interpret the result of the calculations. In the coming post we’ll look at how to assess whether the regression line is a good fit.

[1] It’s important to not confuse the stochastic error term with the residual . The stochastic error term represents the inherent variability of data whereas the residual represents the difference between the predicted and actual values of .

The first topic we’ll look at is linear regression. As mentioned previously, regression is about understanding dependence of one variable on another. We’ll start by building an intuitive understanding of what it is before we dive into the math-y details of it. We’ll cover the theory, learn to solve some problems by hand, look at the assumptions supporting it, then look at what happens if these assumptions aren’t held. Once this is done we’ll look at how we can do the same by using gradient descent.

What is Regression Analysis?

Regression analysis is the study of the dependence of one variable (dependent variable, regressand) over one or more other variables (explanatory variables, regressors). The aim is to see how the dependent variable changes, on average, if there is a change in the explanatory variables. This is better explained with an example.

The plot shown here is of daily income and daily expenditure. The dependent variable here is the expenditure which depends on the explanatory variable income. As we can see from the plot, for every income level we have a range of expenditures. However, as the regression line shows, on average as income goes up, the expenditure goes up, too.

Why do we say that the income goes up on average? Let’s look at the raw data when daily income is 80 or 100.

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income (X)  expenditure (Y)  average
        80               55     65.0
        80               60
        80               65
        80               70
        80               75
       100               65     77.0
       100               70
       100               74
       100               80
       100               85
       100               88

The highest expenditure when income is 80 (75) is greater than the lowest expenditure when income level is 100 (65). If we look at the averages, the expenditure has gone up. There are a number of factors which may have caused the person with income 80 to expend 75 but as a whole, people with his daily income tend to expend less than those who earn 100 a day. The regression line passes through these averages.

That’s the basic idea behind regression analysis.

The Math behind Regression Analysis

The regression line passes through the average expenditure for every given income level. In other words, it passes through the conditional expected value — . This is read as “the expected value of expenditure (Y) given an income level ()”. Therefore, is a function of i.e. . The question now is: what is the function ?

We can start by assuming to be a straight line function which means that we’re assuming expenditure to be linearly related to income. So, . This is the slope-intercept form of a line where is the intercept and is the slope of the line. Both and are called the regression coefficients (or parameters). The function is called the population regression function.

Now given that we know how to calculate the average expenditure for a given income level, how do we calculate the expenditure of each individual at that income level? We can say that the individual’s expenditure is “off the average expenditure by a certain margin”. This can be written as . The term denotes how far off an individual’s expenditure is from the average and is called the stochastic error term.

No matter how good our regression model is, there is always going to be some inherent variability. There are factors other than income which define a person’s expenditure. Factors like gender, age, etc. affect the expenditure. All of these factors are subsumed into the stochastic error term.

When it comes to applying regression analysis, we won’t have access to the population data. Rather, we have access to sample data. Our task is to come up with sample regression coefficients such that they’ll be as close as possible to the population regression coefficients. The sample regression function is defined as:

Here , , , and are estimators of , , , and respectively. An estimator is simply a formula which tells us how to estimate the population parameter from the information provided by the sample data available to us.

What it means to be ‘Linear’

Linearity can be defined as being either linear in parameters or linear in explanatory variables. To be linear in parameters means that all your parameters in the regression function will have power 1. By that definition, is linear. In contrast, is linear in explanatory variables since has power 1. We’ll be looking at models that are linear in parameters. Henceforth, “linear” would mean “linear in parameters”.

That’s it. This is what sets the foundation for further studying linear regression.

A while ago I’d started writing a series on machine learning titled ML for Newbies which I never got around to finishing. This series of posts is a redoing of the older series with greater mathematical, and theoretical rigor. Some of the content will be ported over from that series but most of it will be new content.

The aim of the series is to summarize everything that I know about machine learning in an approachable, intuituve manner, and also be notes for myself.

What is Machine Learning?

ML is all about automated learning. It is about creating algorithms that can “learn” from the input given to them. These algorithms can then do a number of tasks such as filter out spam emails, predict the weather, etc. More formally, it’s about converting experience into expertise. The input to the algorithm become its “experience” and the task that it performs becomes its “expertise”.

Although ML is a sub-field of AI (Artificial Intelligence), they’re different. AI focuses on replicating human intelligence whereas ML focuses on complementing human intelligence by doing tasks that fall beyond human capabilities like looking for patterns in massive data sets.

How does a Machine Learn?

Like I mentioned earlier, a machine learns from the input data given to it. The way the machine learns can also be classified as supervised, unsupervised, active, or passive.

Supervised Learning

Let’s say your task is to build a spam filter. You have a large data set of emails, each of which is marked as either “spam” or “not spam”. You’ll feed this data set to your algorithm and it’ll learn from it. Later, you’ll input an email and it’ll tell you if it is spam or not. This type of learning is called supervised learning. It is called “supervised” because the algorithm was first taught what is spam and what isn’t by a data set generated manually. You had to supervise its learning process.

Unsupervised Learning

Let’s stay with the spam filter example. Let’s now suppose that the emails in the data set are not marked as “spam” or “not spam”. It’s the task of the algorithm to figure it out somehow. Such learning is called unsupervised learning.

Passive Learning

In passive learning, the algorithm only observes the data provided to it and learns from it, without influencing it in anyway. Suppose now that the spam filter starts out blank with no notion of what is and what isn’t spam. It’s up to the user to mark the incoming emails so that the algorithm can learn from it. This is passive learning.

Active Learning

In contrast, let’s say that the spam filter now looked at the email first, and if it found something suspicious, asked the user to mark it as spam or not. This is active learning.

What can ML do?

There are a number of ML algorithms, each focusing on a different task. These algorithms can do a number of things like:

Clustering

In simple terms, clustering is grouping. It’s an unsupervised form of learning in which the algorithm looks at the data and groups the similar pieces of together based on some similarity measure. In the spam filter examples above, when the algorithm, on its own, figured out which of the emails are spam, it clustered them together.

Classification

Classification is about finding out which group a piece of data belongs to. It’s a form of supervised learning because the algorithm needs to know beforehand what each of the group looks like before it can be asked to classify a piece of data. When the spam filter was given a data set with marked emails, all it had to do was to answer the question “is this email spam?”. Basically, it was asked to classify that email into a group.

Regression

Regression is about understanding relationships between variables. If one variable changes, how would it affect the other? For example, if the amount of power needed for a phone to stay on increased, how would it affect battery life?

Association Rules

Finding out association rules in data sets is also about finding out relationships between variables. Suppose you have a large data set from an online store, association rules is about finding out what items you are likely to buy based on what you’ve previously bought. For example, if you buy milk and sugar, you’re likely to buy eggs. Your rule then becomes {milk, sugar} => eggs.

For now, don’t worry about the theoretical rigor of these. We’ll get to that in subsequent posts. That’s all for now. I hope this post was motivation enough for you to start learning ML. :)

So, I am taking a break from my JVM JIT series to write this post about a table summarization algorithm that I had implemented way back in my college days. Simply put, the algorithm finds the most descriptive patterns in the table that succinctly convey the meaning of the rows contained i.e. summarize it. This post will focus on what the algorithm is and how it’s implemented in JavaScript. Take this post with a grain of salt because the implementation was a part of my college assignment and hasn’t been thoroughly checked for correctness.

What is TSum?

TSum is the algorithm published by Google Research ^{[link to the pdf]}. To quote the abstract:

Given a table where rows correspond to records and columns correspond to attributes, we want to find a small number of patterns that succinctly summarize the dataset. … TSum, a method that provides a sequence of patterns ordered by their “representativeness.” It can decide both which these patterns are, as well as how many are necessary to properly summarize the data.

An algorithm like this is useful in situations where the volume of data is so large that it’d be impossible for a human to deduce anything from it simply by looking at the data. For example, given patient records, a description like “most patients are middle-aged men with cholestrol, followed by child patients with chickenpox” provides a very human-readable summary of the data.

The algorithm views finding the best pattern as a compression problem — the best pattern will give the best compression.

Definitions

Pattern and Pattern Size

A pattern is a tuple where can can be either a specific value or a “don’t care” value represented by . The number of matching attributes in a pattern is called the size of the pattern and is denoted by . An attribute is considered matching if and only if its value doesn’t equal “don’t care” value.

Pattern List and Pattern Set

A pattern list is an ordered sequence of patterns while a pattern set is a set of patterns with no order.

Compression Saving

Compression saving of a pattern on a table , denoted as is the amount of compression it can achieve. Specifically, it is the difference of bits between he uncompressed and compressed representations of the data records covered by the pattern . Let be the number of records covered by pattern and be the number of attributes in . Then,

where

and

is the average number of bits in the i^th attribute.
is the number of attributes.

Residue Data Table

Given a table and a pattern collection , the residue data table contains the records that are not covered by any of the patterns in .

Pattern Marhsalling

Given a set of patterns , the pattern marshalling algorithm picks a pattern from which has the highest compression saving. After every iteration, the records in the table which have been covered by the patterns chosen so far will be removed from consideration.

Generating Patterns - Local Expansion

Local expansion is a pattern generation strategy that tries to “grow” the patterns to increase the compression savings on the data record. In this approach, the algorithm will start with single attributes first and and find a single-condition pattern that has the best compression saving, and then expand the pattern by adding other conditions until the compression cost cannot be improved. To find the next pattern, the same procedure is repeated, but only on the residue data table - the part of the table not covered by any of the patterns found so far.

Code

We’ll start with simpler code first. Let’s start by writing a function to calculate the benefit.

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"use strict";
const _ = require("lodash");
/**
 * The benefit of using this pattern to represent the table
 * @param T the table containing rows
 * @param pattern the pattern to be applied
 */
function benefit(pattern, T){
    if(pattern == null) return 0;

    // a function for internal use.
    // find out the bits needed to encode the value
    function bitsFor(value) {
        if( typeof(value) === "string" || value instanceof String){
            // JS uses UTF-16
            return value.length * 16;
        }
        if( typeof(value) === "number" || value instanceof Number){
            // 64-bit for each number
            return 64;
        }
        return 0;
    }

    let N = _.filter(T, pattern).length;
    let W = 0;
    let attributes = Object.keys(pattern);

    attributes.forEach(attribute => {
        W += bitsFor(pattern[attribute]);
    });

    return (N-1) * W;
}
module.exports = benefit;

The actual translation of formula to code happens from line #25. We start by finding N which is the number of records in the table that match the pattern. Im using lodash‘s filter function to avoid the boilerplate of having to find the matching records myself. W is the accumulator in which I will sum the number of bits each attribute in the pattern take. attributes are the attributes in the pattern. Then for each attribute, we use bitsFor and add it up with the value of W. Finally, we return the value according to the formula.

Simply put, benefit is times the total number of bits each attribute in the pattern would take.

Next, let’s write code to find

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"use strict";
/**
 * Calculates log*(N).
 * @param N number of rows in the table
 * @returns {number}
 */
function log(N){
    return 2 * Math.log2( N+2 );
}
module.exports = log;

This is a fairly straightforward translation of the formula to code and needs no explanation.

We now have all the code we need to calculate overhead so let’s do that.

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"use strict";
const _ = require("lodash");
const log = require("./log");
/**
 * The overhead of using this pattern to represent this table
 * @param T the table containing rows
 * @param pattern the pattern to be applied
 */
function overhead(pattern, T){
    let N = _.filter(T, pattern).length;
    let D = Object.keys(T[0]).length; // number of attributes
    return D + log(N);
}
module.exports = overhead;

Notice that I start by require-ing the log.js module we just saw. I am using filter to find the number of rows in the table that match the pattern. On the next line I find the number of attributes in a given record of the table. Since the algorithm assumes no null / empty values for any attribute, we can safely pick up the 0^th record and see how many attributes it contains.

Now that we have benefit and overhead in place, let’s calculate saving

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"use strict";
const benefit = require("./benefit");
const overhead = require("./overhead");
/**
 * The compression saving of a pattern P on a data table T,
 * denoted by saving(P,T) is the amount of compression it can achieve.
 * @param T the table containing rows
 * @param pattern the pattern to be applied
 */
function saving(pattern, T){
    return benefit(pattern, T) - overhead(pattern, T);
}
module.exports = saving;

Now let’s write code for pattern marshalling.

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"use strict";
const _ = require("lodash");
const saving = require("./saving");
const coverage = require("./coverage");

function patternMarshalling(S, T){
    // chosen patterns
    let patterns = []; // empty = don't care for every attribute
    let remainingPatterns = _.cloneDeep(S);

    while( remainingPatterns.length > 0 ){
        // select the pattern with the top incremental compression saving
        let bTop = Number.NEGATIVE_INFINITY;
        let pTop;
        let residualTable = residue(patterns, T);

        remainingPatterns.forEach(pattern => {
            let compression = saving(pattern, residualTable);
            if( compression > bTop ) {
                bTop = compression;
                pTop = pattern;
            }
        });

        if( bTop > 0 ){
            patterns.push({
                "pattern" : pTop,
                "saving" : bTop,
                "coverage" : coverage(pTop,T)
            });
        }

        remainingPatterns = _.difference(remainingPatterns, _.filter(remainingPatterns, pTop));
    }

    return patterns;
}

function residue(patterns, T){
    patterns.forEach(pattern => {
        T = _.difference(T, _.filter(T, pattern.pattern));
    });
    return T;
}

module.exports = patternMarshalling;

Given a set of patterns and a table , we start by making a copy of the patterns passed to us. The aim is to select a pattern which gives us non-negative compression. If we find such a pattern, we add it to our patterns list and remove the chosen pattern from the current list of patterns. We continue the next itreation on the residue table. We repeat this until we have no more patterns to consider.

The next piece of the puzzle is to write code for local expansion.

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"use strict";
const _ = require("lodash");
const expand = require("./expand");

/**
 * Local expansion pattern generation strategy that directly looks
 * for patterns that could minimize the compression cost
 * @param T an array of JSON objects
 * @returns {Array} of patterns that best summarize the data
 */
function localExpansion(T){
    let patterns = []; // final list of patterns to return

    // while we still have rows
    while( T.length > 0 ){
        // expand from an empty pattern (Algorithm 4)
        let pattern = expand(T, {});
        // stop if we cannot achieve more compression saving
        if( _.isEqual(pattern, {}) ){
            break;
        }
        // found a new pattern
        patterns.push( pattern );
        // remove all the rows that match the pattern i.e.residual table for the pattern
        T = _.difference(T, _.filter(T, pattern));
    }

    return patterns;
}

module.exports = localExpansion;

As mentioned in the definitions, local expansion grows a pattern, starting with an empty pattern. The expansion is done in expand function.

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const _ = require("lodash");
const saving = require("./saving");

/**
 * Expands a pattern to improve compression saving. See algorithm 4.
 * @param T
 * @param pattern
 * @returns {*}
 */
function expand(T, pattern){
    // find the attributes not included in the pattern
    let allAttributes = Object.keys(T[0]);
    let matchedAttributes = Object.keys(pattern);
    let attributes = _.difference(allAttributes, matchedAttributes);

    let bestPattern;
    let highestCompressionSaving = 0;

    attributes.forEach( attribute => {
        // find all the unique values for the current attribute
        let values = _.map(T, attribute);
        values = _.uniq(values);

        values.forEach( value => {
            // an expanded pattern created by appending the current
            // attribute and its value to the existing pattern
            let newPattern = _.cloneDeep(pattern);
            newPattern[attribute] = value;

            let compressionSaving = saving(newPattern, T);

            if(compressionSaving > highestCompressionSaving){
                highestCompressionSaving = compressionSaving;
                bestPattern = newPattern;
            }
        });
    });

    if( saving(bestPattern, T) > saving(pattern, T) ){
        return expand(T, bestPattern);
    }else{
        return pattern;
    }
}

module.exports = expand;

The final piece of code we need to look at is to calculate the coverage i.e. how much of the data is described by a given pattern.

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"use strict";

const _ = require("lodash");
function coverage(pattern,T){
    let matchingRows = _.filter(T,pattern);
    let coverage = matchingRows.length / T.length;
    return coverage * 100; // % of coverage
}

module.exports = coverage;

Now that we have all the machinery in place, let’s write a simple test. We’ll take the patients example given in the paper and turn it into JSON objects. Then we’ll write a simple script to run our code using this data and check whether the results make sense.

Here’s the data:

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[
  {"gender":"M","age":"adult","blood_pressure":"normal"},
  {"gender":"M","age":"adult","blood_pressure":"low"},
  {"gender":"M","age":"adult","blood_pressure":"normal"},
  {"gender":"M","age":"adult","blood_pressure":"high"},
  {"gender":"M","age":"adult","blood_pressure":"low"},

  {"gender":"F","age":"child","blood_pressure":"low"},
  {"gender":"M","age":"child","blood_pressure":"low"},
  {"gender":"F","age":"child","blood_pressure":"low"},

  {"gender":"M","age":"teen","blood_pressure":"high"},
  {"gender":"F","age":"child","blood_pressure":"normal"}
]

Here’s the test:

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"use strict";

const tsum = require("../index");
const table = require("./data/patients.json");
const _ = require("lodash");

let patterns = tsum.localExpansion( table );
let sorted = tsum.patternMarshalling(patterns,table);

patterns = _.shuffle(patterns);
console.log( sorted );

Now let’s run this:

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$ node test/test1.js
[ { pattern: { age: 'adult', gender: 'M' },
    saving: 375.3852901558848,
    coverage: 50 },
  { pattern: { age: 'child', blood_pressure: 'low' },
    saving: 248.35614381022526,
    coverage: 30 } ]

The output says that the most descriptive pattern is “adult male” which makes up 50% of the rows followed by “children with low blood pressure” which make up 30% of the rows. if we look at the sample data, out of the 10 rows, 5 are “adult male” and 3 are “children with low blood pressure”. So the output of the algorithm checks out.

Finito.

In the previous post we looked at how JVM generates assembly code and how we can take a look at it. In this post we will look at yet another optimization technique called loop unrolling.

What is loop unrolling?

To quote The Java HotSpot Performance Engine Architecture,

the Server VM features loop unrolling, a standard compiler optimization that enables faster loop execution. Loop unrolling increases the loop body size while simultaneously decreasing the number of iterations.

Okay so it is an optimization that makes our loops faster by increasing the body size, and reducing the iterations. This is much better explained by example.

A loop like this:

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for(int i = 0; i < N; i++) {
    S(i);
}

can be unrolled into this:

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for(int i = 0; i < N; i += 4) {
    S(i);
    S(i+1);
    S(i+2);
    S(i+3);
}

So the size of the loop has increased because instead of calling method S just once per iteration, we call it 4x. The iterations have reduced because we now have a stride of 4 (i += 4). This is a space-time tradeoff because you gain speed by increasing the size of the program. If this for loop were to be a part of a hot method that got compiled to assembly, more code cache would be consumed because of unrolling.

Loop unrolling, however, wins at a different front — speed. By increasing the stride of the loop, we’re reducing the number of jumps that the CPU has to make. Jumps are costly and a reduction in jumps is a huge performance boost.

Loop unrolling in action

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import org.openjdk.jmh.annotations.*;
import java.util.concurrent.TimeUnit;

@Warmup(iterations = 10, time = 1, timeUnit = TimeUnit.SECONDS)
@Measurement(iterations = 5, time = 1, timeUnit = TimeUnit.SECONDS)
@Fork(value = 1, jvmArgsPrepend = {"-XX:+UnlockDiagnosticVMOptions", "-XX:+PrintAssembly"})
@BenchmarkMode(Mode.AverageTime)
@OutputTimeUnit(TimeUnit.NANOSECONDS)
public class MyBenchmark {
    @Benchmark
    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    public void testMethod() {
        int[] nums = new int[10];

        for (int i = 0; i < nums.length; i++) {
            nums[i] = 0x42;
        }
    }
}

The code above is a JMH benchmark which will fork 1 JVM (@Fork), warm it for 10 iterations (@Warmup), then run our benchmark for 5 iterations (@Measurement). The generated assembly looks like this:

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0x00000001296ac845: cmp    %r10d,%ebp
0x00000001296ac848: jae    0x00000001296ac8b4
0x00000001296ac84a: movl   $0x42,0x10(%rbx,%rbp,4)
0x00000001296ac852: inc    %ebp               
0x00000001296ac854: cmp    %r11d,%ebp
0x00000001296ac857: jl     0x00000001296ac845

...

0x00000001296ac880: vmovdqu %xmm0,0x10(%rbx,%rbp,4)  
0x00000001296ac886: add    $0x4,%ebp          
0x00000001296ac889: cmp    %r8d,%ebp
0x00000001296ac88c: jl     0x00000001296ac880 

So the first block of code is moving 0x42 to some memory location (line #3), incrementing the counter %ebp (line #4), and jumping back to the start of the loop for the next iteration (line #6). Then there is the second block which increments the counter by 4 (line #11) and uses vmovdqu (line #10).

In the second block, vmovdqu moves %xmm0 (a 128-bit register) to some memory location (line #10) i.e. we’re writing 4 32-bit numbers in one go. This is equivalent of writing nums[0 .. 3] in one iteration.

This means that our entire loop which writes 10 array elements will not run for 10 iterations at all. It will run for 4 — 2 iterations to write 8 elements in groups of 4 each (block 2), and 2 more to write the remaining 2 elements (block 1).

What we saw above was loop unrolling to enable vectorization. Since this happened without any effort on our part, it’s called auto-vectorization.

What is vectorization?

There are certain CPU instructions which are capable of operating on multiple data elements simultaneously. Such instructions are called SIMD instructions — Single Instruction Multiple Data or vectorized instructions. In the example above, all 4 elements were packed into a single register XMM0 and then moved to their memory location with vmovdqu. This results in faster processing by saving jumps and clock cycles.

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+--------|--------|--------|--------+
|  0x42  |  0x42  |  0x42  |  0x42  | xmm0
+--------|--------|--------|--------+
     ↓       ↓         ↓        ↓     

Auto-vectorization is when the compiler converts the scalar instructions (which operate on a single data element at a time) to vector instructions (which operate on multiple data elements at a time) without any effort on the part of the programmer.

Vectorizing the code enables superword level parallelism (SLP). SLP is a type of SIMD parallelism in which source and result operands are packed in a storage location.

The example shows statement packing in which the operands have been packed into registers and the scalar addition and multiplication have been replaced by their vectorized counterparts.

How does loop unrolling enable SLP?

Consider this loop:

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for (i=0; i<16; i++) {
    localdiff = ref[i] - curr[i];
}

This is a loop containing isomorphic statements — statements that contain same operations in the same order. Such statements are easy to unroll. So the loop above can be transformed as follows:

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for (i=0; i<16; i+=4) {
    localdiff = ref[i+0] - curr[i+0];
    diff += abs(localdiff);
    
    localdiff = ref[i+1] - curr[i+1];
    diff += abs(localdiff);

    localdiff = ref[i+2] - curr[i+2];
    diff += abs(localdiff);

    localdiff = ref[i+3] - curr[i+3];
    diff += abs(localdiff);
}

This can be further transformed as:

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for (i=0; i<16; i+=4) {
    localdiff0 = ref[i+0] - curr[i+0];
    localdiff1 = ref[i+1] - curr[i+1];
    localdiff2 = ref[i+2] - curr[i+2];
    localdiff3 = ref[i+3] - curr[i+3];

    diff += abs(localdiff0);
    diff += abs(localdiff1);
    diff += abs(localdiff2);
    diff += abs(localdiff3);
}

Now it becomes easy to see how SLP can be used to speed up the execution of the loop. Loop unrolling sets the stage for SLP.

Which flags control loop unrolling and SLP?

There are a couple of flags - -XX:LoopUnrollLimit and -XX:+UseSuperWord. LoopUnrollLimit controls how many times your loop will be unrolled and UseSuperWord controls the transformation of scalar operations into vectorized operations.

Conclusion

Loop unrolling makes your code faster by doing more per iteration and increasing the stride. It’s a space-time trade off where you make your code larger so that the execution takes less time. Loop unrolling can be taken a step further to make scalar instructions into vector instructions. These instructions operate on packed data and achieve more per instruction than their scalar counterparts.

The conversion from scalar instructions to vector instructions is called vectorization. Since this happens transparently to the programmer, it is called auto-vectorization.

Recommended reading

There are a few blog posts that I recommend you read.

A fundamental introduction to x86 assembly programming
This post provides a brief introduction to x86 assembly programming. Having studied assembly programming in university a long time ago, the post is quick refresher on the basic concepts of assembly programming like addresing modes, registers, etc. I keep it handy and refer to it when I have to read the assembly code generated by JVM.

JMH - Java Microbenchmark Harness
This post provides a brief introduction to JMH (Java Mirobenchmark Harness). What does JMH do?

JMH is a Java harness for building, running, and analysing nano/micro/milli/macro benchmarks written in Java and other languages targetting the JVM.

The post will show you how to write a simple benchmark, how to avoid common pitfalls, etc. It is a good read regardless of this series for two reasons: one, microbenchmarking is coming to Java 9 out-of-the-box and two, microbenchmarking is hard to get right.

Exploiting Superword Level Parallelism with Multimedia Instruction Sets
This is the research paper on superword level parallelism. I’ve borrowed examples and diagrams from this paper. The paper will introduce you to what SLP is, and also provide an algorithm on how to find parts of code which are amenable to SLP.

In the previous post we saw how JIT inlining works. We also saw how the JVM performs OSR to replace the interpreted version of the method to the compiled version on the fly. In this post we’ll dig even deeper and see the assembly code that is generated when the method gets compiled.

Prerequisites

The flag which enables us to see assembly code is -XX:+PrintAssembly. However, viewing assembly code does not work out of the box. You’ll need to have the disassembler on your path. You’ll need to get hsdis (HotSpot Disassembler) and build it for your system. There’s a prebuilt version available for Mac and that’s the one I am going to use.

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git clone https://github.com/liuzhengyang/hsdis.git

Once we have that, we’ll add it to LD_LIBRARY_PATH.

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export LD_LIBRARY_PATH=./hsdis/build/macosx-amd64

Now we’re all set to see how JVM generates assembly code.

Printing assembly code

We’ll reuse the same inlining code from last time:

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public class Inline {
    public static void main(String[] args) {
        long upto = Long.parseLong(args[0]);

        for(int i = 0; i < upto; i++) {
            int x = inline1();
        }
    }

    public static int inline1() {
        return inline2();
    }

    public static int inline2() {
        return inline3();
    }

    public static int inline3() {
        return 4;
    }
}

-XX:+PrintAssembly is a diagnostic flag so we’ll need to unlock JVM’s disgnostic options first. Here’s how:

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java -XX:+UnlockDiagnosticVMOptions -XX:+PrintAssembly Inline 100000

This will generate a lot of assembly code. We will, however, look at the assembly code generated for inline1.

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Decoding compiled method 0x000000010d6095d0:
Code:
[Disassembling for mach='i386:x86-64']
[Entry Point]
[Verified Entry Point]
[Constants]
  # {method} 'inline1' '()I' in 'Inline'
  #           [sp+0x20]  (sp of caller)
  0x000000010d609700: sub    $0x18,%rsp
  0x000000010d609707: mov    %rbp,0x10(%rsp)    ;*synchronization entry
                                                ; - Inline::inline1@-1 (line 11)
  0x000000010d60970c: mov    $0x4,%eax
  0x000000010d609711: add    $0x10,%rsp
  0x000000010d609715: pop    %rbp
  0x000000010d609716: test   %eax,-0x496b71c(%rip)        # 0x0000000108c9e000
                                                ;   {poll_return}
  0x000000010d60971c: retq
  0x000000010d60971d: hlt
  0x000000010d60971e: hlt
  0x000000010d60971f: hlt
[Exception Handler]
[Stub Code]
  0x000000010d609720: jmpq   0x000000010d6050a0  ;   {no_reloc}
[Deopt Handler Code]
  0x000000010d609725: callq  0x000000010d60972a
  0x000000010d60972a: subq   $0x5,(%rsp)
  0x000000010d60972f: jmpq   0x000000010d5deb00  ;   {runtime_call}
  0x000000010d609734: hlt
  0x000000010d609735: hlt
  0x000000010d609736: hlt
  0x000000010d609737: hlt    Decoding compiled method 0x000000010d6064d0:

So this is the assembly code that we get when we run the program. It’s a lot to grok in one go so let’s break it down.

Line #7 and #8 are self explanatory; they show which method we’re looking at. Line #9 and #10 (and #13 to #17) are for thread synchronization. The JVM can get rid of thread synchronization if it sees that there is no need for it (lock eliding) but since we are using static methods here, it needs to add code for synchronization. It doesn’t know that we only have only one thread running.

Our actual program is on line #11 where we are moving the value 4 to %eax register. This is the register which holds, by convention, the return value for our methods. This shows that the JVM has optimized our code. Our call chain was inline1 ⟶ inline2 ⟶ inline3 and it was inline3 which returned 4. However, JVM is smart enough to see that these method calls are superfluous and decided to get rid of them. Very nifty!

Line #21 to #23 has code to handle exceptions. We know there won’t be any exceptions but the JVM doesn’t so it has to be prepared to deal with that.

And finally, there’s code to deoptimize. In addition to static optimizations, there are some optimizations that the JVM makes which are speculative. This means that the JVM generates assembly code expecting things to go a certain way after it has profiled the interpreted code. However, if the speculation is wrong, the JVM can go back to running the interpreted version.

Which flags control compilation?

-XX:CompileThreshold is the flag which controls the number of call / branch invocations after which the JVM compiles bytecodes to assembly. You can use -XX:+PrintFlagsFinal to see the value. By default it is 10000.

Compiling a method to assembly depends on two factors: the number of times that method has been invoked (method entry counter) and the number of times a loop has been executed (back-edge counter). Once the sum of the two counters is above CompileThreshold, the method will be compiled to assembly.

Maintaining the two counters separately is very useful. If the back-edge counter alone exceeds the threshold, the JVM can compile just the loop (and not the entire method) to assembly. It will perform an OSR and start using the compiled version of the loop while the loop is executing instead of waiting for the next method invocation. When the method is invoked the next time around, it’ll use the compiled version of the code.

So since compiled code is better than interpreted code, and CompileThreshold controls when a method will be compiled to assembly, reducing the CompileThreshold would mean we have a lot more assembly code.

There is one advantage to reducing the CompileThreshold - it will reduce the time taken for the branches / methods to be deemed hot i.e. reduce the JVM warmup time.

In older JDKs, there was another reason to reduce CompileThreshold. The method entry and back-edge counters would decay at every safepoint. This would mean that some methods would not compile to assembly since the counters kept decaying. These are the “lukewarm” methods that never became hot. With JDK 8+, the counters no longer decay at safepoints so there won’t be any lukewarm methods.

In addition, JDK 8+ come with tiered compilation enabled and the CompileThreshold is ignored. The idea of there being a “compile threshold”, though, does not change. I’m defering the topic of tiered compilation for the sake of simplicity.

Where is the compiled code stored?

The compiled code is stored in JVM’s code cache. As more methods become hot, the cache starts to get filled. Once the cache is filled, the JVM can no longer compile anything to assembly and will resort to purely interpreteting the bytecodes.

The size of code cache is platform dependent.

Also, JVM ensures that the access to cache is optimized. The hlt instructions in the assembly code exist for aligning the addresses. It is much more efficient for the CPU to read from even addresses than it is to read from odd addresses in memory. The hlt instructions ensure that the code is at an even address in memory.

Which flags control code cache size?

There are two flags which are important in setting the code cache size - InitialCodeCacheSize and ReservedCodeCacheSize. The first flag indicates the code cache size the JVM will start with and the latter indicates the size to which the code cache can grow. With JDK 8+, ReservedCodeCacheSize is large enough so you don’t need to set it explicitly. On my machine it is 240 MB (5x what it is for Java 7, 48 MB).

Conclusion

The JVM compiles hot code to assembly and stores it at even addresses in it’s code cache for faster access. Executing assembly code is much more efficient than interpreting the bytecodes. You don’t really need to look at the assembly code generated everyday but knowing what is generated as your code executes gives you an insight into what the JVM does to make your code run faster.

In the previous post we looked at how interpreted and compiled languages work. To recap, an interpreter works by generating assembly code for every bytecode it encounters. This is a very simple way to execute a program and also a very slow one. It ends up redoing a lot of translation from bytecode to assembly. Also, this simplistic approach means that the interpreter cannot do optimizations as it executes the bytecodes. Then there are compilers which produce assembly ahead-of-time. This overcomes having to generate assembly again and again but once the assembly is generated it cannot be changed on the fly.

JVM comes with both an interpreter and a compiler. When the execution of the code begins, the bytecodes are interpreted. For the sake of this series, I’ll be looking at Oracle HotSpot JVM which looks for “hot spots” in the code as the bytecodes get interpreted. These are the parts of the code which are most frequently executed and the performance of the application depends on these. Once the code is identified as “hot”, JVM can go from interpreting the code to compiling it to assembly i.e. the code is compiled “just-in-time”. In addition, since the code is being profiled as it is run, the compiled code is optimized.

In this post we’ll look at one such optimization: inlining.

Inlining

Inlining is an optimization where the call to a method is replaced by the body of the called method i.e. at the call site, the caller and the callee are melded together. When a method is called, the JVM has to push a stack frame so that it can resume from where it left off after the called method has finished executing. Inlining improves performance since JVM will not have to push a stack frame.

I’ll start with a simple example to demonstrate how inlining works.

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public class Inline {
    public static void main(String[] args) {
        long upto = Long.parseLong(args[0]);

        for(int i = 0; i < upto; i++) {
            int x = inline1();
        }
    }

    public static int inline1() {
        return inline2();
    }

    public static int inline2() {
        return inline3();
    }

    public static int inline3() {
        return 4;
    }
}

Next, let’s compile and run the code.

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java -XX:+PrintCompilation -XX:+UnlockDiagnosticVMOptions -XX:+PrintInlining Inline 100000

Output:

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63    1             Inline::inline1 (4 bytes)
63    2             Inline::inline2 (4 bytes)
                       @ 0   Inline::inline3 (2 bytes)   inline (hot)
                       @ 0   Inline::inline2 (4 bytes)   inline (hot)
                         @ 0   Inline::inline3 (2 bytes)   inline (hot)
66    3             Inline::inline3 (2 bytes)
66    4 %           Inline::main @ 9 (28 bytes)
                       @ 16   Inline::inline1 (4 bytes)   inline (hot)
                         @ 0   Inline::inline2 (4 bytes)   inline (hot)
                           @ 0   Inline::inline3 (2 bytes)   inline (hot)
66    4 %           Inline::main @ -2 (28 bytes)   made not entrant

Line #1 shows that inline1 was compiled to assembly. Line #2 and Line #6 show that inline2 and inline3 were also compiled to assembly. Line #3 to line #5 show inlining. We can see that inline3 was merged into inline2. Similarly, line #8 and #9 show that inline2 was merged into inline1. So basically, all the methods were inlined into inline1. This means that once a certain threshold is crossed, we’ll no longer be making methods calls at all. This gives a significant performance boost.

Which flags control inlining?

When you run a Java program, you can view the flags with which it ran using -XX:+PrintFlagsFinal. Let’s do that and look at a few flags of interest.

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java -XX:+PrintFlagsFinal Inline 10000

You’ll see a bunch of flags and their default values. The ones we are interested in are CompileThreshold, MaxInlineLevel, MaxInlineSize, and FreqInlineSize.

CompileThreshold is the number of invocations before compiling a method to native.
MaxInlineLevel is a limit on how deep you’d go before you stop inlining. The default value is 9. This means if we had method calls like inline1 ⟶ inline2 … ⟶ inline20, we’d only inline upto inline10. There after, we’d invoke inline11.
MaxInlineSize decides the maximum size of a method, in bytecodes, to be inlined. The default value is 35. This means that if the method to be inlined has mre than 35 bytecodes, it will not be inlined.
FreqInlineSize, in contrast, decides the maximum size of a hot method, in bytecodes, to be inlined. This is a platform-dependent value and on my machine it is 325.

You can tweak these flags to change how inlining behaves for your program.

What is On Stack Replacement (OSR)?

When we make a method call, JVM pushes a stack frame. When a method is deemed hot, the JVM replaces the intrepreted version with the compiled version by replacing the old stack frame with a new one. This is done while the method is running. We saw OSR being indicated in our example. The % indicates that an OSR was made.

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66    4 %           Inline::main @ -2 (28 bytes)   made not entrant

Let’s write some code to see OSR in action once again.

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import java.lang.ref.WeakReference;

public class OSR {
    public static void main(String[] args) {
        Object unused = new Object();
        WeakReference<Object> ref = new WeakReference<>(unused);

        int x = 0;

        while( ref.get() != null ) {
            x += 1;
            System.out.println(x);
        }

        System.out.println("Finished!");
    }
}

So this is a loop that will never terminate, right? Let’s run the program and see.

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...
434062
  16828   59 %           OSR::main @ -2 (48 bytes)   made not entrant
Finished!

What just happened? When the JVM decided to perform an OSR, it saw that there was no use for the unused object and decided to set it to null, causing the WeakReference to return null and thus breaking the loop. When an OSR is performed, the method that is invoked doesn’t restart execution from the start. Rather, It continues from the “back-edge”. In our case, it would be the loop. Since the JVM saw that there was no use for the unused object after this back-edge, it was removed and the loop could terminate.

Being able to resume execution from the back-edge is very efficient. This means that once a method has been compiled to native code it can be used rightaway rather than at the next invocation of the method.

Conclusion

To recap, we saw how JVM inlines code. Fusing the caller and the callee provides for improved performance since the overhead of method dispatch is avoided. We saw the flags which control inlining and we saw how JVM performs OSR.

Inlining is a very useful optimization because it forms the basis for other optimizations like escape analysis and dead code elimination.