The Bounded Difference Method
Combining Azuma's inequality with the construction of Doob martingales, we have the powerful Bounded Difference Method for concentration of measures.
For arbitrary random variables
Given a sequence of random variables and a function . The Doob sequence constructs a martingale from them. Combining this construction with Azuma's inequality, we can get a very powerful theorem called "the method of averaged bounded differences" which bounds the concentration for arbitrary function on arbitrary random variables (not necessarily a martingale).
|Theorem (Method of averaged bounded differences)
- Let be arbitrary random variables and let be a function of satisfying that, for all ,
| Define the Doob Martingale sequence by setting and, for , . Then the above theorem is a restatement of the Azuma's inequality holding for .
For independent random variables
The condition of bounded averaged differences is usually hard to check. This severely limits the usefulness of the method. To overcome this, we introduce a property which is much easier to check, called the Lipschitz condition.
|Definition (Lipschitz condition)
- A function satisfies the Lipschitz condition, if for any and any ,
In other words, the function satisfies the Lipschitz condition if an arbitrary change in the value of any one argument does not change the value of the function by more than 1.
The diference of 1 can be replaced by arbitrary constants, which gives a generalized version of Lipschitz condition.
|Definition (Lipschitz condition, general version)
- A function satisfies the Lipschitz condition with constants , , if for any and any ,
The following "method of bounded differences" can be developed for functions satisfying the Lipschitz condition. Unfortunately, in order to imply the condition of averaged bounded differences from the Lipschitz condition, we have to restrict the method to independent random variables.
|Corollary (Method of bounded differences)
- Let be independent random variables and let be a function satisfying the Lipschitz condition with constants , . Then
| For convenience, we denote that for any .
We first show that the Lipschitz condition with constants , , implies another condition called the averaged Lipschitz condition (ALC): for any , ,
And this condition implies the averaged bounded difference condition: for all ,
Then by applying the method of averaged bounded differences, the corollary can be proved.
For any , by the law of total expectation,
Let and , and take the diference. Then
Thus, the Lipschitz condition is transformed to the ALC. We then deduce the averaged bounded difference condition from ALC.
By the law of total expectation,
We can trivially write as
Hence, the difference is
The averaged bounded diference condition is implied. Applying the method of averaged bounded diferences, the corollary follows.
Throwing balls uniformly and independently at random to bins, we ask for the occupancies of bins by the balls. In particular, we are interested in the number of empty bins.
This problem can be described equivalently as follows. Let be a uniform random function from . We ask for the number of that is empty.
For any , let indicate the emptiness of bin . Let be the number of empty bins.
By the linearity of expectation,
We want to know how deviates from this expectation. The complication here is that are not independent. So we alternatively look at a sequence of independent random variables , where represents the bin into which the th ball falls. Clearly is function of .
We than observe that changing the value of any can change the value of by at most 1, because one ball can affect the emptiness of at most one bin.
Thus as a function of independent random variables , satisfies the Lipschitz condition. Apply the method of bounded differences, it holds that
Thus, for sufficiently large and , the number of empty bins is tightly concentrated around
Let be a sequence of characters chosen independently and uniformly at random from an alphabet , where . Let be an arbitrarily fixed string of characters from , called a pattern. Let be the number of occurrences of the pattern as a substring of the random string .
By the linearity of expectation, it is obvious that
We now look at the concentration of . The complication again lies in the dependencies between the matches. Yet we will see that is well tightly concentrated around its expectation if is relatively small compared to .
For a fixed pattern , the random variable is a function of the independent random variables . Any character participates in no more than matches, thus changing the value of any can affect the value of by at most . satisfies the Lipschitz condition with constant . Apply the method of bounded differences,
Combining unit vectors
Let be unit vectors from some normed space. That is, for any , where denote the vector norm (e.g. ) of the space.
Let be independently chosen and .
This kind of construction is very useful in combinatorial proofs of metric problems. We will show that by this construction, the random variable is well concentrated around its mean.
is a function of independent random variables .
By the triangle inequality for norms, it is easy to verify that changing the sign of a unit vector can only change the value of for at most 2, thus satisfies the Lipschitz condition with constant 2. The concentration result follows by applying the method of bounded differences:
Consider a problem as follows: We have a set of points in a high-dimensional Euclidean space . We want to project the points onto a space of low dimension in such a way that pairwise distances of the points are approximately the same as before.
Formally, we are looking for a map such that for any pair of original points , distorts little from , where is the Euclidean norm, i.e. is the distance between and in Euclidean space.
This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensional space, however, computations on high-dimensional data are usually hard due to the infamous "curse of dimensionality". The computational tasks can be greatly eased if we can project the data points onto a space of low dimension while the pairwise relations between the points are approximately preserved.
The Johnson-Lindenstrauss Theorem states that it is possible to project points in a space of arbitrarily high dimension onto an -dimensional space, such that the pairwise distances between the points are approximately preserved.
- For any and any positive integer , let be a positive integer such that
- Then for any set of points in , there is a map such that for all ,
- Furthermore, this map can be found in expected polynomial time.
The random projections
The map is done by random projection. There are several ways of applying the random projection. We adopt the one in the original Johnson-Lindenstrauss paper.
|The projection (due to Johnson-Lindenstrauss)
- Let be a random matrix that projects onto a uniform random k-dimensional subspace.
- Multiply by a fixed scalar . For every , is mapped to .
The projected point is a vector in .
The purpose of multiplying the scalar is to guarantee that .
Besides the uniform random subspace, there are other choices of random projections known to have good performances, including:
- A matrix whose entries follow i.i.d. normal distributions. (Due to Indyk-Motwani)
- A matrix whose entries are i.i.d. . (Due to Achlioptas)
In both cases, the matrix is also multiplied by a fixed scalar for normalization.
A proof of the Theorem
We present a proof due to Dasgupta-Gupta, which is much simpler than the original proof of Johnson-Lindenstrauss. The proof is for the projection onto uniform random subspace. The idea of the proof is outlined as follows:
- To bound the distortions to pairwise distances, it is sufficient to bound the distortions to the length of unit vectors.
- A uniform random subspace of a fixed unit vector is identically distributed as a fixed subspace of a uniform random unit vector. We can fix the subspace as the first k coordinates of the vector, thus it is sufficient to bound the length (norm) of the first k coordinates of a uniform random unit vector.
- Prove that for a uniform random unit vector, the length of its first k coordinates is concentrated to the expectation.
From pairwise distances to norms of unit vectors
Let be a vector in the original space, the random matrix projects onto a uniformly random k-dimensional subspace of . We only need to show that
Think of as a for some . Then by applying the union bound to all pairs of the points in , the random projection violates the distortion requirement with probability at most
so has the desirable low-distortion with probability at least . Thus, the low-distortion embedding can be found by trying for expected times (recalling the analysis fo geometric distribution).
We can further simplify the problem by normalizing the . Note that for nonzero 's, the statement that
is equivalent to that
Thus, we only need to bound the distortions for the unit vectors, i.e. the vectors that . The rest of the proof is to prove the following lemma for the unit vector in .
- For any unit vector , it holds that
As we argued above, this lemma implies the Johnson-Lindenstrauss Theorem.
Random projection of fixed unit vector fixed projection of random unit vector
Let be a fixed unit vector in . Let be a random matrix which projects the points in onto a uniformly random -dimensional subspace of .
Let be a uniformly random unit vector in . Let be such a fixed matrix which extracts the first coordinates of the vectors in , i.e. for any , .
In other words, is a random projection of a fixed unit vector; and is a fixed projection of a uniformly random unit vector.
A key observation is that:
- The distribution of is the same as the distribution of .
The proof of this observation is omitted here.
With this observation, it is sufficient to work on the subspace of the first coordinates of the uniformly random unit vector . Our task is now reduced to the following lemma.
- Let be a uniformly random unit vector in . Let be the projection of to the subspace of the first -coordinates of .
Due to the above observation, Lemma 3.2 implies Lemma 3.1 and thus proves the Johnson-Lindenstrauss theorem.
Note that . Due to the linearity of expectations,
Since is a uniform random unit vector, it holds that . And due to the symmetry, all 's are equal. Thus, for all . Therefore,
Lemma 3.2 actually states that is well-concentrated to its expectation.
Concentration of the norm of the first entries of uniform random unit vector
We now prove Lemma 3.2. Specifically, we will prove the direction:
The direction is proved with the same argument.
Due to the discussion in the last section, this can be interpreted as a concentration bound for , which is a sum of . This hints us to use Chernoff-like bounds. However, for uniformly random unit vector , 's are not independent (because of the constraint that ). We overcome this by generating uniform unit vectors from independent normal distributions.
The following is a very useful fact regarding the generation of uniform unit vectors.
|Generating uniform unit vector
- Let be i.i.d. random variables, each drawn from the normal distribution . Let . Then
- is a uniformly random unit vector.
Then for ,
To avoid writing a lot of 's. We write . The first inequality (the lower tail) of Lemma 3.2 can be written as:
The probability is a tail probability of the sum of independent variables. The 's are not 0-1 variables, thus we cannot directly apply the Chernoff bounds. However, the following two key ingredients of the Chernoff bounds are satisfiable for the above sum:
- The 's are independent.
- Because 's are normal, it is known that the moment generating functions for 's can be computed as follows:
- If follows the normal distribution , then , for
Therefore, we can re-apply the technique of the Chernoff bound (applying Markov's inequality to the moment generating function and optimizing the parameter ) to bound the probability :
The last term is minimized when
which is is for the choice of k in the Johnson-Lindenstrauss theorem that
So we have proved that
With the same argument, the other direction can be proved so that
which is also for .
Lemma 3.2 is proved. As we discussed in the previous sections, Lemma 3.2 implies Lemma 3.1, which implies the Johnson-Lindenstrauss theorem.