Small-bias sample space

In theoretical computer science, a small-bias sample space (also known as $\epsilon$-biased sample space, $\epsilon$-biased generator, or small-bias probability space) is a probability distribution that fools parity functions. In other words, no parity function can distinguish between a small-bias sample space and the uniform distribution with high probability, and hence, small-bias sample spaces naturally give rise to pseudorandom generators for parity functions.

The main useful property of small-bias sample spaces is that they need far fewer truly random bits than the uniform distribution to fool parities. Efficient constructions of small-bias sample spaces have found many applications in computer science, some of which are derandomization, error-correcting codes, and probabilistically checkable proofs. The connection with error-correcting codes is in fact very strong since $\epsilon$-biased sample spaces are equivalent to $\epsilon$-balanced error-correcting codes.

Definition

Bias

Let $X$ be a probability distribution over $\{0,1\}^{n}$. The bias of $X$ with respect to a set of indices ${\displaystyle I\subseteq \{1,\dots ,n\}}$ is defined as^[1]

${\displaystyle {\text{bias}}_{I}(X)=\left|\Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=0\right)-\Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=1\right)\right|=\left|2\cdot \Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=0\right)-1\right|\,,}$

where the sum is taken over $\mathbb F_2$, the finite field with two elements. In other words, the sum ${\displaystyle \sum _{i\in I}x_{i}}$ equals ${\displaystyle 0}$ if the number of ones in the sample $x\in \{0,1\}^{n}$ at the positions defined by $I$ is even, and otherwise, the sum equals $1$. For ${\displaystyle I=\emptyset }$, the empty sum is defined to be zero, and hence ${\displaystyle {\text{bias}}_{\emptyset }(X)=1}$.

ϵ-biased sample space

A probability distribution $X$ over $\{0,1\}^{n}$ is called an $\epsilon$-biased sample space if ${\displaystyle {\text{bias}}_{I}(X)\leq \epsilon }$ holds for all non-empty subsets ${\displaystyle I\subseteq \{1,2,\ldots ,n\}}$.

ϵ-biased set

An $\epsilon$-biased sample space $X$ that is generated by picking a uniform element from a multiset ${\displaystyle X\subseteq \{0,1\}^{n}}$ is called $\epsilon$-biased set. The size $s$ of an $\epsilon$-biased set $X$ is the size of the multiset that generates the sample space.

ϵ-biased generator

An $\epsilon$-biased generator ${\displaystyle G:\{0,1\}^{\ell }\to \{0,1\}^{n}}$ is a function that maps strings of length $\ell$ to strings of length $n$ such that the multiset ${\displaystyle X_{G}=\{G(y)\;\vert \;y\in \{0,1\}^{\ell }\}}$ is an $\epsilon$-biased set. The seed length of the generator is the number $\ell$ and is related to the size of the $\epsilon$-biased set $X_{G}$ via the equation ${\displaystyle s=2^{\ell }}$.

Connection with epsilon-balanced error-correcting codes

There is a close connection between $\epsilon$-biased sets and $\epsilon$-balanced linear error-correcting codes. A linear code ${\displaystyle C:\{0,1\}^{n}\to \{0,1\}^{s}}$ of message length $n$ and block length $s$ is $\epsilon$-balanced if the Hamming weight of every nonzero codeword $C(x)$ is between ${\displaystyle ({\frac {1}{2}}-\epsilon )s}$ and ${\displaystyle ({\frac {1}{2}}+\epsilon )s}$. Since $C$ is a linear code, its generator matrix is an ${\displaystyle (n\times s)}$-matrix $A$ over $\mathbb F_2$ with ${\displaystyle C(x)=x\cdot A}$.

Then it holds that a multiset ${\displaystyle X\subset \{0,1\}^{n}}$ is $\epsilon$-biased if and only if the linear code $C_{X}$, whose columns are exactly elements of $X$, is $\epsilon$-balanced.^[2]

Constructions of small epsilon-biased sets

Usually the goal is to find $\epsilon$-biased sets that have a small size $s$ relative to the parameters $n$ and $\epsilon$. This is because a smaller size $s$ means that the amount of randomness needed to pick a random element from the set is smaller, and so the set can be used to fool parities using few random bits.

Theoretical bounds

The probabilistic method gives a non-explicit construction that achieves size ${\displaystyle s=O(n/\epsilon ^{2})}$.^[2] The construction is non-explicit in the sense that finding the $\epsilon$-biased set requires a lot of true randomness, which does not help towards the goal of reducing the overall randomness. However, this non-explicit construction is useful because it shows that these efficient codes exist. On the other hand, the best known lower bound for the size of $\epsilon$-biased sets is ${\displaystyle s=\Omega (n/(\epsilon ^{2}\log(1/\epsilon ))}$, that is, in order for a set to be $\epsilon$-biased, it must be at least that big.^[2]

Explicit constructions

There are many explicit, i.e., deterministic constructions of $\epsilon$-biased sets with various parameter settings:

• Naor & Naor (1990) achieve ${\displaystyle \displaystyle s={\frac {n}{{\text{poly}}(\epsilon )}}}$. The construction makes use of Justesen codes (which is a concatenation of Reed–Solomon codes with the Wozencraft ensemble) as well as expander walk sampling.
• Alon et al. (1992) achieve ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon \log(n/\epsilon )}}\right)^{2}}$. One of their constructions is the concatenation of Reed–Solomon codes with the Hadamard code; this concatenation turns out to be an $\epsilon$-balanced code, which gives rise to an $\epsilon$-biased sample space via the connection mentioned above.
• Concatenating Algebraic geometric codes with the Hadamard code gives an $\epsilon$-balanced code with ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon ^{3}\log(1/\epsilon )}}\right)}$.^[2]
• Ben-Aroya & Ta-Shma (2009) achieves ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon ^{2}\log(1/\epsilon )}}\right)^{5/4}}$.

These bounds are mutually incomparable. In particular, none of these constructions yields the smallest $\epsilon$-biased sets for all settings of $\epsilon$ and $n$.

Application: almost k-wise independence

An important application of small-bias sets lies in the construction of almost k-wise independent sample spaces.

k-wise independent spaces

A random variable $Y$ over $\{0,1\}^{n}$ is a k-wise independent space if, for all index sets ${\displaystyle I\subseteq \{1,\dots ,n\}}$ of size $k$, the marginal distribution ${\displaystyle Y|_{I}}$ is exactly equal to the uniform distribution over $\{0,1\}^{k}$. That is, for all such $I$ and all strings ${\displaystyle z\in \{0,1\}^{k}}$, the distribution $Y$ satisfies ${\displaystyle \Pr _{Y}(Y|_{I}=z)=2^{-k}}$.

Constructions and bounds

k-wise independent spaces are fairly well-understood.

• A simple construction by Joffe (1974) achieves size $n^{k}$.
• Alon, Babai & Itai (1986) construct a k-wise independent space whose size is ${\displaystyle n^{k/2}}$.
• Chor et al. (1985) prove that no k-wise independent space can be significantly smaller than ${\displaystyle n^{k/2}}$.

Joffe's construction

Joffe (1974) constructs a $k$-wise independent space $Y$ over the finite field with some prime number ${\displaystyle n>k}$ of elements, i.e., $Y$ is a distribution over ${\displaystyle \mathbb {F} _{n}^{n}}$. The initial $k$ marginals of the distribution are drawn independently and uniformly at random:

${\displaystyle (Y_{0},\dots ,Y_{k-1})\sim \mathbb {F} _{n}^{k}}$.

For each $i$ with ${\displaystyle k\leq i<n}$, the marginal distribution of $Y_{i}$ is then defined as

${\displaystyle Y_{i}=Y_{0}+Y_{1}\cdot i+Y_{2}\cdot i^{2}+\dots +Y_{k-1}\cdot i^{k-1}\,,}$

where the calculation is done in ${\displaystyle \mathbb {F} _{n}}$. Joffe (1974) proves that the distribution $Y$ constructed in this way is $k$-wise independent as a distribution over ${\displaystyle \mathbb {F} _{n}^{n}}$. The distribution $Y$ is uniform on its support, and hence, the support of $Y$ forms a $k$-wise independent set. It contains all $n^{k}$ strings in ${\displaystyle \mathbb {F} _{n}^{k}}$ that have been extended to strings of length $n$ using the deterministic rule above.

Almost k-wise independent spaces

A random variable $Y$ over $\{0,1\}^{n}$ is a $\delta$-almost k-wise independent space if, for all index sets ${\displaystyle I\subseteq \{1,\dots ,n\}}$ of size $k$, the restricted distribution ${\displaystyle Y|_{I}}$ and the uniform distribution $U_{k}$ on $\{0,1\}^{k}$ are $\delta$-close in 1-norm, i.e., ${\displaystyle {\Big \|}Y|_{I}-U_{k}{\Big \|}_{1}\leq \delta }$.

Constructions

Naor & Naor (1990) give a general framework for combining small k-wise independent spaces with small $\epsilon$-biased spaces to obtain $\delta$-almost k-wise independent spaces of even smaller size. In particular, let ${\displaystyle G_{1}:\{0,1\}^{h}\to \{0,1\}^{n}}$ be a linear mapping that generates a k-wise independent space and let ${\displaystyle G_{2}:\{0,1\}^{\ell }\to \{0,1\}^{h}}$ be a generator of an $\epsilon$-biased set over ${\displaystyle \{0,1\}^{h}}$. That is, when given a uniformly random input, the output of $G_{1}$ is a k-wise independent space, and the output of $G_{2}$ is $\epsilon$-biased. Then ${\displaystyle G:\{0,1\}^{\ell }\to \{0,1\}^{n}}$ with ${\displaystyle G(x)=G_{1}(G_{2}(x))}$ is a generator of an $\delta$-almost $k$-wise independent space, where ${\displaystyle \delta =2^{k/2}\epsilon }$.^[3]

As mentioned above, Alon, Babai & Itai (1986) construct a generator $G_{1}$ with ${\displaystyle h={\tfrac {k}{2}}\log n}$, and Naor & Naor (1990) construct a generator $G_{2}$ with ${\displaystyle \ell =\log s=\log h+O(\log(\epsilon ^{-1}))}$. Hence, the concatenation $G$ of $G_{1}$ and $G_{2}$ has seed length ${\displaystyle \ell =\log k+\log \log n+O(\log(\epsilon ^{-1}))}$. In order for $G$ to yield a $\delta$-almost k-wise independent space, we need to set ${\displaystyle \epsilon =\delta 2^{-k/2}}$, which leads to a seed length of ${\displaystyle \ell =\log \log n+O(k+\log(\delta ^{-1}))}$ and a sample space of total size ${\displaystyle 2^{\ell }\leq \log n\cdot {\text{poly}}(2^{k}\cdot \delta ^{-1})}$.

Notes

1. cf., e.g., Goldreich (2001)
2. cf., e.g., p. 2 of Ben-Aroya & Ta-Shma (2009)
3. Section 4 in Naor & Naor (1990)

References

• Alon, Noga; Babai, László; Itai, Alon (1986), "A fast and simple randomized parallel algorithm for the maximal independent set problem" (PDF), Journal of Algorithms, 7 (4): 567–583, doi:10.1016/0196-6774(86)90019-2
• Alon, Noga; Goldreich, Oded; Håstad, Johan; Peralta, René (1992), "Simple Constructions of Almost k-wise Independent Random Variables" (PDF), Random Structures & Algorithms, 3 (3): 289–304, doi:10.1002/rsa.3240030308
• Ben-Aroya, Avraham; Ta-Shma, Amnon (2009), "Constructing Small-Bias Sets from Algebraic-Geometric Codes" (PDF), Proceedings of the 50th Annual Symposium on Foundations of Computer Science, FOCS 2009: 191–197, doi:10.1109/FOCS.2009.44, ISBN 978-1-4244-5116-6
• Chor, Benny; Goldreich, Oded; Håstad, Johan; Freidmann, Joel; Rudich, Steven; Smolensky, Roman (1985), "The bit extraction problem or t-resilient functions", Proceedings of the 26th Annual Symposium on Foundations of Computer Science, FOCS 1985: 396–407, doi:10.1109/SFCS.1985.55, ISBN 0-8186-0644-4
• Goldreich, Oded (2001), Lecture 7: Small bias sample spaces
• Joffe, Anatole (1974), "On a Set of Almost Deterministic k-Independent Random Variables", Annals of Probability, 2 (1): 161–162, doi:10.1214/aop/1176996762
• Naor, Joseph; Naor, Moni (1990), "Small-bias Probability Spaces: efficient constructions and Applications", Proceedings of the 22nd annual ACM symposium on Theory of computing, STOC 1990: 213–223, doi:10.1145/100216.100244, ISBN 0897913612