A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm

Introduction

Concentration (large deviation) inequalities are one of the most important subjects of study in probability theory. A class of distributions for which sharp concentration inequalities have been developed is the class of subGaussian distributions.

A random variable $X\in R$ is subGaussian, if there exists $\sigma\in R$ so that:

A random vector $\mathbf{X}\in R^{d}$ is subGaussian, if there exists $\sigma\in R$ so that:

The concentration bounds of subGaussian random vectors/variables depends on the parameter $\sigma$ – smaller the $\sigma$ better the concentration bounds. While subGaussian distributions arise naturally in several applications, there are settings where the random vectors have nice concentration properties but the subGaussian parameter $\sigma$ is very large (so that applying concentration bounds for general subGaussian random vectors gives loose bounds). In this short note, we consider a related but different class of distributions, called norm-subGaussian random vectors and establish tighter concentration bounds for them.

Organization: In Section 2, we introduce norm subGaussian random vectors and some of their properties and we prove our main results in Section 3. We conclude in Section 4.

Norm SubGaussian Random Vector

The norm subGaussian random vector is defined as follows.

Norm subGaussian includes both subGaussian (with a smaller $\sigma$ parameter) and bounded norm random vectors as special cases.

There exists absolute constant $c$ so that following random vectors are all $\text{nSG}(c\cdot\sigma)$.

Then let $\mathbf{v}(\mathbf{X})=\mathbf{X}/\|{\mathbf{X}}\|$, since $\{\mathbf{v}_{i}\}$ is a $1/2$-cover, there always exists a $j(\mathbf{X})$ so that $\mathbf{v}_{j(\mathbf{X})}$ in cover and $\|{\mathbf{v}(\mathbf{X})-\mathbf{v}_{j(\mathbf{X})}}\|\leq 1/2$. Therefore, we have:

Now we are ready to check the second claim of Lemma 1, when $t^{2}\leq 8\sigma^{2}\ln 4$, we have,

when $t^{2}>8\sigma^{2}\ln 4$, we let $t^{2}=8\sigma^{2}\ln 4+s$ where $s>0$, then:

In sum, this proves that $\mathbf{X}$ is $\text{nSG}(2\sqrt{2}\cdot\sigma)$. ∎

The following lemma gives equivalent characterizations of norm subGaussian in terms of moments and moment generating function (MGF).

Note $\|{\mathbf{X}}\|$ is a 1-dimensional random variable. This lemma directly follows from the equivalent properties of $1$-dimensional subGaussian, for instance, Lemma 5.5 in (Vershynin, 2010). ∎

The following lemma says that if a random vector is $\text{nSG}(\sigma)$, then its norm squared is subexponential and its projection on any direction a is subGaussian random variable.

The undesirable thing about the MGF characterization in Lemma 2 is that even if $\mathbf{X}$ is a zero mean random vector, $\|{\mathbf{X}}\|$ is not zero mean, so it is difficult to directly work with MGF of $\|{\mathbf{X}}\|$. Instead, we first convert the random vector $\mathbf{X}$ to a matrix $\mathbf{Y}$ and characterize the MGF of $\mathbf{Y}$.

There is an absolute constant $c$, if random vector $\mathbf{X}\in R^{d}$ is zero-mean $\text{nSG}(\sigma)$, then let

where in the last inequality we used the fact that $\frac{p^{p}}{(2p)!}\leq\frac{1}{p!}$, this finishes the proof. ∎

Vector Martingales with SubGaussian Norm

In this section, we will prove our main result (Lemma 6, Corollaries 7 and 8) giving concentration bounds for norm subGaussian random vectors. The main tool we use is Lieb’s concavity theorem.

Let $\mathbf{A}$ be a fixed symmetric matrix, and let $\mathbf{Y}$ be a random symmetric matrix. Then,

We will prove our concentration result for norm subGaussian random vectors in a general setting where the subGaussian parameter $\sigma_{i}$ for the $i^{\textrm{th}}$ vector can itself be a random variable.

where step (1) is due to Theorem 5, and step (2) used the fact that if matrix $\mathbf{A}\preceq\mathbf{B}$, then $e^{\mathbf{C}+\mathbf{A}}\preceq e^{\mathbf{C}+\mathbf{B}}$. On the other hand, since identity matrix commutes with any matrix, we know:

Therefore, for any $t\geq 0$, $\theta\geq 0$, by Markov’s inequality, we have:

where step (1) is because $\sum_{i=1}^{n}\mathbf{Y}_{i}$ is a rank-2 matrix whose eigenvalues are $\|{\sum_{i=1}^{n}\mathbf{X}_{i}}\|,-\|{\sum_{i=1}^{n}\mathbf{X}_{i}}\|$; step (2) is due to all preconditions are symmetric with respect to 0. Finally, setting RHS equal to $\delta$, we finish the proof. ∎

Since now $\{\sigma_{i}\}$ are fixed which are not random, we can pick $\theta$ in Lemma 6 as a function of $\{\sigma_{i}\}$. Indeed, pick $\theta=\sqrt{\frac{1}{\sum_{i=1}^{n}\sigma_{i}^{2}}\log\frac{2d}{\delta}}$ finishes the proof. ∎

For simplicity, denote log factor $\iota\mathrel{\mathop{:}}=\log\frac{2d}{\delta}+\log\log\frac{B}{b}$ By Lemma 6, we know for any fixed $\theta$, with probability $1-\delta\cdot\log^{-1}(B/b)$, we have:

Construct two sets of $\Psi=\{\psi_{1},\ldots,\psi_{s}\}$ and $\Theta=\{\theta_{1},\ldots,\theta_{s}\}$, where $\psi_{j}=2^{j-1}\cdot b$ and $\theta_{j}=\sqrt{\frac{\iota}{\psi_{j}}}$ with last element $\psi_{s}\leq B$, $2\psi_{s}>B$. It is easy to see $|\Psi|=|\Theta|\leq\log(B/b)$. By union bound, we have with probability $1-\delta$:

Consider following two cases: (1) $\sum_{i=1}^{n}\sigma_{i}^{2}\in[b,B]$. Then, there exists $j\in[s]$ such that $\psi_{j}\leq\sum_{i=1}^{n}\sigma_{i}^{2}<2\psi_{j}$:

(2) $\sum_{i=1}^{n}\sigma_{i}^{2}\in[0,b)$. In this case we know $\psi_{1}=b$ and:

Conclusion

In this short note, we introduced the notion of norm subGaussian random vectors, which include subGaussian random vectors and bounded random vectors as special cases. While it is true that $\textrm{subGaussian}\left(\frac{\sigma}{\sqrt{d}}\right)\subseteq\text{nSG}(\sigma)\subseteq\textrm{subGaussian}(\sigma)$, applying concentration bounds for $\textrm{subGaussian}(\sigma)$ would yield bounds which have at least linear dependence on $d$. In contrast, the bounds we develop (in Lemma 6 and Corollaries 7 and 8) have only logarithmic dependence on $d$. It is not clear if this logarithmic dependence is tight – totally eliminating this dependence is an interesting open problem.

Acknowledgements

We thank Gabor Lugosi and Nilesh Tripuraneni for helpful discussions.

References