A tail inequality for quadratic forms of subgaussian random vectors (1110.2842v1)

Abstract: We prove an exponential probability tail inequality for positive semidefinite quadratic forms in a subgaussian random vector. The bound is analogous to one that holds when the vector has independent Gaussian entries.

• The paper provides an exponential tail inequality for quadratic forms in subgaussian random vectors.
• It leverages moment generating functions and martingale techniques to extend classical Gaussian tail bounds.
• The findings enhance high-dimensional statistical analysis and fixed-design regression by modeling realistic subgaussian noise.

A Tail Inequality for Quadratic Forms of Subgaussian Random Vectors

This paper presents a significant contribution to the field of probability theory by offering an exponential probability tail inequality for quadratic forms in subgaussian random vectors. It extends a well-known result for Gaussian vectors to a broader class of random vectors, those exhibiting subgaussian behavior.

The authors Daniel Hsu, Sham M. Kakade, and Tong Zhang focus on the quadratic form $\|Ax\|^2 = x^\top (A^\top A) x$, where $A$ is a fixed matrix and $x$ is a subgaussian random vector. This paper considers the deviation of this form from its expected value. This problem is analogous to the Gaussian case, which is characterized by Proposition 1 in the paper—a tail inequality for quadratic forms with standard Gaussian entries.

Paper Synopsis

The main theorem (Theorem 1) provides a sharp upper tail bound for quadratic forms where the entries of the random vector $x$ are not necessarily independent but exhibit subgaussian behavior. Specifically, the paper shows:

• A precise analog of the Gaussian case's tail inequality is achievable for subgaussian vectors when $\sigma = 1$.
• Under the assumption that the random vector $x$ follows a subgaussian random vector distribution, the authors derive an explicit exponential bound on the probability that the quadratic form substantially exceeds its mean.

The results rely on technical expansions of existing inequalities, including leveraging the properties of subgaussian vectors and extending martingale techniques. One method employed is the characterization of the moment generating function of quadratic forms in Gaussian random variables, using the properties of subgaussian distributions to obtain analogous bounds.

Results and Implications

Quantitatively, the paper's bound on $\Pr\left[\|Ax\|^2 > \text{threshold}\right]$ matches the existing bounds when $x$ consists of standard Gaussian variables. The theorem also extends known martingale bounds, such as the vector Bernstein inequality, to a more general setting, providing a sharper estimate.

Practically, these findings are crucial for high-dimensional statistics where subgaussian assumptions provide a more realistic model of data compared to purely Gaussian assumptions. Applications span areas like fixed-design regression, where the analysis of subgaussian noise can lead to more robust estimates and predictions.

Theoretically, these results enhance the understanding of concentration inequalities and provide a tool for analyzing the concentration of random quadratic forms. It paves the way for future research in deriving similar inequalities for more generalized forms of random vectors beyond the subgaussian category.

Concluding Remarks and Future Work

While this paper delivers a comprehensive result for subgaussian vectors, it suggests several avenues for further research. Extending these techniques to handle even broader classes of distributions, such as heavy-tailed or anisotropic distributions, could significantly impact adjacent fields. Furthermore, practical experiments validating the theoretical guarantees in various statistical or machine learning contexts would be a valuable extension.

In conclusion, this work makes a critical step in expanding classical inequality results to encompass more complex and realistic data settings, enhancing both theoretical and practical tools available for statistical analysis and machine learning applications.