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Sub-Gaussian score bound under fixed-design incoherence

Statement

Let $Y = X θ^{*} + ε$ be as in Fixed-design regression with sub-Gaussian noise, with $n \geq 1$ , $d \geq 1$ , noise proxy $σ^{2} \geq 0$ , and $X$ satisfying the incoherence condition of Fixed-design incoherence condition at some $1 \leq s \leq d$ . For $δ \in (0, 1)$ define $τ = 4 σ (\sqrt{\frac{\log (2 d)}{n}} + \sqrt{\frac{\log (1 / δ)}{n}}) .$ Then $P ⁣ ({∥ X^{T} ε / n ∥}_{\infty} \leq τ / 2) \geq 1 - δ .$

Facts & Assumptions

Given: The fixed-design sub-Gaussian model, $n \geq 1$ , $d \geq 1$ , $1 \leq s \leq d$ , $δ \in (0, 1)$ , and incoherence at sparsity $s$ .

[F1]

Every linear functional $⟨ u, ε ⟩$ is sub-Gaussian with proxy $σ^{2} ∥ u ∥_{2}^{2}$ (Fixed-design regression with sub-Gaussian noise).

[F2]

A maximum of finitely many centered sub-Gaussian variables with common proxy $v > 0$ obeys $P (\max_{j} ∣ Z_{j} ∣ \geq t) \leq 2 m e^{- t^{2} / (2 v)}$ for $t > 0$ , without an independence assumption; for proxy $v = 0$ , this probability is zero (Finite sub-Gaussian maxima and epsilon-net reduction).

[F3]

Incoherence gives $X_{j}^{T} X_{j} / n \leq 1 + 1 / (14 s) \leq 15 / 14$ (Fixed-design incoherence condition).

[F4]

Probability is countably subadditive (Basic identities for a probability measure).

Proof

1.1F1F3algebra

For each $j$ , $X_{j}^{T} ε / n$ is a centered sub-Gaussian variable with proxy $σ^{2} ∥ X_{j} ∥_{2}^{2} / n^{2} \leq 15 σ^{2} / (14 n)$ by [F1] and [F3]. The $d$ variables need not be independent.

2.1F2F4step 1.1algebra∎

If $σ = 0$ , then for every positive integer $m$ , [F2] gives $P (∥ X^{T} ε / n ∥_{\infty} \geq 1 / m) = 0$ . Since the event that this finite maximum is positive is the countable union of these events, [F4] shows that the score is zero almost surely, and the conclusion follows. Suppose $σ > 0$ . Apply [F2] at $t = τ / 2$ to obtain $P ⁣ ({∥ X^{T} ε / n ∥}_{\infty} > τ / 2) \leq 2 d \exp ⁣ (- \frac{28}{15} {[\sqrt{\log (2 d)} + \sqrt{\log (1 / δ)}]}^{2}) \leq δ,$ because the squared bracket is at least $\log (2 d / δ)$ and $28 / 15 > 1$ . This proves the claim.

Depends on

Used by

Lasso prediction and coefficient bounds under incoherence Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Philippe Rigollet, 18.S997 High-Dimensional Statistics, Theorem 2.18 proof (standard reference, not scraped)
Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Theorem 2.18 proof (standard reference, not scraped)
Peter J. Bickel, Ya'acov Ritov and Alexandre B. Tsybakov, Simultaneous Analysis of Lasso and Dantzig Selector, Theorem 7.2 (standard reference, not scraped)