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Isotropic ridgeless risk limit when the signal is zero

Statement

Assume the Axiom of Choice. Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite iid array of real random variables with mean zero, variance one, and finite fourth moment. Let $p_{n} \geq 1$ with $p_{n} / n \to γ \in (0, \infty) ∖ {1}$ , and let $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq p_{n}}$ be its upper-left block. Let $ε_{n}$ have independent centered components of variance $σ^{2}$ , independent of $X_{n}$ , and let $z_{0, n}$ be independent of $(X_{n}, ε_{n})$ , centered, and have covariance $I_{p_{n}}$ . Set $S_{n} = X_{n}^{T} X_{n} / n$ , $y_{n} = ε_{n}$ , ${\hat{β}}_{n} = X_{n}^{+} y_{n}$ , and $R_{n} : = E_{ε_{n}, z_{0, n}} [(z_{0, n}^{T} {\hat{β}}_{n})^{2} ∣ X_{n}] .$ Then almost surely $R_{n} ⟶ {\begin{cases} \frac{σ^{2} γ}{1 - γ}, & 0 < γ < 1, \\ \frac{σ^{2}}{γ - 1}, & γ > 1. \end{cases}$

Statement references

Exact bias and noise decomposition of isotropic prediction risk

Isotropic minimum-norm risk limits under a positive spectral gap

Facts & Assumptions

Given: The nested real iid design, positive aspect ratio $γ \neq 1$ , independent centered noise with covariance $σ^{2} I_{n}$ , centered test feature with covariance $I_{p_{n}}$ , and the Axiom of Choice from the Statement.

[A1]

The Axiom of Choice permits a choice function for every family of nonempty sets (The Axiom of Choice).

[F1]

For the nested real iid finite-fourth-moment design and every deterministic signal sequence of fixed norm $r > 0$ , the design-conditional risk tends almost surely to $σ^{2} γ / (1 - γ)$ for $γ < 1$ and to $r^{2} (1 - 1 / γ) + σ^{2} / (γ - 1)$ for $γ > 1$ (Isotropic minimum-norm risk limits under a positive spectral gap).

[F2]

The conditional risk for a deterministic signal is its squared projection onto $\ker X_{n}$ plus the signal-independent noise term (Exact bias and noise decomposition of isotropic prediction risk).

[F3]

Iid entries are independent and share one common distribution (Identical distribution and IID families).

Proof

1.1givenchoose

For each $n$ , let $e_{1, n}$ be the first standard basis vector in $R^{p_{n}}$ . For $a \in {1, 2}$ , let $R_{n}^{(a)}$ denote the conditional risk with signal $β_{n}^{(a)} = a e_{1, n}$ , and put $B_{n} = e_{1, n}^{T} (I - X_{n}^{+} X_{n}) e_{1, n}$ and $V_{n} = (σ^{2} / n) tr (S_{n}^{+})$ .

2.1A1F1F2F3step 1.1algebra

Applying [F2] to each auxiliary signal gives $R_{n}^{(1)} = B_{n} + V_{n}$ and $R_{n}^{(2)} = 4 B_{n} + V_{n}$ . Applying [F1] with norms $1$ and $2$ gives almost-sure limits for both risks; the intersection of these two probability-one events still has probability one. The iid law is common in both applications by [F3].

3.1F1step 2.1algebra

The exact identities in step 2.1 give $V_{n} = (4 R_{n}^{(1)} - R_{n}^{(2)}) / 3$ . If $γ < 1$ , both risk limits in [F1] equal $σ^{2} γ / (1 - γ)$ , so $V_{n}$ tends to that quantity. If $γ > 1$ , their limits are respectively $1 - 1 / γ + σ^{2} / (γ - 1)$ and $4 (1 - 1 / γ) + σ^{2} / (γ - 1)$ ; their indicated combination is $σ^{2} / (γ - 1)$ .

4.1givenF2step 3.1∎

For the zero signal in the Statement, [F2] identifies $R_{n}$ with the same $V_{n}$ . Step 3.1 therefore gives both asserted limits.

Depends on

Used by

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Sources

Hastie, Montanari, Rosset, and Tibshirani, Surprises in High-Dimensional Ridgeless Least Squares Interpolation (standard reference, not scraped)