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Isotropic ridgeless least-squares risk on both sides of interpolation

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 $β_{n} \in R^{p_{n}}$ be deterministic with $∥ β_{n} ∥_{2} = r$ for a fixed $r \geq 0$ . 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 $y_{n} = X_{n} β_{n} + ε_{n}$ , ${\hat{β}}_{n} = X_{n}^{+} y_{n}$ , and define the design-conditional prediction risk $R_{n} : = E_{ε_{n}, z_{0, n}} [(z_{0, n}^{T} ({\hat{β}}_{n} - β_{n}))^{2} ∣ X_{n}] .$ Then almost surely $R_{n} ⟶ {\begin{cases} \frac{σ^{2} γ}{1 - γ}, & 0 < γ < 1, \\ r^{2} (1 - \frac{1}{γ}) + \frac{σ^{2}}{γ - 1}, & γ > 1. \end{cases}$ This result gives no limit assertion for sequences with $p_{n} / n \to 1$ .

Statement references

Ridgeless minimum-norm least-squares regression

Isotropic ridgeless risk limit when the signal is zero

Isotropic minimum-norm risk limits under a positive spectral gap

Hastie, Montanari, Rosset, and Tibshirani, Theorem 1

Facts & Assumptions

Given: The nested real iid design, $p_{n} / n \to γ \neq 1$ , fixed signal norm $r \geq 0$ , independent centered noise with covariance $σ^{2} I_{n}$ , centered test features with covariance $I_{p_{n}}$ , and the Axiom of Choice.

[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 model with fixed signal norm $r > 0$ and the same noise and test-feature assumptions, the almost-sure risk limits are $σ^{2} γ / (1 - γ)$ for $γ < 1$ and $r^{2} (1 - 1 / γ) + σ^{2} / (γ - 1)$ for $γ > 1$ (Isotropic minimum-norm risk limits under a positive spectral gap).

[F2]

When $r = 0$ , the same model has almost-sure risk limits $σ^{2} γ / (1 - γ)$ for $γ < 1$ and $σ^{2} / (γ - 1)$ for $γ > 1$ (Isotropic ridgeless risk limit when the signal is zero).

[F3]

The estimator $X_{n}^{+} y_{n}$ is the ridgeless minimum-norm least-squares estimator (Ridgeless minimum-norm least-squares regression).

[F4]

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

Proof

1.1givenA1F1F4

If $r > 0$ , the design, signal, noise, and test-feature hypotheses in the Statement are exactly those of [F1]; [F4] fixes the iid convention and [A1] supplies its choice assumption, so the two limits apply.

1.2givenA1F2F4algebra

If $r = 0$ , then $β_{n} = 0$ for every $n$ , and [F2] gives the limits in the Statement under [A1] and the iid convention [F4].

2.1givenF3step 1.1step 1.2algebra∎

Since $r \geq 0$ , the cases in steps 1.1 and 1.2 exhaust the signal norms. The estimator and conditional risk are those defined in [F3], so the resulting limits are precisely the displayed formula.

Depends on

Used by

Separate isotropic ridgeless risk into limiting bias and variance Lemma

Dependency tree · two levels

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Sources

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