LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Separate isotropic ridgeless risk into limiting bias and variance

Statement

Under the hypotheses and notation of the isotropic ridgeless risk theorem, put $S_{n} = X_{n}^{T} X_{n} / n$ , $B_{n} : = ∥ (I_{p_{n}} - X_{n}^{+} X_{n}) β_{n} ∥_{2}^{2}, V_{n} : = \frac{σ^{2}}{n} tr (S_{n}^{+}) .$ Then almost surely, if $0 < γ < 1$ , $B_{n} ⟶ 0, V_{n} ⟶ \frac{σ^{2} γ}{1 - γ},$ and if $γ > 1$ , $B_{n} ⟶ r^{2} (1 - \frac{1}{γ}), V_{n} ⟶ \frac{σ^{2}}{γ - 1} .$

Statement references

Exact bias and noise decomposition of isotropic prediction risk

Isotropic ridgeless risk limit when the signal is zero

Isotropic ridgeless least-squares risk on both sides of interpolation

Facts & Assumptions

Given: The model, parameters, and risk notation in the Statement.

[F1]

The exact conditional risk is $R_{n} = B_{n} + V_{n}$ (Exact bias and noise decomposition of isotropic prediction risk).

[F2]

For zero signal, the risk equals $V_{n}$ and has the limits stated in the zero-signal lemma (Isotropic ridgeless risk limit when the signal is zero).

[F3]

The total risk has the two limits in the isotropic ridgeless risk theorem (Isotropic ridgeless least-squares risk on both sides of interpolation).

Proof

1.1F1F2algebra

By [F1], for every $n$ the risk decomposes exactly as $R_{n} = B_{n} + V_{n}$ . For the zero signal the bias term is zero, so the zero-signal risk is exactly this same design-dependent $V_{n}$ ; [F2] therefore gives the almost-sure limits of $V_{n}$ .

2.1F2F3step 1.1algebra∎

If $γ < 1$ , [F3] and [F2] give the same limit for $R_{n}$ and $V_{n}$ , so their difference $B_{n}$ tends to zero. If $γ > 1$ , subtracting $σ^{2} / (γ - 1)$ , the limit of $V_{n}$ in [F2], from the total-risk limit in [F3] leaves $r^{2} (1 - 1 / γ)$ for $B_{n}$ . The probability-one events in [F2] and [F3] may be intersected.

Depends on

Used by

Bias, variance, and signal-to-noise regimes of the isotropic risk curve Example

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Sources

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