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Polynomial and exponential minimax estimation rates

Statement

Assume Countable Choice, $Q > 0$ , and $p, κ > 0$ . As $ϵ ↓ 0$ , the minimax RMS error over $Θ_{a} (Q)$ has the following order-sharp rates: if $s_{j} ≍ j^{- p}$ and $a_{j} ≍ j^{κ}$ , then $R_{*} (ϵ) ≍ ϵ^{κ / (κ + p + 1 / 2)}$ ; if $s_{j} ≍ e^{- p j}$ and $a_{j} ≍ j^{κ}$ , then $R_{*} (ϵ) ≍ [\log (1 / ϵ)]^{- κ}$ ; if $s_{j} ≍ j^{- p}$ and $a_{j} ≍ e^{κ j}$ , then $R_{*} (ϵ) ≍ ϵ [\log (1 / ϵ)]^{p + 1 / 2}$ ; and if $s_{j} ≍ e^{- p j}$ and $a_{j} ≍ e^{κ j}$ , then $R_{*} (ϵ) ≍ ϵ^{κ / (p + κ)}$ . Here $u_{j} ≍ v_{j}$ as $j \to \infty$ or $f (ϵ) ≍ g (ϵ)$ as $ϵ ↓ 0$ means the ratio is bounded above and below by positive constants on the indicated tail; the constants may depend on $Q, p, κ$ and the axis comparison constants.

Statement references

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, §3 and Table 1

Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, §§4.5–4.6

Facts & Assumptions

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited through the compact SVD coordinate model.

[F1]

Gaussian sequence inverse model and Weighted ellipsoid class for sequence inversion: The observation coordinates are $Z_{j} = θ_{j} + ϵ s_{j}^{- 1} ξ_{j}$ and the parameter set is $Θ_{a} (Q) = {θ \in ℓ^{2} : \sum_{j} a_{j}^{2} θ_{j}^{2} \leq Q^{2}}$ with positive nondecreasing weights.

[F2]

Exact worst-case risk of the truncated estimator: At cutoff $D \in N_{0}$ , the exact worst-case expected squared risk is $B_{D} = Q^{2} / a_{D + 1}^{2} + ϵ^{2} \sum_{j = 1}^{D} s_{j}^{- 2}$ .

[F3]

Cutoff risk and minimax RMS error over statistical ellipsoids: The minimax RMS error satisfies $R_{*} (ϵ) \leq (\inf_{D} B_{D})^{1 / 2} \leq 2.2 R_{*} (ϵ)$ .

[F4]

The natural logarithm as the inverse of the exponential function and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is the inverse of $\exp$ on positive reals, is increasing, and obeys $\log (x y) = \log x + \log y$ .

[F5]

Real powers for positive bases, with the zero-base positive-exponent convention, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, and Continuity and derivatives of positive-base real powers: Positive-base real powers obey the exponent laws and $x^{α}$ is increasing for $α > 0$ .

[F6]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges and The exponential function is strictly increasing: For $p > 0$ , the geometric sums of $e^{2 p j}$ are bounded above and below by positive constants times their final term.

[F7]

The exponential dominates every fixed nonnegative integer power at $+ \infty$ : $x / \exp (x) \to 0$ as $x \to + \infty$ .

Given: $Q > 0$ , $p, κ > 0$ , positive nonincreasing singular values $s_{j} \to 0$ , positive nondecreasing weights $a_{j}$ , and a positive noise level $ϵ$ tending to zero.

Proof

1.1F5F6algebra

For $q > 0$ and integer $D \geq 2$ , $\sum_{j = 1}^{D} j^{q} \leq D^{q + 1}$ . For a lower bound, the at least $D / 2$ indices with $D / 2 \leq j \leq D$ each satisfy $j^{q} \geq (D / 2)^{q}$ , so $\sum_{j = 1}^{D} j^{q} \geq (D / 2)^{q + 1}$ . Thus $\sum_{j \leq D} j^{q} ≍ D^{q + 1}$ . Also the finite geometric-sum identity gives $\sum_{j \leq D} e^{2 p j} ≍ e^{2 p D}$ for each fixed $p > 0$ .

2.1givenF1F2F5step 1.1algebra

Suppose $s_{j} ≍ j^{- p}$ and $a_{j} ≍ j^{κ}$ . With $H = ϵ^{- 2 / (2 κ + 2 p + 1)}$ and $D_{0} = ⌊ H ⌋$ , step 1.1 and [F2] show that both terms of $B_{D_{0}}$ are $O (H^{- 2 κ})$ . For any $D \leq D_{0}$ , monotonicity of $a_{j}$ gives $Q^{2} / a_{D + 1}^{2} \geq Q^{2} / a_{D_{0} + 1}^{2} ≍ H^{- 2 κ}$ ; for $D > D_{0}$ , the variance is at least $ϵ^{2} \sum_{j \leq D_{0}} j^{2 p} ≍ ϵ^{2} H^{2 p + 1} = H^{- 2 κ}$ . Therefore $\inf_{D} B_{D} ≍ H^{- 2 κ}$ and the cutoff RMS error is $≍ ϵ^{κ / (κ + p + 1 / 2)}$ .

2.2givenF1F2F4F5F6F7step 1.1algebra

Suppose $s_{j} ≍ e^{- p j}$ and $a_{j} ≍ j^{κ}$ . Put $L = \log (1 / ϵ)$ and $D_{0} = ⌊ (L - κ \log L) / p ⌋$ . By [F7], $\log L / L \to 0$ , so $D_{0} ≍ L$ . At $D_{0}$ , the squared bias and variance in [F2] are both $O (L^{- 2 κ})$ . For $D \leq D_{0}$ , the bias is at least $Q^{2} / a_{D_{0} + 1}^{2} ≍ L^{- 2 κ}$ ; for $D > D_{0}$ , the geometric variance is at least $ϵ^{2} e^{2 p D_{0}} ≍ e^{- 2 L} e^{2 L} L^{- 2 κ} = L^{- 2 κ}$ . Hence $\inf_{D} B_{D} ≍ L^{- 2 κ}$ and the cutoff RMS error is $≍ L^{- κ}$ .

2.3givenF1F2F4F5F7step 1.1algebra

Suppose $s_{j} ≍ j^{- p}$ and $a_{j} ≍ e^{κ j}$ . Put $L = \log (1 / ϵ)$ and $D_{0} = ⌊ (L - (p + \frac{1}{2}) \log L) / κ ⌋$ . Again [F7] gives $D_{0} ≍ L$ . At $D_{0}$ , the squared bias and variance are both $O (ϵ^{2} L^{2 p + 1})$ . For $D \leq D_{0}$ , monotonicity of $a_{j}$ gives bias at least $Q^{2} / a_{D_{0} + 1}^{2} ≍ e^{- 2 κ D_{0}} ≍ ϵ^{2} L^{2 p + 1}$ ; for $D > D_{0}$ , the polynomial variance is at least $ϵ^{2} \sum_{j \leq D_{0}} j^{2 p} ≍ ϵ^{2} L^{2 p + 1}$ . Thus $\inf_{D} B_{D} ≍ ϵ^{2} L^{2 p + 1}$ and the cutoff RMS error is $≍ ϵ L^{p + 1 / 2}$ .

2.4givenF1F2F4F5F6step 1.1algebra

Suppose $s_{j} ≍ e^{- p j}$ and $a_{j} ≍ e^{κ j}$ . Let $L = \log (1 / ϵ)$ and $D_{0} = ⌊ L / (p + κ) ⌋$ . Then $D_{0} \to \infty$ ; at this cutoff the squared bias and geometric variance are both $O (ϵ^{2 κ / (p + κ)})$ . For $D \leq D_{0}$ , monotonicity of $a_{j}$ gives bias at least $Q^{2} / a_{D_{0} + 1}^{2} ≍ e^{- 2 κ D_{0}} ≍ ϵ^{2 κ / (p + κ)}$ ; for $D > D_{0}$ , the variance is at least $ϵ^{2} e^{2 p D_{0}} ≍ ϵ^{2 κ / (p + κ)}$ . Hence $\inf_{D} B_{D} ≍ ϵ^{2 κ / (p + κ)}$ and the cutoff RMS error is $≍ ϵ^{κ / (p + κ)}$ .

3.1A1F3step 2.1step 2.2step 2.3step 2.4∎

Steps 2.1–2.4 establish the four rates for $(\inf_{D} B_{D})^{1 / 2}$ . By [A1], the compact singular-system coordinate model and its minimax comparison theorem are used under Countable Choice. By [F3], this quantity and the minimax RMS error differ by at most the fixed factor $2.2$ , so the same four order-sharp rates hold for $R_{*} (ϵ)$ .

Remarks

The four cases are indexed separately by singular-value decay and ellipsoid-axis growth. Ding–Mathé call polynomial operator singular-value decay moderately ill-posed and exponential decay severely ill-posed; this operator terminology is distinct from the statistical minimax-rate description. The comparison is order-sharp up to constants, not Pinsker sharp-constant optimality.

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Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, §3 and Table 1, PDF pp. 5–6 (standard reference, not scraped)
Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, 2011 draft, §4.5, Proposition 4.16, and §4.6, Theorem 4.17 proof, printed pp. 109–113 (standard reference, not scraped)