Square Root Saving Error Bounds

• Square root saving error bounds are a framework that improves error rates from linear O(nu) to square-root O(√n u) performance using statistical independence and normalization.
• They are applied in settings such as sparse regression, stochastic rounding in numerical analysis, and lattice reduction in computational geometry to yield robust, adaptive guarantees.
• These bounds facilitate optimal convergence, σ-free tuning, and enhanced algorithmic stability in high-dimensional statistics, numerical integrations, and privacy-preserving matrix factorizations.

Square root saving error bounds describe a range of phenomena across numerical analysis, optimization, statistics, probability, and algorithmic geometry, where error rates, convergence, or upper bounds improve from a linear dependence (e.g., O(nu)) to a rate involving the square root (e.g., O(√n u)), or where the necessity of certain parameter dependencies such as variance estimation is eliminated by a square-root normalization. This "square root saving" leverages statistical independence, special normalization, or optimal metric properties to yield sharper or adaptive performance guarantees. The following sections synthesize the key technical settings, mechanisms, and implications for square root saving error bounds, as expounded in the cited research literature.

1. Statistical Estimation: Square-Root Lasso, Slope, and Extensions

The square-root Lasso and square-root Slope estimators are pivotal instances wherein the square-root loss brings marked benefits in sparse linear regression. Both are formulated to be independent of the unknown noise variance σ, conferring "pivotality":

• Square-root Lasso:

$$\widehat{\beta}^{\mathrm{SQL}} \in \arg\min_{\beta \in \mathbb{R}^p} \left\{ (1/\sqrt{n})\|Y - X\beta\|_2 + \lambda \|\beta\|_1 \right\}$$

• Square-root Slope:

$$\widehat{\beta}^{\mathrm{SQS}} \in \arg\min_{\beta \in \mathbb{R}^p} \left\{ (1/\sqrt{n})\|Y - X\beta\|_2 + \|\beta\|_* \right\}$$

where $\|\beta\|_* = \sum_{j=1}^p \lambda_j |\beta|_{(j)}$ (the ordered $\ell_1$-norm).

Error bounds and adaptivity: Both methods achieve the minimax optimal prediction and estimation rates up to constants, with non-asymptotic (finite-sample, high-probability) guarantees. The key "square root saving" here is:

• The optimal prediction rate for sparse regression:

$(\frac{s}{n}) \log\left(\frac{p}{s}\right)$

is achieved with error bounds that do not require prior knowledge of σ. Further, square-root Slope is additionally adaptive to unknown sparsity $s$.

Model generalization: The graph-based square-root estimation (GSRE) (Li et al., 19 Nov 2024) generalizes this principle, incorporating a square-root loss with a structured node-wise regularizer leveraging predictor graphs. The resulting bounds retain "σ-free" tuning, with error rates of the form:

• Prediction error:

$$\frac{\|X(\widehat{\beta} - \beta^*)\|_2}{\sqrt{n}} \lesssim \frac{\sigma \lambda \sqrt{s^*}}{n \kappa}$$

under compatibility conditions; again, no knowledge of σ needed.

Implications: The square-root normalization decouples λ from σ, allowing robust error control and adaptation with respect to noise and sparsity—this is central to the "saving" effect on error bounds in high-dimensional statistics (Derumigny, 2017, Li et al., 19 Nov 2024).

2. Deterministic and Probabilistic Numerical Error: Stochastic Rounding and Matrix Computations

Stochastic Rounding (SR):

SR is an unbiased rounding mode in floating-point arithmetic that, under the martingale property, yields error bounds with "square root saving":

• Instead of the standard deterministic worst-case $O(nu)$ error (u = unit roundoff), SR gives a probabilistic bound of $O(\sqrt{n} u)$ for forward error in linear algebraic kernels such as inner products and polynomial evaluation (Arar et al., 2022):

$$|\hat{y} - y| / |y| \leq \mathcal{K}_1 u \sqrt{n \ln(2n/\delta)}$$

(with high probability 1–δ), leveraging variance propagation and concentration inequalities.

Bienaymé-Chebyshev approach:

SR error variance $V(\hat{y})$ satisfies $V(\hat{y}) \leq y^2 \mathcal{K}_1^2 n u^2$, leading to a probabilistic guarantee:

$$|\hat{y} - y|/|y| \leq \mathcal{K}_1 u \sqrt{n} / \sqrt{\lambda}$$

which is often tighter than martingale-based bounds for practical λ (Arar et al., 2022). This order reduction—from O(nu) to O(√n u)—is the "square root saving effect" for error accumulation.

Matrix Square Roots via Krylov Methods:

A priori error bounds for approximating $f(M)\mathbf{b}$ (here $f(x) = x^{1/2}$) using the Arnoldi process translate residual control in linear system solves into error bounds for matrix square roots (Adler et al., 27 Jun 2025):

$$\|M^{1/2}\mathbf{b} - \mathrm{Arn}_k(x^{1/2}; M, \mathbf{b})\| / \|M^{1/2}\mathbf{b}\| \leq \frac{4\sqrt{2}\,\kappa(M)^{7/2}}{(k - 1/2)^{3/4}}$$

(for a $k$-step Arnoldi approximation in the non-Hermitian case), indicating sublinear decay in error with increasing $k$, i.e., a "saving" due to iterative methods. For large positive definite matrices, data-sparse approximations combined with this bound ensure feasible computation with reliable error control.

3. Combinatorial and Computational Geometry: Lattice Reduction for Sums of Square Roots

In computational geometry, problems such as distance comparison often reduce to evaluating the minimum positive value:

$$R(n,k) = \min_{e_i, s_i, t} |e_1\sqrt{s_1} + \cdots + e_k\sqrt{s_k} - t|$$

where $e_i \in \{-1,0,1\}$, $s_i \leq n$, $t \in \mathbb{Z}$ (0905.4487). Establishing tight lower bounds for $R(n,k)$ enables robust geometric computation.

Lattice reduction approach:

By associating $R(n,k)$ with the length of the shortest vector in a specially constructed integral lattice $L(s_1,...,s_k;N)$ and applying BKZ lattice reduction:

• The method achieves lower bounds exponentially better than root separation techniques, particularly when $k$ is large.
• There is a conjecture (Conjecture 1) asserting that the shortest nonzero vector has length $>N^{1-1/k}/k$, which, if proved, improves the lower bound to $R(o(k),k)\geq 1/(20(k)k^3)^k$, and implies polynomial-time algorithms for comparing such sums.

Constructive upper bounds:

For $n \ll 2^{2k}$, constructive bounds are obtained by directly constructing short lattice vectors, providing explicit coefficients attaining the error bounds.

Significance:

These results directly impact the design of geometric algorithms, as they facilitate robust radical comparison; algorithmic advances here stem from the "saving" embedded in the connection to lattice geometry.

4. Optimal Rates in Numerical Integration and Compressed Sensing

Quasi-Monte Carlo integration:

Convergence rates for the root mean square error (RMSE) of Monte Carlo and randomized QMC algorithms exhibit a hierarchy:

• Standard MC: RMSE $= O(N^{-1/2})$.
• Standard RQMC (Owen's scrambling): $O(N^{-3/2+\varepsilon})$ under bounded variation.
• Higher order RQMC: $O(N^{-\alpha-1/2+\varepsilon})$ when the integrand has square-integrable mixed partial derivatives up to order $\alpha>1$ (Dick, 2010), matching known lower bounds and delivering "square root saving" over MC as smoothness increases.

Compressed sensing under Poisson noise:

Standard compressed sensing error analyses fail under Poisson noise due to signal-dependent variance and the non-metric property of the generalized Kullback–Leibler divergence.

• By replacing the negative log-likelihood fidelity term with the square root of the Jensen–Shannon divergence (SQJSD), which is a metric, one can obtain relative error bounds of the form (Patil et al., 2016):

$$\|\theta - \theta^*\|_2/I \leq \mathcal{O}(N/\sqrt{I}) + \mathcal{O}(\|\theta-\theta_s\|_1/(I\sqrt{s}))$$

• This formulation enables statistically motivated, parameter-free thresholding and principled error control, leveraging the square-root metric property for a "saving" over less direct approaches.

5. Bounds in Quadrature and Analytic Function Approximation

Worst-case error in quadrature:

Lower bounds on the integration error over Hardy spaces for quadrature rules reveal that while many rules (e.g., trapezoidal, scaled Gauss–Legendre, Clenshaw–Curtis) attain nearly exponential decay $O(e^{-c n^{\alpha}})$ with $\alpha > 1/2$, the Gauss–Hermite quadrature is provably suboptimal—its error decays as $O(e^{-c\sqrt{n}}/n^2)$ (Goda et al., 14 Jan 2024). This is a square root loss in the exponent:

• The error decay rate loses a "square root" in $n$ compared to optimal rules.
• The sharpness of these lower bounds demonstrates that, apart from a polynomial factor, the upper bounds for the best rules are unimprovable.

Implication:

The suboptimality of certain classical quadrature rules under analytic decay conditions becomes clear only via these square-root-sensitive lower bounds, guiding both theoretical understanding and practical rule selection.

6. Matrix Factorizations for Differential Privacy

Multi-epoch matrix factorizations:

In differentially private training, matrix factorization mechanisms must minimize the error due to privacy noise. The Banded Inverse Square Root (BISR) method for multi-epoch matrix factorization achieves tight (asymptotically optimal) error bounds (Kalinin et al., 17 May 2025).

• The error bound is:

$$O(\sqrt{k}\log n + k) \text{ (for } \alpha = 1),\qquad O(\sqrt{k}) \text{ (for } \alpha < 1)$$

where $k$ is the participation count. This matches the corresponding lower bounds, signifying that no competing method can do better up to constants.

Key insight:

Imposing a banded structure on the inverse correlation matrix (rather than the square root itself) and utilizing explicit formulas allows explicit, easy-to-analyze error guarantees with square-root scaling in $k$.

7. Algorithmic Robustness and Kalman Filtering

Square root unscented Kalman filtering for graph signals:

The robust square root unscented Kalman filter (SRUKF) combines double square root updates for covariance matrices with robust M-estimation loss functions, improving both numerical stability and resistance to non-Gaussian noise (Hu et al., 11 Sep 2024):

• The error covariance update uses double square root decomposition to maintain positive definiteness.
• The convergence of the mean square error is established via Lyapunov-type arguments, with the error evolution equation:

$$\delta_i = (I-K H_i)F \delta_{i-1}F^T(I-K H_i)^T + (I-K H_i)Q(I-K H_i)^T - K R K^T$$

leading to steady-state boundedness formalized via a discrete Lyapunov equation.

• Robust cost functions (Huber, Cauchy, general M-estimation) further suppress the impact of outliers.

Empirical validation:

Simulations under various noise regimes confirm significant improvements in RMSE, robustness, and stability relative to standard UKF variants, owing directly to the square root updating and robust error handling.

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Collectively, the square root saving error bounds documented herein constitute a pervasive structural phenomenon across disciplines—statistical estimation, numerical analysis, computational geometry, and privacy-preserving algorithms—enabling practitioners to attain provably tighter, more adaptive, and robust performance guarantees by exploiting square-root normalization, statistical properties, or specialized problem structure.