High-Probability Deviation Bounds

• High-Probability Deviation Bounds are probabilistic tools that precisely quantify how much an estimator deviates from its target with a user-specified confidence level.
• They rely on refined tail inequalities and robust estimation methods to achieve tight, non-asymptotic guarantees even under heavy-tailed or high-dimensional settings.
• Applications include finite-sample statistical procedures, optimization algorithms, and privacy-aware estimation, ensuring reliable inference across diverse scenarios.

High-probability deviation bounds precisely quantify the probability that a random variable or estimator deviates from its target (e.g., mean, minimizer, or model parameter) by a given amount, with probability directly controlled by a user-specified level. They underpin many advances in theoretical statistics, learning theory, optimization, and high-dimensional probability, offering guarantees that hold uniformly over the sample, rather than merely in expectation. The paper of such bounds encompasses refined tail expansions, robust estimation procedures, sharp non-asymptotic inequalities, and information-theoretic lower bounds, facilitating a nuanced understanding of estimator reliability and the fundamental difficulty of inference tasks.

1. Definitions and Conceptual Framework

Let $X_1, \ldots, X_n$ be independent (or weakly dependent) random variables, and let $\hat{\theta}_n$ denote an estimator for the target parameter $\theta$ (such as a mean or minimizer). A high-probability deviation bound asserts that

$$\mathbb{P}\big\{ |\hat{\theta}_n - \theta| \geq \eta \big\} \leq \delta$$

for prescribed confidence level $\delta \in (0,1)$. The function $$\eta = \eta(\delta, n, \text{distribution parameters})$$ quantifies the rate and sharpness of concentration.

This “quantile” (or tail) control is fundamentally distinct from risk bounds (which control expectations), as large deviations may dominate in heavy-tailed or robust settings. Recent conceptual shifts (Ma et al., 19 Jun 2024) formalize the minimax $(1-\delta)$-quantile: $$\mathcal{M}(\delta, \mathcal{P}_\Theta, L) \coloneqq \inf_{\hat{\theta}} \sup_{\theta \in \Theta,\, P \in \mathcal{P}_\theta} \mathrm{Quantile}(1-\delta; L(\hat{\theta}(X), \theta))$$ which encapsulates the minimal worst-case loss achievable with $1-\delta$ confidence.

2. Classical and Modern Inequalities

2.1. Exponential Tail Bounds for Sums

For bounded independent random variables $\xi_i$ (e.g., $\xi_i \leq 1$, mean-zero), the probability of large deviations can be captured sharply. The seminal results of (Fan et al., 2012) establish that, for $S_n = \sum_{i=1}^n \xi_i$ with variance $\sigma^2$,

$$\mathbb{P}(S_n > x\sigma) = \big(\Theta(x) + \theta \varepsilon_x\big) \inf_{\lambda\geq 0} \mathbb{E}\exp\{\lambda(S_n - x\sigma)\}$$

with

$\Theta(x) = (1 - \Phi(x))e^{x^2/2}$

explicit Mill's ratio, and error $\varepsilon_x = O(B/\sigma)$ for absolute bound $B$.

This expansion “completes” the classic Chernoff–Hoeffding exponential tilting by identifying the missing polynomial prefactor, ensuring Gaussian-type tails with explicit finite-sample corrections. In the i.i.d. regime or under weak moment conditions, the expansion is tight: $$\frac{\mathbb{P}(S_n \geq x\sigma)}{\Theta(x)\inf_{\lambda \geq 0}\mathbb{E}e^{\lambda(S_n - x\sigma)}} = 1 + o(1)$$ connecting directly to Cramér, Bahadur–Rao, and Sakhanenko large deviation theory (Fan et al., 2012).

2.2. Bernstein-Type and Sharp Unbounded Sums

With possibly unbounded summands but under Bernstein’s moment condition (i.e., $$\mathbb{E}|\xi_i|^k \leq \frac{1}{2}k!\varepsilon^{k-2}\mathbb{E}\xi_i^2$$), sharp bounds (Fan et al., 2012) yield: $$\mathbb{P}(S_n > x\sigma) \leq (1-\Phi(\tilde{x}))[1 + C_\delta (1+\tilde{x})(\varepsilon/\sigma)]$$ with

$$\tilde{x} = \frac{2x}{1 + \sqrt{1 + 2(1+\delta)x\varepsilon/\sigma}}$$

showing that both the exponential decay and multiplicative factor extend to broader settings, improving the classical Bennett and Hoeffding inequalities by including the correct Mill's ratio.

2.3. Large Deviations for Heavy-Tailed Sums

For i.i.d. variables with heavy tails $P(|X| > x) \sim L(x)x^{-\alpha}$, when $x$ grows faster than the CLT scale, (Vogel, 2022) shows

$$\mathbb{P}\Big( \sum_{i=1}^n X_i > x \Big) = n \mathbb{P}(X_1 > x) (1 + o(1))$$

with explicit control of the error, quantifying the “one-big-jump” principle for large deviation events. High-probability deviation bounds thus reflect the heavy-tail regime’s fundamental distinctness from the light-tailed (Gaussian-like) setting.

2.4. Minimax Quantile Lower Bounds

The high-probability minimax framework (Ma et al., 19 Jun 2024) “lifts” Le Cam and Fano methods to quantile bounds. For example, for robust mean estimation with covariance $\Sigma$: $$\mathcal{M}(\delta, \mathcal{P}_\Theta, \|\cdot\|_2^2) \asymp \frac{\operatorname{tr}(\Sigma)}{n} + \frac{\|\Sigma\|_{\mathrm{op}} \log(1/\delta)}{n}$$ which shows that deviation control necessitates a $\log(1/\delta)/n$ additive price, sharp for all $\delta \in (0,1)$. Similar results hold for high-dimensional regression, density estimation, and more, universally demonstrating an additive (or occasionally square-root) dependence on $1/\delta$.

3. High-Probability Bounds for Complex/Tail-Sensitive Estimation

3.1. Robust and Heavy-Tailed Estimation

High-probability deviation bounds for robust estimators often rely on “truncated” or “M–estimator” constructions. (Catoni, 2010) introduces M–estimators $\hat{\theta}_\alpha$ defined implicitly by

$$\sum_{i=1}^n \psi(\alpha(Y_i - \hat{\theta}_\alpha)) = 0$$

Where $\psi$ is a nondecreasing influence function, the solution satisfies

$$|\hat{\theta}_\alpha - m| \leq \sqrt{\frac{2v \log(1/\epsilon)}{n (1 - 2\log(1/\epsilon)/n)}}$$

with probability $\geq 1-2\epsilon$, for known or estimated variance $v$. These estimators achieve minimax deviation optimality under weak moment assumptions and provide substantially shorter high-probability confidence intervals than the empirical mean, especially under heavy tails.

3.2. Subgradient Methods under Weak Assumptions

Optimization algorithms with heavy-tailed noise require new techniques for high-probability deviation control. (Parletta et al., 2022) analyzes a “clipped” stochastic subgradient method where

$$x_{k+1} = P_{X}\big(x_k - \gamma_k \cdot CLIP(\bar{u}_k, \lambda_k)\big)$$

and shows that, for averaged iterates,

$$f(\bar{x}_K) - f^* = O\bigg(\frac{\sqrt{\log(1/\delta)}}{\sqrt{K}}\bigg)$$

with only finite variance of subgradient noise, using martingale- and truncation-based error control.

4. Structural Results: Oracle Inequalities and Function Estimation

4.1. Principal Component Analysis in Infinite Dimensions

(Milbradt et al., 2019) establishes oracle-type high-probability bounds for the reconstruction error in PCA: $$R(P) \leq \big(1 + C_1 \frac{d'}{n}\big) \min_{P \in \mathcal{P}_{d'}} R(P)$$ with probability at least $1 - \exp(-t)$ for $d' \sim d$ and sample size $n$. For polynomial or exponential eigenvalue decays of the covariance operator, these bounds adapt to the correct error rate without requiring spectral gap assumptions.

4.2. High-Probability Minimax Lower Bounds

Across estimation tasks, the high-probability minimax lower bounds framework (Ma et al., 19 Jun 2024) establishes general tools to “boost” risk lower bounds to quantile bounds, yielding explicit additive price in $\log(1/\delta)$ or $\sqrt{\log(1/\delta)}$. Examples include covariance estimation (operator norm), sparse regression, nonparametric estimation (Hölder/Besov), and isotonic regression.

5. Applications in Statistical Learning and Optimization

5.1. Finite-Sample Procedures

High-probability deviation bounds enable the design of procedures with non-asymptotic error guarantees. In online change point detection, (Ye et al., 5 Aug 2025) uses computable strong approximation (KMT-type) inequalities to set adaptive thresholds for CUSUM tests, controlling false alarms at any time point with pre-specified error, even when variance parameters are unknown.

5.2. Joint Source–Channel Coding

In coding theory, finite-length bounds (Yaguchi et al., 2017) provide rates for the error probability in terms of finite-blocklength Rényi entropy spectral quantities. In the moderate deviations regime, the error decays at a subexponential rate, with leading constant determined by information-dispersion quantities, allowing practitioners to precisely guarantee performance for the required confidence level and code length.

5.3. Distributed and Privacy-aware Estimation

When mechanisms must provide local differential privacy, high-probability $\ell_2$-error bounds (Aliakbarpour et al., 13 Oct 2025) are sharp for heterogeneous user privacy levels, scaling as

$$O\big(\frac{\log(1/\beta)}{\sum_{i=1}^n \varepsilon_i^2}\big)$$

with probability $1-\beta$, quantifying privacy-utility trade-offs in decentralized data collection.

6. Technical Methods: Inequalities, Lifting, and Adaptivity

6.1. Lifting Risk Bounds to Deviation Inequalities

The iterative “boosting” from global risk to quantile bounds (Ma et al., 19 Jun 2024) relies on high-probability versions of Le Cam's method,

$\mathcal{M}_-(\delta) \geq g(\eta)$

and Fano-type inequalities for a finite family of hypotheses: $$\mathcal{M}_-(\delta) \gtrsim \eta \quad\text{for}\quad \delta < \mathrm{threshold}$$ thus converting classical minimax rates into fine-grained high-confidence lower bounds.

6.2. Adaptivity and Robustness

Several procedures achieve “adaptive” deviation control: the Lepski method for unknown variance (Catoni, 2010), median-of-means for regression under heavy tails (Ben-Hamou et al., 2023), and parallelization/averaging for stochastic convex optimization (Dvurechensky et al., 2017). These approaches allow practical high-probability guarantees with minimal prior parameter knowledge.

7. Summary Table: Key High-Probability Deviation Bounds

Setting	Bound Formulation	Reference
Sums of bounded i.i.d. RVs	$$P(S_n > x\sigma) = \Theta(x)\cdot \inf_\lambda \mathbb{E}e^{\lambda(S_n-x\sigma)}$$	(Fan et al., 2012)
Centered sum (Bernstein cond.)	$$P(S_n > x\sigma)\leq (1-\Phi(\tilde{x}))[1+C_\delta(1+\tilde{x})(\varepsilon/\sigma)]$$	(Fan et al., 2012)
Heavy-tailed sum	$P(S_n > x) \sim nP(X_1 > x)$, with quantified error	(Vogel, 2022)
Robust mean M-estimator	$$|\hat{\theta}_\alpha - m| \leq \sqrt{ \frac{2v \log(1/\epsilon)}{ n (1-(2\log(1/\epsilon))/n ) } }$$	(Catoni, 2010)
Sharpened PCA reconstruction	$R(P) \leq (1 + C_1 d'/n) \min_{P} R(P)$ (oracle bound)	(Milbradt et al., 2019)
Nonparametric regression, heavy tails	$$P( |\hat{r}_{\mathrm{MOM}}(x) - r(x)| \geq \eta ) \leq \delta$$, with $$\eta = c ( \frac{\sigma^2 \log(1/\delta)}{n} )^{1/(d+2)}$$	(Ben-Hamou et al., 2023)
Minimax quantile, e.g., regression	$$\mathcal{M}(\delta, \mathcal{P}_\Theta, L) \asymp \mathrm{risk}(n) + C\cdot \frac{\log(1/\delta)}{n}$$	(Ma et al., 19 Jun 2024)

8. Implications and Directions

High-probability deviation bounds provide a sharp, quantifiable, and fine-grained analysis of estimator performance and problem complexity, crucial where reliability, robustness to outliers, or tail control are required. Classical and modern tools—exponential inequalities, PAC-Bayesian theory, information-theoretic methods, and innovative robust statistics—converge to furnish both tight upper and lower deviation guarantees. Ongoing developments target weakening structural assumptions (e.g., dependence, heavy-tail noise), improved adaptive algorithms, and explicit characterization of the price for high-confidence in complex, high-dimensional, and privacy-sensitive statistical settings.