Minimax-Optimal Misclassification Rate

• Minimax-optimal misclassification rate is defined as the optimal worst-case convergence speed of a classifier’s excess risk to the Bayes risk.
• It integrates key structural parameters like smoothness, margin conditions, and dimensionality to determine achievable convergence rates.
• Algorithms, including localized deep neural networks and adaptive methods, are designed to exploit these properties to attain the minimax rate.

The minimax-optimal misclassification rate quantifies the optimal worst-case convergence rate of the excess risk for classifiers over specified function and distribution classes. This concept is foundational in statistical learning theory and provides a benchmark for the efficacy of learning algorithms under regularity and margin conditions. The minimax-optimal rate details how fast the misclassification probability of the best possible estimator converges to that of the Bayes oracle, as the sample size increases, given constraints on smoothness, margin, dimension, structure, and other problem-dependent characteristics.

1. Formal Definition and General Framework

Given input space $\mathcal X\subseteq\RR^d$, binary labels $y\in\{-1,1\}$ (or $\{0,1\}$), and regression function $\eta(x)=\Pr(Y=1\mid X=x)$, the Bayes classifier is $f^*(x)=\text{sign}(\eta(x)-\tfrac12)$. For any classifier $f:\mathcal X\to\RR$, the induced classification risk is:

$R(f) = \mathbb E[\mathbb I\{f(x)\cdot y < 0\}],$

with excess risk (misclassification rate above Bayes): $\mathcal E(f) = R(f) - R(f^*).$

The minimax-optimal misclassification rate is: $$\inf_{\hat f}\;\sup_{P\in\mathcal P}\,\mathbb E_P[R(\hat f) - R(f^*)],$$ where $\mathcal P$ is a specified model class (parametric or nonparametric), and $\hat f$ runs over all estimators allowed to use $n$ labeled samples.

This rate gives the sharp exponent or convergence order, potentially up to polylogarithmic factors, for the worst-case excess risk over $\mathcal P$.

2. Canonical Minimax-Optimal Rates and Structural Parameters

For nonparametric binary classification under regularity/margin assumptions—such as H\"older smoothness of the regression or boundary and Tsybakov margin (noise) conditions—the minimax rate depends critically on (a) smoothness $\beta$, (b) ambient/effective intrinsic feature dimension $d$, and (c) margin exponent $\kappa$ (or $\alpha$) (Hu et al., 2022, Zhao et al., 2019, Ryu et al., 2022).

The prototypical minimax-optimal excess risk rate is: $$n^{-\frac{\beta(\kappa+1)}{\beta(\kappa+2)+(d-1)\kappa}},$$ where:

• $n$ = sample size,
• $\beta$ = H\"older smoothness of the Bayes boundary or regression,
• $d$ = intrinsic (possibly effective) dimension,
• $\kappa$ (or $\alpha$) = margin (noise) exponent.

This rate arises in smooth decision boundary problems (localized margin framework) and is achieved exactly (up to log factors) by divide-and-conquer deep neural network classifiers constructed locally and aggregated globally (Hu et al., 2022).

The general phenomenon is that smoother boundaries and larger margins (higher $\beta$, higher $\kappa$) yield faster rates, while larger intrinsic dimension $d$ slows convergence, exemplifying the curse of dimensionality unless compositional or low-dimensional structure exists.

3. Achievability and Construction of Minimax-Optimal Classifiers

Optimality is achieved by algorithms that exploit structural properties of the Bayes boundary and the regression function. In the classical smooth boundary regime, generic deep neural networks are rate suboptimal. The divide-and-conquer DNN approach partitions the domain and fits local networks on each cell, which are then merged to form a global classifier. For the boundary model where $G^*=\{x:\ x_d\leq f^*(x_{-d})\}$, a local ReLU network is trained per base cell $D_{\bj}$, then extended and aggregated (Hu et al., 2022).

Main steps for upper bound tightness include:

• Deep ReLU approximation for local $f^*$ in each partition (approximation error analysis).
• Empirical process and bracketing entropy control of ERM risk in localized classes (estimation error).
• Aggregation via selection of local minimizers.
• Balancing approximation and estimation errors to saturate the minimax rate.

The matching lower bound is proved via construction of a well-separated local hypercube of Bayes boundaries, using Fano's or Assouad's lemma, leveraging the localized margin exponent $\kappa^-$.

Simulation studies support theoretical predictions: the slope of empirical excess risk on log–log plots aligns with the minimax exponent $$n^{-\beta(\kappa^-+1)/(\beta(\kappa^-+2)+(d-1)\kappa^-)}$$ (Hu et al., 2022).

4. Adaptation and Circumventing the Curse of Dimensionality

If the Bayes boundary function admits a deep compositional or modular structure—i.e., factorization $f^*=h_\ell\circ\cdots\circ h_0$ where each $h_i$ depends on few variables—then the effective smoothness $\beta^*$ and effective dimension $d^*$ replace the ambient $d$ in the minimax rate:

$$n^{-\frac{\beta^*(\kappa+1)}{\beta^*(\kappa+2)+d^*\,\kappa}},$$

enabling adaptive rates that can be much faster when the intrinsic structure is low-dimensional (Hu et al., 2022, Wang et al., 2023). Deep neural networks constructed to mirror this modularity can break the curse of dimensionality, achieving rates not possible with non-compositional (shallow) methods.

In the high-dimensional regime, if the regression function indeed possesses modular structure, DNN classifiers attain the explicit nonasymptotic minimax rate (in sample size $n$ and $\log d$) (Wang et al., 2023):

$$\Bigl(\frac{\log d}{n}\Bigr)^{s_0}, \text{ where } s_0 = \min_{0\le u\le q} \frac{\beta_u^*(\alpha+1)}{\beta_u^*(\alpha+2)+t_u}.$$

5. Influence of Margin/Noise Exponents

The minimax rate is intimately linked to the local noise or separation around the Bayes boundary, formalized by Tsybakov's noise condition and refined via localized margin exponent $K(x)$:

$$m_x(t) = |p(x_{-d},f^*(x_{-d})+t) - q(x_{-d},f^*(x_{-d})+t)| \asymp |t|^{1/K(x)}.$$

With $K(x)\equiv\kappa$, this reproduces the standard margin exponent. Fast rates (including exponential) occur when the conditional distributions or boundaries are well-separated, while slow rates are determined by local flatness or ambiguity of the class conditional densities (Jiao et al., 2011).

6. Minimax-Optimal Rates Beyond Classical Settings

Extensions include:

• Transfer learning and private distributed settings, where minimax rates are governed by a trade-off between smoothness, margin, sample size, privacy noise, and heterogeneity among sources (Auddy et al., 28 Jun 2024).
• Community detection, with minimax exponential rates determined by Rényi-type divergences, taking the form $\exp(-(1+o(1))nI^*)$ in (hyper)graph models (Gao et al., 2015, Chien et al., 2018).
• Covariate shift/posterior drift settings, where the minimax rate can exhibit phase transitions depending on smoothness and alignment among source and target distributions (Liu et al., 2020).

Algorithmic strategies for achieving minimax-optimal misclassification rates include localized DNNs, adaptive nearest neighbors, kernel aggregation, and two-step refinement with weakly consistent initializers, with each method tailored to the structural/statistical assumptions of the scenario.

7. Summary Table: Minimax-Optimal Misclassification Rate Exponents

Setting	Minimax-Optimal Excess Risk Rate	Key References
Smooth boundary/classification, $\beta,\kappa$	$$n^{-\frac{\beta(\kappa+1)}{\beta(\kappa+2)+(d-1)\kappa}}$$	(Hu et al., 2022)
Compositional boundary, effective $(\beta^*, d^*)$	$$n^{-\frac{\beta^*(\kappa+1)}{\beta^*(\kappa+2)+d^*\kappa}}$$	(Hu et al., 2022, Wang et al., 2023)
Standard $k$–NN (fixed $k$)	$$N^{-\min\{\beta(\alpha+1)/(2\beta+\alpha+1),2\beta(\alpha+1)/(\beta d+2(\alpha+2\beta))\}}$$	(Zhao et al., 2019)
Adaptive $k$–NN	$$N^{-\min\{\beta,2\beta(\alpha+1)/(\beta d + 2(\alpha+2\beta))\}}$$	(Zhao et al., 2019)
High-dim DNN, modular regression	$(\log d/n)^{s_0}$	(Wang et al., 2023)
Prior probability estimation (detection)	$n^{-1/2}$ to $n^{-(1+\alpha)/2}$ to $\exp(-cn)$	(Jiao et al., 2011)
Community detection SBM/hSBM	$\exp(-cnI^*)$	(Gao et al., 2015, Chien et al., 2018)

These rates are sharp (up to logarithmic factors) under their respective modeling assumptions and are attainable by explicit algorithmic constructions found in the referenced works.

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References:

(Hu et al., 2022): Minimax Optimal Deep Neural Network Classifiers Under Smooth Decision Boundary (Zhao et al., 2019): Minimax Rate Optimal Adaptive Nearest Neighbor Classification and Regression (Wang et al., 2023): Minimax optimal high-dimensional classification using deep neural networks (Jiao et al., 2011): Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities (Gao et al., 2015): Achieving Optimal Misclassification Proportion in Stochastic Block Model (Chien et al., 2018): On the Minimax Misclassification Ratio of Hypergraph Community Detection (Liu et al., 2020): A Computationally Efficient Classification Algorithm in Posterior Drift Model: Phase Transition and Minimax Adaptivity (Auddy et al., 28 Jun 2024): Minimax And Adaptive Transfer Learning for Nonparametric Classification under Distributed Differential Privacy Constraints (Ryu et al., 2022): Minimax Optimal Algorithms with Fixed-$k$-Nearest Neighbors