Info-Theoretic Minimax Problem

• Information-Theoretic Minimax Problem is a framework that defines performance limits in decision-making using measures such as entropy, mutual information, and divergence.
• It employs duality principles and minimax theorems to derive optimal strategies against worst-case adversarial or resource-constrained scenarios.
• Applications span adaptive data analysis, robust learning, reinforcement learning, and channel coding, establishing tight bounds and guiding algorithmic design.

An information-theoretic minimax problem seeks fundamental performance limits or optimal strategies in statistical learning, inference, testing, estimation, control, or sequential decision-making under adversarial, model-uncertain, or resource-constrained regimes, with explicit focus on information-theoretic quantities such as entropy, mutual information, divergence, channel capacity, and rate-distortion. This area is characterized by the use of minimax (min-max or max-min) criteria: a decision maker (learner/statistician/agent) minimizes worst-case expected loss, risk, or regret against the most challenging environment/distribution/adversary, often under information constraints.

1. Formal Structure and Prototypical Problem Statements

A canonical information-theoretic minimax problem is represented as

$$\inf_{A \in \mathbb{A}} \sup_{w \in \mathcal{W}} \mathbb{E}^w[\ell(A, w)]$$

where:

• $A$ is an action/estimator/algorithm selected from a class $\mathbb{A}$;
• $w$ indexes a family of data-generating distributions or model parameters $\mathcal{W}$;
• $\ell$ is a loss or regret function;
• $\mathbb{E}^w$ denotes expectation under $w$.

This generic template encompasses estimation, hypothesis testing, online learning, quantization, and channel coding. The information-theoretic character arises when either (i) the loss or risk is an information quantity (e.g., Kullback–Leibler divergence, entropy, mutual information), or (ii) the feasible sets incorporate information constraints (communication, disclosure, privacy, or resource bounds).

Duality principle: Many such minimax problems admit equivalent dual/maximin Bayesian formulations, often via minimax theorems (e.g., Sion’s, von Neumann’s, generalizations to infinite spaces), e.g.,

$$\inf_{A} \sup_{w} \mathbb{E}^w[\ell(A, w)] = \sup_{P_W \in \mathcal{P}(\mathcal{W})} \inf_A \mathbb{E}_{w \sim P_W}[\ell(A, w)]$$

under regularity conditions.

2. Principal Domains and Paradigms

2.1 Adaptive Data Analysis

In adaptive data analysis, the minimax problem quantifies the optimal tradeoff between accuracy and information leakage in a sequence of adaptively-chosen queries. The risk is defined as the maximal (worst-case) expected squared error over $k$ adaptive queries, and sharp bounds are obtained by analyzing the fundamental limit: $$F_k \geq \Omega\left(\frac{\sqrt{k}\, \sigma^2}{n}\right)$$ where $k$ is the number of queries, $\sigma^2$ is the variance bound for queries, and $n$ is the sample size (Wang et al., 2016). The lower bound construction leverages information-theoretic reduction—specifically, optimal obfuscation of Gaussian signs—to demonstrate that adaptive query selection amplifies noise by a factor of $\sqrt{k}$. The matching upper bound uses independent Gaussian noise addition, thus establishing minimax optimality up to constants. The tight rates reveal that information-theoretic quantities (mutual information, maximal correlation) fundamentally govern the price of adaptivity.

2.2 Minimax Regret in Reinforcement Learning and Partial Monitoring

Minimax regret in Markov Decision Processes (MDPs) or partial monitoring settings is characterized by

$$\mathfrak{M}_\mathcal{M} = \inf_{\mathbb{P}_\Pi} \sup_{\theta \in \mathcal{O}} \mathbb{E}_\Pi\, [\mathfrak{R}_\mathcal{M}(\Pi, \theta)]$$

where policy $\mathbb{P}_\Pi$ is randomized over actions, and $\mathfrak{R}_\mathcal{M}$ is the regret compared to the optimal policy in environment parameterized by $\theta$ (Bongole et al., 21 Oct 2024, Lattimore et al., 2019). Minimax duality theorems equate this with maximizing the minimum achievable Bayesian regret, enabling the application of information-theoretic bounds—via mutual information, KL-divergence, or Wasserstein distance—on cumulative regret. Explicit rates in bandit, linear, and contextual bandit settings are

$$O\left(\sqrt{|\mathcal{A}| \log|\mathcal{A}|\, T}\right), \quad O(d \sqrt{T \log T}),\quad O(\sqrt{|\mathcal{A}|\, T \log|\mathcal{O}|})$$

highlighting the role of intrinsic information as a minimax bottleneck for decision making.

2.3 Robust Learning and Distributional Shift

Information-theoretic minimax formulations underpin robust supervised learning under distributional uncertainty. The learner optimizes

$$\min_\omega \max_{p: KL(p \| q) \leq C} \int L(x, \omega) p(x) dx$$

where $q$ is the training distribution, $p$ is the adversarial test distribution, and the KL-ball models a set of plausible shifts (Zhang et al., 2023). Using Lagrangian relaxation and importance sampling, the minimax solution reduces to minimizing a softmax (ISloss) or $p$-norm loss (with temperature parameter $T = 1/p$), bridging ERM and robust optimization. This approach reveals a clean connection between risk-robustness tradeoffs and information-theoretic divergence.

2.4 Minimax Estimation and Sequential Decision

In estimation settings, minimax estimators minimize worst-case risk over all admissible distributions or parameters. For quantum state estimation under general Bregman divergences, every estimator is outperformed (asymptotically) by a suitable sequence of Bayes estimators, and covariant measurements (matching the symmetry of the state space) are universally minimax (Quadeer et al., 2018). For classical statistical models, minimax estimation can be recast as a Nash equilibrium in a zero-sum game between the estimator and Nature, and can be approximately computed via online learning with access to Bayes and worst-case risk oracles (Gupta et al., 2020).

2.5 Channel Coding and Control under Adversity

In adversarial channel coding, the minimax error at blocklength $n$ and rate $R$ is analyzed as

$$\bar{\vartheta}(n; R) = \min_{\text{codes}} \max_{q} \mathbb{E}_q[\text{error}]$$

with $q$ a distribution over channel states (Jose et al., 2018). Minimax and maximin values (compound vs mixed channel) converge asymptotically except at finitely many rates, and the limiting value is characterized via stepwise functions involving channel capacities of state subsets. Similar ideas appear in finite-time queueing control (Liu et al., 23 Jun 2025), where scheduling strategies are optimized to minimize worst-case queue backlog under arrival and service constraints, and sharp information-theoretic lower bounds are established.

3. Methodological Tools and Information-Theoretic Quantities

3.1 Core Information Measures

Key measures include:

• Mutual Information ($I(X; Y)$): Quantifies information between variables, controls generalization, excess risk, and regret.
• Kullback–Leibler Divergence ($D_{KL}(P\|Q)$): Measures discrepancy between probability measures, crucial in robust learning, estimation, and Markov chain approximation.
• Entropy and conditional entropy: Appear in prior selection (Bayesimax priors (Vangala, 21 Aug 2025)), risk-information cost, and remote prediction.
• Channel capacity ($C$): In minimax estimation/filtering, minimax regret equals channel capacity, and the minimax estimator is Bayesian under the capacity-achieving prior (No et al., 2013).
• Rate-distortion functions: Emerge in optimal quantization under bit constraints (Saha et al., 2022).

3.2 Duality, Game Theory, and Nash Equilibria

Minimax optimization is intimately connected to game-theoretic duality. Many problems are recast as Nash equilibria between learners and adversaries or between estimator and Nature. Existence and computation of minimax estimators, least-favorable priors, or robust algorithms often rely on strong and weak minimax theorems.

3.3 Reduction to One-Shot or Classical Information Theory

Results often reduce non-commutative (quantum) or adaptive problems to simpler commutative forms, enabling the derivation of tight one-shot entropy inequalities (Anshu et al., 2019), or leveraging conditional mutual information to upper bound minimax excess risk in learning (Hafez-Kolahi et al., 2022).

4. Representative Results and Rates

Setting	Minimax Rate/Bound	Notes
Adaptive queries	$\Theta(\sqrt{k} \sigma^2/ n)$	No adaptive mechanism beats i.i.d. noise
RL/bandits	$O(\sqrt{|\mathcal{A}| \log|\mathcal{A}|\, T})$	Matches lower bounds up to $\log$ factors
Quantization	$O(2^{-2B})$ excess risk (per-param)	$B$ bits per param for learning risk
Hyp. test (indep.)	$n \gg \sqrt{pq}/\|\Sigma_{XY}\|_F^2$	Min sample for power $> \alpha$
Channel coding	Stepwise in $R$ via $\vartheta(R)$ (capacity sets)	Limiting value of min-max/max-min

These are sharp, sometimes matching upper and lower bounds, and typically independent of algorithmic details (algorithm-agnostic, or "oracle" difficulty).

5. Structural Consequences and Practical Implications

Information-theoretic minimax analysis precisely delineates the boundary between possible and impossible inference or decision performance under uncertainty, adversarial interaction, or resource constraints. Core consequences:

• Optimality of simple/oblivious strategies: E.g., i.i.d. Gaussian noise for adaptive data analysis.
• Fundamental information-theoretic bottlenecks: Proven lower bounds illuminate intrinsic computational or statistical difficulty, unaffected by computational tractability or model choice.
• Algorithm design: Practical algorithms (e.g., Hadamard quantization for linear models, Lyapunov-based scheduling in queueing) attain risk/regret within negligible factors of the minimax limit.
• Limitations of standard IT bounds: For some regime (e.g., gradient descent generalization), mutual information–based bounds fundamentally fail to yield minimax rates (Haghifam et al., 2022), indicating need for new theory.
• Bayesian–frequentist bridge: Minimax duality often enables translation between worst-case (frequentist) and average-case (Bayesian) perspectives, unifying understanding and approaches.

6. Extensions, Open Problems, and Outlook

Research continues into generalizing minimax frameworks:

• To non-Gaussian, non-linear or non-convex regimes (robust learning, quantum tomography, arbitrary oracles).
• To more complex statistical tasks: structure learning, distributed learning, and minimax regret under networked, privacy-constrained, or information-limited settings.
• Establishment of minimax rates in new domains (e.g., optimal integral estimation for smooth but non-cubic functions remains unresolved (Adams et al., 2021)).
• Development of tight, information-theoretic minimax analyses for algorithm-dependent and adaptive scenarios where mutual information is suboptimal.

The information-theoretic minimax paradigm serves as a rigorous, unifying lens for understanding the possibility and impossibility frontiers in statistical learning, inference, control, optimization, and beyond.