Information-Theoretic Lower Bounds

An information-theoretic lower bound is a rigorous constraint—formulated in terms of entropy, mutual information, or related quantities—on the best achievable performance of an algorithm or estimator in a given information-processing task. Such bounds arise in communication, learning theory, signal processing, optimization, statistics, and quantum computation, and play a central role in identifying the limitations of any algorithm or protocol, regardless of computational resources or implementation.

1. Foundational Principles and General Frameworks

Information-theoretic lower bounds are typically derived by considering the minimal uncertainty or error that remains after observing data, subject to various physical or structural constraints. The most fundamental objects are entropy, mutual information, and related divergence measures. The lower bound is often expressed as a function of the information that can be extracted about a hidden variable or structure from available observations.

For example, in sparse recovery problems, the sample complexity $T$ (i.e., the required number of measurements or queries) for support recovery is lower-bounded by

$T \geq \max_{\tilde{S} \subset S_0} \frac{\log\binom{N-|\tilde{S}|}{K-|\tilde{S}|}}{\bar{I}_{\tilde{S}}}$

where $\bar{I}_{\tilde{S}}$ is the average mutual information per measurement about the "unknown" coordinates conditional on prior knowledge, and the numerator quantifies the remaining uncertainty over all possible supports (Aksoylar et al., 2014 ).

A general strategy for establishing such bounds is to relate the probability of error or estimation risk to the Kullback–Leibler divergence or mutual information between the true underlying structure (e.g., a hypothesis, parameter, or function value) and the observations, often via Fano's inequality or its generalizations.

2. Applications Across Domains

Sparse Signal Recovery

In adaptive and nonadaptive compressive sensing, group testing, and related settings, a unified information-theoretic lower bound is provided by the mutual information between unknown coordinates and the observations. The formula applies to both linear models (e.g., $Y = X\beta + W$) and nonlinear models (e.g., Boolean group testing, 1-bit compressed sensing), encompassing adaptive measurement scenarios as well (Aksoylar et al., 2014 ).

Learning Theory, Estimation, and PAC Framework

In statistical learning and estimation, information-theoretic lower bounds on sample complexity or excess risk can be derived via the rate-distortion function,

$R(D) := \inf_{Q_{\widehat{W}|W}} I(W; \widehat{W}) \;\;\text{subject to}\;\; \mathbb{E}[\ell(Z; \widehat{W}) - L^*] \leq D,$

which quantifies the minimal average information about the model parameters $W$ that must be learned to keep risk below $D$. This is coupled with an upper bound on the mutual information between the sample and the parameters (Nokleby et al., 2021 ). When these match (as for VC classes under 0-1 loss), risk lower bounds scale as $\Omega(d_{\rm vc}/n)$.

In the context of quantum PAC learning, information-theoretic methods can certify that quantum sample complexity cannot be substantially smaller than the classical, up to constant or logarithmic factors (Hadiashar et al., 2023 , Angrisani et al., 2021 ).

Distributed Function Computation and Decentralized Estimation

In distributed scenarios, the minimum computation time or minimal risk for function computation/estimation is lower-bounded in terms of the conditional mutual information between the function of interest and the estimate, given local partial observations. Small ball probabilities (a.k.a. Lévy concentration functions) and strong data processing inequalities yield explicit expressions for these lower bounds, which capture the effect of topology (e.g., network diameter), communication channel noise, and resource distribution (Xu et al., 2015 , Xu et al., 2016 ).

Bayesian Optimization and Black-box Methods

In Bayesian optimization, acquisition functions based on lower bounds of mutual information between new queries and the objective supremum set a principled floor on expected gain from a query. These bounds account for feasibility constraints and estimator variance, ensuring robust, non-negative acquisition scores (Takeno et al., 2021 ).

Zero-Order Oracle Models and Integral Estimation

For derivative-free optimization and numerical integration using oracles, the best-possible estimation accuracy is bounded below by information-theoretic quantities,

$\epsilon^* \geq c \sqrt{d/T}$

for $d$-dimensional gradient estimation using $T$ queries (Alabdulkareem et al., 2020 ), or, in integrating functions over high-dimensional domains,

$\epsilon^*(\mathcal{F}, \phi) \geq c\,2^d r^{d+1} \sqrt{d/T}$

where $r$ is the domain size, and $c$ is a constant (Adams et al., 2021 ).

Communication Complexity and Information Complexity

In two-party communication problems, information-theoretic lower bounds are captured by information complexity ($IC$), Rényi information complexity ($IC_\infty$), and partition complexity. $IC_\infty$ forms a spectrum bridging Shannon mutual information-based lower bounds and combinatorial partition bounds, unifying approaches and improving equivalence between the two methodologies (Prabhakaran et al., 2015 ). For interactive protocols, lower bounds on information cost directly limit achievable communication rates (Rajakrishnan et al., 2016 ).

3. Methodological Features and Key Ingredients

Fano-Type Inequalities and Information Contraction

Fano's inequality and its extensions are foundational in converting mutual information bounds into lower bounds on error probabilities, sample complexity, or risk. In distributed and decentralized architectures, strong data processing inequalities (SDPI) capture the loss of information as data passes through noisy channels, providing tight non-asymptotic lower bounds for achievable performance (Xu et al., 2015 , Xu et al., 2016 ).

Role of Concentration Functions and Small Ball Probabilities

In distributed function computation, small ball probabilities or Lévy concentration functions precisely capture the uncertainty that remains about a (possibly linear) function after observing parts of the data. Tight bounds for sums of independent variables yield strictly better lower bounds for computation time compared to previous techniques, especially under accuracy requirements that become stringent as $\varepsilon \to 0$ (Xu et al., 2015 ).

Rate-Distortion and Variational Formulations

Rate-distortion formulations naturally encode tradeoffs between accuracy and information: $\text{Lower bound on NLL} \geq \text{model NLL} - \sup_z \log \left( \frac{1}{N} \sum_{i=1}^N \frac{\ell(x_i|z)}{p(x_i)} \right)$ demonstrating that improvements by optimizing over priors or likelihoods are fundamentally limited by information-theoretic constraints (Lastras, 2019 ).

4. Consequences for Adaptivity, Structure, and Achievability

A central theme is the tightness and achievability of lower bounds:

• Optimality: In several settings, such as support recovery in compressive sensing or learning Gaussian graphical models, new algorithms have been proposed that nearly achieve the information-theoretic lower bound without additional assumptions, closing the gap with earlier nonoptimal scalable methods (Misra et al., 2017 ).
• Limits of Adaptivity: In group testing and 1-bit compressed sensing, adaptivity (i.e., adaptively choosing next queries/measures) cannot reduce the minimum sample complexity below the nonadaptive bound; only in compressive sensing with sublinear sparsity can mild gains be achieved (Aksoylar et al., 2014 ).
• Network Effects: In distributed/parallel settings, information dissipates across multiple network cutsets, leading to time lower bounds proportional to the network diameter and contraction per hop, which does not arise in single-cutset or centralized bounds (Xu et al., 2015 ).

5. Practical and Theoretical Implications

Information-theoretic lower bounds inform several practical and conceptual areas:

• Algorithm Benchmarking: They specify performance thresholds for algorithms in estimation, optimization, learning, and control, justifying or refuting the possibility of "breaking" current performance levels with better methods.
• System Design Guidance: In active noise cancellation, the theoretical minimum normalized mean squared error (NMSE) is given by the maximum of an information-processing limit (mutual information captured by the anti-noise channel) and a physical (support) limit due to uncancellable frequencies. These inform both algorithm design and hardware configuration (Derrida et al., 23 May 2025 ).
• Limits of Existing Theories: Mutual information-based generalization bounds, while popular, are proven to be fundamentally dimension-dependent in convex optimization and thus insufficient to explain generalization in overparameterized or high-dimensional learning, highlighting the need for alternative analytical tools (Livni, 2023 ).

6. Summary Table: Representative Lower Bound Forms

7. Open Questions and Future Directions

Despite the breadth and depth of current results, key research challenges remain:

• Closure under Continuous-Time Models: For continuous-time network inference or diffusion models, the tight lower bound matches known upper bounds only in discrete settings; finding optimal algorithms in the continuous setting remains unresolved (Park et al., 2016 ).
• Beyond Classical Measures: Mutual information and entropy bounds may fail to capture all aspects of generalization or identifiability, especially in quantum, adversarial, or deep learning models (Livni, 2023 ).
• Extensions to Non-i.i.d. or Adaptive Regimes: Most lower bounds rely on i.i.d. assumptions and regularity; refining or generalizing these to adaptive/nonparametric settings is an active area.
• Interaction with Physical Constraints: Understanding how physical or engineering constraints—such as path support in ANC (Derrida et al., 23 May 2025 )—interact with information-theoretic limits continues to present both theoretical and practical challenges.

Information-theoretic lower bounds thus provide a unifying conceptual and mathematical framework for understanding the ultimate limits of inference, learning, computation, and communication. Their continued refinement and application are central to both advancing theory and guiding the practical development of algorithms and systems.