Information-Based Complexity

• Information-Based Complexity is a framework that defines the minimum information required to achieve accurate solutions in computational problems using measures such as Shannon entropy and mutual information.
• It employs information-theoretic quantities to derive lower bounds on queries and resources, guiding the design and evaluation of algorithms in convex optimization and learning theory.
• IBC bridges computational and statistical challenges by formalizing the cost of acquiring information, influencing fields like numerical analysis, stochastic optimization, and decision-making under uncertainty.

Information-Based Complexity (IBC) is a quantitative framework for analyzing the minimal information requirements for solving mathematical problems to a specified degree of accuracy. Rooted in computational mathematics, information theory, and theoretical computer science, IBC aims to characterize the intrinsic hardness of problems in terms of the information accessible about the inputs and the cost associated with acquiring and utilizing this information. IBC formalizes and generalizes concepts such as oracle complexity, sample complexity, and statistical complexity, unifying them through the lens of information measures—most notably Shannon entropy and mutual information—across a diverse range of problem domains, including numerical analysis, learning theory, stochastic optimization, and decision-making under uncertainty.

1. Foundations and General Framework

At its core, IBC proceeds from the premise that exact solutions to many problems are typically unattainable, and practical algorithms must operate with partial, often noisy or indirect, information about the input. The basic setup considers an agent (algorithm) seeking to accomplish a well-defined task under uncertainty, where the agent’s performance is measured by a loss functional $L(I)$ dependent on the information $I$ acquired. The “quasi-quantity” $Q(I)$ quantifies the cost or difficulty associated with acquiring or representing $I$, which may be Shannon entropy, the number of oracle queries, or other metrics tailored to the information environment.

The $\varepsilon$-complexity $C(\varepsilon)$ is defined as

$$C(\varepsilon) = \min_{\{I : L(I) \leq \varepsilon\}} Q(I)$$

expressing the least information cost needed to guarantee loss at most $\varepsilon$ (Perevalov et al., 2013). Unlike classical complexity theory—which typically counts computational steps—IBC focuses on the fundamental information bottleneck, decoupling achievable accuracy from computational resources, and therefore directly interfaces with the limits of algorithmic performance in the presence of uncertainty.

2. Mutual Information and Lower Bounds

A central methodological contribution of IBC is the use of information-theoretic quantities, especially mutual information, to derive lower bounds on the resources required for problem-solving. For a search problem $f(x) = y$ approached via queries to an auxiliary function $g(x)$, the expected number of queries is lower bounded by

$$\mathbb{E}[\text{\#queries}] \geq \frac{I(f = y)}{I(f; g)}$$

where $I(f=y)$ is the self-information of the target event, and $I(f; g)$ is the mutual information between $f$ and $g$ (Zhao, 2016). This principle applies not only to black-box search and decision problems but extends to PAC learning (quantifying the information revealed by hypotheses), numerical approximation, and convex optimization.

In randomized settings, IBC distinguishes between adaptive and non-adaptive algorithms based on their information acquisition strategies. Recent results show that adaptivity confers polynomial advantages in mean computation problems for certain norms, with adaptive errors outperforming non-adaptive errors by a factor as large as $n^{1/4}$ in particular regimes (Heinrich, 2024).

3. IBC in Convex Optimization and Oracle Models

In convex programming, IBC inspects the minimal number of queries necessary to approximate the minimum of a convex function within a given tolerance, under specified oracle feedback models. In the "oracle model," the agent queries for function values, subgradients, or other partial information—possibly noisy—and assimilates the responses sequentially.

Lower bounds are constructed by recasting optimization as a sequential hypothesis testing problem, where the required information gain to distinguish between a packing of $N$ candidate functions yields a minimax rate: $$T \geq \frac{(1-\delta)\log N - \log 2} {\max_{i, j, x} D(P(\cdot \mid f_i,x)\|P(\cdot \mid f_j,x))}$$ where $D(\cdot\|\cdot)$ is the Kullback-Leibler divergence between oracle outputs (Raginsky et al., 2010). For example, with Gaussian first-order oracles, the $\varepsilon$-information complexity for Lipschitz convex functions is $\Omega(n \sigma^2 / \varepsilon^{2/r})$, aligning the informational and classical (oracle) complexity views.

Extensions to mixed-integer convex optimization demonstrate that the information complexity exhibits an exponential dependence on the number of integer variables. The “transfer theorem” asserts that lower bounds for continuous settings can be transferred to mixed-integer cases by an exponential blow-up in query complexity (Basu et al., 2023). Furthermore, the informational power of oracles is stratified: binary-value oracles are provably less informative than full first-order oracles, as partial information prolongs the ambiguity maintained by adversarial input distributions.

4. IBC in Learning Theory

The application of IBC to learning theory introduces the notion of "information complexity" of a hypothesis class $H$ with respect to PAC learning tasks. For sample $S$ and a proper, consistent algorithm $A$, the mutual information $I(S;A(S))$ quantifies the number of bits about the random input sample that must be revealed by the learned hypothesis. The information complexity of $H$ is

$$IC(H) = \sup_m \inf_{A}\sup_{p \in \Delta_H} I(S;A(S))$$

where $p$ ranges over all distributions consistent with $H$ (Nachum et al., 2018). Sharp lower bounds demonstrate that, even for classes of constant VC dimension $d$, the information complexity scales as $\Omega\left(d \log \log\frac{|X|}{d}\right)$.

This framework not only connects to sample compression schemes, minimum description length (MDL), and Occam's razor, but also interacts with privacy and generalization: lower information leakage implies better generalizability and potential guarantees of differential privacy. Direct-sum theorems establish additivity—combining multiple hypothesis classes increases the required information cost linearly in the number of components.

5. Extensions to Stochastic Processes and Decision-Making

Beyond optimization and learning, IBC unifies statistical prediction, experimental design, and decision-making under uncertainty. For stationary discrete-time stochastic processes, the information complexity specializes to the statistical complexity of $\varepsilon$-machines—the Shannon entropy of the causal state distribution—connecting predictive modeling and information storage (Perevalov et al., 2013). For stochastic optimization, information-related complexity is the minimal Shannon entropy (over partitions of the uncertainty space) required to reduce the expected loss to a given threshold.

The framework subsumes classical statistical complexity as a special case, and, by explicitly modeling the loss function and information cost, tailors complexity analysis to specific inference, estimation, and sequential decision tasks.

6. Impact, Open Problems, and Future Directions

IBC has significantly sharpened understanding of the fundamental limits for a variety of algorithmic problems: it shows when information bottlenecks, not raw computational effort, dominate the difficulty. The demonstration that adaptivity provides polynomial improvement over non-adaptivity in randomized settings closes open questions regarding the power of algorithm design (Heinrich, 2024). In mixed-integer optimization, IBC exposes the exponential complexity barrier as intrinsic, not an artifact of existing algorithms (Basu et al., 2023).

Open directions include characterizing conditions under which low information complexity coincides with PAC learnability, extending lower bounds to improper or inconsistent learners, and adapting the general framework to encompass noisy and robust oracles. Furthermore, the interplay between information-based and worst-case (computational) complexity for FNP-type problems remains a frontier, with entropy-based lower bounds providing a promising lens for longstanding questions such as $P \neq NP$ (Zhao, 2016).

Overall, Information-Based Complexity continues to supply a universal, rigorous quantification of the informational limits governing computational and statistical procedures, with implications spanning mathematics, computer science, and engineering.