Relative Information Gain

• Relative Information Gain is a metric that compares incremental reductions in uncertainty across different partitions, parameterizations, or sequential updates.
• It is used in algorithmic optimization, feature selection, and hyperparameter tuning to enhance decision-making in various learning systems.
• The measure bridges information theory and effective dimensionality, providing tight complexity bounds in kernel learning and Gaussian process regression.

Relative information gain extends the foundational notion of information gain in information theory and learning, quantifying the additional information acquired when algorithmic choices or system parameters (such as attribute groupings, noise levels, or regularization hyperparameters) are varied. Unlike absolute information gain, which measures uncertainty reduction from a single split, observation, or measurement, relative information gain contextualizes that reduction across alternative partitions, parameterizations, or sequential updates. This enables the assessment of comparative or marginal informativeness, supports algorithmic optimization (e.g., in feature selection or hyperparameter tuning), and yields nuanced complexity measures, particularly in nonparametric or kernel-based settings.

1. Fundamental Concepts and Definitions

Classically, information gain in supervised learning—especially in decision tree induction—measures the decrease in entropy (uncertainty) in class labels $T$ due to partitioning on an attribute $X$:

$\text{Gain}(X) = H(T) - H(T|X),$

where $H(T)$ is the Shannon entropy of the class label distribution, and $H(T|X)$ is the conditional entropy after splitting on $X$.

Relative information gain broadens this by:

• Comparing the reduction in entropy or error rate obtained by one attribute, group, or system setting versus another.
• Quantifying the incremental or marginal gain in complex or sequential settings, such as multi-valued attribute groupings (Dabhade, 2011), changes in observation noise (Flynn, 5 Oct 2025), or successive experimental updates (Yu et al., 2023).
• Using difference or KL divergence-based frameworks to express the informativeness of a refinement relative to a baseline.

In kernel learning and Gaussian process regression, the canonical (absolute) information gain is defined via the log-determinant of the regularized kernel matrix:

$$\gamma_n(\eta) = \tfrac{1}{2} \log \det(\eta K_n + I).$$

Relative information gain in this context is the difference between log-determinants under two regularization/noise regimes $\eta > \beta \geq 0$:

$$\gamma_n(\eta, \beta) = \gamma_n(\eta) - \gamma_n(\beta) = \tfrac{1}{2} \sum_{i} \log \frac{1 + \eta \lambda_i}{1 + \beta \lambda_i},$$

where $\lambda_i$ are the kernel eigenvalues (Flynn, 5 Oct 2025).

2. Information Gain in Decision Trees: Role of Multivalued Subsets

Traditional decision tree algorithms (e.g., ID3) evaluate individual attribute values for their information gain, but this restriction can underperform when attribute-value groupings are more predictive. The formation of multivalued subsets, as explored in (Dabhade, 2011), extends the information gain paradigm:

• By considering groupings of categorical values, the split can more effectively reduce class entropy.
• The information gain for a partition based on a multivalued subset $S$ is:

$$\text{Gain}(S) = H(T) - \Big(P_S H(T|S) + P_{\bar S} H(T|\bar S)\Big),$$

where $P_S$ is the probability mass on subset $S$ and $H(T|S)$ the conditional entropy given $X \in S$.

• Relative information gain, in this framework, refers to the difference between the gain achieved by a superset partition $S$ versus a baseline singleton or other grouping.
• Empirical results demonstrate that optimizing for relative information gain via heuristic search (e.g., adaptive simulated annealing) can lower classification error in feature selection, underscoring the utility of evaluating group-based splits (Dabhade, 2011).

3. Relative Information Gain in Kernel Learning

In Gaussian process regression and related kernel methods, both sample complexity and excess risk bounds have long been linked to the "information gain" ${\gamma_n(\eta)}$ and the "effective dimension" $$d_n(\eta) = \sum_i \frac{\eta \lambda_i}{1 + \eta \lambda_i}$$, with $\eta$ interpreting as inverse noise variance.

Relative information gain is defined as:

$$\gamma_n(\eta, \beta) = \gamma_n(\eta) - \gamma_n(\beta) = \frac{1}{2} \sum_i \log \left( \frac{1 + \eta \lambda_i}{1 + \beta \lambda_i} \right)$$

and interpolates smoothly between $2\gamma_n(\eta)$ (when $\beta = 0$) and $d_n(\eta)$ (in the limit $\beta \uparrow \eta$). It thus functions as a finite-difference approximation to the derivative of information gain with respect to noise, providing a complexity measure that both:

• Retains the information-theoretic meaning of mutual information (for larger noise changes), and
• Matches the minimax-convergence-improving behavior of the effective dimension (for small noise reductions or fine regularization settings).

The paper (Flynn, 5 Oct 2025) proves that localization terms in PAC-Bayesian excess risk bounds for GP regression naturally adopt the form $2\gamma_n(2\eta\alpha,2\beta\alpha)$, resulting in minimax-optimal rates under standard spectral decay conditions.

Table: Complexity measures in kernel methods

Complexity quantity	Definition	Interpretation
Information gain	$$\gamma_n(\eta) = \frac{1}{2}\sum_i \log(1 + \eta\lambda_i)$$	Mutual information (response vs latent function)
Effective dimension	$$d_n(\eta) = \sum_i \frac{\eta\lambda_i}{1+\eta\lambda_i}$$	Effective number of parameters
Relative information gain	$\gamma_n(\eta, \beta)$ as above	Sensitivity of info gain to noise/regularization; interpolates between the two

4. Experimental Design, Hypothesis Testing, and Sequential Update Contexts

Relative information gain naturally appears in Bayesian inference and sequential experiment analysis. For a posterior update from prior $\pi(\theta)$ to posterior $P(\theta|D)$, the KL divergence:

$$D_{\mathrm{KL}}[P(\theta|D) \| \pi(\theta)] = \int P(\theta|D)\log\frac{P(\theta|D)}{\pi(\theta)} d\theta$$

quantifies the absolute information gain.

In sequential settings, relative information gain is the KL divergence between the updated posterior after an extra datum and the posterior before:

$$I_{\mathrm{rel}} = D_{\mathrm{KL}}[P(p|N+1, T_{N+1}, I)\;\|\; P(p|N, T_N, I)]$$

This measure is always nonnegative (Yu et al., 2023), guaranteeing monotonic "knowledge gain" even in the presence of surprising or "black swan" data, and is asymptotically prior-independent.

By contrast, the differential information gain (difference in KL divergence to the prior) can be negative in exceptional cases. The paper argues for the operational relevance of both measures, while the monotonicity and robustness of relative information gain are notable properties in experimental physics, machine learning updates, and some quantum settings (Yu et al., 2023).

5. Feature Selection, Model Selection, and Algorithmic Applications

Relative information gain provides a direct framework for algorithmic choices, especially in:

• Filter-based feature selection, where ranking features by the incremental gain when added to a set enables more effective subset selection (Dabhade, 2011).
• Retrieval Augmented Generation in LLM pipelines, where relevant information gain is used as an optimization objective for passage selection. Here, the gain is computed over sets, not individual items, and measures the expected coverage of all possible information targets—a design that organically encourages both relevance and diversity (Pickett et al., 16 Jul 2024).
• Comparisons across system states, hyperparameters, or regularization levels, where relative information gain offers a natural criterion for guiding search or early stopping.
• Automatic differentiation and planning, where smooth relative information gain metrics can be optimized for trajectories or subsets, as seen in differentiable exploration methods (Deng et al., 2020).

6. Theoretical and PAC-Bayesian Complexity Bounds

Relative information gain has a formal role in excess risk, generalization, and sample complexity results:

• In GP regression, the key complexity term in PAC-Bayesian excess risk bounds is formulated as a function of relative information gain, leading to minimax-optimal rates when combined with spectral decay constraints (Flynn, 5 Oct 2025).
• It provides a bridge between the information-theoretic approach (favoring log-det and KL-based terms) and capacity-control via effective dimension, interpolating between the two and inheriting favorable growth-rate properties.
• Sandwich inequalities demonstrated in (Flynn, 5 Oct 2025)

$$d_n(\eta) \leq \frac{2\eta}{\eta-\beta}\gamma_n(\eta, \beta) \leq \frac{\eta}{\beta}d_n(\eta)$$

formalize this interpolation, providing tight, practical complexity envelopes for algorithm design and theoretical analysis.

7. Broader Implications and Related Concepts

Relative information gain has implications and conceptual resonance in domains beyond classical supervised learning:

• In quantum information, the principle of monotonic relative gain structuring how total system uncertainty about measurement outcomes is strictly reduced in the process of decoherence or controlled measurement, subject to the operational parameters (Zhu et al., 2012, Brody et al., 26 Feb 2024).
• In feature evaluation, ranking, and subset selection, relative information gain enables the discovery of latent structure or combinatorial effects that are invisible to singleton-based criteria (Dabhade, 2011).
• In theoretical machine learning, measures such as critical information gain are closely connected to the eluder dimension and elliptic potential quantities that characterize sample efficiency in linear and nonparametric settings (Huang et al., 2021).

Relative information gain, uniquely situated between absolute and differential notions of informativeness, thus provides a mathematically robust, interpretable, and operationally useful metric for comparing, selecting, and optimizing in inference, learning, and decision-making systems.