Eluder Dimension: Adaptive Learning Complexity

• Eluder dimension is a sequential complexity measure that captures the number of adaptive queries needed to identify functions within a hypothesis class.
• It underpins theoretical bounds by quantifying sample complexity and regret in adaptive settings like bandit optimization and reinforcement learning.
• Its analysis differentiates sequential exploration from static measures like the VC dimension, informing optimal algorithm design for complex function classes.

The eluder dimension is a sequential combinatorial parameter capturing the intrinsic difficulty of identifying unknown functions within a hypothesis class via adaptive queries. Unlike the VC dimension, which governs worst-case identifiability in static (i.i.d.) learning, the eluder dimension characterizes minimax sample complexity in adaptive, exploration-driven settings such as bandit optimization, reinforcement learning, contextual decision processes, policy learning, and certain algorithmic unlearning scenarios. Its definition and properties sharpen the understanding of statistical hardness in function approximation, prompt optimal design of exploration algorithms, and rigorously delineate lower bounds on sample or memory requirements in sequential learning.

1. Definition and Formalism

Given a function class $\mathcal{F}\subseteq\{f:X\to\mathbb{R}\}$ and a tolerance parameter $\epsilon>0$, the notion of $\epsilon$-dependence is central. For a point $x\in X$ and a sequence $S=\{x_1,\dots,x_n\}$, $x$ is said to be $\epsilon$-dependent on $S$ (w.r.t.\ $\mathcal{F}$) if for every $f,f'\in\mathcal{F}$,

$$\sqrt{\sum_{i=1}^n |f(x_i)-f'(x_i)|^2} \le \epsilon \implies |f(x)-f'(x)| \le \epsilon.$$

Otherwise, $x$ is $\epsilon$-independent of $S$. An $\epsilon$-eluder sequence for $\mathcal{F}$ is a sequence $(x_1,x_2,\dots,x_d)$ such that for each $k$, $x_k$ is $\epsilon$-independent of its predecessors $\{x_1,\dots,x_{k-1}\}$. The $\epsilon$-eluder dimension, $\dim_E(\mathcal{F},\epsilon)$, is the largest length of such an independent sequence. For binary-valued hypothesis classes $\mathcal{H}\subseteq\{h:X\to\{0,1\}\}$, eluder sequences are defined analogously, with dependence on labeled sequences and version spaces, as in the explicit construction of (Cherapanamjeri et al., 16 Jun 2025).

2. Intuitive Interpretation, Key Examples, and Parameter Regimes

The eluder dimension quantifies the number of sequential probes (input points) one can make before all further queries become nearly determined (to within $\epsilon$) by previous data, given the hypothesis class. It thus upper-bounds the complexity of "surprise" in adaptive sampling, and, dually, lower-bounds the number of queries required to identify a function in $\mathcal{F}$. Crucially, it captures "sequential" complexity, which can be much higher than static complexities like VC dimension.

Examples:

• For thresholds on $\mathbb{R}$, $$\mathcal{H}_\text{th}=\{x\mapsto\mathbb{I}[x>a]: a\in\mathbb{R}\}$$, the eluder dimension is 1: after labeling at any $x_1$, the class is separated into two half-lines, so no further independence is possible (Cherapanamjeri et al., 16 Jun 2025).
• For linear functions in $\mathbb{R}^d$, $$\mathcal{F}=\{x\mapsto \theta^\top x: \|\theta\|_2\le L\}$$, $\dim_E(\mathcal{F},\epsilon)=O(d\log(L/\epsilon))$ (Pacchiano et al., 10 Jan 2024, Grant et al., 2020, Osband et al., 2014).
• For finite classes, $\dim_E(\mathcal{F},\epsilon)\le |\mathcal{F}|-1$.
• For many generalized linear or kernel classes, the dimension interpolates between linear and much higher complexities, depending on smoothness and effective feature dimension.

This dimension can substantially exceed the VC dimension. For example, certain halfspace classes have finite VC dimension yet infinite eluder dimension, illustrating separation in sequential settings (Cherapanamjeri et al., 16 Jun 2025).

3. Central Roles in Statistical and Information-Theoretic Bounds

The eluder dimension is the dominant term in minimax sample and regret complexity for various learning settings:

• Adaptive bandit and contextual bandit regret: Regret bounds via optimism-based or Thompson sampling algorithms scale as $$O\bigl(\sqrt{\dim_E(\mathcal{F},\epsilon) T}\,\bigr)$$ or are otherwise directly controlled by the eluder dimension at scale $\epsilon$, with $\epsilon$ chosen as a function of horizon or tolerance (Pacchiano et al., 10 Jan 2024, Grant et al., 2020, Osband et al., 2014).
• Model-based RL and PSRL: In reinforcement learning, the regret bound for Posterior Sampling for Reinforcement Learning (PSRL) is $\widetilde{O}(\sqrt{d_K d_E T})$, where $d_K$ is Kolmogorov dimension (covering number growth) and $d_E$ is eluder dimension (Osband et al., 2014, Wang et al., 2020).
• Sample complexity in policy class generalization: Policy eluder dimension, a natural extension to policy spaces, yields near-optimal bounds linear in the dimension, independently of state/action cardinality (Mou et al., 2020).
• Unlearning and information deletion: The minimum memory required for exact unlearning (realizability testing) in a central memory model is lower bounded by the eluder dimension of the hypothesis class (Cherapanamjeri et al., 16 Jun 2025).

Lower and upper bound proofs universally exploit sequential information arguments: long eluder sequences allow adversaries or information-theoretic encodings to force high memory, exploration, or sample complexity.

4. Extensions, Generalizations, and Relatives

Metric and distributional generalizations: The eluder dimension has been extended from absolute value metrics to arbitrary bounded metric spaces, e.g., using Hellinger or $\ell_2$ distances on probability distributions in contextual MDPs (Levy et al., 2022). In RL, the Bellman–Eluder (BE) dimension extends the notion to classes of Bellman residuals under families of distributions, governing sample or regret complexity in the presence of function approximation and nonlinearity (Feng et al., 29 May 2025, Jin et al., 2021). The transfer eluder dimension further generalizes the notion to the "information geometry" shaped by external feedback (e.g., language feedback) and can be strictly smaller or larger than the classic form, depending on the informativeness of side-channel supervision (Xu et al., 12 Jun 2025).

Relations to other complexity measures: For finite or Hilbert ball classes, the (critical) information gain in Gaussian process bandits is equivalent (up to absolute constants) to the eluder dimension (Huang et al., 2021). Kolmogorov dimension and covering numbers typically appear as log factors or in multiplicative terms, controlling the non-sequential (batch) entropy of the function class (Osband et al., 2014, Wang et al., 2020).

Algorithmic framework unification: The generalized eluder coefficient (GEC) subsumes the classical and Bellman eluder dimensions, as well as Bellman rank, witness rank, and bilinear, kernel, or PSR-specific definitions, yielding a universal framework for regret bounds in both fully and partially observable reinforcement learning settings (Zhong et al., 2022).

5. Impact in Algorithm Analysis and Model Selection

Sharp sequential complexity analysis via the eluder dimension enables the design of sample- and regret-efficient algorithms in challenging settings:

• Optimism-driven planning and adaptive exploration: Utilization of the eluder dimension allows for theoretically justified confidence bonus schedules, yielding near-minimax exploration algorithms (Pacchiano et al., 10 Jan 2024, Wang et al., 2020, Osband et al., 2014).
• Nonparametric and structured function classes: For smoother-than-Lipschitz classes, eluder dimension analysis reveals improved regret rates as smoothness increases, drawing connections with classical regularization and interpolation theory (Grant et al., 2020).
• Offline and adaptive decision making under constraints: Algorithmic strategies minimizing regret or sample complexity under adaptivity (switching or batching) can attain optimal tradeoffs parametrized by eluder-type complexity (Xiong et al., 2023).
• Principled lower bounds and separations: The existence of hypothesis classes with small VC dimension yet arbitrarily large eluder dimension mandates that sequential adaptivity is fundamentally more challenging (or costly) than batch identification in those settings (Cherapanamjeri et al., 16 Jun 2025).

6. Open Problems, Practical Computation, and Structural Properties

Despite a decade of development, several open questions remain:

• Computation and bounding: For high-dimensional or combinatorial classes (e.g., deep networks), explicit computation or tight bounding of the eluder dimension is challenging. Upper bounds are known for parametric, GLM, and kernelized classes, but nonparametric and structured spaces often require specialized arguments or reductions.
• Robustness to misspecification: The behavior of the eluder dimension under approximate realizability or model misspecification is unresolved. There are ongoing efforts to define robust or "approximate eluder" analogues that interpolate between worst-case and benign regimes (Roy et al., 2019).
• Relationship to alternative ranks: How the eluder dimension interacts, overlaps, or separates from Bellman rank, witness rank, or other policy-oriented complexity metrics continues to be a topic of active research, especially in new RL paradigms and settings with rich side information (Jin et al., 2021, Zhong et al., 2022).
• Practical estimation and model selection: The practical use of eluder dimension as a regularity or model selection criterion is an emerging direction, particularly as algorithmic frameworks are extended to exploit function class properties adaptively.

In sum, the eluder dimension is a sequential, representation-sensitive complexity measure that governs sample, regret, or memory requirements in a breadth of adaptive learning settings, with ongoing work extending its applicability and interpretability to new function spaces, feedback channels, and learning paradigms.