Representation Condition Beyond Realizability

• Representation condition beyond realizability is a framework that relaxes the exact inclusion requirement, allowing models to approximate critical properties within defined bounds.
• It enables tractable learning and control in complex settings such as contextual bandits, active learning, reinforcement learning, quantum models, and geometric probability.
• The approach balances bias-variance trade-offs by quantifying misspecification through explicit measures, leading to robust algorithmic strategies even under partial model specification.

The representation condition beyond realizability generalizes classical realizability regimes by relaxing the requirement that the true system, predictor, or distribution resides exactly within a specified model class. Instead, these frameworks only require that the model class adequately approximates or "represents" critical properties—such as conditional means, decision margins, correlation measures, quantum commutators, or policy values—subject to well-characterized bounds. This conceptual shift enables tractable learning, inference, or control in misspecified or partially specified domains, and has led to substantial theoretical developments in contextual bandits, multiclass classification, reinforcement learning with function approximation, quantum stochastic models, and random geometric structures.

1. Formal Definition and Canonical Examples

Realizability traditionally requires that a target object (e.g., $f^*$, $q^*$, or a probability law) exactly belongs to a function class $\mathcal{F}$ or satisfies a structural equation. The representation condition beyond realizability, by contrast, allows for misspecification, quantified by explicit measures of approximation or algebraic constraints. Key instantiations are:

• Contextual bandits: The best-in-class approximation $$f^* = \arg\min_{f \in \mathcal{F}} \mathbb{E}_{x,a} [(f(x,a) - f^*(x,a))^2]$$ with misspecification error $b = \mathbb{E}_{x,a}[(f^*(x,a) - f^*(x,a))^2]$ (Krishnamurthy et al., 2020).
• Active learning: For every region $Q \subseteq \mathcal{X}$, the in-class minimizer $f^*_Q$ must preserve margin separation up to a nondecreasing transformation $\psi$ of the true margin, expressed as $$\Pr[\operatorname{gap}(\phi(f^*_Q(x)),c) \geq \psi(\operatorname{gap}(\phi(f_\eta(x)),c))] = 1$$ (Ganju et al., 31 May 2025).
• RL with function approximation: Partial $q^\pi$-realizability requires that all policies $\pi \in \Pi$ admit linearly-realizable value functions, interpolating between $q^*$-realizability ($\Pi = \{\pi^*\}$) and full $q^\pi$-realizability ($\Pi =$ all policies) (Karimi et al., 24 Oct 2025).
• Quantum models: Commutation-preservation via bilinear matrix constraints ensures Pauli commutator structure, irrespective of existence of a Hamiltonian/coupling operator representation (Espinosa et al., 2012).
• Point process and random set theory: Regularity conditions, such as existence of a regularity modulus $\chi$ and positive extension of $\Phi$ to a function lattice, guarantee the countable additivity needed for probability law realization (Lachieze-Rey et al., 2011).

These representations allow for rigorous analysis and algorithmic progress where the classical realizability regime is either too restrictive or fails to capture operational properties.

2. Algorithmic Strategies under Representation Conditions

Across domains, the representation condition enables algorithmic approaches that explicitly accommodate model misalignment, often by batchwise or local constrained optimization:

• Contextual bandits: Epsilon-FALCON employs epoch-wise constrained regression, balancing passive (uniform) exploration—required for identifiability in the absence of realizability—with active kernel-based sampling (Krishnamurthy et al., 2020). The model fit is constrained to be close to the uniform-sampling empirical minimizer, yielding robust control of bias from misspecification.
• Active learning: The proposed epoch-based surrogate minimization algorithm fits a model to all queried data each epoch, with local version space refinement based only on observed data and surrogate risk (Ganju et al., 31 May 2025). No global version space tracking is possible beyond realizability, so classification output is improper and aggregated from locally unanimous solutions.
• Quantum feedback control: Representation constraints are encoded as explicit matrix equations (commutation-preserving conditions) to check whether proposed stochastic dynamics can arise from an actual two-level quantum system (Espinosa et al., 2012).
• Geometric probability: Constructive extension theorems leverage regularity modulus bounds and positivity checks for linear functionals on small subalgebras, sidestepping the full Riesz–Markov framework (Lachieze-Rey et al., 2011).

These procedural innovations enable tractable inference, learning, or control even when the target law or system is not exactly realizable.

3. Regret, Complexity, and Misspecification Bounds

Performance in the representation conditioned regime is formally characterized by decomposing guarantees into variance and bias terms, with explicit dependence on the degree of misspecification:

Domain	Regret/Complexity Bound Structure	Misspecification Term
Contextual bandits (Krishnamurthy et al., 2020)	$$O(\sqrt{KTD} + KT\sqrt{b/\sqrt{\epsilon}} + T\epsilon)$$	$KT\sqrt{b/\sqrt{\epsilon}}$
Active learning (Ganju et al., 31 May 2025)	$\widetilde{O}(\sup_{a} \frac{a}{\psi(a)^2})$ multiplier on sample/label complexity	Factor $\sup_a \frac{a}{\psi(a)^2}$ when $\psi \ne \mathrm{id}$
RL w/ partial $q^\pi$-realizability (Karimi et al., 24 Oct 2025)	No efficient (polytime) algorithm in general; NP-hardness/exponential-time lower bounds	—
Quantum realizability (Espinosa et al., 2012)	Hierarchy: commutation-preservation $\subset$ physical realizability	Models preserving commutators need not admit operator representations
Geometric probability (Lachieze-Rey et al., 2011)	Existence iff $\sup_{g\leq \chi} \Phi(g) < \infty$	Regularity modulus bound $\Phi(\chi) < \infty$

The bias–variance trade-off is explicit: as model misalignment ($b$, $\psi$-gap, non-unique extensions) increases, excess risk, regret, or hardness grows proportionally, often optimally balanced by tuning exploration or sample allocation.

4. Theoretical Implications and Relation to Classical Realizability

The representation condition subsumes classical realizability as a special case, but demonstrates fundamentally different behaviors when mispecification is present:

• Robustness: Algorithms like Epsilon-FALCON gracefully degrade to agnostic rates, maintaining tractability up to an additive misspecification penalty, while classical algorithms may suffer linear regret with vanishingly small $b > 0$ (Krishnamurthy et al., 2020).
• Expressivity: In surrogate-based active learning, the representation condition with general $\psi$ enables label complexity bounds comparable to the realizable case, but prior methods reliant on exact realizability or global version spaces fail completely (Ganju et al., 31 May 2025).
• Computational hardness: In RL, expanding the policy class $\Pi$ for partial $q^\pi$-realizability does not circumvent NP-hardness or exponential-time barriers—these are tied intrinsically to combinatorial richness, not only to optimality restrictions (Karimi et al., 24 Oct 2025).
• Quantum mechanics: Commutation-preserving models are necessary preconditions for physical realization, but not sufficient. The existence of operators $H, L$ (quantum Hamiltonian, coupling) corresponds to linear realizability; the more general representation constraints may be satisfied by non-physical models (Espinosa et al., 2012).
• Geometric context: Regularity moduli and extension theorems permit constructive realization of point processes and random sets beyond functional positivity, tying existence to integrability/tightness (Lachieze-Rey et al., 2011).

This suggests that relaxation to representation conditions can preserve desirable properties of realizable models, provided that approximation error, algebraic constraints, or regularity bounds are explicitly incorporated in analysis and design.

5. Domain-Specific Instantiations and Case Studies

Specific domains leverage the representation condition in structurally distinct ways:

• Bandits: Replacement of $f^* \in \mathcal{F}$ by $b = \mathbb{E}[(f^* - f^*(x,a))^2]<\infty$ enables FALCON-type methods to retain tractable batch regression formulations (Krishnamurthy et al., 2020).
• Multiclass classification: Surrogate minimization under the active classification assumption ensures that the version space maintains reliable margin separation, with output constructed from unanimous epochwise proxies (Ganju et al., 31 May 2025).
• Reinforcement learning: Partial $q^\pi$-realizability with parametric policy families leads to inherent computational barriers, even with linear approximability for every $\pi \in \Pi$ (Karimi et al., 24 Oct 2025).
• Quantum open systems: Preservation of the Pauli commutator algebra under stochastic evolution maps to three explicit matrix conditions on drift and diffusion (the “$\Theta$-constraints”), which must be checked before physical realization is considered (Espinosa et al., 2012).
• Random sets/point processes: Extension of positive linear functionals with regularity modulus yields practical realization criteria for correlation measures, two-point covering functions, and constraints like local finiteness, stationarity, or isotropy (Lachieze-Rey et al., 2011).

These case studies illustrate the effectiveness and limitations of the representation condition as an analytical and algorithmic tool.

6. Outlook and Further Structural Assumptions

A plausible implication is that, while representation conditions beyond realizability permit significant flexibility and tractability, escaping computational intractability (particularly in RL with partial $q^\pi$-realizability) requires additional structure, such as unified feature maps, sandwich assumptions, sparsity, small suboptimality gaps, or specialized oracles for cross-policy generalization (Karimi et al., 24 Oct 2025). In geometric probability, constructive algorithms depend critically on the choice of regularity modulus and the verification of tightness, compactness, or invariance conditions (Lachieze-Rey et al., 2011).

The representation condition continues to inform theoretical progress across domains, clarifying the separation between structural approximability and operational computability, and providing a foundation for robust misspecified statistical learning, quantum control, combinatorial optimization, and spatial process modeling.