Task-Sufficient Contraction: Source Selection for Machine Information Interfaces

1 Introduction

• For a turbine’s high-wind cut-out decision, average wind speed over the relevant period can suffice even though a forecast cannot determine every local weather consequence.The example illustrates that sufficiency is relative to the declared machine decision, not to weather in general.Source: §1 · p. 1
• Task-Sufficient Contraction asks whether a task can determine a reduced source before downstream choices, with the same result as optimizing on the richer source.The reduction is fixed before the later rate, distortion target, codebook, or optimizer is chosen.Source: §1 · p. 2
• The paper identifies a consumer-specific regret-complete source and an exact quadratic-loss source based on directions in which feasible actions can differ.For finite action sets, the regret reduction preserves the whole one-step rate-regret curve; under fixed-energy scheduling, the quadratic source is centered load, while the optimal water-filled action is too coarse.Source: §1 · p. 2
• Earlier Information Bottleneck, semantic rate-distortion, and goal-oriented quantization results are compared as exact, architecture-conditioned, approximate, failed, or corrected reductions—not as three new contributions.The framework groups them as instances of a reduce-first problem.Source: §1 · p. 2
• The theory is static: it does not establish that a reduction remains valid after interaction changes the state, learner, candidate set, action menu, criterion, or dependence structure.Recursive questions about changing decision problems and representation resolution are left outside the paper.Source: §1 · pp. 2–3

2 Contraction before optimization

• The reduced source depends on the problem: it may be an Information Bottleneck sufficient statistic, a TRI contract, GC, or the load component that can change a feasible action.These are distinct task-specific reductions, not one universal formula.Source: §2 · p. 3
• A certification requires fixing the reduction before the operating point, matching full-source solutions with no worse resource use, pulling reduced solutions back, and preserving claimed constraints.These four checks establish the two-way correspondence and constraint preservation.Source: §2 · p. 3
• Proofs use different mechanisms: encoder averaging in Information Bottleneck, regret-equivalent states for rate-regret, architecture and loss for semantic rate-distortion, and feasible-action differences for constrained GOQ.The valid reduction follows from the structure of each problem.Source: §2 · p. 3
• The TRI contract preserves everything the source tells a receiver about a criterion, while a particular consumer may need less; the two reductions can be nested without requiring one first.The nesting relationship is presented in Figure 1.Source: §2 · p. 3

3 Consumer-specific task-sufficient sources

• A machine consumer is specified by its admissible actions and evaluation loss; the main definitions assume finite source support, and the rate-regret theorem also assumes finitely many actions.The resulting regret profile depends on the conditional law of Y given X and on the declared consumer.Source: §3 · pp. 3–4

3.1 Optimal action and the regret-complete source

• Under unique Bayes actions, regret-equivalent states share an optimal action, but BC keeps only that action while GC retains the costs of all possible mistakes.The two quotients coincide only when the complete regret profile is determined by the Bayes action.Source: §3.1 · pp. 5–6
• With finite source and action sets, GC is the coarsest pointwise regret-preserving quotient and preserves the entire one-step rate-regret curve before any distortion target is selected.States are grouped when every available action has the same regret in both, even if their absolute baseline losses differ.Source: §3.1 · pp. 5–6

3.2 Simple examples

• Squared-error regret is determined by the conditional mean, log-loss quotients coincide with posterior equivalence, and under zero–one loss equal Bayes labels can still hide distinct regret profiles.Thus, in classification, the complete regret vector can retain more than the maximizing class.Source: §3.2 · pp. 6–7

3.3 Relation to the full posterior about the criterion

• The posterior SY retains everything X tells the receiver about Y, GC retains distinctions that can change relative action costs, and BC retains what identifies the best action.These objects form a deterministic hierarchy, but TSC need not begin by constructing SY.Source: §3.3 · p. 7

3.4 Loss decomposition

• Expected loss separates into full-information Bayes loss and expected regret; an information interface can affect only the extra loss from choosing a non-optimal action.The decomposition motivates asking whether that extra loss can be evaluated without retaining the full source X.Source: §3.4 · p. 7

4 Task-Sufficient Contraction

• For every achievable distortion target, the consumer-specific source GC yields the same one-step rate-regret curve as the richer source X.The source is fixed before the target is selected.Source: §4 · p. 8
• Averaging an action channel within each regret-equivalence fibre preserves expected regret, while data processing bounds mutual information by that of X.Conversely, any GC-channel can be pulled back to X with identical information and expected regret.Source: §4 · p. 8
• Under unique Bayes actions, the Bayes quotient captures the zero-regret endpoint, but GC may retain distinctions needed to preserve every feasible action’s regret.The result does not claim GC is minimal among all statistics preserving the optimized curve.Source: §4 · p. 8
• The rate-regret invariance theorem assumes a finite action set; continuous-action quadratic results instead compare full regret functions directly.Source: §4 · p. 8

4.1 Five contraction regimes

• A regret-complete task variable contracts exactly to GC; a relabelling or refinement remains exact but need not be minimal.Source: §4.1 · p. 8
• An arbitrary Q gives an architecture-conditioned reduction only when downstream channels are restricted to factor through Q; this does not show discarded information is unusable by the consumer.The induced distortion also applies to stochastic encoders and intermediate representations generated from Q.Source: §4.1 · p. 9
• A surrogate loss on Q yields only a bounded approximation, while merging states with different admissible-action costs fails exact contraction; another reduction may still correct the failure.Source: §4.1 · p. 9

5 Neighbouring theories and attribution

• Classical sufficiency and Blackwell comparison concern preservation of decision value, while functional compression preserves information needed for a source function.Indirect rate-distortion addresses cases where the encoder observes a different variable from the one used to judge performance.Source: §5 · p. 9
• Bruhacs (2026) studies task-relative preservation under projection and least task-family-sufficient quotients; this paper develops certification for stated optimization families and two consumer-specific characterizations.The paper does not claim novelty for the general idea that tasks determine which distinctions survive.Source: §5 · p. 9
• Sevetlidis (2026a, 2026b) studies Bayes sufficiency and loss shift, whereas this rate-regret result fixes the consumer and tests one reduction across distortion targets.Walsh (2026b) identifies policy equivalence as the exact zero-regret endpoint of a single-cycle rate-regret problem.Source: §5 · pp. 9–10
• Hyeon et al. (2026, Appendix I) preserve the optimal Q-function or action under a tie-breaking rule; this construction preserves relative costs of every feasible action.Source: §5 · p. 10
• For matched candidate sets and cost criteria, Wei et al. (2026) relate absolute cost profiles to regret profiles, while Theorem 1 certifies the complete one-step rate-regret curve with one fixed source.Myhill–Nerode, observational equivalence, and bisimulation offer analogies, but their dynamic closure requirements are outside this static theory.Source: §5 · p. 10

6 Three earlier reductions

• The three reductions are earlier results; their comparison identifies a shared reduce-first pattern rather than three new contributions.Source: §6 · p. 10

6.1 Information Bottleneck: exact pre-optimization contraction

• Under log loss, source states are equivalent exactly when they induce the same posterior over C, and the complete IB optimization family is unchanged on the sufficient statistic (Armstrong, 2026c).The equivalence supports preserving the complete curve, Lagrangian infima, minimisers, and finite-block problem.Source: §6.1 · pp. 10–11

6.2 Semantic rate-distortion: architecture-conditioned contraction

• Semantic marginal rate-distortion can be solved exactly on the chosen variable ϕ*(T), using its induced loss; that variable is the smallest exact source for the unrestricted consumer.If verification depends on whole-pool occupancy, the problem becomes constrained feasibility instead.Source: §6.2 · pp. 11–12

6.3 Goal-oriented quantization: failure of the obvious source and exact correction

• For isotropic quadratic loss, the oracle action preserves the full regret profile and reduces the task problem to MSE coding; smooth interior conditions yield bounds, not exact sufficiency.Source: §6.3 · p. 12
• For affine-action quadratic regret, two states have identical regret profiles exactly when their difference lies orthogonal to feasible-action differences; projection onto those directions is a canonical quotient representative.A component orthogonal to every feasible-action difference shifts all feasible action values equally without changing their relative costs.Source: §6.3 · p. 13
• Under fixed-energy scheduling, centered load is exact, while states with the same water-filling optimum can still give another feasible action different regret (Armstrong, 2026b).Thus the oracle action alone is too coarse once non-optimal feasible actions matter.Source: §6.3 · pp. 12–14

• A certified task interface may let heterogeneous producers serve the same receiver without making their internal representations mutually intelligible.Source: §7 · p. 15
• A reduced external interface does not make internal representations irrelevant or prove that computing the interface is cheap; a rich latent representation may still be best internally.Source: §7 · p. 15
• Representation alignment can support interoperability, but it need not be necessary when the receiving task admits a shared task-sufficient interface.Source: §7 · p. 15
• The paper stops before testing proposed systems benefits such as task performance, transmitted bits or tokens, latency, compute, memory, and energy.Source: §7 · p. 15

• A multi-consumer interface may need distinctions that any one task could ignore, and the result does not cover tasks absent from its specification.Source: §8 · pp. 15–16
• If the declared actions or loss omit a consequential factor, the reduced source may discard information the real problem needs.Source: §8 · p. 16
• A designer-imposed reduced variable can correctly describe a restricted system without showing that the consumer itself had no use for discarded information.Source: §8 · p. 16
• A simpler reduction such as smooth GOQ may provide useful bounds without preserving exact regret, so it should not be called exact sufficiency.Source: §8 · p. 16
• Removing external task distinctions does not establish that the cheapest implementation avoids computing or storing them internally.Source: §8 · p. 16
• Because the theory is static, acting through an interface may change the learner, state distribution, criterion, candidate set, action menu, or dependence structure and invalidate the current reduction.The paper does not define a dynamic update rule for the task-sufficient source.Source: §8 · p. 16

• A forecast may be insufficient to resolve every physical consequence yet still contain what a particular turbine cut-out decision requires.Source: §9 · p. 16
• As heterogeneous machine handoffs grow, a certified task-sufficient interface may avoid sharing or aligning complete internal representations at every handoff.Source: §9 · p. 16
• Task-Sufficient Contraction fixes the reduction before later optimization, and exactness means returning to the richer source cannot improve covered operating points.Source: §9 · pp. 16–17
• The examples distinguish consumer-specific regret preservation from a criterion’s information and from distinctions needed only for the best full-information action.Source: §9 · p. 17
• External information needs should follow the receiving problem rather than the producer’s full representation, but the paper does not answer how those needs change after action.Source: §9 · p. 17