Task-Sufficient Contraction: Source Selection for Machine Information Interfaces

The gist The paper studies when one such reduction preserves the complete downstream problem family, a property termed Task-Sufficient Contraction.

Task-Sufficient Contraction

This concept investigates whether a declared task can determine a reduced source before the later operating point is selected, such that solving the downstream problem after the reduction gives the same result as solving it on the richer source The central question is therefore: Can a declared task determine a reduced source before the downstream encoder, codebook, rate, distortion target, or optimizer is chosen, such that solving the downstream problem after the reduction gives the same result as solving it on the richer source?

Key Findings and Results

The paper makes three connected contributions First, it develops a common certification framework for task-determined source reductions that remain exact across a declared downstream optimization family, with the reduction fixed before the later operating point is chosen. Second, for a fixed consumer, it identifies the consumer-specific regret-complete source GC and proves that, for finite action sets, replacing the richer source by GC leaves the entire one-step rate-regret curve unchanged. Third, for quadratic loss on an affine feasible-action set, the paper characterizes the exact source directly from the directions in which feasible actions can differ

Types of Reductions and Regimes

The framework distinguishes between several types of reductions:

1. Exact consumer-complete contraction: This occurs if a proposed variable Q is regret-complete, meaning gX(a) = dQ(Q, a) ∀a If Q is merely a relabelling or refinement of that quotient, solving on Q is exact but not minimal

2. Architecture-conditioned contraction: This occurs when the architecture deliberately restricts the downstream action channel to factor through Q, leading to E[gX(A)] = E[dQ(Q, A)] This includes stochastic encoders and intermediate representations generated from Q

3. Surrogate contraction: A proposed Q may support a simpler approximate loss de(Q, a) satisfying α de(Q, a) ≤ gX(a) ≤ β de(Q, a) in a stated regime Working on Q then comes with a bound, but it is not an exact reduction

4. Candidate reduction failure: A proposed task variable may merge states that still assign different costs to some admissible action, meaning the variable is then too coarse for exact contraction

5. Corrected contraction: Failure of the obvious task variable need not mean that the full source is irreducible, and a different reduced source may still preserve the downstream problem exactly

Consumer-Specific Source Construction

For a fixed consumer, it identifies the consumer-specific regret-complete source GC This is defined as GC:= [X]∼G where x ∼G x' ⇐⇒ gx(a) = gx'(a) for every a ∈ A It records what each possible mistake would cost, while BC records what the consumer should do with full information The chain SY 7→ GC: consumer-specific coarsening is one application of TSC, not a required sequence for every problem

Relationship to Prior Work

The framework grew out of several earlier reductions, including an Information Bottleneck result, semantic rate-distortion, and goal-oriented quantization results These earlier results are revisited as evidence for the framework rather than three new contributions The comparison distinguishes exact pre-optimization contraction from architecture-conditioned reduction, approximate reduction, candidate failure, and corrected contraction

Systems Implication

Task-Sufficient Contraction suggests another possibility where an external interface can be built around a certified reduced source rather than either party’s complete internal representation This distinction is particularly natural for agent protocols in which the remote system is intentionally opaque The systems implication is narrower: Representation alignment can be sufficient for interoperability, but it need not be necessary when the receiving task admits a shared task-sufficient interface

Scope and Boundaries

The reduction is only as good as the consumer specification, meaning if the declared actions or loss leave out a consequence that actually matters, the reduced source may discard information the real problem needs The present theory is static, meaning it does not compare source requirements across changing decision problems or define a dynamic update rule for the task-sufficient source The results here do not compare source requirements across such changes or define a dynamic update rule for the task-sufficient source

Conclusion

Task-Sufficient Contraction says that the reduction can sometimes be made before the later optimization begins An exact result means that returning to the richer source cannot improve any of the operating points covered by the theorem The consumer-specific regret construction gives another exact case, keeping every distinction that can change the relative cost of an available action and discarding distinctions that cannot The chain separates three questions: What information does X provide about Y? Which of those distinctions can change the cost of an action available to this consumer? Which distinctions are still needed if only the best full-information action matters

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1. Identify task-specific source reductions before optimizing downstream parameters to preserve problem fidelity, as described by Task-Sufficient Contraction. This allows a machine to act on a reduced source before the later information problem is optimized, ensuring that the reduced source remains valid across the range of operating points covered by the result.

2. Implement consumer-specific source selection by merging states only when every available action has the same regret in both, leading to the identification of a consumer-specific source GC which proves that for finite action sets, it leaves the entire one-step rate-regret curve unchanged.

3. Characterize exact reductions for quadratic loss on affine feasible-action sets by using the projection onto the directions in which feasible actions can differ, yielding a canonical reduced source like centered load under fixed-energy scheduling.

4. Employ a certification framework that ensures Every feasible full-source solution has a reduced counterpart that preserves the task-relevant quantity and is no worse in the optimized resource, guaranteeing exactness across the declared optimization family.

5. Utilize a nested source hierarchy where SY keeps everything X tells the receiver about Y, which can be refined to "GC (preserving relative action costs) and finally to BC" (retaining only the optimal-action endpoint).