Outcome-Conditioned Compression

• Outcome-conditioned compression is a method that leverages auxiliary outcomes to condition encoding, thereby reducing uncertainty and focusing on decision-relevant information.
• It uses diverse conditioning variables—such as low-resolution images, semantic labels, or predictive sets—to guide residual coding and enhance reconstruction fidelity.
• The approach yields improved compression rates and semantic consistency, with practical implementations in multiscale, online, and perceptual coding scenarios.

Outcome-conditioned compression denotes a family of compression and summarization strategies in which the encoded representation, reconstruction criterion, or decision rule is conditioned on an auxiliary outcome, side-information variable, or inferential target. In the cited literature, that conditioning variable may be a low-resolution image used to predict missing detail, a lossy reconstruction used to model residuals, a user-defined semantic variable $Y$, a calibrated prediction set for online communication, or a retained $\sigma$-field used to preserve a target $Z$ rather than the full realized data path (Cao et al., 2020, Mentzer et al., 2020, Xu et al., 2023, Ganesan et al., 11 Mar 2025, Chang, 26 May 2026). The common principle is that compression is evaluated relative to what must be preserved: exact pixel recovery, conditional perceptual fidelity, bounded outage, or decision-relevant target coherence.

1. Conceptual scope and recurring design pattern

Across the literature, the “outcome” in outcome-conditioned compression is not fixed to a single mathematical object. It can be a coarse image, a semantic label, a predictive uncertainty set, or a target-conditioned summary. What remains consistent is that the compressor does not model the source unconditionally if a structured conditioning variable can make the remaining uncertainty smaller or more relevant.

Setting	Conditioning variable	Compressed remainder
SReC	low-resolution image	high-resolution detail
BPG-conditioned lossless coding	lossy reconstruction $x_l$	residual $r=x-x_l$
Conditional perceptual compression	semantic side information $Y$	image code $M=f(X,Y)$
Online conformal compression	calibrated prediction set	transmitted symbols vs outages
Target-oriented statistical compression	retained information $\mathcal G_n$	target projection $M_n$

A two-part antecedent appears in “Critical Data Compression,” which separates a message into significant bits $A^n$, compressed losslessly, and a residual $\sigma$0, compressed statistically or with a lossy codec. The split is selected by a criticality criterion derived from finite-difference complexity measures, including

$\sigma$1

and

$\sigma$2

with near-maximal values allowed in practice (Scoville, 2011). This suggests an early formulation of the same structural idea: preserve the part regarded as signal or inferentially useful, and condition the remaining code on that retained component.

2. Formal objectives: conditionality in coding and in inference

A direct information-theoretic formalization appears in conditional perceptual compression. The unconditional perceptual divergence $\sigma$3 is replaced by the conditional divergence $\sigma$4, and the corresponding rate-distortion-perception function becomes

$\sigma$5

The framework inherits monotonicity and convexity properties, has one-shot and asymptotic achievability results, proves that $\sigma$6, and establishes the conditional analog of the MSE trade-off

$\sigma$7

In this formulation, the relevant fidelity criterion is not merely realism of $\sigma$8, but realism within each semantic slice $\sigma$9 (Xu et al., 2023).

A broader statistical formulation appears in “Target-Oriented Statistical Compression.” There the basic object is the triple

$Z$0

where $Z$1 is the compression map, $Z$2 is the retained information field, and $Z$3 is the conditional target process. With a decreasing filtration,

$Z$4

the process

$Z$5

is a reverse martingale satisfying

$Z$6

almost surely and in $Z$7. Exact sufficiency is treated as lossless compression, whereas approximate summaries are described through the reverse quasi-martingale defect

$Z$8

The observable diagnostic

$Z$9

is used as a stability proxy rather than as an unbiased estimator of $x_l$0 (Chang, 26 May 2026).

3. Residual, multiscale, and two-part codecs

Lossless image coding provides several concrete engineering realizations of outcome-conditioned compression. In SReC, the full image $x_l$1 is recursively average-pooled and rounded to produce a multiscale pyramid. The coarsest image $x_l$2 is stored directly as raw pixels; rounding residues $x_l$3 for $x_l$4 are stored with two bits per pixel/channel; and the higher-resolution levels are compressed one at a time with a learned conditional super-resolution model. For each $x_l$5 step, only three pixels in each $x_l$6 block must be encoded, because the fourth is recovered exactly from the averaging constraint: $x_l$7 The channel model uses a discretized mixture of logistics with $x_l$8 mixture components per channel, and training minimizes

$x_l$9

The paper reports that moving from a one-level to a three-level system improves Open Images PNG compression from 3.87 bpsp to about 2.69 bpsp, while a four-level version yields only marginal additional gain; the full benchmark includes 4.29 bpsp on ImageNet64 and 2.70 bpsp on Open Images PNG with about 4.20M parameters (Cao et al., 2020).

A related strategy uses a lossy codec as the conditioning source for a lossless residual coder. “Learning Better Lossless Compression Using Lossy Compression” first applies BPG, obtaining a lossy reconstruction $r=x-x_l$0, then encodes the exact residual

$r=x-x_l$1

with a neural residual compressor modeling

$r=x-x_l$2

The model factorizes spatially and uses a weak autoregressive factorization across RGB channels with $r=x-x_l$3 discretized logistic mixture components. Arithmetic coding yields a fully lossless scheme even if the residual model is inaccurate. The paper further introduces a Q-Classifier predicting $r=x-x_l$4 and a per-image $r=x-x_l$5-optimization that rescales logistic scales via $r=x-x_l$6. On Open Images, it reports 2.790 bpsp for the residual-conditioned method versus 2.991 bpsp for L3C, 4.005 bpsp for PNG, 3.055 bpsp for JPEG, 3.047 bpsp for WebP, and 2.867 bpsp for FLIF; about 42% of total bits are used to store $r=x-x_l$7 (Mentzer et al., 2020).

These systems are closely related to the earlier two-part signal/noise decomposition of critical data compression. There, the retained component $r=x-x_l$8 is encoded exactly, while $r=x-x_l$9 is either modeled statistically or compressed with JPEG or JPEG2000. The method is explicitly motivated by the claim that the most significant bits are typically more compressible than the least significant bits, and it treats the residual as a “noise function” to be sampled or decoded and then added back during reconstruction (Scoville, 2011). This suggests a persistent architectural motif: transmit a structurally informative macrostate first, then code only the remaining uncertainty conditional on that macrostate.

4. Semantic conditioning, perceptual quality, and randomness

Outcome-conditioned compression is not restricted to lossless recovery of exact data. In conditional perceptual compression, $Y$0 represents user-defined semantics such as digit identity in MNIST or scene layout in Cityscapes. The central motivation is that unconditional perceptual codecs can produce reconstructions that are visually plausible but semantically wrong: a “3” may become a plausible “7,” or one bedroom layout may become another. The proposed codec therefore losslessly encodes $Y$1, encodes the residual message $Y$2 with a conditional entropy model, and decodes with $Y$3, with total expected rate $Y$4. Under mild assumptions and when $Y$5 is deterministic from $Y$6, the scheme satisfies

$Y$7

The two-stage training procedure first optimizes

$Y$8

and then trains a conditional perceptual decoder $Y$9 under

$M=f(X,Y)$0

yielding

$M=f(X,Y)$1

Empirically, the paper evaluates MSE, FID, ConFID, classification accuracy, and mIoU, and reports that the conditional codec preserves both perceptual realism and semantic consistency across bitrates (Xu et al., 2023).

A central controversy concerns stochasticity in perceptual decoders. The same paper provides lower bounds on common randomness: $M=f(X,Y)$2 for the unconditional case and

$M=f(X,Y)$3

for the conditional case. The stated implication is that randomness is theoretically necessary for perfect perceptual quality at sufficiently low rates, and that conditioning on $M=f(X,Y)$4 reduces the randomness burden because the decoder already knows $M=f(X,Y)$5 (Xu et al., 2023). A common misconception is therefore that side information alone eliminates the need for stochastic generation; the cited lower bound does not support that conclusion.

5. Predictor-aware and zero-delay outcome adaptation

Online conformal compression applies the same conditioning principle to real-time sequential communication. The source sequence $M=f(X,Y)$6 is discrete, the alphabet $M=f(X,Y)$7 is finite, the decoder must reconstruct $M=f(X,Y)$8 with zero delay, and both encoder and decoder share a pre-trained autoregressive predictor

$M=f(X,Y)$9

The distortion is 0-1 loss,

$\mathcal G_n$0

so errors are treated as outages. The system constructs a calibrated prediction set

$\mathcal G_n$1

with online update

$\mathcal G_n$2

If $\mathcal G_n$3, the encoder transmits enough information to recover it losslessly using the restricted distribution

$\mathcal G_n$4

If $\mathcal G_n$5, the encoder allows an outage; in the asynchronous variant it signals an erasure with

$\mathcal G_n$6

The main guarantee is deterministic, sequence-wise, anytime, and distribution-free: $\mathcal G_n$7 The experiments compare OCC with Dropout-LLMZip and with the idealized offline benchmark BCC, and report a compression rate comparable to BCC despite the online setting (Ganesan et al., 11 Mar 2025). The guarantee, however, is specifically about outage frequency rather than semantic fidelity or general distortion measures.

6. Sufficiency, coherence loss, and boundary declarations

In the target-oriented statistical literature, outcome-conditioned compression becomes a criterion for judging summaries relative to an inferential objective rather than a code-length objective. Exact sufficiency is treated as lossless compression: in Bernoulli and Poisson models the count $\mathcal G_n$8 is sufficient, and in the normal model $\mathcal G_n$9 or $M_n$0 are sufficient depending on the target. Approximate summaries—including penalized estimators, principal components, learned hidden states, and ridge logistic predictions—need not preserve the exact reverse-martingale identity and may exhibit nonzero coherence defect $M_n$1. The practical diagnostic is $M_n$2, interpreted as a stability signal: small $M_n$3 suggests stabilization of the conditional target, while persistent large $M_n$4 indicates slow convergence or an inadequate/coarsened representation (Chang, 26 May 2026).

The same framework is applied to sequential boundary monitoring. For a binary target, exact boundary degeneracy is a statement about the limit $M_n$5, not about any finite $M_n$6. Practical declarations therefore require joint assessment of boundary closeness

$M_n$7

uncertainty width $M_n$8, and trajectory stability $M_n$9, through the three-condition rule

$A^n$0

When the summary is exactly sufficient and $A^n$1, the rule collapses to the two-condition version

$A^n$2

This directly addresses a recurrent misconception: finite near-zero or near-one estimates are only symptoms, not proofs, of boundary degeneracy. In Bernoulli all-failure runs, for example,

$A^n$3

under a $A^n$4 prior shows practical boundary pressure, not exact zero. In logistic regression, separation can drive coefficients toward infinity, but finite fitted probabilities remain interior; the relevant target is the stabilized predictive probability surface $A^n$5, not the diverging coefficient vector. Gaussian and Poisson examples serve as exact-sufficiency calibration cases with $A^n$6, whereas the logistic case remains approximate and requires the additional stability screen (Chang, 26 May 2026).