The Arbitrary-Placement Problem in Entropy-Minimizing Selection, and a Residual-Entropy Formulation

Alyssa H. Shin; Claire H. Shin

cs.LGOct 5, 2026

The Arbitrary-Placement Problem in Entropy-Minimizing Selection, and a Residual-Entropy Formulation

Authors: Alyssa H. Shin, Claire H. Shin

Organizations: Division of Biology and Biological Engineering California Institute of Technology · Robert Frederick Smith School of Chemical and Biomolecular Engineering Cornell University

Abstract

Entropy-based selection objectives suffer from a fundamental degeneracy: minimizing Shannon entropy $H(p_A)$ rewards confident selection regardless of whether the selected candidate is informative. We address this limitation with the residual entropy $D = H(p_A) - H(p_β)$ , where $p_β$ is induced by candidate trust weights. We prove the exact identity $D = -\mathrm{KL}(p_A\Vert p_β) - Δ$ , where $Δ$ measures whether the score-induced distribution and trust profile favor the same candidates. Boundary cases establish basic safety: under uniform trust, $D\leq0$ automatically, so an equal-trust, non-starving state is never penalized, while at any one-hot limit, $D\to0$ regardless of the selected candidate. For the intermediate regime where selection occurs, we prove that $D\leq0$ when candidate ordering by trust agrees pairwise with ordering by informativeness, and derive a tighter certificate based on the leading candidate's margin over its competitors. These results are independent of the candidate-scoring function and apply to both stationary and dynamically changing information. Experiments with a gradient-based mixture-of-experts router confirm that the ordering conditions can hold during real optimization and show that correct ordering improves downstream performance when candidates are non-interchangeable and selections are used directly rather than averaged. Beyond routing, margin-based reweighting matches or outperforms fixed-strength baselines in a class-imbalance task, while informative selection in a production video-prediction system reduces MSE by approximately 20 $\%$ and transfers to a related species. Residual entropy, therefore, provides a safety criterion for selection and a usable signal for deciding when that selection is informative.

Figures & tables

Figure 1: From entropy degeneracy to residual-entropy correction. Propositions 1–2 characterize the problem and correction and Corollaries 1–4 provide boundary results and sufficient sign certificates.

Study / target	Entropy alone / alternatives	Residual entropy
(i) Initialization: objective separation	0.0047 / 0.0050; ratio 0.93	$-0.3641$ / $-0.0134$ ; ratio 27.11
(ii) Functional form: scoring / normalization	Ratios 0.64–0.93	Rational / log: 27.11 / 23.67; sigmoid / softmax: 1.00 / 1.00
(iii) Alternatives: separation / collapse	Penalty ratios: 0.62–0.88; KL variant: 90/90 retain initial window	$-0.3641$ / $-0.0134$ ; lower objective at informative initialization
(iv) Class imbalance: identification	Selection rate 0.775	0.938 ( $1.21\times$ ; $p=0.003$ )
(v) Full-batch MoE: participation boundary	5/6 lower-bounded at floor $>0$ ; no transition	Threshold 0.07–0.10; 0/6 below, 4/6 above
(vi) Streaming MoE: boundary robustness	5/6 lower-bounded at floor $>0$ ; no transition	Threshold 0.062–0.070; 0/6 below, 4/6 above

Table 1: Seven studies: questions and quantitative evidence. Objective pairs are informative / elsewhere initialization means.

Setting	Uniform	Fixed $5\times$	Conf.-scaled	Hurt (fixed / conf.)	$p$ (conf. vs. fixed)
Digits $T{=}5$ , 8% ( $n{=}80$ )	0.8466	0.8727	0.8732	2/80 / 0/80	0.159
Digits $T{=}5$ , 2% ( $n{=}15$ )	0.6776	0.6909	0.7018	5/15 / 0/15	0.023

Table 2: Confidence-modulated reweighting at both scarcity levels: mean held-out minority-class recall after retraining with each oversampling scheme. “Hurt” counts seeds below uniform training; the final column compares confidence-scaled with fixed

5\times

weighting. The same

s{=}0.25

is used.

Noise	Threshold	Clamped acc.	Peaked acc.	Winner
0.0	0.10–0.12	0.960	0.964	peaked ( $+0.3$ pt)
0.5	0.09–0.10	0.924	0.918	clamped ( $+0.6$ pt)
1.0	0.07–0.08	0.803	0.783	clamped ( $+2.0$ pt)
1.5	0.02–0.05	0.660	0.645	clamped ( $+1.5$ pt)
$\geq 2.0$	vanished	–	$0.37$ – $0.54$	n/a (single regime)

Table 3: Floor threshold and accuracy vs. task noise (study (v), residual, fair protocol,

n{=}15

seeds/noise level). “Threshold” separates clamped (

\gamma_{f}\approx 0

) from peaked (

\gamma_{f}\approx 4

); “Winner” has higher test acc. (Appendix D.2 ).

Figure 2: Real-pipeline architecture. An encoder-only upstream model identifies a sweet-spot window among five candidates. That window is then oversampled when training a separate encoder-decoder predictor. The exact upstream score is derived in Appendix F.1 .

Arm	Mean MSE	SD
sweetspot ( $\mu^{\ast}$ , ours)	92.4	82.0
uniform (no oversampling)	104.0	82.5
mc_var (BALD-style)	106.5	91.7
random	116.1	85.3
raw_var	139.9	115.6

Table 4: Bacillus, all five windows pooled (13 evaluation points: 5 held-out synthetic + 5 video1 + 3 video2 windows; lower MSE is better).

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Synthetic landscape: objective is blind to real signal under $H(p_{A})$ alone, but not under the residual form.

Family	Objective	obj(info.)	obj(else.)	ratio	$\gamma$ (i/e)
Rational (main)	$H(p_{A})$ alone	+0.0047	+0.0050	0.93	7.43/7.87
Rational (main)	$H(p_{A})-H(p_{\beta})$	$-0.3641$	$-0.0134$	27.11	0.88/6.58
Sigmoid	$H(p_{A})$ alone	+1.1109	+1.3517	0.82	6.52/6.37
Sigmoid	$H(p_{A})-H(p_{\beta})$	$-0.2502$	$-0.2502$	$1.00$	0.16/0.16
Log	$H(p_{A})$ alone	+0.0042	+0.0046	0.91	7.33/7.74
Log	$H(p_{A})-H(p_{\beta})$	$-0.4139$	$-0.0175$	23.67	0.86/7.05

Appendix

Table 5: Full functional-form and aggregation-scheme generality table (study ii).

Objective	obj(info.)	obj(else.)	ratio	$\gamma$ (i/e)
$H(p_{A})$ alone (reference)	+0.0047	+0.0050	0.93	7.43/7.87
$H(p_{A})-H(p_{\beta})$ (ours, reference)	$-0.3641$	$-0.0134$	27.11	0.88/6.58
$H(p_{A})+1.0\,(\gamma/\gamma_{\max})^{2}$	+0.1004	+0.1137	0.88	5.27/5.65
$H(p_{A})$ , tighter $\gamma$ clamp ( $\gamma_{\max}{=}2.0$ vs. $20.0$ )	+0.3830	+0.4543	0.84	2.00/2.00 ‡
$H(p_{A})+5.0\sum_{t}(p_{\beta,t}-1/T)^{2}$	+1.3691	+2.1906	0.62	0.15/5.51
$H(p_{A})-\mathrm{KL}(p_{A}\\|p_{\beta})$	+0.0040	+0.0048	0.83	7.56/7.91

Appendix

Table 6: Full alternative-regularizer table (study iii), including the KL-divergence variant. Same landscape and protocol as Table 5 (30 seeds

\times

\mu

-inits).

Two-proportion $z$ -test on raw selection rate: $z=2.929$ , $p=0.003$ (significant)
Phase 1 (outer loop)	$H(p_{A})$ alone	Residual (ours)
Relative selection improvement (true minority class)	$1.00\times$ (0.775)	$1.21\times$ (0.938)
Phase 2 (downstream)	Minority recall	Accuracy
Uniform (no emphasis)	0.8466	0.9635
Residual-selected class, fixed $5\times$	0.8727	0.9687
$H(p_{A})$ -selected class, fixed $5\times$	0.8670	0.9675

Appendix

Table 7: Digits (

T{=}5

; minority class retained at 8%, approximately 10 training images; 80 seeds and five

\mu

-initializations per objective). Phase 1 reports the true-minority selection rate relative to

H(p_{A})

alone. Phase 2 reports minority recall and overall accuracy after separately retraining with a fixed

5\times

weight on the class selected by each objective. The 2% variant appears in Appendix C.3 .

floor	lower bdd. (of 6)	mean $\gamma_{f}$	lower bdd. (of 6)	mean $\gamma_{f}$	clip effect
	$H(p_{A})$ alone		Residual (ours)
$0.00$	$4.00$	$6.76$	$0.00$	$0.01$	literal freeze
$0.05$ – $0.07$	$5.00$	$6.77$	$0.00$	$0.01$	trickle
$0.10$ – $0.30$	$5.00$	$6.79$	$4.00$	$\approx 4.06$	trickle
$1.00$	$5.00$	$6.82$	$4.00$	$4.08$	trickle

Appendix

Table 8: Study (v), floor sweep under the fair protocol (one

\mu

-init per candidate window, best-of-6 by lowest final objective, 9 seeds per floor,

E_{\text{TRUE}}=2

, noisy hard-task variant). An expert is lower-bounded when its learning-rate multiplier

\mathrm{lr\_mult}_{e}=\mathrm{clip}(\beta_{e}/\bar{\beta},\,\mathrm{floor},\,5.0)

sits at the lower bound, i.e.

\mathrm{lr\_mult}_{e}=\mathrm{floor}

. Only floor

=0

is literal starvation (exact zero gradient). For floor

>0

these experts still train, rate-capped at floor.

Noise	static_low	static_high	low $\to$ high@75	high $\to$ low@75	Winner
$0.0$	$0.957$	$0.963$	$0.959$	$0.963$	high ( $+0.5$ pt)
$0.5$	$0.922$	$0.916$	$0.922$	$0.916$	low ( $+0.5$ pt)
$1.0$	$0.804$	$0.783$	$0.804$	$0.783$	low ( $+2.1$ pt)
$1.5$	$0.644$	$0.643$	$0.644$	$0.643$	low ( $+0.1$ pt)
$2.0$	$0.537$	$0.537$	$0.537$	$0.537$	tie
$2.5$	$0.442$	$0.441$	$0.442$	$0.441$	low ( $+0.1$ pt)

Appendix

Table 9: Study (v), floor-adjustment strategy comparison across seven noise levels (residual objective, fair protocol,

n{=}25

per cell). floor

{}_{\text{low}}=0.05

, floor

{}_{\text{high}}=1.0

. Both schedules switch at step

75

150

	Lower bdd. (of 6)	Ensemble test acc.
$H(p_{A})$ alone	5.00	$0.9628\pm 0.0092$
Switch-style aux ( $\alpha{=}0.01$ )	5.00	$0.9631\pm 0.0094$
Residual (ours)	0.00	$0.9570\pm 0.0103$

Appendix

Table 10: Study (v) extended with a Switch-Transformer-style load-balancing loss (App. D.3 ), 6-candidate fair protocol (25 seeds, best-of-6 by lowest final objective), same protocol as Table 8 .

$\alpha$	Lower bdd. (of 6)
0.01	5.00
0.1	5.00
1.0	0.00
10.0	0.00

Appendix

Table 11: Load-balancing coefficient sweep, 6-candidate fair protocol (10 seeds each, best-of-6 by lowest final objective), floor

=0.05

(never

0

, so counts below are lower-bounded, not literal starvation). Lower-bounding is fixed monotonically by a large enough

\alpha

floor	lower bdd. (of 6)	mean $\gamma_{f}$	lower bdd. (of 6)	mean $\gamma_{f}$	clip effect
	$H(p_{A})$ alone		Residual (ours)
$0.000$	$4.00$	$6.76$	$0.00$	$0.01$	literal freeze
$0.062$	$5.00$	$6.77$	$0.44$	$0.46$	trickle
$0.070$	$5.00$	$6.77$	$3.56$	$3.61$	trickle
$0.100$ – $1.000$	$5.00$	$6.78$ – $6.82$	$4.00$	$\approx 4.07$ – $4.08$	trickle

Appendix

Table 12: Study (vi), floor sweep under the fair protocol (one

\mu

-init per candidate window, best-of-6 by lowest final objective, 9 seeds per floor,

E_{\text{TRUE}}{=}2

, noisy hard-task variant): lower-bounded-candidate count and mean converged

\gamma_{f}

, both objectives. Same lower-bounded/literal-starvation distinction as Table 8 (floor

=0

only).

	$H(p_{A})$ alone	Residual (ours)
Default (floor $=0.05$ , clean)	$0.960\pm 0.008$	$0.960\pm 0.010$
Hard variant (floor $=0$ , noisy)	$0.756\pm 0.016$	$0.805\pm 0.017$
Floor $=1$ (clean)	$0.960\pm 0.008$	$0.962\pm 0.007$

Appendix

Table 13: Study (vi), fair protocol (one init per window, best-of-6 by lowest final objective, 9 seeds). Same three floor/noise conditions as study (v)’s corresponding ablation (Appendix D.2 ).

gap	mean diff	95% CI	$p$	$d_{z}$
$0.00$	$-0.0029$	$[-0.0056,-0.0003]$	$0.042$	$-0.30$
$0.05$	$+0.0069$	$[+0.0035,+0.0103]$	$0.0003$	$+0.55$
$0.10$	$+0.0144$	$[+0.0108,+0.0180]$	$<10^{-4}$	$+1.08$
$0.20$	$+0.0358$	$[+0.0314,+0.0404]$	$<10^{-4}$	$+2.20$
$0.35$	$+0.0756$	$[+0.0704,+0.0810]$	$<10^{-4}$	$+3.88$
$0.50$	$+0.1414$	$[+0.1359,+0.1470]$	$<10^{-4}$	$+7.04$

Appendix

Table 14: Ordering payoff as a function of candidate interchangeability: invest in the chosen expert, then deploy it. Results are paired within run. 25 seeds per cell, floors pooled (

n{=}50

d_{z}

is the paired effect size relative to its across-seed variation. Values around 1 or larger are conventionally large.

Domain	Domain distance	Result
Bacillus (native)	native	$\sim$ 19–20% MSE improvement over mc_var ; 66/90 pairwise wins (5-seed re-validation); Table 4
Tetraselmis sp. (microalga)	diff. species, same modality	unanimous vs. uniform (3/3 seeds, 6/6 videos); only $+2.5\%$ , 4/6 vs. mc_var (1 seed)
DIC-C2DH-HeLa (mammalian)	diff. organism & modality	mixed vs. uniform (2/3, n.s., $p{=}0.36$ ); mc_var wins vs. sweetspot , $-5.2\%$ , 1/6 (1 seed)

Appendix

Table 15: Downstream case study: does the correctly-identified sweet-spot window improve a separately-developed video predictor?

Arm	W1	W2	W3	W4	W5
uniform	208.3	66.7	219.6	265.2	189.7
sweetspot ( $\mu^{\ast}{=}$ W3)	231.9	53.8	221.3	217.2	171.9
mc_var ( $=$ W2)	246.5	80.1	200.6	309.9	150.9

Appendix

Table 16: Study (vii) downstream results on the Bacillus held out synthetic subset (Tier A), reporting per window MSE for three arms.

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