JANUS: Online Jacobian-Aligned Infill for Black-Box Optimization

Hongyuan Yu; Pufan Xu; Jiaojiao Yi; Yiding Tian; Mingrui Sun; Jiayuan Lu; Changyuan Wen

cs.NEAug 24, 2026

JANUS: Online Jacobian-Aligned Infill for Black-Box Optimization

Authors: Hongyuan Yu, Pufan Xu, Jiaojiao Yi, Yiding Tian, Mingrui Sun, Jiayuan Lu, Changyuan Wen

Organizations: Multimedia Department, Xiaomi Inc. · School of Integrated Circuits, Tsinghua University

Abstract

Population optimizers such as CMA-ES, DE, and multi-objective evolutionary algorithms drive search mainly through selection signals that are scalar or rank based: such a signal indicates that one candidate outperforms another, but not the local direction responsible for the improvement. JANUS (Jacobian-Aligned Newton-Unified Search) is a plug-and-play infill module that extracts this missing local geometric signal without replacing the host optimizer. It estimates a local Jacobian from the recent evaluation trace; the same Jacobian yields both a damped Gauss--Newton exploitation candidate and a trace-preserving exploration metric, reserving a fraction of the host's per-generation candidate slots for geometry-guided infill rather than spending evaluations on top of the host's budget. Unlike MetaBBO methods, JANUS needs no offline training or task distribution, estimating this geometry on the fly from the current run alone, while the host keeps full control of selection, survival, covariance adaptation, and step-size control. Under same-protocol comparisons, JANUS improves the CMA-ES host on 11--15/16 BBOB functions across $d\in\{30,100,500\}$ . It also attains the best mean error on 13 of the 16 functions at $d{=}500$ in the complete NN-BBO/MetaBBO baseline comparison, with no training cost, and yields a $936\times$ geometric-mean improvement over the host on a $d{=}1000$ BBOB subset. On structured and multi-objective tasks, JANUS gives the best mean cost on 1135-dimensional UAV path planning ( $-12.8\%$ vs.\ the strongest baseline), and it improves SMS-EMOA/AGE-MOEA2 hosts on 12/38 multi-objective tasks with zero significant regressions. Code is available at https://github.com/hongyuanyu/JANUS.

Figures & tables

Figure 1: JANUS at a glance. Headline BBOB, UAV, and MOO results.

Symbol	Meaning
$x,f(x)$	Candidate and evaluated black-box value.
$\mathcal{W}$	Recent evaluated window used for local fitting.
$r(x)$	Observed components/objective vector when available.
$s(x)$	Residual coordinates, observed or scalar-derived.
$J$	Online residual Jacobian fitted on $\mathcal{W}$ .
$G=J^{\top}J$	Gauss–Newton Hessian approximation; metric for GN exploitation and CMIX exploration.

Table 1: Core notation used in the method.

Method	Mean	Median	Rank-1
JANUS	9,600	6,644	15
DE	11,006	9,261	1
JDE21	11,017	8,579	1
RLDEAFL	11,411	8,197	0
CMA-ES	14,486	10,908	0

Table 2: UAV main comparison at 2,500 FEs (

d{=}1135

). Lower is better; full 12-method, 28-terrain table and path visualizations are in App. B.3 .

	Host	NN-BBO / MetaBBO baselines				Ours
Func	CMA-ES	RS	SYMBOL	RLEPSO	LDE	JANUS-GN	JANUS
$f_{4}$	4.2e4	3.7e5	1.2e5	2.5e4	1.6e4	1.6e4	5992
$f_{6}$	1.8e6	1.6e7	5.6e6	9.2e5	9.9e5	7103	2113
$f_{7}$	1.5e4	3.9e4	1.8e4	1.2e4	4824	–	988
$f_{8}$	5.9e8	5.9e8	4.7e7	1.5e7	1.3e7	3072	1426
$f_{9}$	6.1e8	4.1e8	2.4e6	9.5e5	1.8e5	508.7	1323

Table 3: BBOB

d{=}500

: mean final error, 15 held-out functions with baseline coverage (

[-5,5]^{d}

, 20,000 FEs;

f_{20}

omitted, no baseline data). NN-BBO baselines and JANUS rerun under our protocol (10 seeds); JANUS-GN from the host-controlled run (30 seeds). Blue

=

best, green

=

second. Lower is better. Full 10-baseline comparison in App. B.1 , Table 12 . The Wins row counts best (blue) cells within this reduced 7-column slice , in which the ablation column JANUS-GN can hold the best cell; this differs from the “13/16” figure in the text, which counts functions where full JANUS is best among all baselines over the complete 16-function set. Sig. : Bonferroni-corrected wins/ties/losses vs JANUS (

p<0.05

), by a paired Wilcoxon signed-rank test over the 15 per-function final errors (each function’s mean over the first 10 JANUS seeds, paired against the corresponding baseline mean); the pairing unit is the function identity, so this is a cross-function comparison of aggregated final errors rather than a within-function seed test.

Figure 2: Per-function BBOB host comparison across $d\in\{30,100,500\}$ . Color encodes $\log_{10}(\text{JANUS variant}/\text{CMA-ES host})$ under the same protocol; blue $=$ JANUS better, red $=$ worse.

Panel of Figure 3 — Figure 3: UAV convergence and path quality on two representative terrains. JANUS separates early and produces smooth low-cost paths; JANUS-GN follows more conservatively, while other baselines often plateau higher under the 2,500-FE budget.

Regime	Method	Avg. Rank	Norm. HV	$+/{\approx}/-$
Low	JANUS-SMS 5%	2.00	0.994	3/15/0
(2–3 obj.)	JANUS-SMS 2%	2.50	0.993	–
	SMS-EMOA	3.11	0.972	–
	NSGA-II	3.61	0.967	–
	MOEA/D	5.00	0.926	–
	Random	6.50	0.756	–

Table 4: MOO main results. 20,000 evaluations, 10 runs/task (180 low-, 200 many-objective), unified post-hoc HV reference points. Blue cell

=

best, green cell

=

second per regime.

+/{\approx}/-

: Mann–Whitney significant wins/ties/losses vs strongest host.

Variant	GN	CMIX	BBOB $\downarrow$	UAV $\downarrow$	MOO $\uparrow$
Base host			1.00 $\times$	1.00 $\times$	1.00 $\times$
GN only	✓		0.85 $\times$	0.91 $\times$	1.02 $\times$
CMIX only		✓	0.92 $\times$	0.88 $\times$	0.97 $\times$
JANUS	✓	✓	0.83 $\times$	0.89 $\times$	1.01 $\times$

Table 5: Component ablation across representative domains. Ratios are relative to the base host; lower is better for BBOB/UAV and higher is better for MOO.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Value	Meaning
window	300	Sliding window of recent $(x,s)$ pairs for Jacobian regression
$\mu$	$10^{-3}$	Ridge regularization for residual-gradient regression
$\lambda$	$[1,10]$	Levenberg–Marquardt damping (stable across range)
$\kappa$	1.0	Bounded-injection clip factor
$w_{\text{base}}$	0.5	Base mixing weight before dimension scaling
$[w_{\min},w_{\max}]$	$[0.02,0.6]$	Clip range for the adaptive mixing weight

Appendix

Table 6: JANUS hyperparameters shared across all experiments. The sole per-setting exception is the BBOB fuse cadence ( assembly period ), tuned once for the scalar BBOB setting (

d\in\{30,100\}

) and reused verbatim elsewhere.

Notation
Symbol	Meaning	Where used
$x\in\mathcal{X}$	Candidate solution in the native search space	All benchmarks
$f(x)$	Scalar black-box objective or scalarized cost	BBOB/UAV
$r(x)$	Observed component/objective vector when available	Structured/MOO tasks
$s(x)$	Residual coordinates used for local Jacobian fitting	All
$\mathcal{W}$	Recent evaluated window for local fitting	Online adaptation

Appendix

Table 7: Notation and JANUS instantiations.

Method	$d{=}30$	$d{=}100$
JANUS-GN	0.1	14.7
CMA-ES host	0.8	20.1
BOBYQA	1351	2112
COBYLA	1447	2524
DFO-LS	1864	2641
NEWUOA	2419	5477

Appendix

Table 8: Extended baseline checks (Sec. 5.1 ). Lower is better except win rate. Blue cell

=

best, green cell

=

second.

	$d{=}30$		$d{=}100$
Method	Geo-Mean	Wins	Geo-Mean	Wins
CMA-ES (host, full-rank)	14.8	16	332	16
LM-CMA (LPCMA)	67.4	0	4.98e3	0
VkD-CMA (VLPCMA)	2.33e3	0	1.96e4	0
Random Search	3.63e3	0	1.90e4	0

Appendix

Table 9: BBOB restricted- vs. full-rank host comparison: 16 held-out functions, 10 seeds, 5,000 FEs,

[-5,5]^{d}

. Geometric-mean final error across 16 functions, and per-function wins (count of functions on which each method attains the lowest error). Both LPCMA (LM-CMA) and VLPCMA (VkD-CMA), which adapt covariance in a fixed low-dimensional basis, underperform the full-rank CMA-ES host at

d\in\{30,100\}

. This is compatible with the structural limitation of Prop. 1 — restricted-covariance hosts cannot express an arbitrary residual-aligned metric — though the proposition concerns expressiveness and does not by itself predict this performance ordering. The full-rank CMA-ES host is

4.6\times

better than LM-CMA at 30D and

15.0\times

better at 100D. Blue cell

=

best, green cell

=

second-best.

Function	30D	100D	500D	Wins	Mean log-ratio	Verdict
F4	-0.75	-1.08	-0.23	3/3	-0.69	robust win
F6	-9.39	-4.54	-0.30	3/3	-4.74	robust win
F7	-1.84	-2.11	-0.67	3/3	-1.54	robust win
F8	-1.53	-1.57	-1.66	3/3	-1.59	robust win
F9	-3.25	-3.05	-2.04	3/3	-2.78	robust win
F10	-3.30	-3.48	-0.52	3/3	-2.43	robust win

Appendix

Table 10: Dense BBOB function strip. Entries are

\log_{10}(\mathrm{JANUS}/\mathrm{ABOM})

final-value ratios; negative cells favor JANUS.

	$d=30$			$d=100$
Func	JANUS-GN	JANUS	CMA-ES	JANUS-GN	JANUS	CMA-ES
$f_{4}$	35.12 $\pm$ 7.39	70.01 $\pm$ 14.79	66.59 $\pm$ 13.86	979 $\pm$ 78.20	1014 $\pm$ 74.55	2685 $\pm$ 689
$f_{6}$	2.44e-07 $\pm$ 3.8e-08	2.99e-08 $\pm$ 3.7e-08	5.11e-08 $\pm$ 7.4e-08	3140 $\pm$ 659	3416 $\pm$ 1031	1.80e+04 $\pm$ 2845
$f_{7}$	0.276 $\pm$ 0.53	8.36 $\pm$ 2.84	9.82 $\pm$ 8.33	229 $\pm$ 84.01	301 $\pm$ 95.92	5101 $\pm$ 1841
$f_{8}$	3.15 $\pm$ 2.26	13.13 $\pm$ 1.60	16.75 $\pm$ 12.93	2101 $\pm$ 3846	1922 $\pm$ 3465	3075 $\pm$ 4851
$f_{9}$	1.71 $\pm$ 1.31	11.50 $\pm$ 1.70	11.69 $\pm$ 1.27	792 $\pm$ 1751	670 $\pm$ 1817	9.69e+04 $\pm$ 2.8e+05

Appendix

Table 11: BBOB

d=30

and

d=100

: mean

\pm

std; lower is better. 30D JANUS uses tuned configuration (fuse_every=50, 10 seeds); JANUS-GN 30D and all 100D from the host-controlled run (30 seeds). Blue cell

=

best, green cell

=

second-best. W/T/L : wins/ties/losses vs CMA-ES per function (mean comparison).

Host

NN-BBO / MetaBBO baselines

Ours

Func

CMA-ES

GLHF

SYMBOL

LES

RLEPSO

GLEET

LDE

SynCMA

DiBO

JANUS-GN

JANUS

f_{4}

4.2e4

3.7e5

4.7e5

1.2e5

2.4e5

2.5e4

3.1e4

1.6e4

3.4e4

2.1e5

1.6e4

5992

f_{6}

1.8e6

1.6e7

1.8e7

5.6e6

9.3e6

9.2e5

1.5e6

9.9e5

5.8e5

9.8e6

7103

2113

f_{7}

1.5e4

3.9e4

4.7e4

1.8e4

2.3e4

1.2e4

4824

1.5e4

3.5e4

–

988

f_{8}

5.9e8

7.1e8

4.7e7

5.1e7

1.5e7

3.1e7

1.3e7

9.5e7

5.1e8

3072

1426

f_{9}

6.1e8

4.1e8

4.7e8

2.4e6

4543

9.5e5

4.4e6

1.8e5

7.0e6

4.6e8

508.7

1323

Appendix

Table 12: BBOB

d{=}500

, full 10-baseline comparison: mean final error, 16 held-out functions (

[-5,5]^{d}

, 20,000 FEs). NN-BBO baselines and JANUS rerun under our protocol (10 seeds). JANUS-GN shown from the host-controlled run (30 seeds). Blue cell

=

best, green cell

=

second. Lower is better. B2Opt: N/A (network fixed at 10D). Sig. row: Bonferroni-corrected wins/ties/losses vs JANUS (

p<0.05

), by a paired Wilcoxon signed-rank test over the 16 per-function final errors (each function’s mean over the first 10 JANUS seeds, paired against the corresponding baseline mean); the pairing unit is the function identity, so this is a cross-function comparison of aggregated final errors rather than a within-function seed test.

Figure 4: MOO ranks by objective count. JANUS-SMS leads low-objective tasks; JANUS-AGE2 leads many-objective tasks.

Group	Method	Tasks	Avg. rank	Wins	Norm. HV
Low-objective tasks
2	JANUS-SMS 5%	12	1.75	8	1.000
	JANUS-SMS 2%	12	2.33	1	0.996
	SMS-EMOA	12	3.17	0	0.974
3	JANUS-SMS 5%	6	2.50	2	0.982
	JANUS-SMS 2%	6	2.83	3	0.986

Appendix

Table 13: MOO performance by objective count. Top three methods are shown for each group; JANUS variants lead the low-objective SMS regime and the many-objective AGE2 regime. Blue cell

=

best, green cell

=

second-best per group.

Figure 5: Ablation, significance, and cost. (A) JANUS/JANUS-GN ratio. (B) Significance: 0 regressions. (C) CUDA runtime:

1.46\times

1.22\times

Problem	0.05	0.10	0.20	0.30	0.40	0.50	0.60
DTLZ4_d12_n3	72.7	72.9	72.9	72.8	72.9	72.9	72.9
WFG1_d12_n2	8.10	9.23	9.64	9.50	9.63	9.51	9.86
WFG2_d12_n3	117	119	121	124	131	127	128
WFG4_d12_n2	7.02	7.51	7.47	7.99	7.71	8.07	8.74
ZDT1_d30_n2	6.70	6.75	6.81	6.94	7.09	7.07	7.10
ZDT2_d30_n2	5.85	5.90	5.88	5.91	6.12	6.29	6.27

Appendix

Table 14: CMIX candidate fraction ablation (mean HV; higher is better). Blue cell

=

best per row, green cell

=

second-best.

JANUS variant	Host baseline	Tasks	Runs	Slowdown
JANUS-SMS 5%	SMS-EMOA	18	180	$1.46\times$
JANUS-AGE2 5%	AGE-MOEA2	20	200	$1.22\times$

Appendix

Table 15: Runtime overhead with CUDA-accelerated metric estimation. Slowdown is the ratio of mean wall-clock time per independent run for the JANUS variant to its host baseline (lower is better;

1.00\times

= no overhead). Values match Fig. 5 C.

Budget	JANUS	CMA-ES	Gain
2,500	9,600	14,486	$-33.7\%$
10,000	8,488	8,587	$-1.2\%$
20,000	8,019	8,023	$-0.05\%$

Appendix

Table 16: UAV budget sweep over 28 terrains and 30 runs. Lower is better.

#	Method	Family	Mean $\pm$ Std	Median	IQR	Ratio vs JANUS	Terrains rank 1	Sig. vs JANUS
1	JANUS	Ours	9,600 $\pm$ 5,966	6,644	9,579	1.00	15	–
2	DE	Traditional	11,006 $\pm$ 3,865	9,261	7,412	1.15	1	24/3/1
3	JDE21	Adaptive	11,017 $\pm$ 5,604	8,579	2,795	1.15	1	–
4	JANUS-GN	Ablation	11,317 $\pm$ 5,373	9,508	9,815	1.18	8	0/28/0
5	RLDEAFL	MetaBBO	11,411 $\pm$ 5,818	8,197	9,227	1.19	0	–
6	GLEET	MetaBBO	11,603 $\pm$ 7,085	7,455	9,808	1.21	1	–

Appendix

Table 17: UAV path planning: 28 held-out terrains, 30 runs, 2,500 FEs (840 values/method). Blue cell

=

best, green cell

=

second-best. Ratio is mean/JANUS (lower=closer to JANUS). Sig. column: Bonferroni-corrected Mann–Whitney

U

wins/ties/losses vs JANUS (

p<0.05/28

across 28 terrains). The test is unpaired because the CMA-ES baseline runs do not share seeds with the JANUS runs.

Panel of Figure 7 — Figure 7: Additional UAV 3D path visualizations on four terrains (T9/T25/T41/T55). JANUS often finds smooth, low-cost trajectories, but T9/T41 show that this behavior is not universal under the short 2,500-FE budget.

Terrain	JANUS	CMA-ES	Gain
T1	7035	10062	+30.1%
T3	9716	17753	+45.3%
T5	5974	19695	+69.7%
T7	5834	18681	+68.8%
T9	23155	19402	-19.3%
T11	21077	18970	-11.1%

Appendix

Table 18: UAV terrain audit. Every held-out terrain is kept; gain compares JANUS with the CMA-ES host under the identical 2,500-FE protocol. Cell color reflects statistical significance (Bonferroni-corrected Mann–Whitney

U

across 28 terrains, consistent with Table 17 ’s Sig. column): green

=

significant win, amber

=

not significant (tie), red

=

significant loss.

Regime	Evidence	Interpretation
BBOB 30D/100D	CMIX loses more often than JANUS-GN	Scalar pseudo-residual curvature can be noisy in low dimension.
BBOB 500D	Host-control losses on $f_{22}$ / $f_{23}$	Local metric helps broadly but not when basin discovery dominates.
ABOM reference	$f_{19}$ / $f_{20}$ anomalous ratios	Reported-reference comparisons are interpreted qualitatively.
UAV short budget	1 significant host-control loss (T29), 3 ties (T9/11/19); JANUS-GN ranks first on all 4	CMIX can undo a strong GN signal before curvature stabilizes.
MOO saturation	4 tied tasks	Survival already preserves diversity, limiting proposal-geometry gains.

Appendix

Table 19: Negative evidence summary: cases where JANUS underperforms or ties.

Method	@1000	@2000	@3000	@4000	@5000
CMA-ES (baseline)	1.529	1.276	1.177	1.127	1.056
JANUS-GN (exploit only)	1.154	0.968	0.918	0.866	0.837
JANUS (GN + CMIX)	1.200	0.993	0.897	0.863	0.819

Appendix

Table 20: ISP calibration (AWB,

d{=}1135

m{=}111

scenes): running-best MRAE. Lower is better. Single run per method, 5,000 evaluations. Blue cell

=

best, green cell

=

second-best per column.

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