math.OCOct 12, 2026
SaveCalibrated -Barycenters for Multiobjective -Divergence Optimization
Organizations: The University of Hong Kong · Zhongguancun Academy
Abstract
Matching one distribution to many targets can require repeated evaluations of every target. CAL--MODM aggregates a nonlinear predictor into one barycenter, calibrates its residual derivatives, and checks steps against the original -divergence objective. Under smoothness assumptions, calibration controls parameter-step model error without requiring targets to be close to the model. Exact-oracle stationarity and local convergence guarantees are established. At common matching, canonical geometry cancels the leading quadratic iteration error, giving cubic convergence with second-order calibration and sufficiently accurate inner solves. High-precision experiments support this prediction. Sensitivity-based initialization also accelerates preference continuation along regular stationary branches. On 300-target Jensen--Shannon problems, CAL-p2 achieves median paired speedups of over L-BFGS and over second-order Direct-TR, across 23/25 and 24/25 objective-comparable stationary pairs. The savings arise when cheaper predictor queries offset calibration overhead.
Figures & tables
| Notion | Order | Meaning |
|---|---|---|
| Likelihood-ratio normal form | Error as | |
| Calibrated model | Value/gradient error in the step | |
| Global complexity | Certified iterations to first-order stationarity | |
| Local iteration | At least Q- | Error recursion for |
Table 1: Distinct notions of order used in the paper.
Figure 1: Local normal-form scaling in the forward and reverse orientations. Curves show medians over 20 seeds. Dashed references have slopes , , and . Both axes are logarithmic.
Figure 2: E2 amortized cost, , 30 paired repeats. Left: direct (solid) and aggregated (dashed) time per query, with IQRs. Middle: target-kernel (solid) and end-to-end (dashed) speedups, with -fold and unit references. Right: break-even query counts. Gaps indicate a nonpositive 2.5th percentile of paired savings.
Figure 3: Step-local calibrated-model error scaling. Columns correspond to . Rows show value and directional-derivative errors. Curves are medians over three generators, 20 seeds, and two signed rays. Dashed references have slopes and .
Figure 4: E4 robustness and cost, 30 paired seeds. Rows: JS and KL+quartic. Left: success rates and median certified nonacceptance fractions, including failed subproblems and rejected trials (not applicable to L-BFGS). Right: median paired runtime speedups and component-call ratios against Direct-NL-TR in the target-count sweep. Both runs must succeed.
Figure 5: E5 one-step orders: means and one standard deviation over ten seeds and two paired signs (bars smaller than markers, not confidence intervals). Dashed lines mark order . Actual-iterate orders appear in Table 7 .
Figure 6: E7 normalized gradient errors ( C.13 ) at , . Directions and signs are summarized within seeds, then medians/IQRs across 20 seeds. CAL-minus/plus use . All calibrated models and Taylor-p2 match through order two. JS orientations coincide.
Figure 7: E10 one-step errors, seed medians/IQRs, with identical calibration and cubic inner forcing. Canonical CAL-p2 is cubic at and quadratic for positive heterogeneity. Panels show and all six radii. Fits and audits use all four heterogeneities.
| Claim | Observation | Experiment |
|---|---|---|
| Exact aggregation | Max. scaled identity error | E2 |
| Step-based accuracy | Value/directional-derivative exponents | E3 |
| Calibration fidelity | Higher true-gradient success rates than uncalibrated models | E4, E7 |
| Canonical cancellation | CAL-p2 order ; Taylor-p2 order | E10 |
| Quadratic local iteration | Mean fitted orders – | E5, |
| Cubic local iteration | Mean fitted orders – | E5, |
Table 3: Controlled evidence. Local rates require the stated neighborhood and inner-accuracy conditions. The E10 cubic result is at common matching.
Figure 8: E9: measured JS acceleration relative to direct L-BFGS (left) and Direct-TR (right). Values above one favor CAL. Points and bars are seed medians and IQRs of objective-comparable successful pairs. These are the JS panels of the complete comparison in Figure 35 . Table 4 reports CAL-p2 pair coverage.
Figure 9: E9: true-objective cost divided by aggregated-predictor cost, at matched derivative orders. F, FG, and FGH include values, values/gradients, and values/gradients/Hessians. The JS cost separation grows with the target count. KL stays near one because both implementations aggregate exactly. Points and bars are seed medians and IQRs across five seeds.
Figure 10: E9 implementation ablation with identical optimization rules. Left: eager/demand-driven time ratios for CAL-p1/p2. Right: median template-Hessian counts. This compares two implementations of the same method, not CAL against a direct optimizer.
Figure 11: E8: saved decoded mean images for CAL-p2, JS, seed 1, with a fixed decoder. Each preference has nine latent draws, without reranking. All-method samples and complete held-out image diagnostics are in Appendix C.8 .
Figure 12: E8 quadrature and endpoint audits. Dashed lines mark the configured thresholds. The mass panel reports normalization error of the discrete masses. Stored zeros are displayed at without changing the data. Large true gradients of uncalibrated JS endpoints explain their failed audits.
Figure 13: E8 pairwise projections of latent objectives at audited successful endpoints, seed 1. Coincident markers indicate overlapping solutions. The points are sampled stationary trade-offs, not a certified full Pareto front.
Figure 14: E8 held-out image diagnostics: class fractions and image MMD 2 under JS, summarized by seed medians and IQRs. Preference IDs are categorical and follow the saved preference key. Failed endpoints are included. These quantities describe decoded distributions rather than certify optimality.
| Time speedup | True-jet ratio | ||
|---|---|---|---|
| Case | vs Cold | vs Warm | vs Cold |
| JS, F | |||
| JS, R | |||
| KL4, F | |||
| KL4, R | |||
Table 5: E6: Tangent costs at 49 preferences; paired seed medians. Timings include aggregation, initialization, and correction. F/R: forward/reverse; KL4: KL+quartic.
Figure 15: E6 independent prediction errors: medians over signs within each seed, then over ten seeds. Warm uses the base reference. Tangent adds its sensitivity prediction. Dashed slopes one and two nearly overlap the curves. The horizontal axis measures preference displacement, not corrector iteration.
Appendix figures & tables26 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 16: Fitted-exponent deviations for E1, scaled by . Markers show means over 20 seeds, error bars one standard deviation, and the dashed line exact agreement.
Figure 17: E2 aggregation identity errors in both orientations: maximum scaled discrepancy over 20 seeds and eight queries per seed, including the aggregate’s constant and scale. The dashed line marks tolerance . Errors remain at double-precision roundoff.
Figure 18: E3 fitted model-error orders by generator (columns): value errors (top, order ) and directional-derivative errors (bottom, order ). Markers and bars show means and one standard deviation over 20 seeds and two signed rays, not confidence intervals. Blue circles/orange squares denote forward/reverse orientations, offset for visibility. Dashed lines mark theoretical orders.
| Time speedup | Call ratio | |||
|---|---|---|---|---|
| Method | JS | KL+quartic | JS | KL+quartic |
| Direct-LBFGS | 12.87 | 5.77 | 0.128 | 0.227 |
| Direct-TR | 3.36 | 4.15 | 0.119 | 0.143 |
| Direct-NL-TR | 1.00 | 1.00 | 1.000 | 1.000 |
| CAL- - | 1.74 | 0.41 | 0.137 | 0.490 |
| CAL- - | 1.76 | 2.55 | 0.135 | 0.160 |
Table 6: E4 at : median paired ratios against Direct-NL-TR. Each displayed method succeeds in all 60 runs per generator. Uncalibrated fails in all 60, so its paired costs are unavailable. Higher speedups and lower call ratios are better.
Figure 19: E4 true-gradient profiles: JS, , , uniform preferences. Rows separate orientations. Budgets are component-equivalent calls (left) and time (right). Curves are pointwise medians over 30 seeds after log-coordinate interpolation. Final gradients are carried forward, and budgets before the first record are unavailable. The dotted line marks .
Figure 20: E5 independent one-step errors by generator (columns) and orientation (rows). Coordinates are medians over ten seeds and two signs. Dashed slopes two and three largely overlap the curves. Fits use each seed/sign’s prespecified window, not the median curves.
| Generator | Orientation | ||
|---|---|---|---|
| JS | Forward | ||
| JS | Reverse | ||
| KL+quartic | Forward | ||
| KL+quartic | Reverse |
Table 7: E5 actual-iterate orders: mean one standard deviation of the last eligible from each run. Each cell contains ten seeds and two paired signs. JS orientations coincide mathematically.
Figure 21: E5 parameter-error trajectories, seed 1, positive initial direction, 120-digit arithmetic. Panels separate generators and orientations. Table 7 summarizes all runs.
Figure 22: E5 individual condition ablations on the first forward KL+quartic problem. Top: parameter errors and true gradients for standard and four controls. Bottom: a flat parameter direction permits vanishing gradient without convergence to the selected reference. The text reports each control’s outcome.
| Generator | Orientation | Warm | Tangent |
|---|---|---|---|
| JS | Forward | ||
| JS | Reverse | ||
| KL+quartic | Forward | ||
| KL+quartic | Reverse |
Table 8: E6 local prediction orders. Entries are mean one standard deviation across ten seeds, after averaging the two signed slopes within each seed. The reference orders are one for Warm and two for Tangent.
Figure 23: E6 whole-path costs by generator/orientation: runtime speedup against Cold (left) and true-jet pass ratio (right). Paired ratios use median times over three repetitions, then medians over traversals within seeds and medians/IQRs over ten seeds. Initialization and the common first solve are included.
Figure 24: E6 pre-correction gradient norms, seed 1, forward traversal of the 49-point loop, both divergence orientations. All methods share the first cold initialization. The reference line is the corrector tolerance .
Figure 25: E6 corrected objective paths, seed 1, finest grid, forward traversal. The black line connects independent references. Staggered markers show Cold, Warm, and Tangent solutions. All sampled points enter validation. The loop does not cover the full Pareto surface.
Figure 26: E7 normalized value errors with the probes and aggregation of Figure 6 . Canonical median errors improve on JS and KL+chi-square. Calibrated and Taylor curves nearly coincide for KL+quartic at larger heterogeneities. The horizontal axis varies targets, not step size.
Figure 27: E7 paired speedups against Taylor-p2 (above one is faster), using median times over three repetitions and seed medians/IQRs. Each calibrated group has 20 successful, objective-comparable pairs. Uncalibrated pairs exist only for JS and KL+chi-square at . Missing points indicate no eligible pairs.
Figure 28: E7 true-gradient histories, preselected seed 1 and , including rejected trials. These representative histories do not measure asymptotic local order or replace all-seed success statistics.
Figure 29: E8 representation diagnostics for seed 1: training and validation reconstruction loss, validation accuracy of an independent image classifier, and held-out inputs with frozen-decoder reconstructions.
Figure 30: E8 saved decoded samples, four JS preferences, seed 1. Columns compare methods. All displayed uncalibrated endpoints failed to converge. Mosaics retain the saved samples, cropping only surrounding whitespace. CAL-p2 also appears in Figure 11 .
Figure 31: E8 paired runtime and all-run audited success. Timing uses only successful, objective-comparable pairs and reports pair counts. Success uses all attempted cases. N/A denotes absence of eligible timing pairs.
Figure 32: E8 signed objective gaps to independent local best-found references, including failed endpoints. A negative value means that the tested run improved on its reference.
Figure 33: E9 absolute evaluation costs at matched derivative orders, summarized by seed medians and IQRs. FGH includes both gradients and Hessians. Predictor preprocessing is accounted for separately and included in total solves.
Figure 34: E9 total solve times including preference-dependent scalarization setup, over all attempts. Because different methods can reach different local solutions, these marginal medians supplement rather than replace paired speedups.
Figure 35: E9 full paired speedups for JS and the exact-aggregation KL control, against Direct-TR and Direct-LBFGS. Curves summarize successful objective-comparable pairs by seed medians and IQRs. CAL-p2 JS pair coverage is given in Table 4 . Raw records retain every attempted case.
Figure 36: E9 diagnostic cost breakdown at from separate instrumented replays. Each component is summarized by its median. Performance claims use the unprofiled paired timing experiment.
Figure 37: E9 signed objective gaps relative to independent local references, retaining all endpoints. Negative gaps are possible because references are best-found local solutions. These plots report solution variation separately from stationarity success and timing-pair coverage.
Figure 38: E10 expansion audit: subtracting the predicted quadratic vector leaves an approximately constant . Panels show . Fits cover all four heterogeneities. Third derivatives enter only the offline prediction and audit.
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