Kometo Algorithm for Adaptive Multi-fidelity Optimization

• Kometo is a rank-based, optimistic partition-search method designed for adaptive multi-fidelity global optimization, operating without prior knowledge of smoothness or bias parameters.
• It integrates hierarchical exploration with a Zipf-like fidelity schedule and a cross-validation phase to balance exploration depth, evaluation cost, and bias resolution.
• The algorithm achieves regret rates matching minimax lower bounds up to polylog factors across polynomial, stretched-exponential, and finite-cost fidelity regimes, ensuring robust performance.

Kometo is a rank-based, optimistic, partition-search algorithm for adaptive multi-fidelity global optimization under a finite budget. It is formulated for an unknown objective $f:X\to\mathbb{R}$ on a compact domain $X\subset\mathbb{R}^p$, when only zeroth-order evaluations are available and each query may be made at a fidelity $z\in[0,1]$ with known cost $\lambda(z)$ and unknown bias. Its central design combines hierarchical optimistic exploration, fidelity levels indexed by costs of order $e^j$, and a final cross-validation stage that compares candidate points at a common high-cost fidelity. Under polynomial, stretched-exponential, and finite-cost exact-fidelity cost-to-bias regimes, Kometo achieves expected simple-regret rates that match lower bounds up to polylogarithmic factors, while requiring no knowledge of the smoothness parameters, the near-optimality dimension, or the bias-decay law (Fiegel et al., 17 Apr 2026).

1. Problem formulation and regularity model

Kometo is posed in the budgeted global-optimization setting. The target is to identify

$x^*\in \arg\max_{x\in X} f(x),$

where $X\subset\mathbb{R}^p$ is compact and $f$ is unknown. The learner observes only function values, and only through a family of approximations $(f_z)_{z\in Z}$ with $Z=[0,1]$. The budget constraint is cumulative: if the learner queries fidelities $X\subset\mathbb{R}^p$0, then it must stop once $X\subset\mathbb{R}^p$1, and its performance is measured by the simple regret

$X\subset\mathbb{R}^p$2

where $X\subset\mathbb{R}^p$3 is the recommendation returned at budget $X\subset\mathbb{R}^p$4 (Fiegel et al., 17 Apr 2026).

The regularity assumption is local rather than global. Fix a $X\subset\mathbb{R}^p$5-ary hierarchical partition $X\subset\mathbb{R}^p$6 of $X\subset\mathbb{R}^p$7, with root $X\subset\mathbb{R}^p$8, and let $X\subset\mathbb{R}^p$9 denote the unique depth-$z\in[0,1]$0 cell containing a maximizer $z\in[0,1]$1. Assumption 1 states that there exist $z\in[0,1]$2 and $z\in[0,1]$3 such that

$z\in[0,1]$4

This is the local smoothness condition governing how rapidly the objective can deteriorate away from the optimal branch of the partition.

A second structural quantity is the near-optimality dimension $z\in[0,1]$5. For $z\in[0,1]$6, let $z\in[0,1]$7 be the number of depth-$z\in[0,1]$8 cells satisfying $z\in[0,1]$9. Given $\lambda(z)$0, one says that $\lambda(z)$1 is a near-optimality dimension if there exists $\lambda(z)$2 such that

$\lambda(z)$3

The parameter $\lambda(z)$4 quantifies the branching complexity of near-optimal regions. The paper states that, in general, $\lambda(z)$5, and that $\lambda(z)$6 is common in practice (Fiegel et al., 17 Apr 2026).

The multi-fidelity model is deliberately weaker than absolute numerical closeness between $\lambda(z)$7 and $\lambda(z)$8. Kometo assumes the existence of strictly increasing link functions $\lambda(z)$9 such that

$e^j$0

where $e^j$1 is an unknown, nonincreasing bias function. Because $e^j$2 is strictly increasing, ranks are preserved within a fixed fidelity. This rank-preserving model is the reason exploration decisions can be based on within-fidelity comparisons alone.

Cost is summarized through the known map $e^j$3. For any $e^j$4, the learner can request a fidelity $e^j$5 with $e^j$6. The associated cost-to-bias function is

$e^j$7

which upper-bounds the best guaranteed bias available at cost at most $e^j$8. The analysis distinguishes three regimes: polynomial decay $e^j$9, stretched-exponential decay $x^*\in \arg\max_{x\in X} f(x),$0, and finite-cost exact fidelity, meaning $x^*\in \arg\max_{x\in X} f(x),$1 for all $x^*\in \arg\max_{x\in X} f(x),$2 (Fiegel et al., 17 Apr 2026).

2. Lower bounds and the rate regimes

The paper proves lower bounds for budgeted multi-fidelity global optimization under the local-smoothness and cost-to-bias assumptions. These bounds are stated in terms of expected simple regret and show how the attainable rate depends jointly on budget $x^*\in \arg\max_{x\in X} f(x),$3, near-optimality dimension $x^*\in \arg\max_{x\in X} f(x),$4, and the decay regime of $x^*\in \arg\max_{x\in X} f(x),$5 (Fiegel et al., 17 Apr 2026).

Cost-to-bias regime	$x^*\in \arg\max_{x\in X} f(x),$6	$x^*\in \arg\max_{x\in X} f(x),$7
$x^*\in \arg\max_{x\in X} f(x),$8	$x^*\in \arg\max_{x\in X} f(x),$9	$X\subset\mathbb{R}^p$0
$X\subset\mathbb{R}^p$1	$X\subset\mathbb{R}^p$2	$X\subset\mathbb{R}^p$3
$X\subset\mathbb{R}^p$4 for $X\subset\mathbb{R}^p$5	$X\subset\mathbb{R}^p$6	$X\subset\mathbb{R}^p$7

More precisely, Theorem 1 states that for any sufficiently large budget $X\subset\mathbb{R}^p$8 and any possibly randomized policy $X\subset\mathbb{R}^p$9, there exist a target function $f$0, a cost function $f$1, and a family of fidelities consistent with the assumptions such that the expected simple regret satisfies the corresponding lower bound: $f$2 in the polynomial case,

$f$3

when $f$4 under stretched-exponential decay, and

$f$5

when $f$6 under the same regime. In the finite-cost exact-fidelity regime, the lower bounds are

$f$7

for $f$8, and

$f$9

for $(f_z)_{z\in Z}$0, where $(f_z)_{z\in Z}$1 are constants independent of $(f_z)_{z\in Z}$2 and of the algorithm.

The proof intuition given in the paper is combinatorial and adversarial. A truncated subtree of the hierarchical partition is constructed so that only a small number of cells are near-optimal at each depth, with gaps controlled by $(f_z)_{z\in Z}$3. The approximation scheme is then chosen so that distinguishing the near-optimal cell at depth $(f_z)_{z\in Z}$4 requires bias below the depth-$(f_z)_{z\in Z}$5 scale, hence a minimum evaluation cost determined by $(f_z)_{z\in Z}$6. Because the budget constrains how many such openings can be performed, one can choose a depth $(f_z)_{z\in Z}$7 that the algorithm cannot reliably resolve in width or depth. The regret is then lower-bounded by $(f_z)_{z\in Z}$8, which yields the displayed regimes after translating through the cost-to-bias relationship.

These lower bounds are presented as the first general lower bounds for multi-fidelity global optimization in this framework. Their role is not merely descriptive: they define the minimax benchmarks that Kometo subsequently matches up to logarithmic factors.

3. Algorithmic construction

Kometo is a hierarchical optimistic search method equipped with a multi-fidelity schedule. Its state space is indexed by triples $(f_z)_{z\in Z}$9, where $Z=[0,1]$0 is depth in the partition tree, $Z=[0,1]$1 is a cell index, and $Z=[0,1]$2 is a fidelity level corresponding to cost $Z=[0,1]$3. Each cell $Z=[0,1]$4 has a representative point $Z=[0,1]$5, and the indicator

$Z=[0,1]$6

records whether cell $Z=[0,1]$7 is opened at fidelity level $Z=[0,1]$8. If $Z=[0,1]$9, Kometo performs at most one evaluation of

$X\subset\mathbb{R}^p$00

at cost at most $X\subset\mathbb{R}^p$01 (Fiegel et al., 17 Apr 2026).

The exploration rule is progressive in both depth and fidelity. At depth $X\subset\mathbb{R}^p$02, the algorithm opens a decreasing number of cells as the fidelity level $X\subset\mathbb{R}^p$03 increases; the paper characterizes this as a Zipf-like schedule. The purpose is to balance width and accuracy without knowing in advance which fidelity level is required to resolve near-optimal cells at a given depth. If a parent cell is opened at level $X\subset\mathbb{R}^p$04, then for each child Kometo sets

$X\subset\mathbb{R}^p$05

thereby enabling one evaluation per child at each level $X\subset\mathbb{R}^p$06. All exploration decisions at level $X\subset\mathbb{R}^p$07 depend only on comparisons of $X\subset\mathbb{R}^p$08 across cells opened at that same level, which is exactly what the monotone-link model permits.

The high-level procedure is:

1. Compute the internal allocation parameter

$X\subset\mathbb{R}^p$09

1. Initialize the root $X\subset\mathbb{R}^p$10 with a cost schedule tied to $X\subset\mathbb{R}^p$11.
2. For each depth $X\subset\mathbb{R}^p$12, and each

$X\subset\mathbb{R}^p$13

set

$X\subset\mathbb{R}^p$14

Among cells $X\subset\mathbb{R}^p$15 with $X\subset\mathbb{R}^p$16 and not yet opened at level $X\subset\mathbb{R}^p$17, open the one with the largest $X\subset\mathbb{R}^p$18, breaking ties arbitrarily.

1. In the cross-validation phase, for each fidelity level $X\subset\mathbb{R}^p$19, let

$X\subset\mathbb{R}^p$20

be the best candidate found at that level.

2. Evaluate all $X\subset\mathbb{R}^p$21 at a common, large cost $X\subset\mathbb{R}^p$22 proportional to $X\subset\mathbb{R}^p$23, and output

$X\subset\mathbb{R}^p$24

The paper emphasizes a budget-accounting invariant: opening a cell at level $X\subset\mathbb{R}^p$25 induces at most $X\subset\mathbb{R}^p$26 children with per-child fidelities $X\subset\mathbb{R}^p$27, so the total opening cost is at most

$X\subset\mathbb{R}^p$28

With the prescribed choice of $X\subset\mathbb{R}^p$29, the total cost of exploration plus cross-validation is guaranteed to be at most $X\subset\mathbb{R}^p$30.

Two design choices distinguish Kometo from standard multi-fidelity selection rules. First, it never tries to estimate the correct fidelity-depth matching directly; instead it hedges over many fidelity levels at each depth. Second, because ranks are only comparable within a fixed fidelity, it resolves cross-fidelity competition by rescoring one candidate from each fidelity at a common high-cost fidelity. This avoids assuming access to the true target $X\subset\mathbb{R}^p$31 for final selection.

4. Depth–fidelity tradeoff, upper bounds, and adaptivity

The central technical device behind Kometo’s guarantees is a depth–fidelity lemma. Let $X\subset\mathbb{R}^p$32 be any nonincreasing function with $X\subset\mathbb{R}^p$33. If there exist a depth $X\subset\mathbb{R}^p$34 and a fidelity level $X\subset\mathbb{R}^p$35 such that

$X\subset\mathbb{R}^p$36

and

$X\subset\mathbb{R}^p$37

then the regret satisfies

$X\subset\mathbb{R}^p$38

where $X\subset\mathbb{R}^p$39 is the common cost used in cross-validation (Fiegel et al., 17 Apr 2026).

The first inequality is a bias-resolution condition: the fidelity level $X\subset\mathbb{R}^p$40 must be precise enough that its bias is below the scale of depth $X\subset\mathbb{R}^p$41. The second is a width condition: there must be enough budget to open all near-optimal cells at that depth. The resulting bound exhibits the exact tradeoff the algorithm must solve—deeper cells improve localization via $X\subset\mathbb{R}^p$42, but also require more openings and a sufficiently accurate fidelity.

Instantiating $X\subset\mathbb{R}^p$43 with the bounds in the three cost-to-bias regimes yields Theorem 2. Up to polylogarithmic factors in $X\subset\mathbb{R}^p$44, Kometo satisfies

$X\subset\mathbb{R}^p$45

when $X\subset\mathbb{R}^p$46 and $X\subset\mathbb{R}^p$47,

$X\subset\mathbb{R}^p$48

when $X\subset\mathbb{R}^p$49 under the same regime,

$X\subset\mathbb{R}^p$50

when $X\subset\mathbb{R}^p$51 under stretched-exponential bias decay,

$X\subset\mathbb{R}^p$52

when $X\subset\mathbb{R}^p$53 under stretched-exponential decay, and

$X\subset\mathbb{R}^p$54

respectively, for the $X\subset\mathbb{R}^p$55 and $X\subset\mathbb{R}^p$56 finite-cost exact-fidelity cases. The paper states that these rates match the lower bounds up to logarithmic factors in all cases.

The exact finite-$X\subset\mathbb{R}^p$57 expressions are given using the Lambert $X\subset\mathbb{R}^p$58 function. The paper also remarks that, compared with the minimax lower bound, the only exception is the case $X\subset\mathbb{R}^p$59 under polynomial bias decay, where additional logarithmic factors remain.

The adaptivity claim is twofold. Kometo does not require $X\subset\mathbb{R}^p$60, $X\subset\mathbb{R}^p$61, $X\subset\mathbb{R}^p$62, or $X\subset\mathbb{R}^p$63, because the Zipf allocation across depths ensures that enough cells are opened at the unknown relevant depth. It also does not require $X\subset\mathbb{R}^p$64, $X\subset\mathbb{R}^p$65, or the parameters $X\subset\mathbb{R}^p$66, because the fidelity hedging and the final cross-validation automatically realize the needed bias-depth balance. Since all exploration decisions at level $X\subset\mathbb{R}^p$67 depend only on ranks within that level, the algorithm is invariant to the unknown monotone transforms $X\subset\mathbb{R}^p$68.

5. Empirical behavior and implementation considerations

The empirical study reproduces and extends the deterministic experiments of Sen et al. (2018) with identical code and settings. The benchmark suite consists of Curin (2D), Branin (2D), Hartmann-3D, Hartmann-6D, Borehole (8D), and a practical SVM text-classification task with two hyperparameters. In the SVM task, fidelity controls the number of training samples used in 5-fold cross-validation, and the budget is wall-clock time, so both algorithmic overhead and evaluation time are counted (Fiegel et al., 17 Apr 2026).

The baselines are MFPDOO for multi-fidelity optimization and POO and SequOOL for single-fidelity optimization. For multi-fidelity methods, all fidelity levels are usable; for the single-fidelity baselines, only the highest fidelity $X\subset\mathbb{R}^p$69 is used. Performance is measured by regret $X\subset\mathbb{R}^p$70 on the synthetic problems and by accuracy on the SVM task. The horizontal axis in the plots is the effective budget consumed, which reflects algorithms that exceed the allotted budget.

The reported findings are differentiated rather than uniformly favorable. Kometo substantially outperforms MFPDOO on Curin, Branin, Hartmann-3D, and the SVM tuning task, but is slightly worse on Borehole and Hartmann-6D. SequOOL can dominate Kometo on Branin and Hartmann-6D, where high-fidelity evaluations are crucial and lower fidelities are less informative. The paper links this behavior to the theory: in the finite-cost exact-fidelity setting, Kometo carries extra logarithmic factors relative to single-fidelity specialized algorithms, and budget spent on low fidelities can delay progress at the highest fidelity.

The practical guidance is correspondingly concrete. For the partition, one may choose a $X\subset\mathbb{R}^p$71-ary geometric decomposition such as axis-aligned splits, with cell centers as representatives $X\subset\mathbb{R}^p$72. For the fidelity interface, the system must be able to return a fidelity $X\subset\mathbb{R}^p$73 satisfying $X\subset\mathbb{R}^p$74 for any $X\subset\mathbb{R}^p$75; Kometo uses costs $X\subset\mathbb{R}^p$76, though the paper notes that any exponentially spaced grid suffices. Internally, the algorithm sets $X\subset\mathbb{R}^p$77 to meet the budget bound, and a doubling trick can provide anytime behavior.

The computational footprint is modest relative to evaluation cost. The number of opened triples $X\subset\mathbb{R}^p$78 is

$X\subset\mathbb{R}^p$79

and each opening requires at most one evaluation. The paper states that the dominant cost is usually the oracle evaluation itself. The only algorithmic design choices are the branching factor $X\subset\mathbb{R}^p$80, the partition structure, and the fidelity grid $X\subset\mathbb{R}^p$81; the method has no tuning parameters tied to $X\subset\mathbb{R}^p$82.

The primary analysis is deterministic. The paper adds that, in noisy settings, Kometo can be made high-probability correct by averaging multiple evaluations at higher fidelities, thereby inducing an effective cost-to-bias function $X\subset\mathbb{R}^p$83 that decays to zero. This is presented as a route to extending the same analysis via concentration bounds.

6. Position within the literature, limitations, and nomenclature

Kometo is compared most directly with MFPDOO. The paper states that MFPDOO adapts optimistic partitioning to the multi-fidelity setting but assumes stronger knowledge of the bias function and, in some analyses, assumes access to the true function $X\subset\mathbb{R}^p$84 for final cross-validation at finite cost. Kometo differs in three explicit ways: it requires neither $X\subset\mathbb{R}^p$85 nor access to $X\subset\mathbb{R}^p$86; it operates under the strictly more general rank-preserving bias model $X\subset\mathbb{R}^p$87; and it achieves optimal rates up to logarithmic factors without tuning problem-dependent parameters (Fiegel et al., 17 Apr 2026).

The paper also situates Kometo alongside HOO, DOO, SOO, SequOOL, POO, StroquOOL, and GPO. It is described as extending the parameter-free spirit of SequOOL and StroquOOL to the multi-fidelity setting. Under finite-cost exact fidelity, it recovers the classical single-fidelity rates: exponential decay in $X\subset\mathbb{R}^p$88 when $X\subset\mathbb{R}^p$89, and $X\subset\mathbb{R}^p$90 when $X\subset\mathbb{R}^p$91. Compared with methods that require smoothness parameters, such as HOO and DOO, Kometo is adaptive; compared with POO and GPO, it is said to attain the lower bounds without oracle tuning.

The limitations are structural. The guarantees rely on the hierarchical-partition framework and the near-optimality-dimension formalism; large $X\subset\mathbb{R}^p$92 degrades performance, as in other optimistic partition methods. The cost-to-bias function is assumed nonincreasing and must satisfy one of the three specified regimes. The algorithm also assumes the ability to request fidelities at arbitrary costs $X\subset\mathbb{R}^p$93, or at least on a sufficiently fine discrete grid. High-dimensional scaling remains difficult in the standard black-box sense, and the paper identifies additivity and low effective dimension as promising directions for future work. A full finite-sample, high-probability analysis for noisy multi-fidelity feedback is likewise presented as an open extension.

A common source of confusion is nomenclature. Kometo should not be conflated with the automated search algorithm for single, asymmetric exocomet transits in Kepler long-cadence photometry (Kennedy et al., 2018), nor with the Gaussian-process emulator "COMET: Clustering Observables Modelled by Emulated perturbation Theory" for redshift-space galaxy power spectrum multipoles (Eggemeier et al., 2022). The 2026 Kometo algorithm is a multi-fidelity global-optimization method, not an exocomet detector and not a cosmological perturbation-theory emulator.

Within its stated scope, Kometo is presented as the first multi-fidelity global-optimization algorithm that simultaneously works under a general rank-preserving bias model, requires no knowledge of smoothness or fidelity parameters, and achieves minimax-optimal simple-regret rates up to logarithmic factors across multiple cost-to-bias regimes. This suggests that its principal contribution is not a new partition heuristic alone, but an integrated theory of adaptivity across both geometric smoothness and fidelity structure (Fiegel et al., 17 Apr 2026).