KL Before and After Mixing: A Round-Complexity Separation in Parallel Sampling

Tianyi Wu

cs.LGOct 9, 2026

KL Before and After Mixing: A Round-Complexity Separation in Parallel Sampling

Authors: Tianyi Wu

Organizations: Amazon

Abstract

A parallel sampler can choose at random which coordinates to generate together. For a target $p$ and the output law $q_U$ of a schedule $U$ drawn afresh for each output, we compare the schedule-averaged divergence $\mathbb{E}_U D_{\mathrm{KL}}(p\|q_U)$ with the output-law divergence $D_{\mathrm{KL}}(p\|\mathbb{E}_U q_U)$ , in nats. Rounds commit independent draws from exact conditional marginals frozen at the round's start, each output's schedule is drawn before sampling from a finite mixture of schedules, and every scheduled round is charged. For the uniform distribution on binary strings of even length $n$ with $n/2$ ones, the least expected number of rounds at a fixed tolerance $\varepsilon>0$ is $Θ_\varepsilon((\log n)^2)$ under the first criterion but $(e^{-\varepsilon}+o(1))\log_2\ln n$ under the second, and the double-logarithmic order is attainable with the same round count on every run. For this target, a count-state recursion gives the exact least output divergence at every round cap over all causal randomized policies. The separation extends to weighted fixed-count targets with bounded weight ratios and quota fractions bounded away from zero and one, without permutation symmetry, and to the full-support Beta $(1,1)$ --Bernoulli target, where the sharp expected cost under the output-law criterion is $(e^{-\varepsilon}/2+o(1))\log_2\log_2n$ , attained with rare $n$ -round runs. Prior work implies the order of the schedule-averaged cost on the balanced target and its $O_\varepsilon((\log n)^2)$ upper bounds on both extensions.

Figures & tables

Figure 1: The four-bit example. Columns: the six valid strings (two ones) and, pooled, the ten invalid ones; disc areas are unconditional probabilities, and nothing is renormalized. The example implies no rate.

design, criterion, cost	balanced slice	full support	bounded weights
fixed schedule, KL, length	$\Theta((\log n)^{2})$	$\Theta((\log n)^{2})$	$\Theta((\log n)^{2})$
mixture, $H$ , mean	$\Theta((\log n)^{2})$	$\Theta((\log n)^{2})$	$\Theta((\log n)^{2})$
relabelled profile, $F$ , length	$\ell+O(1)$	$\ell+O(1)$	$\ell+O(1)^{\ddagger}$
causal policy, $F$ , cap	$\ell+O(1)$	$\Theta(\ell)^{\dagger}$	$\ell+O(1)$
mixture, $F$ , mean	$(e^{-\varepsilon}+o(1))\ell$	$(\frac{1}{2}e^{-\varepsilon}+o(1))\ell$	$(e^{-\varepsilon}+o(1))\ell$
causal policy, $F$ , $\mathbb{E}\tau$	$(e^{-\varepsilon}+o(1))\ell$	$(\frac{1}{2}e^{-\varepsilon}+o(1))\ell$	$(e^{-\varepsilon}+o(1))\ell$

Table 1: Least number of rounds at a fixed tolerance

\varepsilon>0

, by design, criterion and cost;

\ell=\log_{2}\log_{2}n=\log_{2}\ln n-\log_{2}\ln 2

Figure 2: Balanced slice,

\varepsilon=\ln 2

: round counts of one fixed labelled schedule (circles) and one relabelled profile (squares). Each has one length per run, so circles and squares bound

L_{H}

and

L_{F}

from above; open squares (best-found witnesses) and the kernel-checked

L_{F}\leq 11

n=2^{16}

are upper bounds not shown optimal. No rate is fitted.

Figure 3: Exact grid points and bars; relations schematic. Top: level

c=8

; middle: levels

6

–

11

; bottom: their union, with gaps of order

c_{\min}^{-2}

away from small denominators (shaded: larger gaps near

1/3

1/2

2/3

; not the converse’s set

S^{\star}

). Dots: support, not probability.

Figure 4: One level of the bounded-weight sampler (top, executed) and of the reference law (bottom, analysis only). Both share the kept set

C

and the block values

x_{B}

;

W

compares the two completions of

C

(

W=0

j

is infeasible).

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Statement	Proof	Key step
Lemma 1	Appendix B	invariant laws are uniform on the single orbit $V$
The survival envelope ( Env )	Appendix D , Proposition 9 (its weaker $\kappa_{r}$ form $\Gamma_{r}$ , at the end of that appendix, is used in Appendices E and L )	one-round negative-moment bound, then Cauchy–Schwarz exponent halving
Lemma 3 , Theorem 4 , Corollary 6	Appendix E (total-variation versions in its paragraph Total variation ; the window in Appendix E.1 )	count-state supermartingale bounds success; tail sums charge rounds
Theorem 2 (a),(b)	Appendix B (sketches in Section 3 )	(a) per-reveal entropy decrement; (b) square-root ladder
Theorem 2 (c), fixed-schedule upper bound on $L_{H}$	Appendix C	per-block bound (Step 1) on a geometric schedule
Corollary 5	Appendix E , end, from Theorem 2 , Lemmas 1 (a),(c),(e) and 3 (a), and ( Env )	point masses in Theorem 2 ; ( Env ) caps a profile’s success

Appendix

Table 6

class, criterion, cost	order	status
Balanced slice
fixed schedule, KL, length	$\Theta(\log^{2}n)$	K (Cor. 5 )
one relabelled profile, $F$ , length	$\log_{2}\ln n+O_{\varepsilon}(1)$	K (Cor. 5 )
mixture, $H$ , mean	$\Theta(\log^{2}n)$	K (Thm 2 a,c)
mixture, $F$ , mean	$(e^{-\varepsilon}+o(1))\log_{2}\ln n$	K (Thm 2 b; Cor. 6 b,c)
$\mathcal{A}_{r}$ , $F$ , cap	$\log_{2}\ln n+O_{\varepsilon}(1)$	K (Thm 4 a; Cor. 6 a)

Appendix

Table 2: Classes, criteria and costs (

\varepsilon>0

fixed; for tilts and bounded weights also

\Lambda,\kappa_{0}

;

\mathcal{A}_{r}

is the class of Section 2 , which for two-value tilts is

\mathrm{RAND}_{r}

of Appendix G ). “Mean”: scheduled mean

C(\mu)

; “

\mathbb{E}\tau

”: executed mean; “cap”: hard cap; “length”: rounds of one schedule or relabelled profile, the same on every path. K: kernel-checked (declarations in Appendix Q , whose table also marks the results whose class identification uses one ordinary step; for an asymptotic order, the asymptotic statement); P: prior work (preprint). ‡ One fixed labelled schedule, relabelled uniformly within an equal-weight group of linear size; without one, a common length is open. Finite regime: on the balanced slice, explicit bounds (Table 8 ), kernel-checked bounds and finite computations at

n=2^{16}

(Table 7 ), and an exact least profile length at

n=2^{4}

, computed optima (float64), linear-program brackets, best-found upper bounds and one seeded Monte Carlo estimate (Table 5 , Figure 2 ); on full support, explicit bounds on

L_{H}

(Proposition 14 ) but astronomical constants for

F

, with computed

L_{H}

optima and best-found upper bounds on

L_{F}

(both float64; the latter interval-certified at

n\leq 2^{10}

; Table 6 ); for two-value tilts, explicit but impractical constants (e.g., caps of hundreds to thousands of rounds; Appendix H , §6.9) and, at

n\leq 24

and

r\leq 4

, rigorous enclosures of the optimal output KL, some kernel-checked, beside float64 values (Appendix K ); for bounded weights, existence-only constants in the reservoir-free

F

upper bounds (Appendix I ).

design	one round	rounds
balanced, ladder	unrevealed $M\to\sqrt{M}$	$\log_{2}\log_{2}n$
full support, one profile	level $c\to c^{2}$	$\log_{2}\log_{2}n$
full support, mixture	band $c_{\min}\to c_{\min}^{4}$	$\frac{1}{2}\log_{2}\log_{2}n$

Appendix

Table 3: One round’s growth, up to lower-order factors (

M

: number of unrevealed coordinates;

c

: level;

c_{\min}

: start of a band of levels), and the leading round count: the length of every path for the two profiles, and the scheduled mean

C(\mu)

for the mixture, which has rare paths of length

n

. Mixing in the one-round schedule multiplies the leading term of the scheduled mean by

e^{-\varepsilon}

Symbol	Meaning	Defined in
Interface, criteria and costs
$p$ , $n$	target law on $\{0,1\}^{n}$ ; $n$ the number of coordinates	§ 1 ; § 2
$V$ , $k$	balanced slice $\{x:\sum_{i}x_{i}=k\}$ , $n=2k$ ; $k$ the required number of ones (also: $V(s)$ )	§ 2
$(M,j)$	count state: $M$ coordinates unrevealed, $j$ of them must still be one (also: rung index $j$ ); with a subscript, an unrevealed count at a stage: $M_{t-1}$ before round $t$ (§ 3 ), the rungs $M_{j}$ of Theorem 2 (b)’s ladder and $M_{t}$ of Theorem 4 (b)’s witness (rows $j$ and $L^{\ast}$ ). In § 4.2 the nominal sizes $M_{t}$ are deterministic and the free count is $N=\|\mathcal{C}_{t}\|$ , random around $M_{t}$ ; there the $M$ of $M\geq\rho n$ is the reservoir size of part (b)’s construction (Theorem 8 ’s $j/M$ keeps the count-state meaning)	§ 2
$U$ ; $\|U\|$ (length); $b(U)$ (profile)	labelled schedule $(B_{1},\dots,B_{r})$ , an ordered partition of $[n]$ ; its length (rounds); its profile (block sizes)	§ 2
$\sigma$	a permutation of $[n]$ , as in the uniform relabelling $\sigma(U)$ ; $\sigma(\cdot)$ is also the $\sigma$ -algebra generated, as in $\mathcal{F}_{t-1}$ . In the appendices $\sigma$ also denotes, each defined where used, a standard deviation ( $\sigma(y)$ , $\sigma_{n}$ , $\sigma^{2}$ in Apps. F , K , G ), the logistic function (App. I ) and a state $\sigma_{t}$ (App. H )	§ 2

Appendix

Table 4: Notation index. The first locator is the first place the paper defines the symbol; a second locator gives the formal or per-round definition. “(also: …)” marks a symbol that has another meaning in another row.

$n$	$L_{H}$	$L_{F}$	$A$	shape	$R_{F}^{\mathrm{PROF}}$	TV fixed / mixed
$2^{4}$	2.66	1.999	–	–	2	–
$2^{8}$	8.19	3.28–3.39	9	89%	4	0.412 / 0.005
$2^{10}$	12.56	3.71–3.86	13	95%	4	0.574 / 0.014
$2^{11}$	15.14	3.92–4.04	–	–	$\geq 4$	–
$2^{12}$	18.06	–	19	98%	$\leq 5$	0.673 / 0.015
$2^{16}$	26.20–33	1.99–4.69	33	99%	$\leq 5$	0.831 / 0.022

Appendix

Table 5: Finite comparison at

\varepsilon=\ln 2

(finite computation; details in Appendix K ).

L_{H},L_{F}

: mean-cost optima of the same class (two-length linear programs;

L_{F}

bracketed below by the lower program over the computed

G_{r}

and above by best-found relabelled profiles; at

2^{16}

, where the program’s

d^{*}_{r}

were not computed, the

L_{H}

entry brackets the optimum between the kernel-checked lower bound and the length

A

of one schedule);

A

: least length of one fixed schedule with KL

\leq\ln 2

(computed optimum: float64 evaluation of an exact dynamic program) and the shape share of its divergence;

R_{F}^{\mathrm{PROF}}

: least length of one relabelled profile with

F\leq\ln 2

(exact at

2^{4}

; at

2^{8}

and

2^{10}

a computed optimum, float64 with margins of at least

0.02

from

\frac{1}{2}

; best-found upper bound at

2^{12},2^{16}

;

\geq 4

2^{11}

, since

G_{3}(2048,1024)=0.3145<\frac{1}{2}

in float64 bounds the success of every relabelled profile with at most three rounds); TV: total-variation distance between the target and the output law of the

\delta=\frac{1}{2}

ladder of Theorem 2 (b)’s proof with the Hoeffding–Ville choice

K=\lceil 2\ln(2/\delta)\rceil=3

(the stated bound takes the smaller

K_{\delta}=2

of the Kolmogorov branch; both ladders fail with probability at most

\delta

), whose bulk blocks are

(160,72)

(256,672,72)

(3328,672,72)

and

(40960,23808,672,72)

2^{8}

2^{10}

2^{12}

and

2^{16}

, each followed by

24

singletons; fixed labelling versus relabelled (

=1-s

, with

s

the ladder’s success probability; exact at

2^{8}

–

2^{12}

, within

2\cdot 10^{-57}

2^{16}

); the fixed-labelling TV is a

30

-digit enumeration at

2^{8}

–

2^{12}

(pruned mass below

10^{-31}

) and at

2^{16}

a seeded Monte Carlo estimate of

\mathbb{E}_{P}[(1-Q/P)_{+}]

over

400{,}000

target draws (standard error

0.0004

). Optima, shares and TV values are rounded to nearest, bounds outward; –: not computed.

	$\varepsilon=0.1$			$\varepsilon=\ln 2$
$n$	$L_{H}$	$L_{F}\leq$	ratio $\geq$	$L_{H}$	$L_{F}\leq$	ratio $\geq$
$2^{6}$	14.55	4.36	3.3	4.35	2.29	1.9
$2^{8}$	29.48	5.59	5.2	8.03	2.91	2.7
$2^{10}$	51.61	6.50	7.9	13.10	3.43	3.8
$2^{12}$	81.45	7.12	11.4	19.29	3.82	5.0
$2^{14}$	119.34	7.68	15.5	26.75	4.04	6.6

Appendix

Table 6: Full-support target Beta

(1,1)

–Bernoulli: computed

L_{H}

optima (float64 evaluation of the exact optimization) and best-found upper bounds on

L_{F}

(float64; interval-certified only where stated). Each

L_{F}

bound mixes the best-found profiles with

R

and

R+1

rounds whose

F

values bracket

\varepsilon

, weighted so that the average

F

\varepsilon

; as

\mathrm{KL}(p\|\cdot)

is convex, the mixture has

F\leq\varepsilon

, and the

(R+1)

-round profile alone has

F\leq\varepsilon

, so each bound, rounded up to an integer, is the length of the shortest best-found single profile with

F\leq\varepsilon

. All

L_{F}

entries come from one evaluation pipeline applied to the candidate profiles of two independent optimizers; at

n\leq 2^{10}

a separate implementation, which recomputes each profile’s output count law exactly and encloses

F

in interval arithmetic, certifies these

L_{F}

entries as upper bounds; a second implementation reproduces

L_{H}

n=2^{8}

–

2^{12}

(

\varepsilon=0.1

L_{H}

rounded down, upper bounds up, ratios (computed from the unrounded values) down.

quantity	value
$L_{H}$	$>26.20$	K
$L_{H}$	$\leq 33$ (one schedule)	Fin
$L_{F}$	$\leq 11$ (one relabelled profile)	K
$L_{H}/L_{F}$	$>2$	K
$L_{F}$ (finite brackets)	$[1.996,\,4.681]$	Fin

Appendix

Table 7: Balanced slice,

n=65536

\varepsilon=\ln 2

. K: kernel-checked; Fin: finite computation, of two kinds:

L_{H}\leq 33

is one schedule found by a float64 dynamic program, its KL recomputed at

60

digits; the

L_{F}

bracket comes from exact-rational linear programs over inputs selected by a float64 search. The lower bound on

L_{H}

uses the three-term form of Theorem 2 (a) (the relaxed form gives

23.02

); the

L_{F}\leq 11

certificate relabels one explicit profile (numbers of unrevealed coordinates

65536\to 1735\to 201\to 50\to 19\to 9\to 5

, then five singletons). The kernel check bounds that profile’s failure probability by the second-moment budget

\sum_{t}b_{t}/m_{t}^{2}

over its shrinking blocks (

m_{t}

the coordinates left after block

t

; the singleton tail adds nothing), here

0.4889<\frac{1}{2}

, so its success exceeds

\frac{1}{2}

and

F<\ln 2

. The five-round profile behind the finite computation

F=0.650

has a budget above

7

, which is why the certificate uses eleven rounds.

Figure 5: The mechanism at finite

n

(

\varepsilon=\ln 2

; finite computation, Table 5 ): total variation to

p

of the

K=\lceil 2\ln(2/\delta)\rceil=3

ladder of Theorem 2 (b)’s proof (

\delta=\frac{1}{2}

; the Hoeffding–Ville choice, where the stated bound takes

K_{\delta}=2

; Table 5 lists its blocks) under the fixed labelling (circles) and under its relabelling (squares;

=1-s

, the quota-violation probability). Filled: computed; open: a seeded Monte Carlo estimate (standard error

0.0004

). The least lengths of the same table are plotted in Figure 2 .

$\log_{2}n$	$L_{H}\geq$	$L_{F}\leq$	$L_{F}\geq$	$R_{F}^{\mathrm{adapt}}\geq$
8	5.09	19	1.32	2
16	26.20	20	1.48	2
32	122.43	21	1.88	3
64	530.82	22	1.99	4
128	2211.37	23	2.49	4
1024	$1.46\cdot 10^{5}$	26	3.49	7

Appendix

Table 8: Proved bounds versus

n

(

\varepsilon=\ln 2

). Formula evaluations; lower bounds rounded down, upper bounds up (script hash below). The second and third columns evaluate Theorem 2 (a) and (b), and the two right-hand columns the kernel-checked envelope forms (Appendix Q ), with

\Gamma_{t}(n)

of Appendix D .

Figure 6: Proved bounds as functions of

n

\varepsilon=\ln 2

and the kernel-checked finite certificates at

n=2^{16}

. Curves are formula evaluations, not measurements.

Figure 7: Controlled test with learned conditionals (144 fresh trajectories, 48 independent clusters, six held-out regimes and fresh seeds on two discovery targets; output KL of the complete output law, computed in float64 by dynamic programming over revealed sets (no sampling),

n\leq 16

; predictions frozen before evaluation). (a) Learned regret of the path-optimal profile PATH versus its oracle loss

g_{P}

(identity dashed). (b) Learned versus oracle gap between a fixed labelled order and its relabelling. (c) Learned regret of the exact-oracle terminal-optimal profile TERM against the head’s development-loss excess (“DEV excess”), for cells with oracle margin

0.01

–

0.03

and the EARLY, ES and END checkpoints of each run (descriptive, not a frozen pass/fail item; dotted line at

0.05

, the top boundary of the freeze’s descriptive head-error bands

\leq 0.02

0.02

–

0.05

and

>0.05

; DEV excess, which can be negative, is drawn clipped below at

10^{-5}

). Controlled prediction only.

Result	Criterion; samplers or policies	Round count; target	Bearing on the least rounds under $F\leq\varepsilon$
Bounds on $H$ , which bound $F$ only from above, through $F\leq H$
Chen et al. (2026, Def. 15, Thms. 4, 9, 16)	$H$ , the expected per-schedule KL: exact formula, optimal block sizes, a bound via total correlation; $F\leq H$ . Samplers: random unmasking: prescribed block sizes, uniformly random positions	number of blocks; any target	upper bounds on $F$ through $F\leq H$
Secchi and Zanella (2026, Prop. 4)	factorization error: the $H$ of the exact-conditional sampler; output KL at most it plus a learning error. Samplers: $\tau$ -leaping (random block sizes) versus deterministic block sizes, both with random positions	steps; limit at a fixed ratio of length to steps; under all assumptions of their Prop. 4	on $H$ , random block sizes cost at least as much in the limit; $F$ enters only through $F\leq H$ (exact conditionals) or $F\leq H+\varepsilon_{\mathrm{learn}}$ (learned ones)
Lower bounds on the output law
Feng et al. (2025, Thm. 4.4)	output mass off the support; above $1/2$ it gives $F>\ln 2$ . Samplers: masked-diffusion sampling with any masking schedule, each masked position revealed independently; conditionals exact or with per-token KL strictly below $1/128$	sampling steps; an explicit hidden Markov target	$\Omega(n)$ steps for $F\leq\ln 2$ , also for mixtures of schedules and for $H$ ; does not separate $H$ from $F$
Chen et al. (2026, Def. 22, Thms. 24–25)	output TV. Samplers: randomized adaptive conditional-marginal queries, unrestricted output	at most $T$ queries with probability 1; worst case over a known family, the target in it unknown	family-level; does not constrain a fixed known target

Appendix

Table 17

result	ordinary proof	kernel-checked declarations	not kernel-checked
Lemma 1 (a)–(e)	App. B	(a) A36RW.Relabel.KLx_eq_neg_log_add and, for $q(V)=0$ , A36RW.Lemma1aZero.lemma1a_zero ; (b) A36RW.W11.Fext_le_Hext , A36RW.W11.Hext_eq_add ; (c) AGF.agfv_succMass_eq_snOf ; (d) A36RW.Relabel.KLx_schedLawE_profile ; (e) A36RW.Relabel.lemma1e_balanced	—
( Env )	App. D	A36Sharp.sharp_envelope , A36Sharp.sharpKap_lt_36	—
Lemma 3	App. E	(a) A36RW.Balanced.lemma2a_printed , A36.GrRec_mono ; for fixed schedules: A36.succMass_qSched_le_GrRec ; (b) A36RW.ExecMean.lemma2b_tail , A36RW.Converse.deadline_le_GrRec , A36RW.Converse.margUnit_balanced , A36RW.Converse.policy_outBy_le_succ , A36RW.Converse.policy_outBy_nonneg , A36RW.Converse.policy_doneBy_le_one	†
Theorem 4 (a)	App. E	A36RW.Balanced.hardcap_optimum_outLim ; attained by the randomly ordered rule: A36RW.Attain.sigmaStar_KL , A36RW.Attain.sigmaStar_capAS , A36RW.Converse.outLim_eq_outBy_of_capAS , A36RW.Converse.margUnit_balanced	†
Theorem 4 (b)	Apps. D , E.1	A36RW.Window.window_two_sided , A36RW.Window.thm2b_upper	†
Theorem 2 (a)	App. B	AGFH.Hopt_ge_three_unfolded , AGFCert.Hopt_family_lb	—

Appendix

Table 18

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