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Online Random Sampling with Real Probabilities

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read An online sampler can draw exact samples from any changing discrete distribution while losing only O(log n) bits of entropy after n draws, using O(log n) bits of persistent memory.

desk verdict New local-state recycling scheme gets O(log n) entropy loss with O(log n) persistent space; the main caveat is that the runtime/space bounds rest on 'efficient oracle' assumptions that deserve a precise statement in the theorem headers. read the letter →

arxiv 2607.13828 v1 pith:ZFMMEWK6 submitted 2026-07-15 cs.DS cs.ITmath.ITmath.PRstat.CO

Thomas L. Draper , David G. Harris , Feras A. Saad This is my paper
classification cs.DScs.ITmath.ITmath.PRstat.CO MSC 68W2068Q8794A17
keywords onlinerandomsamplingrandomnessrecyclingentropylossdiscretedistributionsCDForacleexactspacecomplexitycoinflips
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that exact online random sampling—producing a sample Xi from each freshly revealed discrete distribution Fi—can be done essentially for free in entropy: total entropy loss O(log n) after n samples, with O(log n) bits of persistent state. This matters because simulations and randomized algorithms consume random bits, and entropy loss accumulates over many draws. The method replaces the global high-precision interval of classic arithmetic decoding with a small local uniform state that compresses unused randomness from the previous sample. It handles distributions with irrational probabilities and countably infinite supports exactly, under a standard oracle model where CDFs are queried at rational approximations. A user-tunable parameter trades entropy loss against space, down to O(log(1/epsilon)) bits.

What carries the argument

The central object is the discrete uniform state (Z, M), with Z uniformly distributed in {0,...,M-1}, maintained across rounds. The Recycle subroutine re-encodes the relative position of the uniform variate Q inside the interval [F(X-1), F(X)] into a fresh pair (Z',M'), with M' bounded by a parameter Delta_i that grows as i log i up to a cap. The analysis is carried by the telescoping identity E[T - lg M' | M, X] <= lg(1/f(X)) - lg M + O(lg Delta / Delta), which, summed over rounds, yields the final entropy-loss bound.

What would settle it

Run the sampler on a long stream of Bernoulli(p) draws with an irrational p (e.g. p = 1/pi), with epsilon = 0, and measure total coin flips minus n H(p). The theorem predicts this excess stays O(log n); if it grows linearly with n for any irrational p, the central entropy-loss claim fails.

Watch

Extended reading notes

Core claim

For every sequence of discrete distributions with computable-real CDF access, exact online sampling can be achieved with total entropy loss at most epsilon n + O(log n) for any epsilon >= 0, using O(log min{n,1/epsilon}) bits of persistent state; with epsilon = 0 the loss is O(log n). Concretely, the paper proves L_n <= epsilon n + 11 lg(n+2), with persistent space O(log min{n,1/epsilon}), expected temporary space O(log i + E[log X_i]), and oracle calls O(E[log X_i]) per sample; under efficiency assumptions, the runtime per sample is within a constant factor of the oracle cost at the sampled index. The key is that the algorithm recycles randomness locally: after sampling X, the interval [F(X

Load-bearing premise

The algorithm relies on a CDF-oracle model in which every distribution is queried through rational approximations with error < 2^-k, and on the ability to discover a lower bound on each positive atom f(x); if the oracle cannot resolve arbitrarily small atoms at the needed accuracy with controlled expected cost, the O(log n) entropy and space bounds do not follow.

Editorial extensions

If this is right

  • Random-bit consumption per sample approaches the information-theoretic minimum (the Shannon entropy of the target) within an additive O(log n) total, for arbitrary non-stationary distributions.
  • Persistent memory shrinks from linear in n (global interval methods) to logarithmic, so long-running samplers on memory-constrained devices can maintain near-optimal entropy use.
  • The method removes the rational-denominator restriction: distributions with irrational probabilities and countably infinite supports are handled exactly, not approximately.
  • The epsilon-tradeoff yields a practical spectrum: set epsilon small to save entropy at the cost of O(log(1/epsilon)) bits of space, or set epsilon = 0 for pure logarithmic loss.
  • The space lower bound in the paper shows O(log n + E[max log X_i]) is necessary, so the algorithm is optimal up to constants when the means of the target distributions are polynomially bounded in n.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same local-recycling trick could be paired with other sampling primitives beyond dichotomous CDF search, potentially reducing entropy loss in statistical computing libraries that currently use rejection or inversion sampling.
  • The space-entropy tradeoff suggests an interactive-protocol reading: each sample emits a compact certificate of unused randomness, which could be useful in secure multiparty settings where wasting random bits is costly.
  • The telescoping potential-function argument may extend to batching or to streams with a known total length N, where the cap Delta could be tuned to the horizon to shave the log-factor in the entropy loss.
  • A testable extension is to implement the oracle for common families (Poisson, discrete Gaussian) and measure wall-clock entropy savings against alias-table and rejection samplers for long streams.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies online exact sampling of a sequence of discrete distributions F_i from an i.i.d. fair-coin source, in a Word RAM model where each CDF is accessed through rational-approximation oracles. It replaces the global arithmetic-decoding interval of Han--Hoshi with a local discrete uniform state (Z_i, M_i), where Z_i | M_i is uniform on {0,...,M_i-1}. Algorithm 3 (IntervalSample plus Recycle) maintains this invariant and extracts the next state from the scaled interval of the sampled atom. Theorem 4.2 gives entropy loss L_n <= eps n + 11 lg(n+2) and persistent space O(log min{1/eps,n}); Proposition 6.1 telescopes the single-step entropy bound. Runtime bounds are derived in Theorems 5.8--5.10 and Corollary 6.2 under accuracy/index-efficiency assumptions (7)--(8) and a uniform moment bound. Section 7 gives a lower bound of Omega(log n + E[max_i log X_i]) for any CDF-oracle sampler. The paper also compares with Han--Hoshi, Kozen--Soloviev, and Draper--Saad.

Significance. If the results hold as stated, this is a substantial advance: it reduces the persistent state from O(n) bits (the global interval method) to O(log n) bits while retaining near-optimal entropy loss, and it extends randomness recycling beyond finite rational distributions with bounded denominator to arbitrary real probabilities and countably infinite supports. The key invariant and the telescoping entropy accounting are elegant and appear sound. The paper is self-contained and explicit: constants such as 5000 and the Delta schedule are concrete, with no fitted parameters, and the lower-bound argument in Section 7 is an independent contribution. The main caveat is that the algorithmic efficiency claims are conditional on oracle-efficiency and moment assumptions that deserve more prominent statement in the main theorems.

major comments (3)
  1. [§3, Thm 1.2, Cor 6.2, Prop 6.3] The polynomial-runtime and O(log n) overall-space claims are conditional on the accuracy/index-efficiency assumptions (7)--(8) and on a uniform moment bound E[X_i | F_i] <= mu_n. These conditions are not stated in Theorem 1.1/1.2 or Theorem 4.2, and no countably-infinite-support oracle is actually proved to satisfy (7)--(8); Poisson and discretized Gaussian are only named. If C_F(x,nu) is super-polynomial in log nu, or if E[log X_i] is unbounded with n, then Proposition 5.11 and Corollary 6.2 do not give the advertised polynomial runtime/O(log n) space. Please state the conditions in the main theorems and give at least one concrete efficient infinite-support oracle, or explicitly delimit the theorem's scope.
  2. [§7, Lemma 7.2] The proof writes Pr(tilde Y = z) = prod c_i prod (1-c_i), which requires the outputs Y_i to be conditionally independent. The online-sampling correctness condition—conditional exactness given previous outputs—does imply this, but the manuscript does not say so; as written, the factorization is not justified for an arbitrary sampler satisfying only marginal correctness. Please add an explicit derivation of independence from the online-sampling definition. Also, clarify why an oracle query with precision parameter j forces Omega(j) bits of space; this should be tied to the oracle's return representation (a rational with denominator 2^j) in Section 3.
  3. [§5.2, Prop 5.11, Cor 6.2] The temporary-space tail bound in Proposition 5.11 is O(log(X + Delta/f(X))), whose expectation is finite only when E[log X] is finite. Theorem 1.2's statement 'expected temporary space is O(log i + E[log X_i])' is vacuous when E[log X_i] = infinity. The abstract's claim of O(log n) expected overall space under 'standard assumptions' is therefore only meaningful under a finite-moment assumption. Please state this assumption explicitly in the theorem statements and clarify that Corollary 6.2's mu_n is an a.s. uniform bound, not merely a mean bound in the usual sense.
minor comments (6)
  1. [§3, (7)--(8)] Give a compact formal definition of 'efficient oracle' in the text and state that all asymptotic notation in later theorems may depend on the polynomial g in (7)--(8). The current remark at the bottom of page 6 is helpful but should be reflected in the theorem statements.
  2. [Algorithm 3, lines 27--28] The ceil and floor operations produce U^- and U^+ from rational bounds. It may help readers to note explicitly that these are exact integer computations over O(s + lg M) bits, and that the oracle bounds are dyadic rationals with denominator 2^{s+ceil(lg M)}.
  3. [Theorem 5.6, Eq. (27)] The expression for E[T - lg M' | M,X] omits the explicit M'=1 case on recycle failure. The formula is correct because lg M'=0 there, but stating this will prevent confusion.
  4. [Section 7, Theorem 7.3] The lower bound Omega(E[max_i log X_i]) counts the output length as part of the sampler's space usage. This is a modeling choice; please state it explicitly, since some definitions of internal memory exclude the output tape.
  5. [Table 1] The table notes that the [HH97] persistent-space bound holds in expectation while Algorithm 3's is deterministic; this should be a table footnote or stated next to the relevant rows for emphasis.
  6. [General] Minor typos and formatting: 'Theorem 4.2 (Restated)' is redundant; the variable 't' is used both for the number of samples and for the time index in Algorithm 2; please disambiguate. Also, the references to [DS26a,b,c] are self-citations and should be accompanied by a short description of their exact relation to this work.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy and space theorems are derived from explicit uniform-state invariants; self-citations are comparative or support optimality claims and are not load-bearing for the core derivation.

full rationale

The main derivation chain is self-contained: Proposition 5.1 establishes the interval invariants, Proposition 5.3 derives exactness from uniformity of Q, Proposition 5.5 bounds the tail of the search depth, and Theorem 5.6 combines these to bound E[T - lg M'] in terms of lg(1/f(X)) and an explicit O(lg Delta / Delta) error. Proposition 6.1 and Theorem 4.2 then sum this bound using the explicit schedule Delta_i = 5000 i ceil(lg(i+2)); no parameter is fitted to data or to the targeted entropy loss. The runtime bounds are conditional on the stated oracle-efficiency assumptions (7)-(8), which are model assumptions, not imported results. The only same-author citations are used for comparison ([DS26b]) and for an optimality lower bound in the epsilon-tradeoff regime ([DS26a]); neither is needed to prove correctness, entropy loss, or the O(log n) space/entropy claims of Algorithm 2/3, and Section 7 supplies an independent lower bound for the main logarithmic-space regime. I find no step where a claimed prediction is equivalent by construction to an input, fitted value, or self-citation chain.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The algorithm does not fit free parameters to data; the Delta schedule and epsilon are user-selected algorithmic constants with explicit effects on the bounds. The key assumptions are the CDF-oracle model, the efficiency of the oracle, and the i.i.d. fair-coin source. The invented state (Z,M) is the algorithmic mechanism, not a postulated physical entity.

free parameters (2)
  • Delta growth constant (5000 t ceil(lg(t+2))) = 5000 t ceil(lg(t+2))
    Chosen to make the entropy-loss accounting work; used in Algorithm 2 and Theorem 4.2. It is a design constant with explicit bounds, not fitted to data, but the central entropy bound depends on it.
  • Stabilization threshold (m = ceil(0.1/epsilon)) = ceil(0.1/epsilon)
    A user-specified epsilon determines when Delta stops growing. This is a tunable scheduling choice, not fitted to data, but it is a parameter of the method's tradeoff.
assumptions (6)
  • domain assumption Target distributions are accessed through a CDF oracle phi(i,k) with |F(i) - phi(i,k)| < 2^-k and dyadic denominator 2^k.
    This is the computational model of Section 3. The entropy analysis and space bounds are stated relative to this oracle model.
  • domain assumption The entropy source produces an infinite stream of i.i.d. fair coin flips.
    The paper aims to convert such streams into target samples; all entropy accounting is relative to this source.
  • domain assumption The CDF oracle is efficient and satisfies the accuracy-efficient/index-efficient conditions in (7) and (8).
    Used in Corollary 5.10 and Corollary 6.2 to bound expected runtime in terms of CF and means.
  • domain assumption The target distributions have support N and finite mean when runtime bounds are stated.
    The algorithm handles countably infinite support in principle, but runtime and space bounds use E[X] or a uniform bound mu_n on the means.
  • standard math Word RAM arithmetic with word size w; big integers stored in w-bit limbs.
    The complexity model of Section 3, used for all arithmetic-cost claims.
  • standard math Standard properties of Shannon entropy, conditional uniformity, and the data-processing inequality.
    Used throughout to define entropy loss L_n and in proofs such as Proposition 5.3.
invented entities (2)
  • Local discrete uniform state (Z_i, M_i) with Z_i | M_i ~ Unif{0,...,M_i - 1} independent evidence
    purpose: Used to recycle unused randomness from each sampling round, replacing the global interval of arithmetic decoding.
    The state is defined and constructed in the paper, and its behavior is verified by the invariant proofs; it is not an externally observed entity. The reconstruction via the interval [U^-, U^+) has an observable effect on entropy loss and space, which is independently checkable.
  • CDF bound oracles F^- and F^+ independent evidence
    purpose: Derived from the approximation oracle to bracket the real CDF values; used in Algorithm 3 and Recycle.
    The oracles are mathematically defined constructs, not physical entities; their properties follow from the approximation oracle assumption.

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Pith. "Pith review of Online Random Sampling with Real Probabilities." pith.science (2026). https://pith.science/paper/ZFMMEWK6

@misc{pith2026260713828,
  author       = {Pith},
  title        = {Pith review of: Online Random Sampling with Real Probabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFMMEWK6}},
  note         = {Machine review of arXiv:2607.13828}
}
abstract

We develop an efficient online algorithm to sample a sequence of discrete random variables using an entropy source of i.i.d. fair coin flips, in a standard model of real computation where real-valued probabilities are represented by rational approximations. For any sequence $F_1, F_2, \dots$ of probability distributions, our sampler generates $n$ outputs $X_1 \sim F_1, \dots, X_n \sim F_n$ using at most $\mathbb{E}\left[H(F_1) +\dots + H(F_n)\right] + O(\log n)$ coin flips in expectation while carrying $O(\log n)$ bits of persistent space, where $H$ is the Shannon entropy. Under standard assumptions, we prove that the space used by our sampler to achieve this information-theoretically optimal entropy rate is asymptotically optimal. The key idea is to replace the global arithmetic-decoding sampling scheme of Han and Hoshi (1997) with a local discrete uniform state, yielding an exponential reduction in space for a given entropy loss. Our approach applies to distributions with irrational probabilities and countably infinite supports, generalizing recent randomness-recycling methods beyond finite rational distributions with bounded denominator.

Figures

Figures reproduced from arXiv: 2607.13828 by the authors.

Figure 1
Figure 1. Visualization of the randomness recycling mechanism in Algorithm [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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  1. Space-Entropy Lower Bounds for Random Sampling

    cs.CC 2026-07 conditional novelty 8.0 of 10

    Exact entropy-efficient random sampling with per-sample entropy loss ε requires Ω(log(1/ε)) bits of persistent space, with sharp constants for Bernoulli(1/3) and almost all k-outcome distributions.

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    Draper, Thomas L. and Saad, Feras , title =. 2026 , fnote =

    2026

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.