# Reviewer ZHAe
## Strengths
Construction is clean and the key design choice is well motivated: use globally annealed sequence probability as the quality term but propose from locally annealed sampling, then importance-weight to bridge the two. The sketched-distribution abstraction is clear, the Rao-Blackwell interpretation is a nice touch, and the dual-kernel cost analysis is correct. The proofs look internally sound on inspection, but I couldn't verify all of them. DPPs over embeddings are standard, so the originality is in the pool-based estimator and its finite-sample analysis, not the DPP framing itself.

## Weaknesses
1. the theory and the experiments appear somewhat largely disjoint. The convergence guarantee applies only to the sampling mode at low temperature, but every headline result uses the MAP mode at high temperature, where there is no distributional guarantee and greedy MAP is only a (1-1/e) approximation to the pool's best set, not the true optimum. The strongest theory and the strongest results never touch the same object.
2. the bound is asymptotic in the pool size but its constant grows steeply with the set size and embedding dimension. At the set size they actually use, the constant is large enough that the numerical bound is almost certainly vacuous at the pool sizes they run. The result establishes a rate, not a practically reassuring guarantee, and I think it would be important for the paper to say so.
3. the intrinsic FineWeb evaluation is close to circular. Both axes of the main intrinsic plot are components of the objective that MAP directly maximizes, so MAP winning there is nearly definitional and weak evidence on its own. Where I see the most evidence is Ambrosia and GSM8K, and the paper's weight should rest there.
4. On those downstream tasks the gains are real but modest, and the trade-offs are underplayed. Ambrosia coverage is low in absolute terms with overlapping error bars. On GSM8K, all methods are essentially tied at matched temperature; the MAP advantage needs the extreme pure-diversity setting, which raises coverage by a few points but drops per-chain accuracy by about ten points.
5. strong sequence-level set-diversity baselines are missing empirically (e.g. Stochastic Beam Search, Conditional Poisson SBS), as are MMR or MBR reranking with semantic distances, which are the natural comparators. The abstract advertises an LLM-based selection baseline that appears nowhere in the paper. The best temperature ranges enormously across tasks and modes and is swept to trace the frontiers, so picking an operating point needs task labels.

## Questions
1. W2, What does the bound actually evaluate to at your default set size for a realistic importance-ratio bound? If it is vacuous at that set size, please reframe it as a rate result. This affects my confidence and, if the bound is more usable than I estimate, my scores.
2. Could you report the weight distribution (max, variance, effective sample size) per task and say whether any truncation is used? This tests whether the bounded-ratio assumption holds for the local proposal relative to the global target. Well-behaved weights would raise Quality.
3. Is the GSM8K coverage gain preserved at a moderate temperature rather than the extreme one, and are the Ambrosia gains significant, given the overlapping error bars? A significance test or larger sample here directly affects Significance. If the gains fall inside the noise, my score drops.

# Response to Reviewer ZHAe

We thank the really constructive and detailed feedback. The questions have been very helpful in clarifying the paper's contributions and limitations. We address each point below.

## W1. Theory covers sampling, not MAP

We agree, and we will state it explicitly in the revision: the finite-sample guarantee covers PoolDPP-Samp, and greedy MAP is a (1-1/e) approximation to the pool mode. However, MAP inference always leverages the distributional guarantees of the underlying distribution, this is true for PoolDPP and for any MAP inference on maximum-likelihood trained models.

We included MAP as the deterministic counterpart to the deterministic baselines (k-means performing better was raised as a concern internally), but anything that consumes samples should use PoolDPP-Samp. We are all aware of the disadvantages of MAP inference, and it is for that reason that Beam Search or k-means selection are not widely used in practice.

## W2 / Q1. Is the constant vacuous at the default K?

This is a very good point that led us to re-examine the constant. The first thing we did was to measure $\text{KL}(P \| \bar P)\le\log(1+\mathbb{V}_\omega(\hat Z)/Z^{2})$ from Theorem 5.1, since it is estimable: Monte Carlo over pool draws.

We then evaluated our closed-form substitution $\mathbb{V}_\omega(\hat Z) \le C_{K,D} \alpha_{N,K}^{2} \Lambda^{2K-2} \mu_2 / N$ and indeed, it becomes a very loose bound. The looseness comes from bounding $|e_K(\text{eig} \hat C) - e_K(\text{eig} C)|$ additively, through a worst-case Lipschitz constant of $e_K$, while $\hat Z$ concentrates multiplicatively around $Z$.

Working in relative quantities from the start yields a tight closed-form, available in the at the end of this rebuttal. $\mathbb{V}_\omega(\hat Z)/Z^2$ can be written exactly as a weighted sum of chi-square divergences between the set inclusion marginals and the proposal. 

| | Ambrosia, K=3, N=200, tT=0.4 | GSM8K, K=10, N=100, tT=10 | FineWeb, K=10, N=400, tT=0.5 |
|---|---|---|---|
| measured $\mathbb{V}_\omega(\hat Z)/Z^2$ | 0.031 | 1.5 | 0.015 |
| implied KL | <= 0.031 | <= 0.93 | <= 0.015 |
| **new bound** | **0.044** | **5.5** | **0.070** |
| old closed-form bound | 10^10.2 | 10^47.7 | 10^20.5 |
| **N required to bound KL $\le$ 0.1, new** | **109** | **2,222** | **320** |
| N required to bound KL $\le$ 0.1, old | 10^13.4 | 10^50.3 | 10^24.0 |

We include a short version of the new derivation at the end of this rebuttal. Theorem 5.1 and Proposition 5.2 are not affected, only Theorem 5.3 and the discussion of the constant. We thank you again for pointing this out, and really hope that this is reassuring and not too big of a change.

## W3. The intrinsic FineWeb evaluation is close to circular

It is true that it is graded on the axes it optimizes, and we agree the evaluation weight should rest on the downstream tasks. We included it because we believe they capture the right optimization objective, despite their self-referential nature, and serve as an important sanity check. To strengthen this section, we evaluate the same metrics using a different ("held-out") evaluation embedder. In this setting, we measure log-det and mean cosine distance under the held-out embedder, as well as embedder-free metrics (Distinct-n and pairwise Jaccard similarity). Across all of these metrics, every PoolDPP-MAP point remains Pareto non-dominated, and the rank correlation between each metric and the kernel log-det across all 29 method/knob configurations ranges from 0.89 to 0.99. We will add these results as an appendix table and temper the corresponding intrinsic claims accordingly.

## Q2. Weight diagnostics and truncation

All diagnostics are on the self-normalized weights, $w(y) = p(y)^{1/(2tT)} / q_{sT}(y)$ for the scheme FineWeb and Ambrosia run and $w(y) = p(y)^{1/(2tT)}$ for GSM8K, at the anchor each task uses ($sT$ = 1.0, 1.0, 1.2):

| task | tT | max weight (norm) | variance | ESS/N | sup ratio |
|---|---|---|---|---|---|
| FineWeb | 0.3 | 0.718 | 1.6e-3 | 0.011 | 287 |
| FineWeb | 0.45 | 0.209 | 2.8e-4 | 0.102 | 83.5 |
| FineWeb | 0.8 | 0.850 | 2.0e-3 | 0.004 | 340 |
| Ambrosia | 0.3 | 0.033 | 6.7e-5 | 0.323 | 6.7 |
| Ambrosia | 0.45 | 0.009 | 4.4e-6 | 0.856 | 1.7 |
| GSM8K | 2.5 | 0.228 | 1.2e-3 | 0.139 | 22.8 |
| GSM8K | 50 | 0.016 | 1.6e-5 | 0.867 | 1.6 |
| GSM8K | 500 | 0.011 | 1.2e-6 | 0.988 | 1.1 |


On truncation: PoolDPP-Samp applies a floor at 20 nats below the maximum to exactly the weights above (a floor, not a cap: large weights are never truncated). This is done because of the numerical stability of the DPP sampling algorithms. PoolDPP-MAP applies no truncation. We will document both.

On GSM8K and Ambrosia the assumption holds at the reported operating points. On the less constrained FineWeb the pool spans hundreds of nats in log p which disperses the weights. The W2 measurement makes the same statement exact, since Var(Zhat)/Z^2 is the quantity a bounded ratio controls. We will report these diagnostics in the revision.

## W4 / Q3. Are the gains significant, is GSM8K preserved at moderate T?

We generated more pools (30 on Ambrosia, 20 on GSM8K) and paired over them, the unit the error bars use, with a sign-flip permutation test. On Ambrosia the answer is clear: PoolDPP-Samp over ancestral at matched temperature, and PoolDPP-MAP over k-means, are both significant at p < 1e-5.

On GSM8K with the submitted kernel, the gains reach significance only for MAP at high target temperature. However, with the instructed kernel introduced in our response to Gqmg (W1), significance arrives at moderate target temperatures instead: MAP at tT=10 and Samp at tT=25. The frontier plot is also visibly healthier there, with the PoolDPP points and their error bars separated from the baselines across the sweep rather than at one extreme setting. We will update the paper's GSM8K figure to that kernel.

The semantically relevant notion of diversity in GSM8K is very narrow, and low-effort adaptations like the instructed kernel already make a big difference. Taking it even further, appending a one-hot representation of each chain's extracted final answer to the embedding makes every setting significant. We report it as evidence that the kernel is the binding constraint rather than as a headline, since that feature map is very domain-specific.

## Missing baselines

We thank the reviewer for pointing out the missing baselines. We were already implementing SBS and CPSBS, delayed due to the amount of work and validation needed to match the original fairseq implementations. For each method, we sweep their parameters and show the best result. 

We show here the Ambrosia table since it is extrinsic and representative:
| method | recall (coverage) | precision |
|---|---|---|
| SBS | 0.2355 | 0.3915 |
| CPSBS | 0.2068 | 0.4157 |
| MBR | 0.1941 | 0.4010 |
| MMR λ=0.3  | 0.2608 | 0.3344 |
| PoolDPP-MAP tT=10 | 0.2672 | 0.3531 |

GSM8K and FineWeb are ommited due to space constraints and show the same trend: the beam methods and MBR have better precision at the cost of recall, MMR at λ=0.3 is more balanced and competitive in recall but lags behind in precision.

### LLM-based selection baseline

We apologize for the oversight. The experiment was was commented out due to a editing mistake. Will of course be restored in the final text.

The baseline consists of presenting 100 pool items, drawn at sampling temperature 1.0, to Qwen3-8B and ask it to select K=10 diverse, high-quality completions. The model sees only the raw text, with no access to the log-probability or the log-determinant it is implicitly asked to trade off. On the FineWeb intrinsic axes it is dominated on both: quality −72.4 and log-det −6.19, against −40.4 and −4.84 for k-Means at the same operating point.

## Empirical tradeoffs and variance of T among tasks

We will make sure to explain better the tradeoffs and limitations of the empirical experiments in the revision. We acknowledge they are an initial validation of the method, not an systematic exploration on each domain. It is also true that T varies a lot among tasks. Future work could derive optimal temperatures from the model's observed entropy on a given tasks, which is observed during generation and conditions dispersion. We will state this as a limitation.

## Appendix (W2): Compressed tighter bound derivation

Pool $y^{(1)},\dots,y^{(N)}\sim q$ i.i.d., $w=\kappa/q$, $\hat Z=\alpha_{N,K}\sum_{|I|=K}\det(\hat L_I)$.

**(i) $\hat Z$ is a U-statistic.** By eq:quality-diversity $\det(\hat L_I)=\prod_{n\in I}\frac{w^{(n)}}{N}\det(\Phi_I^\top\Phi_I)$, and $\alpha_{N,K}N^{-K}=1/(N)_K=1/[\binom{N}{K}K!]$, so
$$\hat Z=\binom{N}{K}^{-1}\!\!\sum_{|I|=K}h(Y_I),\qquad h(y_1..y_K)=\tfrac1{K!}\textstyle\prod_i w(y_i)\cdot\det(\Phi^\top\Phi):$$
the U-statistic of order $K$ with kernel $h\ge0$. So $\alpha_{N,K}$ *is* its normalizer, and $h=0$ on repeats.

**(ii) Its projections are inclusion ratios.** Fix distinct $y_1..y_c$, $T=\{y_1..y_c\}$. Averaging over $Y_{c+1..K}\sim q$ turns $w(y_j)q(y_j)$ into $\kappa(y_j)$; repeats vanish, each $S\supseteq T$ arises from $(K-c)!$ orderings, and $\prod_{y\in S}\kappa(y)\det(\Phi_S^\top\Phi_S)=\det(L_S)$:
$$\mathbb E[h(y_1..y_c,Y_{c+1}..Y_K)]=\frac{(K-c)!}{K!}\prod_{i\le c}\frac{1}{q(y_i)}\sum_{S\supseteq T}\det(L_S)=Zu_c,\quad u_c\equiv\frac{P(T\subseteq S)}{(K)_c\prod_{i\le c}q(y_i)},$$
and $\mathbb E[u_c]=1$ because $\sum_{|T|=c}P(T\subseteq S)=\binom{K}{c}$ with $c!$ orderings each.

**(iii) Hoeffding.** For $|I\cap J|=c$, conditioning on the shared draws gives $\mathrm{Cov}(h(Y_I),h(Y_J))=\mathbb V(\mathbb E[h\mid Y_{I\cap J}])=Z^2\mathbb V(u_c)$, and $0$ at $c=0$; there are $\binom{N}{K}\binom{K}{c}\binom{N-K}{K-c}$ ordered pairs with $|I\cap J|=c$, so

$$\frac{\mathbb V_\omega(\hat Z)}{Z^{2}}=\sum_{c=1}^{K}a_c\mathbb V_{q^{\otimes c}}(u_c),\qquad a_c=\frac{\binom{K}{c}\binom{N-K}{K-c}}{\binom{N}{K}}.$$

$u_1(y)=P(y\in S)/(Kq(y))$ costs one eigendecomposition of $C$. $\{a_c\}$ is the overlap law of two uniform $K$-subsets of $[N]$ (mean $K^2/N$), so $c=1$ dominates when $K^2\ll N$.

**(iv) Closed form.** $K$-DPPs are strongly Rayleigh, hence negatively associated: $P(T\subseteq S)\le\prod_{y\in T}P(y\in S)$, so $u_c\le\frac{K^c}{(K)_c}r_1^c$ with $r_1=\sup_y u_1(y)$, and $\mathbb V(u_c)\le\sup u_c-1$ since $u_c\ge0$ has mean 1. With $a_c\le\binom{K}{c}(K)_c/(N)_c$, $\binom{K}{c}\le K^c/c!$ and $(N)_c\ge(N-K+1)^c$, every term is $\le\theta^c/c!$ for $\theta=K^2r_1/(N-K+1)$, so $\mathbb V_\omega(\hat Z)/Z^2\le e^{\theta}-1$ and

$$\mathrm{KL}(P\|\bar P)\le\frac{K^{2}r_1}{N-K+1},\qquad\text{so }\mathrm{KL}\le\varepsilon\text{ once }N\ge K^2r_1/\varepsilon+K-1.$$

**(v) Tighter bound.** Bounding only the $c\ge2$ terms and keeping $c=1$ gives the "new bound" row:

$$\mathrm{KL}(P\|\bar P)\le\log\Big(1+\underbrace{a_1\mathbb V_q(u_1)}_{\text{exact}}+\sum_{c\ge2}a_c\Big(\tfrac{K^c}{(K)_c}r_1^c-1\Big)\Big),$$

which needs only $u_1$ and $r_1$, i.e. one eigendecomposition of $C$.