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REVIEW 3 major objections 4 minor 109 references

Direct Prediction Set Minimization via Bilevel Conformal Classifier Training

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that conformal prediction sets can be trained to be smaller by solving a bilevel problem that learns the quantile threshold, with an $O(1/\sqrt{n})$ learning bound and a 20.46% average reduction in set size.

desk verdict A genuinely novel conformal-training formulation whose O(1/√n) theory only applies to the exact lower-level solution, not to the stochastic algorithm that actually runs. read the letter →

arxiv 2506.06599 v1 pith:4B6Y22HZ submitted 2025-06-07 cs.LG stat.ML

Yuanjie Shi , Hooman Shahrokhi , Xuesong Jia , Xiongzhi Chen , Janardhan Rao Doppa , Yan Yan This is my paper
classification cs.LGstat.ML
keywords conformalpredictionsetsizebileveloptimizationquantileregressiontraininguncertaintyquantificationlearningboundmini-batch
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 make conformal prediction sets smaller by training the underlying classifier for that goal, not only calibrating it afterwards. It reformulates conformal training as a bilevel optimization problem in which the upper level minimizes the average prediction-set size and the lower level learns the needed quantile of non-conformity scores by quantile regression. The paper's central theoretical claim is that this achieves a learning bound of $O(1/\sqrt{n})$ in the number of training samples, while prior stochastic-approximation methods have a bound of at least $\Omega(1/s)$ for batch size $s$. If the claim holds, conformal classifiers can be trained end-to-end with the same sample-complexity rate as ordinary supervised learning, and the reported experiments put the average prediction-set-size reduction at 20.46% over the best baseline.

What carries the argument

The machine carrying the argument is a bilevel objective that couples classifier $f$ and a scalar quantile $q$. The lower level minimizes the empirical pinball loss $\hat L_{QR}(f,q)=\frac1n\sum_i \rho_\alpha(q,S_f(X_i,Y_i))$, whose minimizer $q^*_f$ is a $(1-\alpha)$-quantile of conformity scores; the upper level minimizes the classification loss plus the smoothed prediction-set-size loss $\hat L_c^{DM}(f,q)$ conditioned on that $q$. Explicitly parameterizing $q$ decouples the threshold from batch randomness. The proof chain uses concentration of the empirical CDF to show $|Q_f-q^*_f|\le \tilde O(1/\sqrt{n})$, Lipschitzness of the smoothed set-size loss in $q$, and a H\"olderian error bound for the QR loss to connect the penalized problem's solution to the original bilevel problem.

What would settle it

Run DPSM's Algorithm 1 at fixed batch size $s$ with increasing training sizes $n$ and measure the gap between training and test average set size (or between learned $q_t$ and the dataset-level quantile); if that gap decays like $\tilde O(1/\sqrt{s})$ rather than $\tilde O(1/\sqrt{n})$, or if $q_t$ fails to approach the dataset-level quantile, the bound does not describe the implemented method.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that conformal training does not have to estimate the quantile threshold from whatever mini-batch happens to be drawn. By making the quantile an explicit scalar $q$ and training it with the pinball loss in a lower-level subproblem while the upper level minimizes a smoothed average prediction-set size, the alignment loss becomes a function of a learned threshold. Under the paper's bi-Lipschitz assumption on conformity scores, Theorem 4.1 bounds the gap between this implicit conformal loss and the population conformal loss by $\tilde O(1/\sqrt{n})$ with probability at least $1-\delta$, independent of batch size. Experiments on CIFAR-100, Caltech-101, and iNaturalist report a 20.46% average reduction in prediction set size over the best prior conformal training baseline at 90% coverage.

Load-bearing premise

The load-bearing premise is that the stochastic gradient updates in Algorithm 1 drive the scalar quantile to the exact lower-level minimizer $q^*_f$; the paper proves its $O(1/\sqrt{n})$ bound only for that exact minimizer and leaves convergence of the implemented update as an open challenge.

Editorial extensions

If this is right

  • Conformal training can match the $O(1/\sqrt{n})$ generalization rate of standard empirical risk minimization rather than the $\Omega(1/s)$ rate of batch-quantile methods.
  • The learned-quantile formulation gives smaller prediction sets at the same marginal coverage: 20.46% average reduction across CIFAR-100, Caltech-101, and iNaturalist with ResNet and DenseNet at $\alpha=0.1$.
  • The quantile estimation error of DPSM shrinks toward zero during training, giving a tighter empirical learning bound than ConfTr.
  • The method applies across HPS, APS, and RAPS scoring functions, with the largest gains on Caltech-101 and the HPS score.

Reading between the lines

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

  • Extension: the theorem's exact-minimizer assumption leaves open whether the SGD-updated $q_t$ enjoys the same rate; a convergence proof for Algorithm 1 would close the gap and is the natural next step.
  • Extension: the same bilevel template could be applied to other conformal objectives (conditional coverage, conformal risk control) by swapping the smoothed set-size loss in the upper level while retaining the learned quantile.
  • Extension: if the quantile update does converge at a batch-dependent rate, DPSM's practical gain may vary with batch size in ways the current bound does not capture; measuring the training-test gap at fixed $n$ and varying $s$ would test this.
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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 / 4 minor

Summary. This paper proposes DPSM, a bilevel optimization method for training deep classifiers to minimize the expected size of conformal prediction sets. The upper-level objective is a differentiable prediction-set-size loss evaluated at a quantile q of the non-conformity scores, and the lower-level problem learns q by minimizing the pinball loss on the training scores. The authors prove a learning bound of O(1/√n) for the exact lower-level minimizer q*_f, contrasting with an Ω(1/s) lower bound they derive for prior stochastic-approximation (SA) conformal training methods. They present a stochastic first-order algorithm (Algorithm 1) that alternately updates the classifier and q, and they report experiments on CIFAR-100, Caltech-101, and iNaturalist showing an average 20.46% reduction in prediction set size over the best baseline.

Significance. If the O(1/√n) bound were matched to the implemented algorithm, the paper would be a solid contribution to conformal training, since prior analyses degrade with batch size as O(1/√s). The bilevel formulation with quantile regression is a natural way to decouple quantile estimation from mini-batch noise, and the paper is transparent about the open optimization question, which is methodologically honest. However, the current disconnect between Theorem 4.1 and Algorithm 1 prevents the theoretical claim from supporting the method that produced the empirical results, and the empirical headline is not uniform across settings. With these issues addressed, the paper could become a valuable reference for prediction-set-efficient classifier training.

major comments (3)
  1. [§4.2, Algorithm 1 and Theorem 4.1] The learning bound in Theorem 4.1 applies to \bar{L}_c^DM(f) = \hat{L}_c^DM(f, q*_f), where q*_f is an exact minimizer of the full-training-set pinball loss. Algorithm 1 (lines 9–10) updates q by a single stochastic gradient step on a mini-batch pinball loss, and the paper explicitly states in Section 4.2 that proving convergence of such an algorithm is an open challenge left for future work. Therefore the O(1/√n) bound is not established for the implemented method; the quantity that matters for the trained f_T and last iterate q_T is |\hat{L}_c^DM(f_T, q_T) - L_c(f_T)|, which includes the optimization error in q. In addition, the penalty-based reformulation (10) and Corollary 4.6 are not connected to Algorithm 1, since Algorithm 1 does not optimize the penalized objective. The authors should either provide a convergence guarantee for the q-updates (e.g., a bound on |q_T - q*_f|), revise the theorem to bound the actual quantity including the optimization error, or modify the algorithm to provably solve the lower-level problem.
  2. [Appendix B.2, proof of Theorem 3.5] The lower-bound proof of Theorem 3.5 expands (\hat{L}_c^SA(f) - L_c(f))^2 and lower-bounds the cross term 2(E[\hat{ℓ}(f,\hat{q}_f)] - \hat{ℓ}(f,E[\hat{q}_f]))(\hat{ℓ}(f,E[\hat{q}_f]) - ℓ(f,Q_f)) by a positive quantity using Lemma B.3. However, since \hat{ℓ}(f,·) is μ-strongly concave (Assumption 3.2), Jensen's inequality gives \hat{ℓ}(f,E[\hat{q}_f]) - E[\hat{ℓ}(f,\hat{q}_f)] ≥ 0, so E[\hat{ℓ}(f,\hat{q}_f)] - \hat{ℓ}(f,E[\hat{q}_f]) ≤ 0. The sign of this factor is opposite to what the proof assumes, making the product with the nonnegative factor from Lemma B.4 nonpositive. As stated, the proof does not establish the claimed Ω(1/s) lower bound; it needs to control the sign of the total deviation or use a different decomposition.
  3. [Table 1 and Appendix F.1] The headline reduction of 20.46% is an average over all settings and is driven by the Caltech-101 results (e.g., 59.17% reduction for DenseNet with APS). On several settings DPSM is worse than the best baseline: CIFAR-100 DenseNet with APS is 2.64 vs 2.41 for CUT (8.71% increase), iNaturalist ResNet with APS is 79.43 vs 79.00 (0.54% increase), and additional increases appear for RAPS in Tables 5–6 (e.g., CIFAR-100 DenseNet 8.55% increase, iNaturalist ResNet 0.08% increase). The abstract's claim that DPSM 'significantly outperforms' the best baseline is therefore not supported uniformly. The authors should report per-setting improvements with statistical significance and either temper the abstract or restrict the claim to the settings where the improvement is consistent.
minor comments (4)
  1. [Table 5] The entry for iNaturalist ResNet reads '↑-0.54%', which is self-contradictory; it should be either '↑0.54%' or '-0.54%' depending on the intended meaning.
  2. [§4.2] There is a typo in 'Lipscthiz Hessians'; it should be 'Lipschitz Hessians'.
  3. [Figure 3 and Appendix F.1] The notation bQn f is used without a formal definition; it is introduced in the text of Figure 3 but should be defined at first use in Section 3 or the caption.
  4. [§5.2, Figure 2] The convergence justification is based only on CIFAR-100 with DenseNet (and ResNet in the appendix); a statement that this is a representative configuration rather than evidence for all datasets would be helpful.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: DPSM's learning bound is a self-contained concentration result, but it is not yet matched to the implemented stochastic algorithm.

full rationale

The paper's central theoretical claim, Theorem 4.1, bounds |\bar L^{DM}_c(f) - L_c(f)| where \bar L^{DM}_c(f) is the empirical soft set size evaluated at the exact pinball-loss minimizer q*_f and L_c(f) is the population conformal loss at the true quantile Q_f. The proof in Appendix C is self-contained: Lemma C.2 shows |q*_f - Q_f| ≤ O~(1/√n) via Chernoff bounds on the empirical CDF and the quantile property of the QR minimizer, Lemma C.3 gives Lipschitz continuity of the soft set size in q, and Lemma C.4 is a standard Hoeffding generalization bound. This is a genuine concentration result, not a restatement of fitted values and not a consequence of the authors' own prior results. The empirical 20.46% reduction is an experimental comparison, not a fitted quantity renamed as a prediction. The self-citations (e.g., Ghosh et al. 2023a; Shi et al. 2024; Shahrokhi et al. 2025; Yuan et al. 2019) appear only as background or related-work references and are not load-bearing for the derivation. The paper itself flags a real limitation: Theorem 4.1 applies to the exact lower-level minimizer q*_f, while Algorithm 1 updates q by stochastic gradient steps and Section 4.2 states that proving convergence 'is non-trivial and an open challenge.' This mismatch is a correctness or completeness concern, not circularity, because the theorem does not presuppose the algorithm's convergence. Similarly, Figure 3(c) is explicitly described as an approximation of the learning bound via a train-test APSS gap, which does not constitute circular reasoning. Overall, the derivation chain is independent of its conclusions, so no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the exact-solution assumption for the lower level, which is not proven for Algorithm 1, on bi-Lipschitz score order statistics, and on standard i.i.d. and concentration assumptions. No new particles, forces, dimensions, or mediators are introduced; q is a scalar quantile parameter, not an invented entity.

free parameters (4)
  • lambda_regularization = varies by dataset/model, e.g., 0.01, 0.05, 0.1, 1.0 (Table 3)
    Weights the conformal alignment loss against the classification loss in objective (8); selected by hyperparameter search for best predictive efficiency.
  • gamma_lower_level_lr = 0.001 to 0.1 (Table 3)
    Learning rate for the stochastic quantile updates in Algorithm 1; no theoretical convergence guarantee is provided.
  • tau_Sigmoid = not reported in main text
    Temperature of the smoothed indicator that approximates prediction set size; changes the loss surface and the Lipschitz constant K/(4*tau).
  • RAPS_regularization = lambda_RAPS=0.01, k_reg=5
    Used only for RAPS-scored evaluation in Table 6, not for DPSM training; affects the reported set sizes for that scoring function.
assumptions (6)
  • domain assumption Assumption 3.1: conformity scores S_(j) are bi-Lipschitz in normalized rank j/n
    Used in Theorem 3.5, Lemma C.2, and Theorem 4.1; verified empirically only on CIFAR-100 with DenseNet and HPS scores (Figures 4a and 10a).
  • domain assumption Assumption 3.2: smoothed conformal loss b_l(f,q) is locally mu-strongly concave in q around E[bq_f]
    Needed for the Omega(1/s) lower bound on SA methods in Theorem 3.5; verified empirically only near coverage 0.9 (Figures 4b and 10b).
  • domain assumption No ties in non-conformity scores
    Ensures unique empirical quantiles and supports the HEB proof in Lemma 4.5; APS and RAPS break ties by uniform randomization.
  • domain assumption IID or exchangeable samples for training and calibration
    Standard conformal prediction assumption; required by Lemma C.4 and by the coverage guarantee in equation (1).
  • standard math Bounded conformity scores and standard concentration inequalities (Hoeffding, Chernoff, Chebyshev)
    Used to obtain the O(1/sqrt n) and O(1/sqrt s) rates in Lemmas C.2, C.4, and B.5.
  • ad hoc to paper The stochastic q-updates in Algorithm 1 converge to the exact lower-level minimizer q*_f
    This is the unproven link between Theorem 4.1 and the implemented algorithm; the paper explicitly leaves this challenge for future work in Section 4.2.

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Pith. "Pith review of Direct Prediction Set Minimization via Bilevel Conformal Classifier Training." pith.science (2026). https://pith.science/paper/4B6Y22HZ

@misc{pith2026250606599,
  author       = {Pith},
  title        = {Pith review of: Direct Prediction Set Minimization via Bilevel Conformal Classifier Training},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4B6Y22HZ}},
  note         = {Machine review of arXiv:2506.06599}
}
abstract

Conformal prediction (CP) is a promising uncertainty quantification framework which works as a wrapper around a black-box classifier to construct prediction sets (i.e., subset of candidate classes) with provable guarantees. However, standard calibration methods for CP tend to produce large prediction sets which makes them less useful in practice. This paper considers the problem of integrating conformal principles into the training process of deep classifiers to directly minimize the size of prediction sets. We formulate conformal training as a bilevel optimization problem and propose the {\em Direct Prediction Set Minimization (DPSM)} algorithm to solve it. The key insight behind DPSM is to minimize a measure of the prediction set size (upper level) that is conditioned on the learned quantile of conformity scores (lower level). We analyze that DPSM has a learning bound of $O(1/\sqrt{n})$ (with $n$ training samples), while prior conformal training methods based on stochastic approximation for the quantile has a bound of $\Omega(1/s)$ (with batch size $s$ and typically $s \ll \sqrt{n}$). Experiments on various benchmark datasets and deep models show that DPSM significantly outperforms the best prior conformal training baseline with $20.46\%\downarrow$ in the prediction set size and validates our theory.

Figures

Figures reproduced from arXiv: 2506.06599 by the authors.

Figure 1
Figure 1. Box plots of coverage (Top row) and APSS (Bottom row) of all methods using DenseNet and HPS score. DPSM achieves significantly smaller prediction set size while maintaining the valid coverage. (a) Upper loss (b) Lower loss (c) Conformal loss optimization gap (d) QR loss optimization gap [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Justification experiments for the convergence of DPSM on CIFAR-100 using DenseNet and HPS score. (a) Upper level loss (i.e., a combination of classification loss and conformal alignment loss); (b) Lower level loss (i.e., QR loss); (c) Opti￾mization gap of conformal loss, defined as the difference between conformal losses using learned batch-level quantiles and dataset￾level quantiles on the training set; (d) Optimiz… view at source ↗
Figure 3
Figure 3. Justification experiments for the learning bound of DPSM on CIFAR-100 using DenseNet and HPS score. (a) Estimation error between the Qbn f (i.e., the dataset-level quantiles on training data) and qbf (i.e., either the batch-level quantiles evaluated in ConfTr or the learned ones in DPSM); (b) Average soft set size (approximated using Sigmoid function) of DPSM and ConfTr; (c) Approximated learning error comparison be… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Assumption verification on CIFAR-100 using DenseNet and HPS score on the calibration dataset. (a) HPS scores over corresponding normalized index produced by ConfTr. The x-axis is the normalized order, the y-axis is the corresponding conformity score; (b) The soft set s…
Figure 5
Figure 5. Figure 5: Box plots of coverage (Top row) and APSS (Bottom row) of all methods using ResNet and HPS score. DPSM achieves significantly smaller prediction set size while maintaining the valid coverage. (a) Upper loss (b) Lower loss (c) Conformal loss optimization gap (d) QR loss …
Figure 6
Figure 6. Figure 6: Justification experiments for the convergence of DPSM on CIFAR-100 using ResNet and HPS score. (a) Upper level loss (i.e., a combination of classification loss and conformal alignment loss); (b) Lower level loss (i.e., QR loss); (c) Optimization gap of conformal loss, …
Figure 7
Figure 7. Figure 7: Justification experiments for effectiveness of DPSM trained with 40 epochs on CIFAR-100 using DenseNet and HPS score. (a) Upper level loss (i.e., a combination of classification loss and conformal alignment loss) in DPSM; (b) Lower level loss (i.e., QR loss) in DPSM; (…
Figure 8
Figure 8. Figure 8: Justification experiments for effectiveness of DPSM trained with 40 epochs on CIFAR-100 using ResNet and HPS score. (a) Upper level loss (i.e., a combination of classification loss and conformal alignment loss) in DPSM; (b) Lower level loss (i.e., QR loss) in DPSM; (c)…
Figure 9
Figure 9. Figure 9: Justification experiments for the learning bound of DPSM on CIFAR-100 using ResNet and HPS score. (a) Estimation error between the Qbn f (dataset-level quantiles on training data) and qbf (batch-level quantiles evaluated in ConfTr or learned in DPSM); (b) Average soft …
Figure 10
Figure 10. Figure 10: Verification studies on CIFAR-100 with ResNet model using HPS scoring function on calibration dataset. (a): HPS scores over corresponding normalized order produced by ConfTr. The x-axis is the normalized order, the y-axis is the corresponding conformity score; (b): Th…
Figure 11
Figure 11. Figure 11: Class conditional coverage and class-wise prediction set size of all methods on CIFAR-100 with DenseNet and HPS. To better compare the class conditional coverage, we compare the class-conditional coverage between DPSM and 3 baselines in (a), (b) and (c) separately. DP…
Figure 12
Figure 12. Figure 12: Class conditional coverage and class-wise prediction set size of all methods on CIFAR-100 with ResNet and HPS. To better compare the class conditional coverage, we compare the class-conditional coverage between DPSM and 3 baselines in (a), (b) and (c) separately. DPSM…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.