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REVIEW 4 major objections 6 minor 41 references

f-Divergence Variational Inference

T0 review · 4 major / 6 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read By replacing an f-divergence with a surrogate scaled by the inverse evidence, this paper derives a single variational bound that specializes to ELBO, CUBO, and Rényi variational bounds and provides two-sided evidence estimates.

desk verdict A useful but overclaimed unification of f-divergence VI; the mean-field recipe only works for a restricted class, and one of the custom divergences has a convexity error. read the letter →

arxiv 2009.13093 v4 pith:OQKT7FPX submitted 2020-09-28 cs.LG cs.ITmath.ITstat.ML

Neng Wan , Dapeng Li , Naira Hovakimyan This is my paper · ORCID
classification cs.LGcs.ITmath.ITstat.ML
keywords f-divergencevariationalinferencef-variationalboundevidenceboundsELBOCUBORényidivergencemean-fieldapproximation
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

The paper tries to establish that variational inference can be carried out with any f-divergence, not just the Kullback-Leibler divergence, by minimizing a carefully chosen surrogate divergence. The central result is the f-variational bound, which upper-bounds the dual function of the evidence and reproduces ELBO, CUBO, Rényi's variational bound, and other known objectives as special cases. The authors argue that this unifies these methods into one framework that supports both stochastic optimization and mean-field coordinate ascent. A sympathetic reader would care because it offers a standardized toolkit for choosing divergences with properties such as mass covering or tail sensitivity, along with a sandwich estimate of the marginal likelihood.

What carries the argument

The carrying object is the surrogate f-divergence D_{f_{p(D)^{-1}}}, built from the generator f_λ(t) = f(λ t) - f(λ) with λ = p(D)^{-1}; because it shares the same minimizer as the original f-divergence and, for shifted-homogeneous f, scales the divergence by λ^γ, it can replace the original without changing the optimum. The f-variational bound L_f(q,D) = E_q[f*(p(z,D)/q(z))] is the expectation form of that surrogate under the dual function f*, and it forms the unified inequality of Theorem 1. The mean-field updates (15)-(16) rely on the shifted-homogeneity decomposition f(t ilde t) = t^γ f( ilde t) + f(t) ilde t^η of the class F{0,1}, which is what separates one variational factor from the rest.

What would settle it

Take a convex function with f(1)=0 that is not in F{0,1}, such as f(t) = (t-1)^2, and numerically minimize the f-variational bound L_f(q,D) while separately minimizing the original surrogate divergence D_f(q‖p) over the same variational family; if the two optimizers differ on a nontrivial model, the claimed equivalence and the mean-field toolkit do not extend beyond the shifted-homogeneous class. A more direct check is to evaluate the identity D_{f_λ} = λ^γ D_f for this f and observe that it fails.

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Extended reading notes

Core claim

The paper claims that any f-divergence can drive variational inference if it is replaced by a surrogate divergence whose generator is f scaled by the inverse evidence, giving a bound that shares the minimizer of the original divergence. The f-variational bound is the expectation under the variational distribution q of the dual function f*(t) = t f(1/t) evaluated at the importance ratio p(z,D)/q(z), and it satisfies L_f(q,D) ≥ f*(p(D)), with equality at the true posterior. When the dual function is monotonic, composing its inverse with the bound gives an evidence upper or lower bound, and pairing an increasing with a decreasing dual function produces a sandwich estimate of the evidence. For shifted-homogeneous f-functions, the framework also yields coordinate-wise mean-field updates that recover CAVI when f is the KL generator.

Load-bearing premise

For the mean-field component, the argument that minimizing the surrogate equals minimizing the original divergence rests on the f-function being shifted-homogeneous, the class F{0,1}; outside this class, the coordinate-ascent updates are not derived and the equivalence is unproven.

Editorial extensions

If this is right

  • Choosing different f-functions in the f-variational bound reproduces ELBO, CUBO, Rényi's variational bound, and a newly identified evidence upper bound under KL divergence, making existing methods special cases of one inequality.
  • Combining an increasing and a decreasing dual function yields a two-sided sandwich estimate of the evidence, which can improve marginal-likelihood estimation over unilateral bounds.
  • The importance-weighted version of the bound is never below the single-sample bound and converges to f*(p(D)) as the number of samples grows.
  • The stochastic optimization schemes, including score-function, reparameterization, and importance-weighted reparameterization gradients, extend variational inference to divergence choices that previously lacked tractable gradient estimators.

Reading between the lines

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

  • The mean-field coordinate updates are proven only for the shifted-homogeneous class F{0,1}; for a general convex f with f(1)=0, the paper does not show that the surrogate and original divergences are equivalent, so the toolkit's scope in the mean-field setting is narrower than the abstract implies.
  • The observation that a non-monotonic dual function yields local bounds on the monotonic intervals suggests a recipe for constructing new evidence bounds for divergences like total variation distance; this is a direct consequence of the paper's observations but is not explored by the authors.
  • One could test empirically whether minimizing the surrogate bound for an f outside F{0,1} still produces a useful posterior approximation even though the exact equivalence is missing, which would indicate whether the homogeneity condition is sufficient rather than necessary.
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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

4 major / 6 minor

Summary. The paper introduces f-divergence variational inference (f-VI), a framework that minimizes a surrogate f-divergence D_{f_{p(D)^{-1}}}(q || p(z|D)) and derives the general f-variational bound L_f(q,D) = E_q[f*(p(z,D)/q(z))] ≥ f*(p(D)) (Theorem 1, Eq. 8). The bound is shown to specialize to ELBO, CUBO, RVB, and other existing variational bounds, and the paper derives a sandwich estimate of the evidence (Corollary 2). For optimization, the paper proposes a stochastic scheme based on score and reparameterization gradients (Section 3.2) and a mean-field scheme with coordinate ascent updates for f-functions in the shifted-homogeneity class F{0,1} (Section 3.3). Experiments on synthetic data, Bayesian neural networks, and variational autoencoders compare several f-VI instances against KL-, χ-, and α-VI baselines.

Significance. If the claims hold, the paper offers a useful unification of variational inference under f-divergences. The central bound (Theorem 1) is a direct consequence of Jensen's inequality and is correctly stated; the paper's contribution is to organize the existing bounds and to exhibit a tractable class of f-functions (Definition 3) for which coordinate-ascent updates are available. The stochastic gradient estimators are standard but are presented in a unified form. The empirical results are reasonable, though not clearly superior to strong baselines. The main novelty is the scalarization of the surrogate divergence and the identification of the F{0,1} class; however, the paper overstates the scope of the mean-field algorithm and contains a concrete error in one of the custom f-divergences used in the experiments.

major comments (4)
  1. [Abstract; Section 3.3, Eqs. (15)-(16)] The claim of a 'standardized toolkit for VI subject to arbitrary divergences from the f-divergence family' is not supported for the mean-field component. The coordinate-wise update rules (15) and (16) are derived only for f ∈ F{0,1} (or f* ∈ F{0,1}), as stated in the SM derivation. For a general f-divergence (e.g., squared Hellinger f(t) = (√t−1)^2), the homogeneity identity (4) fails, and no coordinate update is provided. The stochastic optimization part is generic, but the abstract and contribution c3 should be qualified to say that mean-field updates are restricted to F{0,1}.
  2. [Section 4.3; SM E.2] The custom dual function f*_{c2}(t) = log 2t + log t is claimed to be convex on t ∈ (0,1), but its second derivative is f*''(t) = −2/t^2 < 0, so it is concave. Consequently the inequality in Theorem 1 is reversed for this function, and the corresponding generator f(t) = t f*(1/t) = t(log 2 − 2 log t) is concave, not convex, so it is not a valid f-divergence generator. This invalidates the fc2-VI as an instance of the proposed framework and should be corrected or the experiment removed.
  3. [SM D.1] The derivation of the mean-field update states that after normalizing m_j to a probability distribution ilde{m}_j with normalization constant c, the f-variational bound becomes c·D_{f*}( ilde{m}_j || q_j). This equality is not correct in general; the calculation yields an expression of the form c^{1−γ} D_{f*}( ilde{m}_j || q_j) plus a term independent of q_j (for f ∈ F1), or the analogous expression for f ∈ F0. The optimality of q_j = ilde{m}_j still follows because the extra terms do not depend on q_j, but the stated equality is formally wrong and should be fixed.
  4. [Proposition 1] Proposition 1 assumes f is uniformly continuous on (0,∞), which excludes common generators such as f(t)=t log t (used for KL divergence) and f(t)=t^α−t. The proof interchanges the limit and integral using uniform continuity, but the convergence D_{f_{λ_n}} → D_f may hold under weaker conditions (e.g., by dominated convergence with suitable bounds). As stated, the proposition is not applicable to the paper's own primary examples. Since Proposition 3 gives an exact equivalence for F{0,1}, Proposition 1 could be removed or substantially weakened without affecting the main results.
minor comments (6)
  1. [Acknowledgments] The grant number contains a typo: 'AFSOR' should be 'AFOSR'.
  2. [SM Section A title] The title 'deviations of the f-variational bound' should be 'derivations of the f-variational bound'.
  3. [Section 3.2, Eq. (10)] The score-function gradient formula uses f'(q_θ/p), but L_f is expressed in terms of f*. The derivation in SM Section C clarifies that L_f = E_p[f(q/p)] and the formula is correct; this equivalence should be stated in the main text.
  4. [Table 2 and SM E.2] The notation for confidence intervals is inconsistent: the paper says results are reported with 95% confidence intervals, but the tables show ± values without defining the standard error; please clarify whether these are standard deviations or confidence intervals.
  5. [Section 3.3] The statement that expectation propagation (EP) can be restored from update rules (15) and (16) is questionable, since EP is a message-passing algorithm, not a coordinate-ascent method; the sentence should be tempered.
  6. [Various] The symbol f*^{-1} is written inconsistently as f∗−1 and (f∗)−1; use one notation throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the f-variational bound is an independent Jensen-inequality derivation, and the mean-field restriction is a scope limitation, not a circular step.

full rationale

The central bound (Theorem 1, Eq. 8) is derived directly from Jensen's inequality applied to the convex perspective function f^*, with no fitted parameter entering the inequality; the evidence p(D) appears only as the fixed right-hand-side value and as a scaling constant in the surrogate divergence, and it does not affect the minimizer over q. The claimed unifications of ELBO, CUBO, and RVB are obtained by substituting explicit generators f(t)=t log t, f(t)=t^{1-n}-t, and f(t)=t^alpha-t into the same inequality, so they are specializations of a genuine derivation rather than renamings of the inputs. The mean-field updates (15)-(16) and Proposition 3 are explicitly proved under the shifted-homogeneity class F{0,1} from Definition 3 and Eq. 4; the abstract's broader 'all f-divergences' promise for the mean-field component is an unsupported generality and a correctness risk, but the derivation does not assume its own conclusion. The paper contains no load-bearing self-citations and invokes no uniqueness theorem from the authors' prior work. The reader-flagged concern that p(D) appears inside the surrogate is not circular: the inequality L_f(q,D) >= f^*(p(D)) holds for every fixed p(D), and optimizing L_f over q does not require knowing p(D).

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on standard convex analysis (Jensen) plus a few domain assumptions common in VI (absolute continuity, reparameterizability, mean-field factorization). The most paper-specific postulate is the shifted-homogeneity class F{0,1}, which drives both the surrogate equivalence and the mean-field updates and is not valid for arbitrary f-divergences. The only fitted free parameter is t0 in the f_c1 custom divergence, which is optimized during training and not reported.

free parameters (1)
  • t0 = not reported (optimized during training)
    The custom f-divergence f_c1 in Section 4.2 and 4.3 defines \tilde{f}*(t) = -1/6 (log t + t0)^3 - 1/2 (log t + t0)^2 - (log t + t0) - 1 with t0 ∈ R stated as 'a parameter to be optimized'. The paper does not report its fitted value or sensitivity.
assumptions (7)
  • standard math Jensen's inequality for convex f*
    Used to prove Theorem 1 and Corollary 1 (Eq. 8 and the importance-weighted bound monotonicity).
  • standard math Uniform continuity of f for the limit in Proposition 1
    Proposition 1 claims D_{f_{λ_n}} → D_f under uniform continuity of f on (0,∞). This excludes common generators like t log t, so the paper must rely on Proposition 3 for practical cases.
  • domain assumption Absolute continuity of q w.r.t. p
    Definition 1 assumes continuous densities and absolute continuity to define the f-divergence.
  • ad hoc to paper Shifted homogeneity f∈F{0,1}
    Definition 3 introduces a structural class F{0,1} used for Proposition 3 and the mean-field coordinate updates (15)-(16). It does not cover all f-divergences.
  • domain assumption Monotonicity and invertibility of f* or g* on the relevant interval
    Corollary 2 sandwich bounds and the EUBO construction require (f*)^{-1} to exist and be monotone, e.g., p(D) ≥ e^{-1} for the KL upper bound in Section B.1.
  • domain assumption Reparameterizability of the variational family
    The reparameterization gradient estimators (12)-(14) require a noise variable ε and a differentiable mapping g_θ such that z = g_θ(ε).
  • domain assumption Mean-field factorization q(z) = ∏_j q_j(z_j)
    Section 3.3 assumes all latent variables are independent to derive the coordinate-wise update rules (15)-(16).

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Cite this review

Pith. "Pith review of f-Divergence Variational Inference." pith.science (2026). https://pith.science/paper/OQKT7FPX

@misc{pith2026200913093,
  author       = {Pith},
  title        = {Pith review of: f-Divergence Variational Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQKT7FPX}},
  note         = {Machine review of arXiv:2009.13093}
}
abstract

This paper introduces the $f$-divergence variational inference ($f$-VI) that generalizes variational inference to all $f$-divergences. Initiated from minimizing a crafty surrogate $f$-divergence that shares the statistical consistency with the $f$-divergence, the $f$-VI framework not only unifies a number of existing VI methods, e.g. Kullback-Leibler VI, R\'{e}nyi's $\alpha$-VI, and $\chi$-VI, but offers a standardized toolkit for VI subject to arbitrary divergences from $f$-divergence family. A general $f$-variational bound is derived and provides a sandwich estimate of marginal likelihood (or evidence). The development of the $f$-VI unfolds with a stochastic optimization scheme that utilizes the reparameterization trick, importance weighting and Monte Carlo approximation; a mean-field approximation scheme that generalizes the well-known coordinate ascent variational inference (CAVI) is also proposed for $f$-VI. Empirical examples, including variational autoencoders and Bayesian neural networks, are provided to demonstrate the effectiveness and the wide applicability of $f$-VI.

Figures

Figures reproduced from arXiv: 2009.13093 by the authors.

Figure 1
Figure 1. f-variational bounds on synthetic data. 4.2 Bayesian neural network We then implement the f-VI for a single-layer neural network for Bayesian linear regression. Our experimental setup generally follows the regression settings in [2], while some parameters vary to adapt to the f-VI framework. The linear regression is performed with twelve datasets from the UCI Machine Learning Repository [36]. Each dataset is randoml… view at source ↗
Figure 2
Figure 2. tlog t and its inverse function t/W(t). forward χ n-divergence Dχn (p||q) = Eq(z) [(p(z, x)/q(z))n − 1] for n ≥ 1, which is different from the reverse χ n-divergence given in [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Evdience and IW-TVB. However, it is still worth noting that numerical issues and biased estimators can contaminate the empirical results or cause the violations of theory, despite the fact that the importance-weighted technique can attenuate these flaws by improving the tightness of bounds and their estimators. The estimation of IW-RVB in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Reconstruction of MNIST handwritten digits. Left column shows the original digits. Right column [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Generation of MNIST handwritten digits. (a) is from IW-ELBO loss. (b) is from IW-CUBO ( [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Generation of Caltech 101 silhouettes. (a) is from IW-ELBO loss. (b) is from IW-CUBO ( [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Generation of Frey Face. (a) is from IW-ELBO loss. (b) is from IW-CUBO ( [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: Generation of Omniglot alphabets. (a) is from IW-ELBO loss. (b) is from IW-CUBO ( [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

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Works this paper leans on

41 extracted references · 37 canonical work pages

  1. [1]

    M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37:183–233, 1999

    1999

  2. [2]

    Li and R

    Y . Li and R. E. Turner. Rényi divergence variational inference. In Advances in Neural Information Processing Systems, 2016

    work page 2016

  3. [3]

    A. B. Dieng, D. Tran, R. Ranganath, J. Paisley, and D. M. Blei. Variational inference viaχ upper bound minimization. In Advances in Neural Information Processing Systems, 2017

    work page 2017

  4. [4]

    Knowles and T

    D. Knowles and T. Minka. Non-conjugate variational message passing formultinomial and binary regression. In Advances in Neural Information Processing Systems, 2011

    work page 2011

  5. [5]

    Wang and D

    C. Wang and D. M. Blei. Variational inference in non-conjugate models. Journal of Machine Learning Research, 614:1005–1031, 2013

    work page 2013

  6. [6]

    M. D. Hoffman, D. M. Blei, C. Wang, and J. Paisley. Stochastic variational inference. Journal of Machine Learning Research, 14(4):1303–1347, 2013

    work page 2013

  7. [7]

    D. P. Kingma and M. Welling. Auto-encoding variational bayes. In International Conference on Learning Representations, 2014. 9

    work page 2014

  8. [8]

    Y . Li, J. M. Hernández-Lobato, and R. E. Turner. Stochastic expectation propagation. In Advances in Neural Information Processing Systems, 2015

    work page 2015

Show all 41 references
  1. [9]

    Burda, R

    Y . Burda, R. Grosse, and R. Salakhutdinov. Importance weighted autoencoders. InInternational Conference on Learning Representations, 2016

    2016

  2. [10]

    C. Tao, L. Chen, R. Zhang, R. Henao, and L. Carin. Variational inference and model selection with generalized evidence bounds. In Proceedings of 35th International Conference on Machine Learning , 2018

    2018

  3. [11]

    D. M. Blei, A. Kucukelbir, and J. D. McAuliffe. Variational inference: A review for statisticans. Journal of the American Statistical Association, 112(518):859–877, 2017

    2017

  4. [12]

    Zhang, J

    C. Zhang, J. Butepage, H. Kjellstrom, and S. Mandt. Advances in variational inference. IEEE Transactions on Pattern Analysis and Machine Intelligence, 41(8):2008–2026, 2019

    2008

  5. [13]

    J. B. Regli and R. Silva. Alpha-beta divergence for variational inference. In arXiv: 1805.01045, 2018

    2018 arXiv

  6. [14]

    Black box variational inference

    Rajesh Ranganath, Sean Gerrish, and David Blei. Black box variational inference. In Proceedings of the 7th International Conference on Artificial Intelligence and Statistics, 2014

    2014

  7. [15]

    C. M. Bishop. Pattern Recognition and Machine Learning. New York: Springer, 2006

    2006

  8. [16]

    T. P. Minka. Expectation propagation for approximate bayesian inference. InProceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, 2001

    2001

  9. [17]

    Sason and S

    I. Sason and S. Verdú.f-divergence inequalities. IEEE Transactions on Information Theory, 62(11):5973– 6006, 2016

    2016

  10. [18]

    I. Sason. On f-divergences: Integral representations, local behavior, and inequalities. Entropy, 20(5):383, 2018

    2018

  11. [19]

    Bamler, C

    R. Bamler, C. Zhang, M. Opper, and S. Mandt. Perturbative black box variational inference. In Advances in Neural Information Processing Systems, 2017

    2017

  12. [20]

    D. Wang, H. Liu, and Q. Liu. Variational inference with tail-adaptivef-divergence. In Advances in Neural Information Processing Systems, 2018

    2018

  13. [21]

    Zhang, T

    M. Zhang, T. Bird, R. Habib, T. Xu, and D. Barber. Variationalf-divergence minimization. In Advances in Neural Information Processing Systems, 2019

    2019

  14. [22]

    Knoblauch, J

    J. Knoblauch, J. Jewson, and T. Damoulas. Generalized variational inference: Three arguments for deriving new posteriors. In arXiv: 1904.02063, 2019

    1904 arXiv

  15. [23]

    K. Murphy. Machine Learning: A Probabilistic Perspective. MIT press, 2012

    2012

  16. [24]

    Boyd and L

    S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004

    2004

  17. [25]

    Nowozin, B

    S. Nowozin, B. Cseke, and R. Tomioka.f-GAN: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems, 2016

    2016

  18. [26]

    Mnih and K

    A. Mnih and K. Gregor. Variational inference and learning in belief networks. In International Conference on Machine Learning, page 1791–1799, 2014

    2014

  19. [27]

    D. R. Cox and D.V . Hinkley. Theoretical Statistics. Chapman and Hall, 1979

    1979

  20. [28]

    R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3-4):229–256, 1992

    1992

  21. [29]

    Paisley and and D

    J. Paisley and and D. M. Blei. Variational bayesian inference with stochastic search. In International Conference on Machine Learning, 2012

    2012

  22. [30]

    F. J. R. Ruiz, M. K. Titsias, and D. M. Blei. The generalized reparameterization gradient. In Advances in Neural Information Processing Systems, 2016

    2016

  23. [31]

    Figurnov, S

    M. Figurnov, S. Mohamed, and A. Mnih. Implicit reparameterization gradients. In Advances in Neural Information Processing Systems, 2018

    2018

  24. [32]

    Jankowiak and F

    M. Jankowiak and F. Obermeyer. Pathwise derivatives beyond the reparameterization trick. InProceedings of 35th International Conference on Machine Learning, 2018

    2018

  25. [33]

    Winn and C

    J. Winn and C. M. Bishop. Variational message passing.Journal of Machine Learning Research, 6:661–694, 2005

    2005

  26. [34]

    M. Wand, J. Ormerod, S. Padoan, and R. Fuhrwirth. Mean field variational bayes for elaborate distributions. Bayesian Analysis, 6(4):847–900, 2011

    2011

  27. [35]

    D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015

    2015

  28. [36]

    https://archive.ics.uci.edu/ml/index.php

  29. [37]

    https://people.cs.umass.edu/~marlin/data.shtml

  30. [38]

    https://cs.nyu.edu/~roweis/data.html

  31. [39]

    http://yann.lecun.com/exdb/mnist/

  32. [40]

    https://github.com/yburda/iwae/tree/master/datasets/OMNIGLOT

  33. [41]

    f-Divergence Variational Inference

    https://www.mathworks.com/help/deeplearning/ug/train-a-variational-autoencoder- vae-to-generate-images.html. 10 Supplementary Material for “f-Divergence Variational Inference” Neng Wan1∗ nengwan2@illinois.edu Dapeng Li 2∗ dapeng.ustc@gmail.com Naira Hovakimyan1 nhovakim@illino...

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Reviewed August 27, 2026 · model on record in the stance chip above.