Pith. sign in

REVIEW 4 major objections 4 minor 55 references

Gradient-based Stochastic Optimization of Utility-based Shortfall Risk

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives finite-sample error bounds for estimating and optimizing utility-based shortfall risk and optimized certainty equivalent risk, including for unbounded losses, and proves a stochastic-gradient convergence rate of O(1/n)…

desk verdict Solid conditional theorems for UBSR/OCE estimation and optimization, but the advertised coverage of entropic risk and VaR is not backed by the assumptions, and the abstract overclaims convex/non-convex rates. read the letter →

arxiv 2506.01101 v2 pith:E4NZL2GF submitted 2025-06-01 cs.CE q-fin.MFstat.CO

Sumedh Gupte , Prashanth L.A. , Sanjay P. Bhat This is my paper
classification cs.CEq-fin.MFstat.CO MSC 90C1591B3062P05
keywords utility-basedshortfallriskoptimizedcertaintyequivalentsampleaverageapproximationstochasticgradientmethodsnon-asymptoticerrorboundsconvexmeasuresportfoliooptimizationestimation
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

Utility-based shortfall risk (UBSR) and optimized certainty equivalent (OCE) risk are two families of convex risk measures with roots in expected-utility theory and financial regulation. This paper tries to show that both can be reliably estimated from i.i.d. samples and optimized by stochastic gradient methods even when the underlying losses are unbounded, a setting where earlier theory often assumed boundedness. The proofs deliver explicit non-asymptotic rates: sample-average estimators with mean-absolute error $O(1/\sqrt{m})$ and mean-squared error $O(1/m)$ for UBSR, gradient estimators with matching rates, and stochastic-gradient iterates converging at $O(1/n)$ under strong convexity. A sympathetic reader would care because these bounds turn risk-sensitive portfolio choice into a problem with quantifiable sample and iteration budgets.

What carries the argument

The load-bearing object is the root function $g_X(t) = \mathbb{E}[l(-X-t)] - \lambda$, whose unique root is the UBSR value; OCE is handled by setting $l = u'$ and $\lambda = 1$, so the same root machinery applies. Differentiation of the root equation via the implicit function theorem yields the gradient formula $\nabla h(\theta) = -\mathbb{E}[l'(-F(\theta,\xi)-h(\theta))\nabla F(\theta,\xi)]/\mathbb{E}[l'(-F(\theta,\xi)-h(\theta))]$, which is turned into a biased double-sample estimator. A general theorem for stochastic gradient methods with biased gradients (Assumption 4) converts estimator bias and variance bounds into last-iterate convergence, while Wasserstein empirical-measure bounds and concentration inequalities for sums of random vectors supply the explicit constants in the estimation rates.

What would settle it

Run UBSR-SB on $m$ independent samples from a distribution with finite variance using the loss $l(x) = x + \arctan(x)$, which satisfies Assumption 2 with $b_1 = 1$ and is smooth, and compare the SAA root to the true UBSR root computed by high-accuracy arithmetic. If the mean-squared error of the SAA estimator does not decay like $C/m$ as $m$ grows, then the claimed $O(1/m)$ MSE bound is false.

Watch

Extended reading notes

Core claim

The central claim is that UBSR and OCE risk measures, traditionally defined for bounded random variables, remain estimable and optimizable with explicit finite-sample rates when the domain is unbounded. For UBSR, the sample-average approximation has mean-absolute error $O(1/\sqrt{m})$ and mean-squared error $O(1/m)$ under a variance condition; for OCE, the SAA estimator achieves $O(1/\sqrt{m})$ MAE and MSE in the Lipschitz-utility case, with slightly weaker rates under variance assumptions. The paper further derives closed-form gradient expressions for both risk measures under smooth parameterizations and constructs biased $m$-sample gradient estimators whose MAE is $O(1/\sqrt{m})$ and MSE is $O(1/m)$. Plugging these into a projected stochastic-gradient scheme with increasing batch sizes $m_k = k$ yields last-iterate mean-squared-error convergence $O(1/n)$, iteration complexity $O(1/\epsilon)$, and sample complexity $O(1/\epsilon^2)$ for strongly convex objectives. The arguments unify VaR, CVaR, entropic risk, expectile risk, monotone mean-variance risk, and quartic risk as special cases.

Load-bearing premise

The rates presuppose that the loss function $l$ (or the utility $u$, via $l=u'$) is globally strongly increasing and smooth, and that $F(\theta,\xi)$ has bounded gradient and variance; the exponential loss of entropic risk, which the paper itself tests on Gaussian data, does not satisfy global strong monotonicity and smoothness on unbounded domains.

Editorial extensions

If this is right

  • With $m$ samples, UBSR can be estimated to mean-absolute error $O(1/\sqrt{m})$ and mean-squared error $O(1/m)$, giving a concrete sample budget for risk measurement.
  • The gradient formulas make UBSR and OCE minimization tractable for vector decision parameters, not just scalar ones, at stochastic-gradient iteration complexity $O(1/\epsilon)$ under strong convexity.
  • The same algorithms and bounds cover VaR, CVaR, entropic, expectile, monotone mean-variance, and quartic risks as special cases, so a single implementation serves several regulatory risk measures.
  • The portfolio experiments indicate that UBSR- and OCE-optimal allocations are comparable to or better than equal-weight and minimum-CVaR benchmarks on market data.
  • The general biased-gradient stochastic-gradient theorem applies to any risk objective whose gradient estimator satisfies the stated bias and variance bounds.

Reading between the lines

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

  • The anytime batch schedule $m_k = k$ avoids needing to know the horizon $n$; the same proof technique likely gives $O(1/n)$ with other increasing schedules, at the cost of different constants.
  • Because the theory requires global strong monotonicity and smoothness, the entropic-risk experiments on Gaussian data fall outside the stated assumptions; if the observed $1/\sqrt{m}$ decay persists for such losses, those assumptions are sufficient but not necessary.
  • The gradient formula suggests second-order and zeroth-order extensions: Newton-type updates could exploit the root structure directly, and derivative-free methods could estimate the same ratio without requiring $l'$.
  • The double sampling in the gradient estimator (separate samples for the root and for the gradient sum) is a natural place to apply common-random-number or control-variate tricks to reduce the constants $D_1$ and $D_2$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops non-asymptotic theory for estimating and optimizing utility-based shortfall risk (UBSR) and optimized certainty equivalent (OCE) risk for possibly unbounded random variables. It provides SAA estimators with MAE/MSE bounds, gradient expressions, biased gradient estimators with MAE/MSE bounds, and stochastic gradient algorithms with O(1/n) last-iterate convergence rates under strong convexity. The paper also reports numerical experiments on entropic risk estimation and optimization and on portfolio allocation.

Significance. If the stated bounds hold, this is a useful contribution: it extends UBSR/OCE analysis beyond bounded random variables, handles vector decision parameters, and gives a general biased-gradient SG theorem with explicit constants that may be of independent interest. The proofs are detailed and the dependence on variance and smoothness assumptions is made explicit. However, the advertised coverage of entropic risk, VaR, and other special cases is broader than the assumptions used in the main theorems, and the reported experiments do not follow the analyzed algorithm configurations, so the significance is conditional on a careful revision of the scope claims.

major comments (4)
  1. [§3.1.2, §5.1.1, §5.2.1, §6.1, Appendix A] The headline examples violate the global regularity assumptions needed for the main rates. For the exponential loss l(x)=e^{βx} used for entropic risk, inf_{y>x}(l(y)-l(x))/(y-x)=0 and sup_{x<y}|l'(y)-l'(x)|/|y-x|=∞, so Assumption 2 and Assumption 9 cannot hold with finite constants. The Heaviside loss used for VaR is not strictly increasing, so Assumption 2 fails as well. Consequently Lemma 11, Lemma 28, and Theorem 33 do not provide the advertised O(1/√m) MAE, O(1/m) MSE, or O(1/n) SG rates for these examples. On the OCE side, the exponential utility u(x)=(e^{βx}-1)/β has u''(x)=β^2 e^{βx}→0 as x→-∞, so no global b2 exists, and the quartic utility in §3.2.3 is not twice continuously differentiable at x=1, so Assumption 12 and Theorem 39 do not cover it. The conditional theorems appear logically sound, but the scope claims and the experiments need to be revised, for example by stating the results on bounded domains or under local strong-convexity/smoothness assumptions.
  2. [§4.1, Algorithm 1 and Algorithm 2] Algorithm 1 can loop forever on ordinary inputs, so Proposition 12 and the approximate-gradient results that rely on it are not supported as written. If \hat g(0)>0, the algorithm initializes high=0, and the loop "while \hat g(high)>0 do high←2*high" leaves high=0 indefinitely; if \hat g(0)<0, the loop on low leaves low=0 indefinitely. The search interval appears to be initialized in the wrong direction: for \hat g(0)>0 the root is to the right of 0, so the initial bracket should put the positive side at high>0. Algorithm 2 shares the same initialization problem. This is a load-bearing issue for the claimed δ-approximation guarantees in Proposition 12 and Proposition 29 and for the implementations used in Section 6.
  3. [Abstract and §8] The abstract promises convergence rates for "strongly convex, convex and non-convex" objectives, but the paper proves only the strongly convex case: Theorems 33 and 39 both require Assumption 10 or Assumption 14, and Section 8 explicitly lists non-convex UBSR/OCE optimization as future work. The abstract's convex and non-convex claims should be removed or the missing results should be supplied.
  4. [§6.1.2, §6.2, Figures 3-5] The experimental hyperparameters differ from those analyzed. Theorem 33 assumes batch size m_k=k and step size α_k=c/k with c≥3/(2μ_1), while Figure 3 uses step size 1/√k, draws 2k samples per iteration, and constructs the UBSR estimate from only the first k samples; the portfolio experiments in Section 6.2 similarly use 2k samples and step size 1/√k. These choices are not covered by the theorem, so the experiments do not empirically validate the analyzed rates. The authors should either run the algorithms with the analyzed hyperparameters or explicitly describe the experiments as illustrative heuristics rather than validations.
minor comments (4)
  1. [§6.2] The FTSE100 and Nasdaq datasets are described as running up to 2026-05-26, which is after the arXiv submission date of 2025-06-01; these dates should be corrected or the data provenance should be clarified.
  2. [Figure 3 caption] The caption in Section 6.1.2 says "Performance of OCE-SAA algorithm" but the figure shows the UBSR-SG and OCE-SG iterates; the caption should be corrected.
  3. [§7.2.6, proof of Proposition 16] In the proof, the definition of \hat s_m uses u'(-z_j-\hat t_m), whereas Algorithm 3 and the proposition statement use u(-z_j-\hat t_m); this is presumably a typo and should be fixed.
  4. [Throughout] There are several typographical errors, including "transpost cost" in Section 2, "Foe" in Section 6.2, and "thies" in the proof of Proposition 7; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the central bounds are derived by error propagation from explicit assumptions and external concentration inequalities; self-citations are comparative, not load-bearing.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The UBSR SAA estimator is compared to the true root SRl,λ(X) through the monotonicity/smoothness of the loss (Assumption 2) and a variance bound, giving MAE/MSE O(1/√m), O(1/m) (Lemma 11); the OCE bounds follow from the OCE-UBSR identity plus Wasserstein concentration of the empirical measure (Lemma 13) or variance assumptions (Lemmas 14–15), not from assuming the conclusion. The gradient expressions (Theorems 26 and 35) are obtained from the implicit function theorem and the defining first-order conditions, and the gradient estimators' bias/variance is propagated from the SAA UBSR/OCE estimation error (Lemmas 28, 36; Propositions 29, 37). The SG convergence rates (Theorems 33, 39) are specializations of Theorem 21, a biased-gradient result proved in the paper, with the estimator bounds used only to verify Assumption 4. Self-citations (Gupte et al., 2024; Prashanth and Bhat, 2022) are used for comparison and context, not as unverified premises; no uniqueness theorem or ansatz is imported from the authors' prior work. The skeptic's concern is a scope/assumption-coverage issue: the exponential and Heaviside losses used in experiments do not satisfy the global regularity assumptions (2/9/12) on unbounded domains, so the advertised rates do not automatically apply to those examples, and the abstract over-promises convex/non-convex rates that are not proved. This is a mismatch between claims and assumptions, not a reduction of a prediction to a fit or to a self-citation, so the circularity score is 0.

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

The central claims rest on a set of integrability, monotonicity, smoothness, and variance assumptions on the random variables and the loss/utility functions, plus strong convexity of the risk objective for the SG rates. No free parameters are fitted to data in the theoretical results; the bounds hold for any problem instance satisfying the assumptions.

assumptions (6)
  • domain assumption The random variables and loss/utility functions are such that E[l(-X-t)] is finite for all t (X in X_l).
    Defines the domain of the risk measures; invoked in Definition 3 and throughout the paper.
  • domain assumption The loss function l is non-constant, increasing, and either continuous or continuous a.e. with continuous CDF of X (Proposition 4/5).
    Ensures the UBSR is the root of the expected loss equation, which underpins estimation and optimization.
  • domain assumption The loss function is strongly increasing (Assumption 2): l(y)-l(x) > b1(y-x) for all y>x.
    Used in Lemma 11 to translate SAA error into estimation error; not satisfied by exponential loss.
  • domain assumption The loss function is S1-smooth (Assumption 9) and the objective F satisfies moment and Lipschitz conditions (Assumptions 5, 6).
    Needed for the UBSR gradient estimator bounds in Lemma 28.
  • domain assumption The risk objective h is strongly convex (Assumptions 10 and 14).
    Required for the O(1/n) SG convergence rates; only verified for the Gaussian portfolio example.
  • domain assumption Variance bounds on the loss and utility (Assumptions 1, 3, 7, 8, 13) hold at the risk root.
    Provide finite constants in the MAE/MSE bounds; these are not verified for the chosen experimental distributions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gradient-based Stochastic Optimization of Utility-based Shortfall Risk." pith.science (2026). https://pith.science/paper/E4NZL2GF

@misc{pith2026250601101,
  author       = {Pith},
  title        = {Pith review of: Gradient-based Stochastic Optimization of Utility-based Shortfall Risk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4NZL2GF}},
  note         = {Machine review of arXiv:2506.01101}
}
read the original abstract

We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR). We extend UBSR to cover possibly unbounded random variables. We cover prominent risk measures such as entropic risk, expectile risk, Value-at-Risk, and quadratic risk as special cases of the UBSR. In the context of estimation, we derive non-asymptotic bounds on the mean absolute error (MAE) and the mean-squared error (MSE) of the classical sample-average approximation (SAA) estimator for the UBSR. In the context of optimization, we derive an expression for the gradient of UBSR under a smooth parameterization. We propose a gradient estimator for the UBSR and derive non-asymptotic bounds on MAE and MSE for this estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) optimization algorithm and derive non-asymptotic bounds on the convergence rate of our SG algorithm for optimizing UBSR under three objectives, namely, strongly convex, convex and non-convex. Finally, we conduct experiments on financial applications to demonstrate the performance of our proposed UBSR estimation and optimization algorithms.

Figures

Figures reproduced from arXiv: 2506.01101 by the authors.

Figure 1
Figure 1. Performance of UBSR-SB algorithm for estimation of entropic risk of a univariate Gaus [PITH_FULL_IMAGE:figures/full_fig_p031_1.png] view at source ↗
Figure 2
Figure 2. The figure shows performance of OCE-SAA algorithm for estimation of entropic risk of [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Performance of OCE-SAA algorithm for estimation of entropic risk of a univariate Gaus [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The figure shows the performance of UBSR-SG algorithm for a variety of UBSR risk [PITH_FULL_IMAGE:figures/full_fig_p034_4.png]
Figure 5
Figure 5. Figure 5: The figure shows the performance of OCE-SG algorithm for a variety of OCE risk mea [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]
Figure 6
Figure 6. Figure 6: The figure shows the error distribution of the [PITH_FULL_IMAGE:figures/full_fig_p066_6.png]
Figure 7
Figure 7. Figure 7: The figure shows the error distribution of the [PITH_FULL_IMAGE:figures/full_fig_p067_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Acerbi and D

    C. Acerbi and D. Tasche. On the coherence of expected shortfall. Journal of Banking & Finance, 26 0 (7): 0 1487--1503, 2002. ISSN 0378-4266

    work page 2002

  2. [2]

    Mathematical analysis; 2nd ed

    Tom M Apostol. Mathematical analysis; 2nd ed. Addison-Wesley series in mathematics. Addison-Wesley, Reading, MA, 1974

    work page 1974

  3. [3]

    Artzner, F

    P. Artzner, F. Delbaen, J. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9 0 (3): 0 203--228, 1999

    work page 1999

  4. [4]

    H. Asi, Y. Carmon, A. Jambulapati, Y. Jin, and A. Sidford. Stochastic bias-reduced gradient methods. In Advances in Neural Information Processing Systems, volume 34, pages 10810--10822, 2021

    work page 2021

  5. [5]

    Basak and A

    S. Basak and A. Shapiro. Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices . The Review of Financial Studies, 14 0 (2): 0 371--405, 06 2015. ISSN 0893-9454

    work page 2015

  6. [6]

    On elicitable risk measures

    Fabio Bellini and Valeria Bignozzi. On elicitable risk measures. Quantitative Finance, 15 0 (5): 0 725--733, 2015

    work page 2015

  7. [7]

    Expected utility, penalty functions, and duality in stochastic nonlinear programming

    Aharon Ben-Tal and Marc Teboulle. Expected utility, penalty functions, and duality in stochastic nonlinear programming. Management Science, 32 0 (11): 0 1445--1466, 1986. ISSN 00251909, 15265501

    work page 1986

  8. [8]

    An Old - New Concept of Convex Risk Measures : The Optimized Certainty Equivalent

    Aharon Ben-Tal and Marc Teboulle. An Old - New Concept of Convex Risk Measures : The Optimized Certainty Equivalent . Mathematical Finance, 17 0 (3): 0 449--476, 2007. ISSN 1467-9965

    work page 2007

Show all 55 references
  1. [9]

    Bhavsar and L

    N. Bhavsar and L. A. Prashanth. Nonasymptotic bounds for stochastic optimization with biased noisy gradient oracles. IEEE Transactions on Automatic Control, 68 0 (3): 0 1628--1641, 2023

    2023

  2. [10]

    Bottou, F

    L. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning. Siam Review, 60 0 (2): 0 223--311, 2018

    2018

  3. [11]

    Closing the Gap : Tighter Analysis of Alternating Stochastic Gradient Methods for Bilevel Problems

    Tianyi Chen, Yuejiao Sun, and Wotao Yin. Closing the Gap : Tighter Analysis of Alternating Stochastic Gradient Methods for Bilevel Problems . In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Syst...

    2021

  4. [12]

    An expectile computation cookbook

    Abdelaati Daouia, Gilles Stupfler, and Antoine Usseglio-Carleve. An expectile computation cookbook. Stat Comput, 34 0 (3): 0 103, March 2024. ISSN 1573-1375

    2024

  5. [13]

    Stochastic first order methods in smooth convex optimization

    Olivier DEVOLDER. Stochastic first order methods in smooth convex optimization . Lidam discussion papers core, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE), December 2011

    2011

  6. [14]

    Duchi, Michael I

    John C. Duchi, Michael I. Jordan, Martin J. Wainwright, and Andre Wibisono. Finite sample convergence rates of zero-order stochastic optimization methods. In Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 1 , NIPS '12, pages ...

    2012

  7. [15]

    Dunkel and S

    J. Dunkel and S. Weber. Stochastic root finding and efficient estimation of convex risk measures. Operations Research, 58 0 (5): 0 1505--1521, 2010

    2010

  8. [16]

    Efficient Monte Carlo methods for convex risk measures in portfolio credit risk models

    Jorn Dunkel and Stefan Weber. Efficient Monte Carlo methods for convex risk measures in portfolio credit risk models. In 2007 Winter Simulation Conference , pages 958--966, December 2007. ISSN: 1558-4305

    2007

  9. [17]

    R. Durrett. Probability: Theory and Examples. Cambridge University Press, 5 edition, 2019

    2019

  10. [18]

    F \"o llmer and A

    H. F \"o llmer and A. Schied. Stochastic finance. de Gruyter, 2016

    2016

  11. [19]

    Föllmer and A

    H. Föllmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time. De Gruyter, Berlin, Boston, 2004. ISBN 9783110212075

    2004

  12. [20]

    The Axiomatic Approach to Risk Measures for Capital Determination

    Hans Föllmer and Stefan Weber. The Axiomatic Approach to Risk Measures for Capital Determination . Annual Review of Financial Economics, 7 0 (1): 0 301--337, 2015

    2015

  13. [21]

    Giesecke, T

    K. Giesecke, T. Schmidt, and S. Weber. Measuring the risk of large losses. Journal of Investment Management, 6, 01 2008

    2008

  14. [22]

    Distributionally robust shortfall risk optimization model and its approximation

    Shaoyan Guo and Huifu Xu. Distributionally robust shortfall risk optimization model and its approximation. Math. Program., 174 0 (1): 0 473--498, March 2019. ISSN 1436-4646

    2019

  15. [23]

    A Prashanth, and Sanjay P

    Sumedh Gupte, L. A Prashanth, and Sanjay P. Bhat. Optimization of utility-based shortfall risk: A non-asymptotic viewpoint. In 2024 IEEE 63rd Conference on Decision and Control (CDC), pages 1075--1080, 2024

    2024

  16. [24]

    Stochastic root finding for optimized certainty equivalents

    Anna-Maria Hamm, Thomas Salfeld, and Stefan Weber. Stochastic root finding for optimized certainty equivalents. In 2013 Winter Simulations Conference ( WSC ) , pages 922--932, 2013

    2013

  17. [25]

    Prashanth, and Krishna Jagannathan

    Vishwajit Hegde, Arvind Satish Menon, L.A. Prashanth, and Krishna Jagannathan. Online estimation and optimization of utility-based shortfall risk. Mathematics of Operations Research, sep 2024. ISSN 1526-5471

    2024

  18. [26]

    Horn and Charles R

    Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cambridge ; New York, 2nd ed edition, 2012. ISBN 978-0-521-83940-2

    2012

  19. [27]

    Biased stochastic first-order methods for conditional stochastic optimization and applications in meta learning

    Yifan Hu, Siqi Zhang, Xin Chen, and Niao He. Biased stochastic first-order methods for conditional stochastic optimization and applications in meta learning. Advances in Neural Information Processing Systems, 33: 0 2759--2770, 2020

    2020

  20. [28]

    On the bias-variance-cost tradeoff of stochastic optimization

    Yifan Hu, Xin Chen, and Niao He. On the bias-variance-cost tradeoff of stochastic optimization. Advances in Neural Information Processing Systems, 34: 0 22119--22131, 2021

    2021

  21. [29]

    Hu and D

    Z. Hu and D. Zhang. Utility-based shortfall risk: Efficient computations via monte carlo. Naval Research Logistics (NRL), 65 0 (5): 0 378--392, 2018

    2018

  22. [30]

    P. Jorion. Value at Risk: The New Benchmark for Controlling Market Risk. McGraw-Hill, 1997. ISBN 9780786308484

    1997

  23. [31]

    Kaina and L

    M. Kaina and L. Rüschendorf. On convex risk measures on Lp -spaces. Math Meth Oper Res, 69 0 (3): 0 475--495, July 2009. ISSN 1432-5217

    2009

  24. [32]

    Karimi, B

    B. Karimi, B. Miasojedow, E. Moulines, and H. Wai. Non-asymptotic analysis of biased stochastic approximation scheme. In Conference on Learning Theory, pages 1944--1974. PMLR, 2019

    1944

  25. [33]

    A. J. Kleywegt, A. Shapiro, and T. Homem de Mello. The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12 0 (2): 0 479--502, 2002

    2002

  26. [34]

    On law invariant coherent risk measures, pages 83--95

    Shigeo Kusuoka. On law invariant coherent risk measures, pages 83--95. Springer Japan, Tokyo, 2001. ISBN 978-4-431-67891-5

    2001

  27. [35]

    Moulines and F

    E. Moulines and F. Bach. Non-asymptotic analysis of stochastic approximation algorithms for machine learning. Advances in neural information processing systems, 24: 0 451--459, 2011

    2011

  28. [36]

    Fournier

    N. Fournier . Convergence of the empirical measure in expected wasserstein distance: non-asymptotic explicit bounds in rd. ESAIM: PS, 27: 0 749--775, 2023

    2023

  29. [37]

    Nemirovski, A

    A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to stochastic programming. SIAM Journal on Optimization, 19 0 (4): 0 1574--1609, 2009

    2009

  30. [38]

    Introductory Lectures on Convex Optimization , volume 87 of Applied Optimization

    Yurii Nesterov. Introductory Lectures on Convex Optimization , volume 87 of Applied Optimization . Springer US, Boston, MA, 2004. ISBN 978-1-4613-4691-3 978-1-4419-8853-9

    2004

  31. [39]

    Newey and James L

    Whitney K. Newey and James L. Powell. Asymmetric Least Squares Estimation and Testing . Econometrica, 55 0 (4): 0 819, July 1987. ISSN 00129682

    1987

  32. [40]

    Panaretos and Yoav Zemel

    Victor M. Panaretos and Yoav Zemel. An invitation to statistics in Wasserstein space . SpringerBriefs in Probability and Mathematical Statistics . SpringerOpen, Cham, 2020. ISBN 978-3-030-38438-8 978-3-030-38437-1

    2020

  33. [41]

    Pasupathy, P

    R. Pasupathy, P. Glynn, S. Ghosh, and F. S. Hashemi. On sampling rates in simulation-based recursions. SIAM Journal on Optimization, 28 0 (1): 0 45--73, 2018. Publisher: SIAM

    2018

  34. [42]

    Pflug and Andrzej Ruszczynski

    Georg Ch. Pflug and Andrzej Ruszczynski. Risk measures for income streams. Stochastic Programming E-Print Series (SPEPS), 2001

    2001

  35. [43]

    Interpreting Expectiles

    Collin Philipps. Interpreting Expectiles . Working Papers 2022-01, Department of Economics and Geosciences, US Air Force Academy, January 2022

    2022

  36. [44]

    Prashanth and S

    L.A. Prashanth and S. P. Bhat. A Wasserstein Distance Approach for Concentration of Empirical Risk Estimates . Journal of Machine Learning Research, 23 0 (238): 0 1--61, 2022

    2022

  37. [45]

    R. T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk . Journal of risk, 2: 0 21--42, 2000

    2000

  38. [46]

    W. Rudin. Principles of mathematical analysis. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953

    1953

  39. [47]

    Lectures on Stochastic Programming: Modeling and Theory, Third Edition

    Alexander Shapiro, Darinka Dentcheva, and Andrzej Ruszczynski. Lectures on Stochastic Programming: Modeling and Theory, Third Edition. Society for Industrial and Applied Mathematics, 2021

    2021

  40. [48]

    S.E. Shreve. Stochastic Calculus for Finance II : Continuous - Time Models . Number v. 11 in Springer Finance Textbooks . Springer, 2004. ISBN 978-0-387-40101-0

    2004

  41. [49]

    Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation

    Achraf Tamtalini, Anis Matoussi, and Sarah Kaakai. Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation . working paper or preprint, October 2022

    2022

  42. [50]

    Joel A. Tropp. The expected norm of a sum of independent random matrices: An elementary approach. In Christian Houdr \'e , David M. Mason, Patricia Reynaud-Bouret, and Jan Rosi \' n ski, editors, High Dimensional Probability VII, pages 173--202, Cham, 2016. Springer Internatio...

    2016

  43. [51]

    Uryasev and R.T

    S. Uryasev and R.T. Rockafellar. Conditional Value-at-Risk: Optimization Approach, pages 411--435. Springer US, Boston, MA, 2001. ISBN 978-1-4757-6594-6

    2001

  44. [52]

    Theory of Games and Economic Behavior

    John von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior. princeton, first edition, 1944

    1944

  45. [53]

    S. Weber. Distribution-invariant risk measures, information, and dynamic consistency. Mathematical Finance, 16 0 (2): 0 419--441, 2006

    2006

  46. [54]

    Johanna F. Ziegel. Coherence And Elicitability . Mathematical Finance, 26 0 (4): 0 901--918, October 2016. ISSN 0960-1627, 1467-9965

    2016

  47. [55]

    On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility

    Aleš Černý, Fabio Maccheroni, Massimo Marinacci, and Aldo Rustichini. On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility. Journal of Mathematical Economics, 48 0 (6): 0 386--395, December 2012. ISSN 0304-4068

    2012

Pith tools

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