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arxiv: 2607.01516 · v1 · pith:OLNTUKMWnew · submitted 2026-07-01 · 💻 cs.DS · cs.GT

Asymmetric Trading Prophets

Gagan Aggarwal , Anupam Gupta , Yifan Wang , Mingfei Zhao This is my paper

Pith reviewed 2026-07-03 17:41 UTC · model grok-4.3

classification 💻 cs.DS cs.GT
keywords asymmetric trading prophetscompetitive analysisonline algorithmsprophet inequalitiescapacity constraintsi.i.d. distributionsinventory managementbuy sell price tuples
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The pith

Online algorithms achieve competitive ratios approaching 1 as capacity grows for asymmetric buy-sell price tuples.

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

The paper introduces the asymmetric trading prophets problem where each timestep presents a trader with a possibly correlated price tuple of buy cost and sell revenue. It supplies the first competitive analysis that characterizes the best achievable expected profit relative to an offline prophet, expressed in terms of total capacity B and starting inventory B0. For the unit-capacity case with B equal to 1 and B0 equal to 1, the algorithms attain constant competitive ratios under both i.i.d. and non-i.i.d. distributions. For larger capacities under i.i.d. arrivals the ratio reaches 1 minus Theta of log B0 over square root of B0, and improves further to 1 minus O of log B over square root of B when buy and sell prices are forced to be identical.

Core claim

We provide the first competitive analysis for this asymmetric trading prophets problem, characterizing the achievable profit based on the trader's capacity B and initial inventory B0. For the unit-capacity case of B=1, we design online algorithms that achieve constant competitive ratios for both i.i.d. and non-i.i.d. distributions on the price tuples, when the trader has one initial copy (B0=1). For the general capacity case where B can be large, we give algorithms for i.i.d. distributions that achieve a competitive ratio of 1 - Theta(log B0/sqrt(B0)). Finally, for the symmetric case (where the price tuple satisfies b_t = s_t), we improve this to get a competitive ratio of 1 - O(log B/sqrt(B

What carries the argument

Online threshold policies that decide at each step whether to buy or sell a unit based on the observed (b_t, s_t) tuple and current inventory level, calibrated to the known or unknown distribution to compete with the offline optimum.

If this is right

  • Constant competitive ratios hold for unit capacity even under arbitrary non-i.i.d. distributions on the price tuples.
  • The competitive ratio improves toward 1 as initial inventory B0 or capacity B grows under i.i.d. arrivals.
  • The symmetric special case where buy and sell prices coincide yields a strictly better ratio of 1 minus O(log B over sqrt(B)).
  • Both the general and symmetric ratios are tight up to logarithmic factors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In practice this implies that traders with larger starting inventories can extract a larger fraction of the offline optimum even when buy and sell opportunities are asymmetric.
  • The same style of analysis may extend to limited-information settings where the distribution is learned on the fly rather than known in advance.
  • Similar inventory-control techniques could be applied to other online problems that combine buying and selling decisions under capacity limits.

Load-bearing premise

Price tuples arrive according to a stochastic model from fixed but possibly unknown distributions, so the competitive ratio is measured in expectation over those draws.

What would settle it

A concrete family of distributions on price tuples for which every online algorithm falls below the stated ratio 1 - c log B0 / sqrt(B0) for some constant c and sufficiently large B0 would falsify the claim.

read the original abstract

The "Trading Prophet" problem challenges an online trader to maximize its profit by buying and selling assets under stochastic prices and capacity constraints, competing against an offline prophet with full foresight. In previous work, each arriving asset was assumed to have a single price $p_t$, and the trader was allowed to either buy a copy at this price (subject to having available capacity), or sell a copy (if it already held at least one copy in hand). However, this abstraction can fail to capture the structural asymmetry of decentralized dealer-based markets, where buying and selling opportunities could be distinct, and driven by individual preferences. To address this, we introduce the Asymmetric Trading Prophets problem, where at each timestep the trader observes a price tuple $(b_t, s_t)$ -- representing the cost to buy, and the revenue from selling at this timestep. Importantly, the $(b_t,s_t)$ tuple could be potentially arbitrarily correlated. We provide the first competitive analysis for this asymmetric trading prophets problem, characterizing the achievable profit based on the trader's capacity $B$ and initial inventory $B_0$. For the unit-capacity case of $B=1$, we design online algorithms that achieve constant competitive ratios for both i.i.d. and non-i.i.d. distributions on the price tuples, when the trader has one initial copy ($B_0=1$). For the general capacity case where $B$ can be large, we give algorithms for i.i.d. distributions that achieve a competitive ratio of $1 - \Theta(\log B_0/\sqrt{B_0})$. Finally, for the symmetric case (where the price tuple satisfies $b_t=s_t$), we improve this to get a competitive ratio of $1 - O(\log B/\sqrt{B})$, demonstrating that the performance approaches optimality as the capacity increases. We show that both ratios are tight up to a logarithmic factor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the Asymmetric Trading Prophets problem, in which an online trader observes possibly correlated price tuples (b_t, s_t) at each timestep and must decide whether to buy or sell subject to capacity B and initial inventory B0, competing against an offline prophet. It gives the first competitive analysis: constant competitive ratios for the unit-capacity case B=1, B0=1 under both i.i.d. and non-i.i.d. distributions; a ratio of 1 - Θ(log B0/√B0) for general B under i.i.d. distributions; and an improved ratio of 1 - O(log B/√B) in the symmetric case b_t = s_t, with both ratios shown tight up to logarithmic factors.

Significance. If the stated ratios and tightness results hold, the work provides the first explicit characterization of achievable profit in an asymmetric trading setting that better models decentralized dealer markets. The extension from single-price to tuple-based decisions, the handling of correlation, and the capacity-dependent bounds (including the asymptotic improvement in the symmetric case) constitute a substantive contribution to online algorithms and prophet inequalities. The explicit tightness claims up to log factors are a strength.

minor comments (3)
  1. The abstract and introduction should explicitly state whether the non-i.i.d. constant-ratio result for B=1 requires any restrictions on the joint distribution of the tuples beyond the stochastic arrival model.
  2. Clarify in §3 or the algorithm description how the online decisions for the B=1 case are implemented without knowledge of the distribution parameters while still achieving the claimed constants.
  3. The tightness constructions for the general-capacity and symmetric cases should be cross-referenced to the exact competitive-ratio expressions to confirm the log-factor gap is unavoidable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the accurate summary of our results on Asymmetric Trading Prophets, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the Asymmetric Trading Prophets problem and derives competitive ratios via explicit online algorithm constructions and stochastic analysis for i.i.d. and non-i.i.d. price tuples. The claimed bounds (constant ratios for B=1, 1-Θ(log B0/√B0) for general B, and symmetric improvements) are obtained through direct mathematical proofs of competitiveness against an offline prophet, expressed in terms of the input parameters B and B0. No steps reduce by construction to fitted parameters, self-definitional equations, or load-bearing self-citations; the analysis is self-contained within the stochastic model and does not rename or smuggle prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on standard definitions from stochastic online algorithms; no free parameters are fitted, no new entities are postulated, and the only background assumptions are the usual probabilistic and competitive-ratio definitions.

axioms (1)
  • domain assumption Competitive ratio is the infimum over input distributions of the ratio of expected online profit to expected offline profit.
    Invoked when the paper characterizes achievable profit for given B and B0.

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Works this paper leans on

183 extracted references · 36 canonical work pages · 1 internal anchor

  1. [1]

    Trading Prophets: How to Trade Multiple Stocks Optimally , booktitle =

    Surbhi Rajput and Ashish Chiplunkar and Rohit Vaish , editor =. Trading Prophets: How to Trade Multiple Stocks Optimally , booktitle =. 2025 , url =. doi:10.1137/1.9781611978315.19 , timestamp =

    work page doi:10.1137/1.9781611978315.19 2025

  2. [2]

    Trading Prophets , booktitle =

    Jos. Trading Prophets , booktitle =. 2023 , url =. doi:10.1145/3580507.3597813 , timestamp =

    work page doi:10.1145/3580507.3597813 2023

  3. [3]

    Approximation Algorithms for Stochastic Minimum-Norm Combinatorial Optimization , booktitle =

    Sharat Ibrahimpur and Chaitanya Swamy , editor =. Approximation Algorithms for Stochastic Minimum-Norm Combinatorial Optimization , booktitle =. 2020 , url =. doi:10.1109/FOCS46700.2020.00094 , timestamp =

    work page doi:10.1109/focs46700.2020.00094 2020

  4. [4]

    Saeed Alaei , title =

  5. [5]

    Theory of computing , volume=

    The multiplicative weights update method: a meta-algorithm and applications , author=. Theory of computing , volume=. 2012 , publisher=

    2012

  6. [6]

    Journal of computer and system sciences , volume=

    A decision-theoretic generalization of on-line learning and an application to boosting , author=. Journal of computer and system sciences , volume=. 1997 , publisher=

    1997

  7. [7]

    Dynamic Placement in Refugee Resettlement , journal =

    Narges Ahani and Paul G. Dynamic Placement in Refugee Resettlement , journal =

  8. [8]

    Stochastic analyses for online combinatorial optimization problems , booktitle =

    Naveen Garg and Anupam Gupta and Stefano Leonardi and Piotr Sankowski , editor =. Stochastic analyses for online combinatorial optimization problems , booktitle =

  9. [9]

    Alberto Vera and Siddhartha Banerjee and Itai Gurvich , title =. Oper. Res. , volume =

  10. [10]

    Kowalski and Piotr Krysta and Jan Olkowski , title =

    Kiarash Banihashem and MohammadTaghi Hajiaghayi and Dariusz R. Kowalski and Piotr Krysta and Jan Olkowski , title =. Proceedings of SODA , pages =

  11. [11]

    Schapire and Aleksandrs Slivkins , title =

    Nicole Immorlica and Karthik Abinav Sankararaman and Robert E. Schapire and Aleksandrs Slivkins , title =. J

  12. [12]

    Ravi , title =

    Marco Molinaro and R. Ravi , title =. Math. Oper. Res. , volume =

  13. [13]

    Huber , title =

    Peter J. Huber , title =. The Annals of Mathematical Statistics , number =

  14. [14]

    47th International Colloquium on Automata, Languages, and Programming,

    Paritosh Garg and Sagar Kale and Lars Rohwedder and Ola Svensson , title =. 47th International Colloquium on Automata, Languages, and Programming,

  15. [15]

    47th International Colloquium on Automata, Languages, and Programming,

    Thomas Kesselheim and Marco Molinaro , title =. 47th International Colloquium on Automata, Languages, and Programming,

  16. [16]

    Echenique and N

    F. Echenique and N. Immorlica and V. V. Vazirani , place=. Online and Matching-Based Market Design , publisher=

  17. [17]

    2023 , publisher=

    Algorithmic high-dimensional robust statistics , author=. 2023 , publisher=

    2023

  18. [18]

    Karp and Umesh V

    Richard M. Karp and Umesh V. Vazirani and Vijay V. Vazirani , editor =. An Optimal Algorithm for On-line Bipartite Matching , booktitle =

  19. [19]

    Journal of Machine Learning Research , volume=

    The two-sided game of googol , author=. Journal of Machine Learning Research , volume=

  20. [20]

    Mathematics of Operations Research , volume=

    Sample-driven optimal stopping: From the secretary problem to the iid prophet inequality , author=. Mathematics of Operations Research , volume=. 2024 , publisher=

    2024

  21. [21]

    Devanur and Kamal Jain , editor =

    Nikhil R. Devanur and Kamal Jain , editor =. Online matching with concave returns , booktitle =

  22. [22]

    Prophet Inequalities Require Only a Constant Number of Samples , booktitle =

    Andr. Prophet Inequalities Require Only a Constant Number of Samples , booktitle =

  23. [23]

    Proceedings of SODA , pages =

    Haim Kaplan and David Naori and Danny Raz , title =. Proceedings of SODA , pages =

  24. [24]

    Single-Sample Prophet Inequalities via Greedy-Ordered Selection , booktitle =

    Constantine Caramanis and Paul D. Single-Sample Prophet Inequalities via Greedy-Ordered Selection , booktitle =

  25. [25]

    Proceedings of SODA 2022 , pages =

    Haim Kaplan and David Naori and Danny Raz , title =. Proceedings of SODA 2022 , pages =

    2022

  26. [26]

    Wang and S

    Aviad Rubinstein and Jack Z. Wang and S. Matthew Weinberg , title =. Proceedings of ITCS , series =

  27. [27]

    Prophet Inequalities for

    Jos. Prophet Inequalities for. Proceedings of EC , pages =

  28. [28]

    C. J. Argue and Alan M. Frieze and Anupam Gupta and Christopher Seiler , title =. Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS , year =

    2022

  29. [29]

    Matthew Weinberg , title =

    Pablo Daniel Azar and Robert Kleinberg and S. Matthew Weinberg , title =. Proceedings of SODA , pages =

  30. [30]

    Data-Driven Algorithm Design , booktitle =

    Maria. Data-Driven Algorithm Design , booktitle =

  31. [31]

    Online Algorithms for Covering and Packing Problems with Convex Objectives , booktitle =

    Yossi Azar and Niv Buchbinder and T. Online Algorithms for Covering and Packing Problems with Convex Objectives , booktitle =

  32. [32]

    Proceedings of

    Niv Buchbinder and Joseph Naor , title =. Proceedings of

  33. [33]

    A Constant Factor Prophet Inequality for Online Combinatorial Auctions , booktitle =

    Jos. A Constant Factor Prophet Inequality for Online Combinatorial Auctions , booktitle =

  34. [34]

    Dynkin, E. B. , journal =

  35. [35]

    Krengel and L

    U. Krengel and L. Sucheston , title =. Bulletin of the American Mathematical Society , volume =

  36. [36]

    Krengel and L

    U. Krengel and L. Sucheston , title =. Advances in Probability and Related Topics , volume =

  37. [37]

    SIGecom Exch

    Brendan Lucier , title =. SIGecom Exch. , volume =

  38. [38]

    Ashwinkumar Badanidiyuru and Robert Kleinberg and Aleksandrs Slivkins , title =. J

  39. [39]

    Devanur , title =

    Shipra Agrawal and Nikhil R. Devanur , title =. Oper. Res. , volume =

  40. [40]

    Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science , pages=

    Throughput-competitive on-line routing , author=. Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science , pages=. 1993 , organization=

    1993

  41. [41]

    Mathematics of Operations Research , volume=

    Online primal-dual algorithms for covering and packing , author=. Mathematics of Operations Research , volume=. 2009 , publisher=

    2009

  42. [42]

    Random-Order Models , booktitle =

    Anupam Gupta and Sahil Singla , editor =. Random-Order Models , booktitle =

  43. [43]

    Balseiro and Haihao Lu and Vahab Mirrokni , title =

    Santiago R. Balseiro and Haihao Lu and Vahab Mirrokni , title =. Oper. Res. , volume =

  44. [44]

    Alberto Vera and Siddhartha Banerjee , title =. Manag. Sci. , volume =

  45. [45]

    2006 , publisher=

    The theory and practice of revenue management , author=. 2006 , publisher=

    2006

  46. [46]

    Operations Research , volume=

    An approximation algorithm for network revenue management under nonstationary arrivals , author=. Operations Research , volume=. 2020 , publisher=

    2020

  47. [47]

    Proceedings of FOCS , year =

    Sepehr Assadi and Sahil Singla , title =. Proceedings of FOCS , year =

  48. [48]

    Proceedings of

    Sepehr Assadi and Thomas Kesselheim and Sahil Singla , title =. Proceedings of

  49. [49]

    An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions , booktitle =

    Paul D. An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions , booktitle =

  50. [50]

    An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions , booktitle =

    Thomas Kesselheim and Klaus Radke and Andreas T. An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions , booktitle =

  51. [51]

    Proceedings of the 13th

    Saeed Alaei and MohammadTaghi Hajiaghayi and Vahid Liaghat , title =. Proceedings of the 13th

  52. [52]

    11th Innovations in Theoretical Computer Science Conference,

    Domagoj Bradac and Anupam Gupta and Sahil Singla and Goran Zuzic , title =. 11th Innovations in Theoretical Computer Science Conference,

  53. [53]

    Foundations and Trends

    The design of competitive online algorithms via a primal--dual approach , author=. Foundations and Trends. 2009 , publisher=

    2009

  54. [54]

    Management Science , volume=

    Real-time optimization of personalized assortments , author=. Management Science , volume=. 2014 , publisher=

    2014

  55. [55]

    2024 , eprint=

    Online Contention Resolution Schemes for Network Revenue Management and Combinatorial Auctions , author=. 2024 , eprint=

    2024

  56. [56]

    Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Nonstochastic Inputs , journal =

    Paul D. Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Nonstochastic Inputs , journal =

  57. [57]

    Mathematical Programming , title =

    Correa, Jos. Mathematical Programming , title =. 2023 , bdsk-url-1 =. doi:10.1007/s10107-023-02027-2 , id =

    work page doi:10.1007/s10107-023-02027-2 2023

  58. [58]

    Aranyak Mehta , title =. Found. Trends Theor. Comput. Sci. , volume =. 2013 , url =. doi:10.1561/0400000057 , timestamp =

    work page doi:10.1561/0400000057 2013

  59. [59]

    Vazirani and Vijay V

    Aranyak Mehta and Amin Saberi and Umesh V. Vazirani and Vijay V. Vazirani , title =. J

  60. [60]

    Anupam Gupta and Marco Molinaro , title =. Math. Oper. Res. , volume =

  61. [61]

    Primal Beats Dual on Online Packing LPs in the Random-Order Model , journal =

    Thomas Kesselheim and Klaus Radke and Andreas T. Primal Beats Dual on Online Packing LPs in the Random-Order Model , journal =

  62. [62]

    Devanur and Thomas P

    Nikhil R. Devanur and Thomas P. Hayes , title =. Proceedings 10th

  63. [63]

    Operations Research , volume=

    A dynamic near-optimal algorithm for online linear programming , author=. Operations Research , volume=. 2014 , publisher=

    2014

  64. [64]

    Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract) , booktitle =

    Gruia C. Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract) , booktitle =. 2007 , url =. doi:10.1007/978-3-540-72792-7\_15 , timestamp =

    work page doi:10.1007/978-3-540-72792-7 2007

  65. [65]

    Online Submodular Maximization under a Matroid Constraint with Application to Learning Assignments

    Daniel Golovin and Andreas Krause and Matthew J. Streeter , title =. CoRR , volume =. 2014 , url =. 1407.1082 , timestamp =

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  66. [66]

    Wang , editor =

    Tim Roughgarden and Joshua R. Wang , editor =. An Optimal Learning Algorithm for Online Unconstrained Submodular Maximization , booktitle =. 2018 , url =

    2018

  67. [67]

    Wang and Fransisca Susan and Ashwinkumar Badanidiyuru , editor =

    Rad Niazadeh and Negin Golrezaei and Joshua R. Wang and Fransisca Susan and Ashwinkumar Badanidiyuru , editor =. Online Learning via Offline Greedy Algorithms: Applications in Market Design and Optimization , booktitle =. 2021 , url =. doi:10.1145/3465456.3467571 , timestamp =

    work page doi:10.1145/3465456.3467571 2021

  68. [68]

    Nicholas J. A. Harvey and Christopher Liaw and Tasuku Soma , editor =. Improved Algorithms for Online Submodular Maximization via First-order Regret Bounds , booktitle =. 2020 , url =

    2020

  69. [69]

    2015 , url =

    Niv Buchbinder and Moran Feldman and Joseph Naor and Roy Schwartz , title =. 2015 , url =. doi:10.1137/130929205 , timestamp =

    work page doi:10.1137/130929205 2015

  70. [70]

    Guess Free Maximization of Submodular and Linear Sums , booktitle =

    Moran Feldman , editor =. Guess Free Maximization of Submodular and Linear Sums , booktitle =. 2019 , url =. doi:10.1007/978-3-030-24766-9\_28 , timestamp =

    work page doi:10.1007/978-3-030-24766-9 2019

  71. [71]

    Vempala , title =

    Adam Tauman Kalai and Santosh S. Vempala , title =. J. Comput. Syst. Sci. , volume =. 2005 , url =. doi:10.1016/j.jcss.2004.10.016 , timestamp =

    work page doi:10.1016/j.jcss.2004.10.016 2005

  72. [72]

    Regret in Online Combinatorial Optimization , journal =

    Jean. Regret in Online Combinatorial Optimization , journal =. 2014 , url =. doi:10.1287/moor.2013.0598 , timestamp =

    work page doi:10.1287/moor.2013.0598 2014

  73. [73]

    Combinatorial bandits , journal =

    Nicol. Combinatorial bandits , journal =. 2012 , url =. doi:10.1016/j.jcss.2012.01.001 , timestamp =

    work page doi:10.1016/j.jcss.2012.01.001 2012

  74. [74]

    Combinatorial Bandits Revisited , booktitle =

    Richard Combes and Mohammad Sadegh Talebi and Alexandre Prouti. Combinatorial Bandits Revisited , booktitle =. 2015 , url =

    2015

  75. [75]

    The Nonstochastic Multiarmed Bandit Problem , journal =

    Peter Auer and Nicol. The Nonstochastic Multiarmed Bandit Problem , journal =. 2002 , url =. doi:10.1137/S0097539701398375 , timestamp =

    work page doi:10.1137/s0097539701398375 2002

  76. [76]

    Papadimitriou and Tristan Pollner and Amin Saberi and David Wajc , editor =

    Christos H. Papadimitriou and Tristan Pollner and Amin Saberi and David Wajc , editor =. Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities , booktitle =. 2021 , url =. doi:10.1145/3465456.3467613 , timestamp =

    work page doi:10.1145/3465456.3467613 2021

  77. [77]

    The Invisible Hand of Dynamic Market Pricing , booktitle =

    Vincent Cohen. The Invisible Hand of Dynamic Market Pricing , booktitle =. 2016 , url =. doi:10.1145/2940716.2940730 , timestamp =

    work page doi:10.1145/2940716.2940730 2016

  78. [78]

    Max-Min Greedy Matching , booktitle =

    Alon Eden and Uriel Feige and Michal Feldman , editor =. Max-Min Greedy Matching , booktitle =. 2019 , url =. doi:10.4230/LIPIcs.APPROX-RANDOM.2019.7 , timestamp =

    work page doi:10.4230/lipics.approx-random.2019.7 2019

  79. [79]

    Max-Weight Online Stochastic Matching: Improved Approximations Against the Online Benchmark , booktitle =

    Mark Braverman and Mahsa Derakhshan and Antonio Molina Lovett , editor =. Max-Weight Online Stochastic Matching: Improved Approximations Against the Online Benchmark , booktitle =. 2022 , url =. doi:10.1145/3490486.3538315 , timestamp =

    work page doi:10.1145/3490486.3538315 2022

  80. [80]

    Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms , pages=

    Fast algorithms for online stochastic convex programming , author=. Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms , pages=. 2014 , organization=

    2014

Showing first 80 references.