Skip to main content

Adaptive online portfolio strategy based on exponential gradient updates


Based on the momentum principle and adaptive learning mechanism, we design online portfolio selection strategies, which are suitable for nonstationary financial market. Firstly, we propose a Moving-window-based Adaptive Exponential Gradient (MAEG) strategy, which updates the learning rate of the EG algorithm by maximizing the recent cumulative return using the price data in a fixed length moving window. Secondly, we consider a special case where all-historical price data is used to design another strategy named Adaptive Exponential Gradient (AEG). Finally, we conduct an extensive numerical analysis using real price data and the empirical results show that the performance of the proposed strategies is steadily superior to some online strategies. In addition, MAEG and AEG are able to withstand certain transaction costs, which further supports their practical applicability in trading applications.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    Comb. 1 contains the following stocks: Alcoa, GM, IBM, Kin Ark, Kodak; Comb. 2 contains the following stocks: Alcoa, American Brands, GM, IBM, Kin Ark, Kodak; Comb. 3 contains the following stocks: Dupont, Ford, GE, GM, IBM, Kin Ark; Comb. 4 contains the following stocks: AHP, Dupont, Ford, GE, GM, IBM, JNJ, Kin Ark, MMM, Sherman Williams; Comb. 5 contains the following stocks: 600757, 600171, 000060, 000830, 000750; Comb. 6 contains the following stocks: 600757, 600171, 000060, 000830, 000750, 000930; Comb. 7 contains the following stocks: 600757, 600171, 000060, 000830, 000750, 000930, 000062, 600060; Comb. 8 contains the following stocks: 000858, 000725, 600585, 600028, 600741, 000898, 600104, 600115, 600027, 600016, 000625, 000069, 600674, 600036, 600663, 000063, 000157, 600066, 000415, 000627, 600271, 600118, 600886, 000768, 600018, 600170, 600068, 600177, 002044.


  1. Agarwal A, Hazan E, Kale S, Schapire RE (2006) Algorithms for portfolio management based on the Newton method. In: ACM: Proceedings of the 23rd international conference on machine learning, pp 9–16

  2. Bean AJ, Singer AC (2012) Universal switching and side information portfolios under transaction costs using factor graphs. IEEE J Select Topics Sig Process 6(4):351–365

    Article  Google Scholar 

  3. Blum A, Kalai A (1999) Universal portfolios with and without transaction costs. Mach Learn 35(3):193–205

    Article  Google Scholar 

  4. Borodin A, El-Yaniv R, Gogan V (2004) Can we learn to beat the best stock. J Artif Intell Res 21:579–594

    MathSciNet  Article  Google Scholar 

  5. Cai X, Ye ZK (2019) Gaussian weighting reversion strategy for accurate online portfolio selection. IEEE Trans Sig Process 67(21):5558–5570

    Article  Google Scholar 

  6. Chai X, Li WH, Yuan H, Wang LB (2020) Online scheduling on a single machine with linear deteriorating processing times and delivery times. J Combin Optim.

  7. Chu G, Zhang W, Sun GF, Zhang XT (2019) A new online portfolio selection algorithm based on Kalman Filter and anti-correlation. Physi A Stat Mech Appl 536(15):Article 120949

  8. Cover TM (1991) Universal portfolios. Math Finance 1(1):1–29

    MathSciNet  Article  Google Scholar 

  9. Cover TM, Ordentlich E (1996) Universal portfolios with side information. IEEE Trans Inform Theory 42(2):348–363

    MathSciNet  Article  Google Scholar 

  10. Dai WQ, Zheng M, Chen X, Yang ZL (2020) Online economic ordering problem for deteriorating items with limited price information. J Combin Optim.

  11. Feng X, Xu YF, Ni GQ, Dai YW (2018) Online leasing problem with price fluctuations under the consumer price index. J Combin Optim 36:493–507

    MathSciNet  Article  Google Scholar 

  12. Gaivoronski AA, Stella F (2000) Stochastic nonstationary optimization for finding universal portfolios. Ann Oper Res 100(1–4):165–188

    MathSciNet  Article  Google Scholar 

  13. Guan H, An ZY (2019) A local adaptive learning system for online portfolio selection. Knowl Based Syst 186:Article 104958

  14. Helmbold DP, Schapire RE, Singer Y, Warmuth MK (1998) On-line portfolio selection using multiplicative updates. Math Finance 8(4):325–347

    Article  Google Scholar 

  15. Huang DJ, Zhou JL, Li B, Hoi SCH, Zhou SG (2016) Robust median reversion strategy for online portfolio selection. IEEE Trans Knowl Data Eng 28(9):2480–2493

    Article  Google Scholar 

  16. Huang DJ, Yu SC, Li B, Hoi SCH, Zhou SG (2018) Combination forecasting reversion strategy for online portfolio selection. ACM Trans Intell Syst Technol 9(5):Article 58

  17. Jegadeesh N (1990) Evidence of predictable behavior of security returns. J Finance 45(3):881–898

    Article  Google Scholar 

  18. Kelly JL (1956) A new interpretation of information rate. Bell Syst Tech J 35(4):917–926

    MathSciNet  Article  Google Scholar 

  19. Khedmati M, Azin P (2020) An online portfolio selection algorithm using clustering approaches and considering transaction costs. Exp Syst Appl 159(30):Article 113546

  20. Lai ZR, Dai DQ, Ren CX, Huang KK (2018a) A peak price tracking-based learning system for portfolio selection. IEEE Trans Neural Netw Learning Syst 29(7):2823–2832

    MathSciNet  Google Scholar 

  21. Lai ZR, Dai DQ, Ren CX, Huang KK (2018b) Radial basis functions with adaptive input and composite trend representation for portfolio selection. IEEE Trans Neural Netw Learn Syst 29(12):6214–6226

    Article  Google Scholar 

  22. Lai ZR, Yang PY, Fang LD, Wu XT (2018c) Short-term sparse portfolio optimization based on alternating direction method of multipliers. J Mach Learn Res 19(1):1–28

    MathSciNet  MATH  Google Scholar 

  23. Lai ZR, Yang PY, Wu XT, Fang LD (2018d) A kernel-based trend pattern tracking system for portfolio optimization. Data Mining Knowl Discov 32:1708–1734

    MathSciNet  Article  Google Scholar 

  24. Lai ZR, Yang PY, Fang LD, Wu XT (2020) Reweighted price relative tracking system for automatic portfolio optimization. IEEE Trans Syst Man Cybernet Syst 50(11):4349–4361

    Article  Google Scholar 

  25. Li B, Hoi SCH, Gopalkrishnan V (2011) CORN: correlation-driven nonparametric learning approach for portfolio selection. ACM Trans Intell Syst Technol 2(3):Article 21

  26. Li B, Zhao PL, Hoi SCH, Gopalkrishnan V (2012) PAMR: passive aggressive mean reversion strategy for portfolio selection. Mach Learn 87(2):221–258

    MathSciNet  Article  Google Scholar 

  27. Li B, Hoi SCH, Sahoo D, Liu ZY (2015) Moving average reversion strategy for on-line portfolio selection. Artif Intell 222:104–123

    MathSciNet  Article  Google Scholar 

  28. Li B, Wang JL, Huang DJ, Hoi SCH (2018) Transaction cost optimization for online portfolio selection. Quant Finance 18(8):1411–1424

    MathSciNet  Article  Google Scholar 

  29. Li TF, Chen K, Feng Y, Ying ZL (2017) Binary switch portfolio. Quant Finance 17(5):763–780

    MathSciNet  Article  Google Scholar 

  30. Lin X, Zhang M, Zhang YF, Gu ZQ, Liu YQ, Ma SP (2017) Boosting moving average reversion strategy for online portfolio selection: a meta-learning approach. In: International conference on database systems for advanced applications (DASFAA 2017), pp 494–510

  31. Lo AW, MacKinlay AC (1990) When are contrarian profits due to stock market overreaction? Rev Finan Stud 3(2):175–205

    Article  Google Scholar 

  32. Ma YL, Han RZ, Wang WZ (2021) Portfolio optimization with return prediction using deep learning and machine learning. Exp Syst Appl 165(1):Article 113973

  33. Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91

    Google Scholar 

  34. O’Sullivan P, Edelman D (2015) Adaptive universal portfolios. Eur J Finance 21(4):337–351

  35. Sezer OB, Gudelek MU, Ozbayoglu AM (2020) Financial time series forecasting with deep learning: a systematic literature review: 2005–2019. Appl Soft Comput 90:Article 106181

  36. Singer Y (1997) Switching portfolios. Int J Neural Syst 8(4):445–455

    Article  Google Scholar 

  37. Vovk VG, Watkins C (1998) Universal portfolio selection. In: Proceedings of the eleventh annual conference on computational learning theory, COLT 1998. Madison, Wisconsin, USA, July 24-26, 1998, pp 12–23

  38. Yang XY, He JA, Lin H, Zhang Y (2020a) Boosting exponential gradient strategy for online portfolio selection: an aggregating experts’ advice method. Comput Econ 55:231–251

  39. Yang XY, He JA, Xian JY, Lin H, Zhang Y (2020b) Aggregating expert advice strategy for online portfolio selection with side information. Soft Comput 24(3):2067–2081

    Article  Google Scholar 

  40. Yu JR, Paul Chiou WJ, Lee WY, Lin SJ (2020) Portfolio models with return forecasting and transaction costs. Int Rev Econ Finance 66:118–130

    Article  Google Scholar 

Download references


This research was supported by the National Natural Science Foundation of China (No. 71501049), the Humanities and Social Science Foundation of the Ministry of Education of China (No. 21YJA630117) and the Philosophy and Social Sciences Planning Project of Guangdong Province (GD19CGL06).

Author information



Corresponding author

Correspondence to Xingyu Yang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Lin, H., Zheng, L. et al. Adaptive online portfolio strategy based on exponential gradient updates. J Comb Optim (2021).

Download citation


  • Online portfolio selection
  • Adaptive strategy
  • Exponential gradient
  • Learning rate
  • Quantitative finance

Mathematics Subject Classification

  • 91G10