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A STOCHASTIC APPROXIMATION ALGORITHM FOR STOCHASTIC SEMIDEFINITE PROGRAMMING

Published online by Cambridge University Press:  18 May 2016

Bruno Gaujal  and
Panayotis Mertikopoulos
Bruno Gaujal
Affiliation:
Inria and Univ. Grenoble Alpes, LIG F-38000 Grenoble, France E-mail:[email protected]://mescal.imag.fr/membres/bruno.gaujal
Panayotis Mertikopoulos
Affiliation:
CNRS (French National Center for Scientific Research), LIG F-38000 Grenoble, France and University Grenoble Alpes, LIG F-38000 Grenoble, France E-mail: [email protected]://mescal.imag.fr/membres/panayotis.mertikopoulos
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Abstract

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Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continuous-time matrix exponential scheme which is further regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an ɛ-approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels. Our theoretical analysis is complemented by extensive numerical simulations, which illustrate the robustness and scalability of the proposed method in realistic network conditions.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 3: Erol Gelenbe's 70th Birthday , July 2016 , pp. 431 - 454
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2016

References

1.Base station (BS) radio transmission and reception. Technical Specification 36.104 V12.4.0, 3GPP, June 2014.Google Scholar
2.User equipment (UE) radio transmission and reception. Technical Specification 36.101 V12.4.0, 3GPP, June 2014.Google Scholar
3.3GPP, User equipment (UE) radio transmission and reception. White paper, June 2014.Google Scholar
4.Anderson, T.W. (2003). An introduction to multivariate statistical analysis. Hoboken, NJ, USA: Wiley–Interscience, 3rd ed.Google Scholar
5.Belmega, E.V., Lasaulce, S., & Debbah, M. (2009). Power allocation games for MIMO multiple access channels with coordination. IEEE Transactions on Wireless Communications 8: 3182–3192.CrossRefGoogle Scholar
6.Belmega, E.V., Lasaulce, S., Debbah, M., & Hjørungnes, A. (2010). Learning distributed power allocation policies in MIMO channels. In EUSIPCO ’10: Proceedings of the 2010 European Signal Processing Conference, Aalborg, Denmark.Google Scholar
7.Benaïm, M. (1999). Dynamics of stochastic approximation algorithms. In Séminaire de Probabilités XXXIII, Azéma, J., Émery, M., Ledoux, M., & Yor, M. (eds.), vol. 1709 of Lecture Notes in Mathematics, Berlin, Heidelberg: Springer, pp. 1–68.Google Scholar
8.Borkar, V.S. (2008). Stochastic approximation: a dynamical systems viewpoint. Cambridge, UK: Cambridge University Press and Hindustan Book Agency.CrossRefGoogle Scholar
9.Boyd, S.P. & Vandenberghe, L. (2004). Convex optimization. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
10.Calcev, G., Chizhik, D., Göransson, B., Howard, S., Huang, H., Kogiantis, A., Molisch, A.F., Moustakas, A.L., Reed, D., & Xu, H. (2007). A wideband spatial channel model for system-wide simulations. IEEE Transactions on Vehicular Technology 56: 389.CrossRefGoogle Scholar
11.Cheng, R.S. & Verdú, S. (1993). Gaussian multiaccess channels with ISI: capacity region and multiuser water-filling. IEEE Transactions on Information Theory 39: 773–785.CrossRefGoogle Scholar
12.Coucheney, P., Gaujal, B., & Mertikopoulos, P. (2014). Distributed optimization in multi-user MIMO systems with imperfect and delayed information. In ISIT ’14: Proceedings of the 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA.Google Scholar
13.Dattorro, J. (2005). Convex optimization & Euclidean distance geometry. Palo Alto, CA, USA: Meboo Publishing.Google Scholar
14.Debreu, G. (1952). A social equilibrium existence theorem. Proceedings of the National Academy of Sciences of the United States of America 38: 886–893.Google Scholar
15.Goldsmith, A.J. & Varaiya, P.P. (1997). Capacity of fading channels with channel side information. IEEE Transactions on Information Theory 43: 1986–1992.CrossRefGoogle Scholar
16.Higham, N.J. (2008). Functions of matrices: theory and computation. Philadelphia, PA: SIAM.CrossRefGoogle Scholar
17.Kakade, S.M., Shalev-Shwartz, S., & Tewari, A. (2012). Regularization techniques for learning with matrices. Journal of Machine Learning Research 13: 1865–1890.Google Scholar
18.Mertikopoulos, P. & Belmega, E.V. (2014). Transmit without regrets: online optimization in MIMO–OFDM cognitive radio systems. IEEE Journal on Selected Areas in Communications 32: 1987–1999.CrossRefGoogle Scholar
19.Mertikopoulos, P., Belmega, E.V., & Moustakas, A.L. (2012). Matrix exponential learning: Distributed optimization in MIMO systems. In ISIT ’12: Proceedings of the 2012 IEEE International Symposium on Information Theory, Cambridge, MA, USA, pp. 3028–3032.CrossRefGoogle Scholar
20.Mertikopoulos, P., Belmega, E.V., Moustakas, A.L., & Lasaulce, S. (2012). Distributed learning policies for power allocation in multiple access channels. IEEE Journal on Selected Areas in Communications 30: 96–106.CrossRefGoogle Scholar
21.Mertikopoulos, P. & Moustakas, A.L. (2016) Learning in an uncertain world: MIMO covariance matrix optimization with imperfect feedback. IEEE Transactions on Signal Processing 64: 5–18.CrossRefGoogle Scholar
22.Moler, C. & van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Reviews 45: 3–49.CrossRefGoogle Scholar
23.Monderer, D. & Shapley, L.S. (1996). Potential games. Games and Economic Behavior 14: 124–143.CrossRefGoogle Scholar
24.Nash, J.F. (1951). Non-cooperative games. Annals of Mathematics 54: 286–295.CrossRefGoogle Scholar
25.Nemirovski, A.S., Juditsky, A., Lan, G.G., & Shapiro, A. (2009). Robust stochastic approximation approach to stochastic programming. SIAM Journal on Optimization 19: 1574–1609.CrossRefGoogle Scholar
26.Nemirovski, A.S. & Yudin, D.B. (1983). Problem complexity and method efficiency in optimization. New York, NY: Wiley.Google Scholar
27.Nesterov, Y. (2009). Primal-dual subgradient methods for convex problems. Mathematical Programming 120: 221–259.CrossRefGoogle Scholar
28.Palomar, D.P., Cioffi, J.M., & Lagunas, M. (2003). Uniform power allocation in MIMO channels: a game-theoretic approach. IEEE Transactions on Information Theory 49: 1707.CrossRefGoogle Scholar
29.Ramana, M. & Goldman, A.J. (1995). Some geometric results in semidefinite programming. Journal of Global Optimization 7: 33–50.CrossRefGoogle Scholar
30.Rosen, J.B. (1965). Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33: 520–534.CrossRefGoogle Scholar
31.Scutari, G., Palomar, D.P., & Barbarossa, S. (2006). Simultaneous iterative water-filling for Gaussian frequency-selective interference channels. In ISIT ’06: Proceedings of the 2006 International Symposium on Information Theory, Seattle, WA, USA, pp. 600–604.CrossRefGoogle Scholar
32.Scutari, G., Palomar, D.P., & Barbarossa, S. (2008). Competitive design of multiuser MIMO systems based on game theory: a unified view. IEEE Journal on Selected Areas in Communications 26: 1089–1103.CrossRefGoogle Scholar
33.Scutari, G., Palomar, D.P., & Barbarossa, S. (2008). Optimal linear precoding strategies for wideband non-cooperative systems based on game theory–part I: Nash equilibria. IEEE Transactions on Signal Processing 56: 1230–1249.CrossRefGoogle Scholar
34.Scutari, G., Palomar, D.P., & Barbarossa, S. (2008). Optimal linear precoding strategies for wideband non-cooperative systems based on game theory–part II: algorithms. IEEE Transactions on Signal Processing 56: 1250–1267.CrossRefGoogle Scholar
35.Scutari, G., Palomar, D.P., & Barbarossa, S. (2009). The MIMO iterative waterfilling algorithm. IEEE Transactions on Signal Processing 57: 1917–1935.CrossRefGoogle Scholar
36.Starr, T., Cioffi, J.M., & Silverman, P.J. (1999). Understanding digital subscriber line technology. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
37.Strassen, V. (1965). The existence of probability measures with given marginals. The Annals of Mathematical Statistics 38: 423–439.CrossRefGoogle Scholar
38.Sun, P. & Freund, R.M. (2004). Computation of minimum-cost covering ellipsoids. Operations Research 52: 690–706.CrossRefGoogle Scholar
39.Telatar, I.E. (1999). Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications and Related Technologies 10: 585–596.CrossRefGoogle Scholar
40.Wilcox, R.M. (1967). Exponential operators and parameter differentiation in quantum physics. Journal of Mathematical Physics 8: 962–982.CrossRefGoogle Scholar
41.Yang, Y., Scutari, G., Palomar, D.P., & Pesavento, M. (2014). A parallel stochastic approximation method for nonconvex multi-agent optimization problems. http://arxiv.org/abs/1410.5076.Google Scholar
42.Yu, W., Rhee, W., Boyd, S., & Cioffi, J.M. (2004). Iterative water-filling for Gaussian vector multiple-access channels. IEEE Transactions on Information Theory 50: 145–152.CrossRefGoogle Scholar
43.Zhu, Y. & Ariyawanasa, K.A. (2011). A preliminary set of applications leading to stochastic semidefinite programs and chance-constrained semidefinite programs. Applied Mathematical Modelling 35: 2425–2442.CrossRefGoogle Scholar