Academic research and selected publications

A complete collection of my research works is available from Google Scholar, ArXiv and SSRN.

A common thread in my academic research has been the combination of techniques from reproducing kernel and approximation theory to investigate the behavior of large systems. My contributions in this area range from purely mathematical studies in system stability and interpolation theory to applications in numerical analysis, the convergence of quantum algorithms and the stability of quantum memories. In recent years deep reinforcement learning (i.e. the combination of reinforcement learning with flexible function approximators) emerged as a new tenet of my research interest.

Here is a list of selected publications sorted according to their level of application with some descriptions.

Pure Mathematics

The below articles illustrate a typical line of my research in pure mathematics. Targeting the construction of matrices whose inverses grow "as fast as possible" (more precisely Schäffer's conjecture, a problem suggested to R. Zarouf and myself by N. Nikolski), we found that an explicit construction could be achieved using relatively standard tools from reproducing kernel Hilbert space (RKHS) theory. Until then the state-of-the-art method by J. Bourgain has been to relate the problem to power sums, while our method is duality/optimization-based. The implementation of our approach required precise asymptotic expansions for the Taylor coefficients of powers of a Blaschke factor, which we derived using new tools from approximation theory.

The new asymptotic formulas allowed us to identify the l_p-norms of their Taylor coefficients and to answer questions about their asymptotic behavior raised by H. Queffelec in 1995 (see page 86, questions 3,4,5 here) in a separate publication. During our work we identified previously unknown forms of asymptotic behavior of Jacobi polynomials and showed that part of the literature on this topic must be revised. Our findings have applications in the study of the Dyson Brownian motion and quantum chaos, see the beautiful article by Peter Forrester, Mario Kieburg, Shi-Hao Li and Jiyuan Zhang.

• l_p-norms of the Taylor coefficients of powers of a Blaschke factor, O. Szehr, R. Zarouf. (2020) J. Analyse. Math. 140
• Explicit counterexamples to Schäffer's conjecture, O. Szehr, R. Zarouf. (2021) J. de Math. Pures et Appl. 146.
• On the asymptotic behavior of Jacobi polynomials with first varying parameter, O. Szehr, R. Zarouf. (2022) J. Approx. Theory. 277.

Applied Mathematics

Quantum Computing

My doctoral studies focused on "numerical aspects" of models for quantum memories and computation. To analyze error stability and convergence of computation models I took inspiration from RKHS techniques in contemporary numerical analysis and transferred them to the area of quantum computation. The following articles illustrate the application of functional calculi (see this work of N. Nikolski) and "analytic programs" (see this work of N. Young) in quantum computing.

• Spectral convergence bounds for classical and quantum Markov processes, O. Szehr, D. Reeb, M. Wolf. (2014) Comm. in Math. Phys.
• Perturbation Theory for Parent Hamiltonians of Matrix Product States, O. Szehr, M. Wolf. (2015) J. Statistic. Phys. 159(4).

Matrix condition numbers and numerical stability

As a byproduct I obtained the strongest spectral resolvent (a key concept in numerical analysis) estimates, improving previous estimates of B.Davies and B.Simon and also the respective estimates by N. Nikolski.

• Perturbation bounds for quantum Markov processes and their fixed points, O Szehr, M Wolf. (2013) J. Math. Phys. 54 (3)
• Eigenvalue estimates for the resolvent of a non-normal matrix, O Szehr. (2014) EMS J. Spectral Theory 4
• Condition numbers of matrices with given spectrum, S. Charpentier, K. Fouchet, O. Szehr, R. Zarouf. (2019) Anal. and Math. Phys. 9

Quantum Information Theory

My work contributed to the introduction of a highly successful entropy measure. My contribution has been to identify the relevant technical framework and to prove data processing inequalities for the new entropy together with F. Dupuis. The form of the new entropic quantity has been the work of M. Tomamichel.

• On quantum Rényi entropies: a new definition and some properties, M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel. (2013) J. Math. Phys. 54(3)

Applied Works

The tremendous success of deep reinforcement learning (deep RL) raised my research interest in learning systems and games. I am leading a research unit at IDSIA that focuses on reinforcement learning and games/ game theory. Currently we are mostly interested in the analysis and implementation of modern deep RL systems, particularly Deepmind's AlphaZero, AlphaStar, MuZero, etc. systems. Developed to tackle combinatorial games likes Chess and Go (AlphaZero) and real-time games like Starcraft (AlphaStar), those algorithms proved successful in a wide variety of planning problems.

It is a common interpretation (originating from a seminal book of G. Shafer and V. Vovk) to view Finance as a "game with the world". Following this interpretation we are particularly interested in the application of game-playing AI in finance.

Machine Learning in Finance

• Overfitting in Portfolio Optimization, M. Maggiolo, O. Szehr. (2023) J. of Risk Model Validation, 17(3).
• Hedging of Financial Derivative Contracts via Monte Carlo Tree Search, O. Szehr. (2023) J. Comput. Finance, 27(2).
• Hedging using reinforcement learning: contextual k-armed bandit versus Q-learning, L. Cannelli, G. Nuti, M. Sala, O. Szehr. (2023) J. Data Science and Finance.

Computer games and bots

• Deep Self-optimizing Artificial Intelligence for Tactical Analysis, Training and Optimization, M. Rüegsegger, M. Sommer, G. Del Rio, O. Szehr. (2021) NATO STO