Georgios Korpas

Johannes Aspman, Georgios Korpas, Jakub Marecek

There has been a great deal of recent interest in binarized neural networks, especially because of their explainability. At the same time, automatic differentiation algorithms such as backpropagation fail for binarized neural networks, which limits their applicability. By reformulating the problem of training binarized neural networks as a subadditive dual of a mixed-integer program, we show that binarized neural networks admit a tame representation. This, in turn, makes it possible to use the framework of Bolte et al. for implicit differentiation, which offers the possibility for practical implementation of backpropagation in the context of binarized neural networks. This approach could also be used for a broader class of mixed-integer programs, beyond the training of binarized neural networks, as encountered in symbolic approaches to AI and beyond.

Harsh Choudhary, Chandan Gupta, Vyacheslav Kungurtsev et al.

Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and in particular Hamiltonian Neural Networks have emerged as a mechanism to incorporate structural inductive bias into the NN model. By ensuring physical invariances are conserved, the models exhibit significantly better sample complexity and out-of-distribution accuracy than standard NNs. Learning the Hamiltonian as a function of its canonical variables, typically position and velocity, from sample observations of the system thus becomes a critical task in system identification and long-term prediction of system behavior. However, to truly preserve the long-run physical conservation properties of Hamiltonian systems, one must use symplectic integrators for a forward pass of the system's simulation. While symplectic schemes have been used in the literature, they are thus far limited to situations when they reduce to explicit algorithms, which include the case of separable Hamiltonians or augmented non-separable Hamiltonians. We extend it to generalized non-separable Hamiltonians, and noting the self-adjoint property of symplectic integrators, we bypass computationally intensive backpropagation through an ODE solver. We show that the method is robust to noise and provides a good approximation of the system Hamiltonian when the state variables are sampled from a noisy observation. In the numerical results, we show the performance of the method concerning Hamiltonian reconstruction and conservation, indicating its particular advantage for non-separable systems.

Martin Krutský, Gustav Šír, Vyacheslav Kungurtsev et al.

Physics-inspired graph neural networks (PI-GNNs) have been utilized as an efficient unsupervised framework for relaxing combinatorial optimization problems encoded through a specific graph structure and loss, reflecting dependencies between the problem's variables. While the framework has yielded promising results in various combinatorial problems, we show that the performance of PI-GNNs systematically plummets with an increasing density of the combinatorial problem graphs. Our analysis reveals an interesting phase transition in the PI-GNNs' training dynamics, associated with degenerate solutions for the denser problems, highlighting a discrepancy between the relaxed, real-valued model outputs and the binary-valued problem solutions. To address the discrepancy, we propose principled alternatives to the naive strategy used in PI-GNNs by building on insights from fuzzy logic and binarized neural networks. Our experiments demonstrate that the portfolio of proposed methods significantly improves the performance of PI-GNNs in increasingly dense settings.

Petr Ryšavý, Pavel Rytíř, Xiaoyu He et al.

In mixed graphs, there are both directed and undirected edges. An extension of acyclicity to this mixed-graph setting is known as maximally ancestral graphs. This extension is of considerable interest in causal learning in the presence of confounders. There, directed edges represent a clear direction of causality, while undirected edges represent confounding. We propose a score-based branch-and-cut algorithm for learning maximally ancestral graphs. The algorithm produces more accurate results than state-of-the-art methods, while being faster to run on small and medium-sized synthetic instances.

Jeremy Kulcsar, Vyacheslav Kungurtsev, Georgios Korpas et al.

In this work we investigate the potential of solving the discrete Optimal Transport (OT) problem with entropy regularization in a federated learning setting. Recall that the celebrated Sinkhorn algorithm transforms the classical OT linear program into strongly convex constrained optimization, facilitating first order methods for otherwise intractably large problems. A common contemporary setting that remains an open problem as far as the application of Sinkhorn is the presence of data spread across clients with distributed inter-communication, either due to clients whose privacy is a concern, or simply by necessity of processing and memory hardware limitations. In this work we investigate various natural procedures, which we refer to as Federated Sinkhorn, that handle distributed environments where data is partitioned across multiple clients. We formulate the problem as minimizing the transport cost with an entropy regularization term, subject to marginal constraints, where block components of the source and target distribution vectors are locally known to clients corresponding to each block. We consider both synchronous and asynchronous variants as well as all-to-all and server-client communication topology protocols. Each procedure allows clients to compute local operations on their data partition while periodically exchanging information with others. We provide theoretical guarantees on convergence for the different variants under different possible conditions. We empirically demonstrate the algorithms performance on synthetic datasets and a real-world financial risk assessment application. The investigation highlights the subtle tradeoffs associated with computation and communication time in different settings and how they depend on problem size and sparsity.

Pavel Rytir, Ales Wodecki, Georgios Korpas et al.

Causal learning from data has received much attention recently. Bayesian networks can be used to capture causal relationships. There, one recovers a weighted directed acyclic graph in which random variables are represented by vertices, and the weights associated with each edge represent the strengths of the causal relationships between them. This concept is extended to capture dynamic effects by introducing a dependency on past data, which may be captured by the structural equation model. This formalism is utilized in the present contribution to propose a score-based learning algorithm. A mixed-integer quadratic program is formulated and an algorithmic solution proposed, in which the pre-generation of exponentially many acyclicity constraints is avoided by utilizing the so-called branch-and-cut (``lazy constraint'') method. Comparing the novel approach to the state-of-the-art, we show that the proposed approach turns out to produce more accurate results when applied to small and medium-sized synthetic instances containing up to 80 time series. Lastly, two interesting applications in bioscience and finance, to which the method is directly applied, further stress the importance of developing highly accurate, globally convergent solvers that can handle instances of modest size.

Zakhar Popovych, Kurt Jacobs, Georgios Korpas et al.

Accurate models of the dynamics of quantum circuits are essential for optimizing and advancing quantum devices. Since first-principles models of environmental noise and dissipation in real quantum systems are often unavailable, deriving accurate models from measured time-series data is critical. However, identifying open quantum systems poses significant challenges: powerful methods from systems engineering can perform poorly beyond weak damping (as we show) because they fail to incorporate essential constraints required for quantum evolution (e.g., positivity). Common methods that can include these constraints are typically multi-step, fitting linear models to physically grounded master equations, often resulting in non-convex functions in which local optimization algorithms get stuck in local extrema (as we show). In this work, we solve these problems by formulating quantum system identification directly from data as a polynomial optimization problem, enabling the use of recently developed global optimization methods. These methods are essentially guaranteed to reach global optima, allowing us for the first time to efficiently obtain the most accurate Markovian model for a given system. In addition to its practical importance, this allows us to take the error of these Markovian models as an alternative (operational) measure of the non-Markovianity of a system. We test our method with the spin-boson model -- a two-level system coupled to a bath of harmonic oscillators -- for which we obtain the exact evolution using matrix-product-state techniques. We show that polynomial optimization using moment/sum-of-squares approaches significantly outperforms traditional optimization algorithms, and we show that even for strong damping Lindblad-form master equations can provide accurate models of the spin-boson system.