arXiv subjects

Jakub Mareček

Publications and source records attributed to Jakub Mareček.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Penalty-Barrier Method for Constrained Machine Learning

Constrained Machine Learning (CML) enables fairness-aware training, physics-informed neural networks, and integration of symbolic domain knowledge into statistical models. In this work, we introduce the Stochastic Penalty-Barrier Method (SPBM) for CML problems. SPBM extends classical penalty and barrier methods by incorporating an exponential averaging of the dual variables, a stabilized penalty schedule, and the Moreau envelope to handle non-smoothness. We analyze the bias that mini-batching introduces in the barrier function and show that the feasible set of the resulting transformed problem is contained within the original one. We compare SPBM with CML baselines across multiple fairness and physics informed neural networks experiments. We find that SPBM is competitive with state-of-the-art methods. We also observe, on our fairness-based computational benchmark, that the per-epoch runtime of CML methods is largely independent of the number of constraints, and within $1.3\times$ of the per-epoch runtime of regularized Adam, for a number of constraints ranging from $90$ to $9900$.

cs.LG↗

arXiv2026

Intersectional Fairness via Mixed-Integer Optimization

The deployment of Artificial Intelligence in high-risk domains, such as finance and healthcare, necessitates models that are both fair and transparent. While regulatory frameworks, including the EU's AI Act, mandate bias mitigation, they are deliberately vague about the definition of bias. In line with existing research, we argue that true fairness requires addressing bias at the intersections of protected groups. We propose a unified framework that leverages Mixed-Integer Optimization (MIO) to train intersectionally fair and intrinsically interpretable classifiers. We prove the equivalence of two measures of intersectional fairness (MSD and SPSF) in detecting the most unfair subgroup and empirically demonstrate that our MIO-based algorithm improves performance in finding bias. We train high-performing, interpretable classifiers that bound intersectional bias below an acceptable threshold, offering a robust solution for regulated industries and beyond.

cs.LG↗

arXiv2026

Penalised and constrained geodesics in geometric control theory

In many problems in optimal control, one seeks to minimise an objective function subject to constraints on the velocity of the system. Imposing these constraints directly -- the ``hard-constrained'' approach -- is often analytically and computationally challenging. A natural alternative is to penalise violations of the constraints, solving a sequence of ``soft-constrained'' problems indexed by a penalty parameter $q$, and hoping that solutions converge to solutions of the hard-constrained problem as $q \to \infty$. We show that this approach is justified when applied to a broad class of geometric control problems on a Riemannian manifold $(M,g)$. We first consider the case where there are no autonomous dynamics, and so the control problem reduces to the problem of finding a curve of minimal length or energy between two points, subject to a nonholonomic velocity constraint/penalty determined by the choice of a bracket-generating subbundle $D$ of $TM$. We show that any sequence of solutions to the soft-constrained problem has an accumulation point which is a solution to the hard-constrained problem. Subsequently, we show how to transform a broad class of optimal control problems to the problem of finding a geodesic, by trivialising the inherent dynamics of the system using a change of coordinates inspired by the interaction picture transformation in quantum mechanics.

math.OC↗

arXiv2026

ExMAG: Learning of Maximally Ancestral Graphs

In mixed graphs, there are both directed and bidirected 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 bidirected edges represent confounding. We propose a branch-and-cut algorithm for learning maximally ancestral graphs using a formulation as a mixed-integer quadratic program. Empirically, our method achieves comparable or improved reconstruction quality while requiring an order of magnitude fewer samples than state-of-the-art approaches.

cs.LG↗

arXiv2026

Deep Learning as the Disciplined Construction of Tame Objects

One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization theory, and deep learning theory and practice. To do so, we gradually introduce the concepts and tools used to build convergence guarantees for stochastic gradient descent in a general nonsmooth nonconvex, but tame, setting. This illustrates some ways in which tame geometry is a natural mathematical framework for the study of AI systems, especially within Deep Learning.

math.OC↗

arXiv2026

On Moment-Based Recovery of Measures with Atomic and Continuous Parts

Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one dimension. A typical strategy when recovering a measure from moments is to verify the flat-extension property, which certifies that the underlying measure is finitely atomic and ultimately leads to recovery. For many classes of measures, however, the flat extension never occurs and thus if one aims to recover the measure corresponding to the moments, assumptions need to be made. We formulate a new kind of recovery problem, where one assumes that the measure has compact support and a fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights, which apply to algorithms widely used for the recovery of atomic measures from moments. Furthermore, the statements proven lead to novel algorithms, which we benchmark, further confirming the theoretical findings.

math.OC↗

arXiv2026

Causal Learning in Biomedical Applications: Krebs Cycle as a Benchmark

Learning causal relationships from time series data is an important but challenging problem. Existing synthetic datasets often contain hidden artifacts that can be exploited by causal discovery methods, reducing their usefulness for benchmarking. We present a new benchmark dataset based on simulations of the Krebs cycle, a key biochemical pathway. The data are generated using a particle-based simulator that models molecular interactions in a controlled environment. Four distinct scenarios are provided, varying in time series length, number of samples, and intervention settings. The benchmark includes ground-truth causal graphs for evaluation. It supports quantitative comparisons using metrics such as Structural Hamming Distance, Structural Intervention Distance, and F1-score. A comprehensive evaluation of 14 causal discovery methods from different modelling paradigms is presented. Performance is compared across datasets using multiple accuracy and efficiency measures. The dataset provides a reproducible, interpretable, and non-trivial benchmark for testing causal learning methods on time series data. It avoids common pitfalls such as residual structural patterns and supports interventions and evaluation with known causal ground truth. This makes it a useful tool for the development and comparison of causal discovery algorithms.

cs.LG↗

arXiv2026

Tractable Probabilistic Models for Investment Planning

Investment planning in power utilities, such as generation and transmission expansion, requires decisions under substantial uncertainty over decade--long horizons for policies, demand, renewable availability, and outages, while maintaining reliability and computational tractability. Conventional approaches approximate uncertainty using finite scenario sets (modeled as a mixture of Diracs in statistical theory terms), which can become computationally intensive as scenario detail increases and provide limited probabilistic resolution for reliability assessment. We propose an alternative based on tractable probabilistic models, using sum--product networks (SPNs) to represent high--dimensional uncertainty in a compact, analytically tractable form that supports exact probabilistic queries (e.g., likelihoods, marginals, and conditionals). This framework enables the direct embedding of chance constraints into mixed--integer linear programming (MILP) models for investment planning to evaluate reliability events and enforce probabilistic feasibility requirements without enumerating large scenario trees. We demonstrate the approach on a representative planning case study and report reliability--cost trade--offs and computational behavior relative to standard scenario--based formulations.

cs.LG↗

arXiv2026

Geometric quantum control and the random Schrödinger equation

Understanding and mitigating noise in quantum systems is a fundamental challenge in achieving scalable and fault-tolerant quantum computation. Error modeling for quantum systems can be formulated in many ways, some of which are very fundamental, but hard to analyze (evolution by general dynamical map) and others perhaps too simplistic to represent physical reality. In this paper, we present an intermediate approach, introducing the random Schrödinger equation, with a noise term given by a time-varying random Hermitian matrix as a means to model noisy quantum systems. We derive bounds on the error of the synthesized unitary in terms of bounds on the norm of the noise, and show that for certain noise processes these bounds are tight. We then show that in certain situations, minimizing the error is equivalent to finding a geodesic on SU(n) with respect to a Riemannian metric encoding the coupling between the control pulse and the noise process, thus connecting our work to the complexity geometry pioneered by Michael Nielsen.

quant-ph↗

arXiv2026

Benchmarking Stochastic Approximation Algorithms for Fairness-Constrained Training of Deep Neural Networks

The ability to train Deep Neural Networks (DNNs) with constraints is instrumental in improving the fairness of modern machine-learning models. Many algorithms have been analysed in recent years, and yet there is no standard, widely accepted method for the constrained training of DNNs. In this paper, we provide a challenging benchmark of real-world large-scale fairness-constrained learning tasks, built on top of the US Census (Folktables). We point out the theoretical challenges of such tasks and review the main approaches in stochastic approximation algorithms. Finally, we demonstrate the use of the benchmark by implementing and comparing three recently proposed, but as-of-yet unimplemented, algorithms both in terms of optimization performance, and fairness improvement. We release the code of the benchmark as a Python package at https://github.com/humancompatible/train.

cs.LG↗

arXiv2025

Unitary Gate Synthesis via Polynomial Optimization

Quantum optimal control plays a crucial role in the development of quantum technologies, particularly in the design and implementation of fast and accurate gates for quantum computing. Here, we present a method to synthesize gates using the Magnus expansion. In particular, we formulate a polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential. The global method we use provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods. By optimizing over Hermitian matrices generating the unitaries, instead of the unitaries themselves, we can reduce the size of the polynomial to optimize, leading to fast convergence and scalability. Numerical experiments comparing our results with CRAB and GRAPE show that we maintain high accuracy while providing globality certificates.

quant-ph↗

arXiv2025

humancompatible.train: Implementing Optimization Algorithms for Stochastically-Constrained Stochastic Optimization Problems

There has been a considerable interest in constrained training of deep neural networks (DNNs) recently for applications such as fairness and safety. Several toolkits have been proposed for this task, yet there is still no industry standard. We present humancompatible.train (https://github.com/humancompatible/train), an easily-extendable PyTorch-based Python package for training DNNs with stochastic constraints. We implement multiple previously unimplemented algorithms for stochastically constrained stochastic optimization. We demonstrate the toolkit use by comparing two algorithms on a deep learning task with fairness constraints.

cs.LG↗

arXiv2025

Time evolution of controlled many-body quantum systems with matrix product operators

We present a method for describing the time evolution of many-body controlled quantum systems using matrix product operators (MPOs). Existing techniques for solving the time-dependent Schrödinger equation (TDSE) with an MPO Hamiltonian often rely on time discretization. In contrast, our approach uses the Magnus expansion and Chebyshev polynomials to model the time evolution, and the MPO representation to efficiently encode the system's dynamics. This results in a scalable method that can be used efficiently for many-body controlled quantum systems. We apply this technique to quantum optimal control, specifically for a gate synthesis problem, demonstrating that it can be used for large-scale optimization problems that are otherwise impractical to formulate in a dense matrix representation.

quant-ph↗

arXiv2025

Plausible Counterfactual Explanations of Recommendations

Explanations play a variety of roles in various recommender systems, from a legally mandated afterthought, through an integral element of user experience, to a key to persuasiveness. A natural and useful form of an explanation is the Counterfactual Explanation (CE). We present a method for generating highly plausible CEs in recommender systems and evaluate it both numerically and with a user study.

cs.LG↗

arXiv2025

Bias Detection via Maximum Subgroup Discrepancy

Bias evaluation is fundamental to trustworthy AI, both in terms of checking data quality and in terms of checking the outputs of AI systems. In testing data quality, for example, one may study the distance of a given dataset, viewed as a distribution, to a given ground-truth reference dataset. However, classical metrics, such as the Total Variation and the Wasserstein distances, are known to have high sample complexities and, therefore, may fail to provide a meaningful distinction in many practical scenarios. In this paper, we propose a new notion of distance, the Maximum Subgroup Discrepancy (MSD). In this metric, two distributions are close if, roughly, discrepancies are low for all feature subgroups. While the number of subgroups may be exponential, we show that the sample complexity is linear in the number of features, thus making it feasible for practical applications. Moreover, we provide a practical algorithm for evaluating the distance based on Mixed-integer optimization (MIO). We also note that the proposed distance is easily interpretable, thus providing clearer paths to fixing the biases once they have been identified. Finally, we describe a natural general bias detection framework, termed MSDD distances, and show that MSD aligns well with this framework. We empirically evaluate MSD by comparing it with other metrics and by demonstrating the above properties of MSD on real-world datasets.

cs.LG↗

arXiv2025

Undecidable problems associated with variational quantum algorithms

Variational Quantum Algorithms (VQAs), such as the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), are widely studied as candidates for near-term quantum advantage. Recent work has shown that training VQAs is NP-hard in general. In this paper, we present a conditional result suggesting that the training of VQAs is undecidable, even in idealized, noiseless settings. We reduce the decision version of the digitized VQA training problem-where circuit parameters are drawn from a discrete set-to the question of whether a universal Diophantine equation (UDE) has a root. This reduction relies on encoding the UDE into the structure of a variational quantum circuit via the matrix exponentials. The central step involves establishing a correspondence between the objective function of the VQA and a known UDE of 58 variables and degree 4. Our main result is conditional on a natural conjecture: that a certain system of structured complex polynomial equations-arising from the inner product of a VQA circuit output and a fixed observable-has at least one solution. We argue this conjecture is plausible based on dimension-counting arguments (degrees of freedom in the Hamiltonians, state vector, and observable), and the generic solvability of such systems in algebraic geometry over the complex numbers. Under this assumption, we suggest that deciding whether a digitized VQA achieves a given energy threshold is undecidable. This links the limitations of variational quantum algorithms to foundational questions in mathematics and logic, extending the known landscape of quantum computational hardness to include uncomputability. Additionally, we establish an unconditional undecidability result for VQA convergence in open quantum systems.

quant-ph↗

arXiv2025

Reserve Provision from Electric Vehicles: Aggregate Boundaries and Stochastic Model Predictive Control

Controlled charging of electric vehicles, EVs, is a major potential source of flexibility to facilitate the integration of variable renewable energy and reduce the need for stationary energy storage. To offer system services from EVs, fleet aggregators must address the uncertainty of individual driving and charging behaviour. This paper introduces a means of forecasting the service volume available from EVs by considering several EV batteries as one conceptual battery with aggregate power and energy boundaries. Aggregation avoids the difficult prediction of individual driving behaviour. The predictability of the boundaries is demonstrated using a multiple linear regression model which achieves a normalised root mean square error of 20% - 40% for a fleet of 1,000 EVs. A two-stage stochastic model predictive control algorithm is used to schedule reserve services on a day-ahead basis addressing risk trade-offs by including Conditional Value-at-Risk in the objective function. A case study with 1.2 million domestic EV charge records from Great Britain illustrates that increasing fleet size improves prediction accuracy, thereby increasing reserve revenues and decreasing an aggregator's operational costs. For fleet sizes of 400 or above, cost reductions plateau at 60% compared to uncontrolled charging, with an average of 1.8kW of reserve provided per vehicle.

eess.SY↗

arXiv2024

Topological quantum compilation of two-qubit gates

We investigate the topological quantum compilation of two-qubit operations within a system of Fibonacci anyons. Our primary goal is to generate gates that are approximately leakage-free and equivalent to the controlled-NOT (CNOT) gate up to single-qubit operations. These gates belong to the local equivalence class [CNOT]. Additionally, we explore which local equivalence classes of two-qubit operations can be naturally generated by braiding Fibonacci anyons. We discovered that most of the generated classes are located near the edges of the Weyl chamber representation of two-qubit gates, specifically between the local equivalence classes of the identity [1] and [CNOT], and between those of the double-controlled-NOT [DCNOT] and [SWAP]. Furthermore, we found a numerically exact implementation of a local equivalent of the SWAP gate using a sequence of only nine elements from the Fibonacci braiding gate set.

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