arXiv subjects

Jakub Marecek

Publications and source records attributed to Jakub Marecek.

At least 37 records · Page 2Linked to original sources

arXiv2025

A Feedback Control Framework for Incentivised Suburban Parking Utilisation and Urban Core Traffic Relief

Urban traffic congestion, exacerbated by inefficient parking management and cruising for parking, significantly hampers mobility and sustainability in smart cities. Drivers often face delays searching for parking spaces, influenced by factors such as accessibility, cost, distance, and available services such as charging facilities in the case of electric vehicles. These inefficiencies contribute to increased urban congestion, fuel consumption, and environmental impact. Addressing these challenges, this paper proposes a feedback control incentivisation-based system that aims to better distribute vehicles between city and suburban parking facilities offering park-and-charge/-ride services. Individual driver behaviours are captured via discrete choice models incorporating factors of importance to parking location choice among drivers, such as distance to work, public transport connectivity, charging infrastructure availability, and amount of incentive offered; and are regulated through principles of ergodic control theory. The proposed framework is applied to an electric vehicle park-and-charge/-ride problem, and demonstrates how predictable long-term behaviour of the system can be guaranteed.

eess.SY↗

arXiv2025

Time-Varying Multi-Objective Optimization: Tradeoff Regret Bounds

Multi-objective optimization studies the process of seeking multiple competing desiderata in some operation. Solution techniques highlight marginal tradeoffs associated with weighing one objective over others. In this paper, we consider time-varying multi-objective optimization, in which the objectives are parametrized by a continuously varying parameter and a prescribed computational budget is available at each time instant to algorithmically adjust the decision variables to accommodate for the changes. We prove regret bounds indicating the relative guarantees on performance for the competing objectives.

math.OC↗

arXiv2025

Truss topology design under harmonic loads: Peak power minimization with semidefinite programming

Designing lightweight yet stiff structures that can withstand vibrations is a crucial task in structural optimization. Here, we present a novel framework for truss topology optimization under undamped harmonic oscillations. Our approach minimizes the peak power of the structure under harmonic loads, overcoming the limitations of single-frequency and in-phase assumptions found in previous methods. For this, we leverage the concept of semidefinite representable (SDr) functions, demonstrating that while compliance readily conforms to an SDr representation, peak power requires a derivation based on the non-negativity of trigonometric functions. Finally, we introduce convex relaxations for the minimization problem and provide promising computational results.

math.OC↗

arXiv2025

Sample Complexity of Bias Detection with Subsampled Point-to-Subspace Distances

Sample complexity of bias estimation is a lower bound on the runtime of any bias detection method. Many regulatory frameworks require the bias to be tested for all subgroups, whose number grows exponentially with the number of protected attributes. Unless one wishes to run a bias detection with a doubly-exponential run-time, one should like to have polynomial complexity of bias detection for a single subgroup. At the same time, the reference data may be based on surveys, and thus come with non-trivial uncertainty. Here, we reformulate bias detection as a point-to-subspace problem on the space of measures and show that, for supremum norm, it can be subsampled efficiently. In particular, our probabilistically approximately correct (PAC) results are corroborated by tests on well-known instances.

cs.LG↗

arXiv2025

Power System Steady-State Estimation Revisited

In power system steady-state estimation (PSSE), one needs to consider (1) the need for robust statistics, (2) the nonconvex transmission constraints, (3) the fast-varying nature of the inputs, and the corresponding need to track optimal trajectories as closely as possible. In combination, these challenges have not been considered, yet. In this paper, we address all three challenges. The need for robustness (1) is addressed by using an approach based on the so-called Huber model. The non-convexity (2) of the problem, which results in first order methods failing to find global minima, is dealt with by applying global methods. One of these methods is based on a mixed integer quadratic formulation, which provides results of several orders of magnitude better than conventional gradient descent. Lastly, the trajectory tracking (3) is discussed by showing under which conditions the trajectory tracking of the SDP relaxations has meaning.

math.OC↗

arXiv2024

Quantum open system identification via global optimization: Optimally accurate Markovian models of open systems from time-series data

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.

quant-ph↗

arXiv2024

Parallel variational quantum algorithms with gradient-informed restart to speed up optimisation in the presence of barren plateaus

Inspired by the Fleming-Viot stochastic process, we propose a parallel implementation of variational quantum algorithms with the aim of reducing the time spent by the algorithm in barren plateaus, where optimization direction is unclear. In the Fleming-Viot tradition, parallel searches are called particles. In the proposed approach, the search by a Fleming-Viot particle is stopped when it encounters a region where the gradient is too small or noisy, suggesting a barren plateau area. The stopped particle continues the search after being regenerated at another location of the parameter space, potentially taking the exploration away from barren plateaus. We first analyze the behavior of the Fleming-Viot particles from a theoretical standpoint. We show that, when simulated annealing optimizers are used as particles, the Fleming-Viot system is expected to find the global optimum faster than a single simulated annealing optimizer, with a relative efficiency that increases proportionally to the percentage of barren plateaus in the domain. This result is corroborated by numerical experiments carried out on synthetic problems as well as on instances of the Max-Cut problem, which show that our method performs better than plain simulated annealing when large barren plateaus are present in the domain.

quant-ph↗

arXiv2024

Challenges and Opportunities in Quantum Optimization

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

quant-ph↗

arXiv2024

Group-blind optimal transport to group parity and its constrained variants

Fairness holds a pivotal role in the realm of machine learning, particularly when it comes to addressing groups categorised by protected attributes, e.g., gender, race. Prevailing algorithms in fair learning predominantly hinge on accessibility or estimations of these protected attributes, at least in the training process. We design a single group-blind projection map that aligns the feature distributions of both groups in the source data, achieving (demographic) group parity, without requiring values of the protected attribute for individual samples in the computation of the map, as well as its use. Instead, our approach utilises the feature distributions of the privileged and unprivileged groups in a boarder population and the essential assumption that the source data are unbiased representation of the population. We present numerical results on synthetic data and real data.

cs.LG↗

arXiv2024

Iteration Complexity of Variational Quantum Algorithms

There has been much recent interest in near-term applications of quantum computers, i.e., using quantum circuits that have short decoherence times due to hardware limitations. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. An enormous breadth of algorithms in this framework have been proposed for solving a range of problems in machine learning, forecasting, applied physics, and combinatorial optimization, among others. In this paper, we analyze the iteration complexity of VQA, that is, the number of steps that VQA requires until its iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, noise makes the evaluations of the objective function via quantum circuits biased. Commonly used optimization procedures, such as SPSA and the parameter shift rule, can thus be seen as derivative-free optimization algorithms with biased function evaluations, for which there are currently no iteration complexity guarantees in the literature. We derive the missing guarantees and find that the rate of convergence is unaffected. However, the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity, i.e., the more bias, the farther one is guaranteed, at best, to reach a stationary point of the VQA objective.

quant-ph↗

arXiv2024

Improving the Validity of Decision Trees as Explanations

In classification and forecasting with tabular data, one often utilizes tree-based models. Those can be competitive with deep neural networks on tabular data and, under some conditions, explainable. The explainability depends on the depth of the tree and the accuracy in each leaf of the tree. We point out that decision trees containing leaves with unbalanced accuracy can provide misleading explanations. Low-accuracy leaves give less valid explanations, which could be interpreted as unfairness among subgroups utilizing these explanations. Here, we train a shallow tree with the objective of minimizing the maximum misclassification error across all leaf nodes. The shallow tree provides a global explanation, while the overall statistical performance of the shallow tree can become comparable to state-of-the-art methods (e.g., well-tuned XGBoost) by extending the leaves with further models.

cs.LG↗

arXiv2024

Piecewise Polynomial Regression of Tame Functions via Integer Programming

Tame functions are a class of nonsmooth, nonconvex functions, which feature in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer programs, or wave functions of small molecules. We consider approximating tame functions with piecewise polynomial functions. We bound the quality of approximation of a tame function by a piecewise polynomial function with a given number of segments on any full-dimensional cube. We also present the first mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.

math.OC↗

arXiv2024

Stochastic Langevin Differential Inclusions with Applications to Machine Learning

Stochastic differential equations of Langevin-diffusion form have received significant attention, thanks to their foundational role in both Bayesian sampling algorithms and optimization in machine learning. In the latter, they serve as a conceptual model of the stochastic gradient flow in training over-parameterized models. However, the literature typically assumes smoothness of the potential, whose gradient is the drift term. Nevertheless, there are many problems for which the potential function is not continuously differentiable, and hence the drift is not Lipschitz continuous everywhere. This is exemplified by robust losses and Rectified Linear Units in regression problems. In this paper, we show some foundational results regarding the flow and asymptotic properties of Langevin-type Stochastic Differential Inclusions under assumptions appropriate to the machine-learning settings. In particular, we show strong existence of the solution, as well as an asymptotic minimization of the canonical free-energy functional.

math.OC↗

arXiv2024

Fairness in AI: challenges in bridging the gap between algorithms and law

In this paper we examine algorithmic fairness from the perspective of law aiming to identify best practices and strategies for the specification and adoption of fairness definitions and algorithms in real-world systems and use cases. We start by providing a brief introduction of current anti-discrimination law in the European Union and the United States and discussing the concepts of bias and fairness from an legal and ethical viewpoint. We then proceed by presenting a set of algorithmic fairness definitions by example, aiming to communicate their objectives to non-technical audiences. Then, we introduce a set of core criteria that need to be taken into account when selecting a specific fairness definition for real-world use case applications. Finally, we enumerate a set of key considerations and best practices for the design and employment of fairness methods on real-world AI applications

cs.CY↗

arXiv2024

Robust Quantum Gate Complexity: Foundations

Optimal control of closed quantum systems is a well studied geometrically elegant set of computational theory and techniques that have proven pivotal in the implementation and understanding of quantum computers. The design of a circuit itself corresponds to an optimal control problem of choosing the appropriate set of gates (which appear as control operands) in order to steer a qubit from an initial, easily prepared state, to one that is informative to the user in some sense, for e.g., an oracle whose evaluation is part of the circuit. However, contemporary devices are known to be noisy, and it is not certain that a circuit will behave as intended. Yet, although the computational tools exist in broader optimal control theory, robustness of adequate operation of a quantum control system with respect to uncertainty and errors has not yet been broadly studied in the literature. In this paper, we propose a new approach inspired by the closed quantum optimal control and its connection to geometric interpretations. To this end, we present the appropriate problem definitions of robustness in the context of quantum control, focusing on its broader implications for gate complexity.

quant-ph↗

arXiv2024

Fairness in Ranking: Robustness through Randomization without the Protected Attribute

There has been great interest in fairness in machine learning, especially in relation to classification problems. In ranking-related problems, such as in online advertising, recommender systems, and HR automation, much work on fairness remains to be done. Two complications arise: first, the protected attribute may not be available in many applications. Second, there are multiple measures of fairness of rankings, and optimization-based methods utilizing a single measure of fairness of rankings may produce rankings that are unfair with respect to other measures. In this work, we propose a randomized method for post-processing rankings, which do not require the availability of the protected attribute. In an extensive numerical study, we show the robustness of our methods with respect to P-Fairness and effectiveness with respect to Normalized Discounted Cumulative Gain (NDCG) from the baseline ranking, improving on previously proposed methods.

cs.LG↗

arXiv2024

Spectral Methods for Quantum Optimal Control: Artificial Boundary Conditions

The problem of quantum state preparation is one of the main challenges in achieving the quantum advantage. Furthermore, classically, for multi-level problems, our ability to solve the corresponding quantum optimal control problems is rather limited. The ability of the latter to feed into the former may result in significant progress in quantum computing. To address this challenge, we propose a formulation of quantum optimal control that makes use of artificial boundary conditions for the Schrödinger equation in combination with spectral methods. The resulting formulations are well suited for investigating periodic potentials and lend themselves to direct numerical treatment using conventional methods for bounded domains.

quant-ph↗

arXiv2024

Learning of Linear Dynamical Systems as a Non-Commutative Polynomial Optimization Problem

There has been much recent progress in forecasting the next observation of a linear dynamical system (LDS), which is known as the improper learning, as well as in the estimation of its system matrices, which is known as the proper learning of LDS. We present an approach to proper learning of LDS, which in spite of the non-convexity of the problem, guarantees global convergence of numerical solutions to a least-squares estimator. We present promising computational results.

math.OC↗