Learning from Samples: Inverse Problems over measures via Sharpened Fenchel-Young Losses (2505.07124v1)

Abstract: Estimating parameters from samples of an optimal probability distribution is essential in applications ranging from socio-economic modeling to biological system analysis. In these settings, the probability distribution arises as the solution to an optimization problem that captures either static interactions among agents or the dynamic evolution of a system over time. Our approach relies on minimizing a new class of loss functions, called sharpened Fenchel-Young losses, which measure the sub-optimality gap of the optimization problem over the space of measures. We study the stability of this estimation method when only a finite number of sample is available. The parameters to be estimated typically correspond to a cost function in static problems and to a potential function in dynamic problems. To analyze stability, we introduce a general methodology that leverages the strong convexity of the loss function together with the sample complexity of the forward optimization problem. Our analysis emphasizes two specific settings in the context of optimal transport, where our method provides explicit stability guarantees: The first is inverse unbalanced optimal transport (iUOT) with entropic regularization, where the parameters to estimate are cost functions that govern transport computations; this method has applications such as link prediction in machine learning. The second is inverse gradient flow (iJKO), where the objective is to recover a potential function that drives the evolution of a probability distribution via the Jordan-Kinderlehrer-Otto (JKO) time-discretization scheme; this is particularly relevant for understanding cell population dynamics in single-cell genomics. Finally, we validate our approach through numerical experiments on Gaussian distributions, where closed-form solutions are available, to demonstrate the practical performance of our methods

Overview of "Learning from Samples: Inverse Problems over Measures via Sharpened Fenchel-Young Losses"

The paper entitled "Learning from Samples: Inverse Problems over Measures via Sharpened Fenchel-Young Losses" by Andrade, Peyre, and Poon provides a comprehensive exploration of inverse problems designed to estimate parameters from optimal probability distributions obtained through sample data. These aspects have wide-ranging applications in socio-economic modeling and biological system analysis, where understanding the driving mechanisms behind distributions is paramount. The study leverages a novel approach through sharpened Fenchel-Young losses, which quantifies the sub-optimality gap in optimization problems over probabilistic measures.

Methodological Insights

The authors introduce a refined class of loss functions, "sharpened Fenchel-Young losses," to solve inverse problems across measures. These functions draw from the Fenchel-Young inequality to frame a convex differentiable method for tackling inverse optimization tasks. The sharpened Fenchel-Young framework is leveraged for analyzing stability and sample complexity in estimating parameters under different settings and finite samples.

Two primary analytical contexts are elaborated upon:

1. Inverse Unbalanced Optimal Transport (iUOT): This concerns estimating cost functions that drive transport computations, with applications extending to link prediction in machine learning. Entropic regularization is incorporated, ensuring the stability and computational feasibility of unbalanced optimal transport problems.
2. Inverse Gradient Flow (iJKO): This pertains to recovering potential functions that influence system state evolution, particularly applicable to cell population dynamics in genomics. The Jordan-Kinderlehrer-Otto (JKO) discretization scheme is applied, highlighting the theoretical advancements in understanding cell trajectory from snapshots.

Results and Contributions

The authors make several theoretical contributions and validate their methods through numerical experiments:

• Stability Analysis: They introduce a general methodology focusing on the strong convexity of loss functions paired with sample complexity results of forward optimization problems to assess empirical estimation stability. A new sample complexity theorem quantifies transport plans in unbalanced settings without demanding smoothness assumptions.
• Optimization Algorithms: Customized algorithms are designed to efficiently address iUOT and iJKO, with the validation on Gaussian distributions demonstrating their effectiveness.
• Model Consistency: Through consistent estimation of the parameter vector $\theta$ for large samples, the research discusses the convergence of empirical Fenchel-Young losses to population versions, thus ensuring manifold identification in structured data contexts such as sparsity and low-rankness.

Practical and Theoretical Implications

Practically, these methodologies are critical for systems where flexibility in modeling unbalanced data is required. For instance, the elucidation of transport mechanisms in economic flows or the analysis of biological trajectories is facilitated by these inverse problem analyses.

Theoretical contributions lie in the sharp estimates and convex formulations afforded by using sharpened Fenchel-Young losses. These provide a robust framework applicable to diverse scientific domains reliant on high-dimensional data models and inverse estimation approaches, ultimately contributing to further advancements in probabilistic modeling and inference.

Future developments in artificial intelligence and data science would benefit from these robust modeling techniques, especially as computational constraints and data complexity continue to evolve. The foundational work laid down could inspire further explorations into generative modeling, offering crisp insights into learning mechanisms from sample data in stochastic environments.