This work examines intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective and describes decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior.
Abstract
Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlying data-generating process, which are formalized in terms of informational bounds. In this work we examine intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective. We analyze minimal achievable error in classification through Fano-type bounds and precision limits in parametric estimation via the Cram\'er-Rao inequality, emphasizing that such limits depend on the underlying model rather than on algorithmic sophistication alone. We further discuss how implicit assumptions, such as independence, ergodicity, and distributional stability, affect the validity of inferential procedures. Building on interaction-based modeling principles, we review typical frameworks such as Markov Random Fields and potential based representations for encoding dependence mechanisms. We also describe decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior. This perspective highlights the importance of having adequate models for the data as a prerequi- site for expanding predictive capability, and situates algorithmic learning within the informational limits imposed by the models.
Evidence that influential samples tend to lie close to the decision boundary, thereby making contact with a standard data selection heuristic in active learning is provided.
Recent proliferation of data-optimization integration has led to a range of methods that aim to improve the statistical performance of data-driven optimization decisions. However, while many of these methods are motivated intuitively from a robustness or regularization perspective, their resulting statistical benefits are often unclear and, even if available, are established on a case-by-case basis. We provide a systematic dissection of data-driven optimization formulations using the view of"directionally perturbed"empirical optimization (EO). Specifically, this umbrella of formulations, which we call"EO+", covers many existing data-driven optimization methods, including regularization, distributionally robust optimization, transfer learning, and analogous methods for contextual optimization. On the one hand, we argue that without additional, correctly specified, side information, any EO+ method can result in at most second-order improvements. This provides a negative conclusion, namely ``no free lunch is possible", on the statistical power of EO+. On the other hand, we show that when leveraging side information that is geometrically effective, achieving first-order improvements is possible by choosing hyperparameters that are significantly larger than what is typically suggested in the literature. Moreover, we construct a principled methodology based on excess risk estimation, via either system knowledge or bootstrap resampling, to maximize the first-order gain. We demonstrate how this gain connects to the control-variate principle, a variance reduction technique in the Monte Carlo simulation literature, which helps explain why geometrically effective side information is necessary.
Substantial research efforts have been devoted to the design of data-driven controllers; however, comparatively less is known about their statistical performance and fundamental limitations. This contribution develops a statistical decision framework for data-driven control, in which a controller is evaluated by its risk, defined as the expected performance degradation relative to the oracle model-based controller, and by its average risk over the parameter space. Within this framework, we propose a collection of design principles for data-driven controllers. We further derive lower bounds on risks by combining the bias-variance decomposition with the Cram\'er-Rao inequality. In particular, the optimal bias that attains the lower bound for the average risk is determined by calculus of variations, thereby making the bias-variance tradeoff in data-driven control explicit. Moreover, the derived bound reveals a ``waterbed''effect in data-driven control: any improvement in risk relative to the lower bound over one region of the parameter space must be compensated by deterioration elsewhere. We illustrate the proposed framework on two canonical data-driven control problems: optimal feedforward control and the linear quadratic regulator benchmark. By comparing several representative data-driven controllers with the derived lower bounds, we sharpen the statistical interpretation of existing methods and reveal quantitative limitations that no controller design can avoid.
Jiabao He, Feiran Zhao, Yushan Li et al.· 0 citations
In this paper, we study contextual stochastic optimization (CSO), where decisions are made under uncertainty and the distribution of random parameters can be partially inferred from covariates observed prior to decision-making. In many practical settings, these distributions also depend on the decisions themselves, a phenomenon known as the
decision-dependent effect
. Most existing studies address this issue by imposing structural assumptions on the relationship between decisions and the underlying distributions. However, such assumptions may lead to model misspecification when the true relationship deviates from the assumed form. A prominent alternative is the weighted sample average approximation (wSAA) method proposed by Bertsimas and Kallus (2019), which adapts sample weights based on their similarity to the current decision–context pair. Nevertheless, because these weights are typically computed using complex machine learning models and depend on the decision variables in decision-dependent settings, solving the resulting optimization problem becomes computationally challenging. To overcome this challenge, we extend the wSAA framework from the loss function to its gradient, leading to the notion of the
contextual gradient
. We show that the contextual gradient serves as a meaningful indicator of optimality and leverage this property to develop the
contextual gradient descent (CGD)
algorithm. Our analysis establishes that CGD converges to a neighborhood of the global optimum when the loss function exhibits sufficient strong convexity. Moreover, the derived bounds reveal a key insight: the strength of convexity in the loss function can compensate for the uncertainty introduced by decision-dependent effects. Extensive numerical experiments on both synthetic and real-world datasets demonstrate that CGD consistently outperforms existing methods for contextual optimization under decision-dependent uncertainty.
Wenxuan Liu, Xiangting Liu, Maoqi Liu et al.· Production and operations ma...· 0 citations
We develop a filtering and optimal-control framework for partially observable stochastic systems in which each observation identifies a class of a finite measurable partition of the hidden state space. This structure covers regional-information mechanisms associated with threshold, quantized, censored, event-triggered, and intermittent observations, and allows observable classes with atomic, continuous, or mixed components. The formulation is constructed first at the level of measures: for each observable class, we define a class-restricted unnormalized conditional measure, and the posterior distribution is obtained by normalizing with its predictive probability. Based on this recursion, we introduce an information state consisting of the observed class and the conditional measure supported on it, thereby transforming the original problem into a fully observable Markov decision process. We formulate the discounted-cost criterion, derive the Bellman equation, and establish conditions for the existence of stationary optimal policies. To address the infinite-dimensional nature of the information space, we propose class-dependent finite-dimensional approximations capable of preserving both continuous components and atomic masses. We also derive an abstract bound linking the error of the approximate filter to the error of the value function. A reference model illustrates the construction through histogram-based approximations
Saul Díaz-Infante Velasco, Yofre H. García, J. Minjárez‐Sosa· 0 citations
Discounted exponential utility provides a principled criterion for risk-sensitive sequential decision-making, but its nonlinear structure complicates reinforcement learning. A recent work \citep{thoppe2026reinforcement} addressed this difficulty by introducing a Bellman-compatible surrogate and two model-free fixed-point algorithms for optimizing it over stationary policies. However, their main convergence results are asymptotic. In this work, we establish finite-time rates of $\tilde{O} (1/\sqrt{n})$ for the aforementioned two algorithms under asynchronous Markovian sampling, where $n$ is the iteration index and $\tilde{O}$ hides logarithmic expressions. Importantly, we employ parameter-free choices for the stepsize parameter to derive these rate results. For the algorithmically simpler one-timescale method, the main challenge is that its update equation is not directly aligned with the contraction geometry of its underlying power-law operator. We overcome this mismatch by exploiting the boundedness, monotonicity, and homogeneity of the operator to obtain a local pseudo-contraction property for the relative-error dynamics. We then use a Moreau-envelope-based Lyapunov function and Polyak--Ruppert averaging to obtain the stated convergence rate with parameter-free stepsizes. For the two-timescale method, the main challenge is to control a tracking error on the faster timescale. These results provide the first finite-time guarantees for model-free discounted exponential-utility reinforcement learning.
Ankur Naskar, A. VivekT, Aditya Kumar et al.· 0 citations