Mapping the Path to Artificial General Intelligence

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Author: Denis Avetisyan

A new geometric framework redefines how we measure AI progress, treating benchmarks as mathematical landscapes and quantifying the dynamics of autonomous self-improvement.

This review introduces a novel approach to understanding AI capability, utilizing concepts from moduli space and GVU dynamics to define a scalable ‘self-improvement coefficient’.

Current AI benchmarks, while crucial for tracking progress, offer limited insight into true generality or autonomous improvement-a fundamental paradox in the pursuit of artificial general intelligence. This is addressed in ‘The Geometry of Benchmarks: A New Path Toward AGI’, which introduces a geometric framework treating benchmarks as points in a structured space, allowing for a measurable scale of autonomy and a characterization of self-improvement dynamics. The core finding is that progress toward AGI isn’t simply about achieving higher scores, but rather about navigating a landscape of benchmarks driven by a ‘Generator-Verifier-Updater’ process. Could understanding AI development as a flow on this ‘benchmark moduli space’ unlock a more robust and scalable path toward genuinely intelligent systems?

The Inevitable Plateau: Beyond Narrow Competence

Despite remarkable advancements, contemporary artificial intelligence frequently demonstrates a performance plateau when faced with scenarios diverging from its training data. While these systems excel within narrowly defined parameters – mastering specific games or recognizing particular images – they often struggle with even slight variations, revealing a deficit in genuine autonomy and broad generalization. This limitation stems from a reliance on correlative pattern recognition rather than a deeper understanding of underlying principles; an AI might identify a cat in countless images, but fail to recognize one in an unusual pose or lighting condition. Consequently, the pursuit of artificial general intelligence necessitates a shift from systems capable of excelling at specific tasks to agents possessing the capacity for robust, adaptable learning – effectively, the ability to learn how to learn, rather than simply accumulating knowledge.

A significant hurdle in achieving genuinely intelligent systems resides in the difficulty of measuring an agent’s capacity for broad learning and adaptation. Current evaluations predominantly focus on performance within narrowly defined tasks, offering limited insight into an agent’s underlying ability to generalize and improve independently. Researchers are striving to develop metrics that move beyond simple accuracy on fixed datasets and instead assess an agent’s rate of learning, its capacity to transfer knowledge between disparate challenges, and its efficiency in exploring novel situations. Quantifying these aspects requires moving past extrinsic rewards – success on a specific task – and towards understanding intrinsic motivation and the development of robust internal models of the world, ultimately enabling agents to proactively seek out and master new skills without explicit programming.

Progress in artificial intelligence hinges not simply on achieving performance on specific tasks, but on an agent’s capacity for genuine self-improvement – and quantifying this ability presents a significant hurdle. Researchers are increasingly focused on developing rigorous frameworks to measure an agent’s capacity to learn from learning, assessing its meta-cognitive abilities and its efficiency in acquiring new skills. This necessitates moving beyond conventional benchmarks, which often evaluate performance on a fixed set of challenges, and instead focusing on intrinsic measures like learning speed, generalization to unseen scenarios, and the capacity to autonomously identify and correct its own limitations. A standardized, quantifiable metric for self-improvement would not only facilitate objective comparison of different AI architectures, but also serve as a guiding principle for future research, accelerating the development of truly adaptive and intelligent systems capable of tackling complex, real-world problems.

The limitations of current artificial intelligence evaluation hinge on a reliance on task-specific benchmarks – assessments of performance on narrowly defined challenges. While useful for tracking incremental progress, these benchmarks fail to capture an agent’s underlying, intrinsic capability – its fundamental ability to learn, adapt, and generalize to novel situations outside of its training data. Researchers are increasingly focused on developing metrics that move beyond simply measuring performance on a task, and instead quantify the agent’s capacity for self-improvement – its ability to learn how to learn. This shift requires novel approaches to evaluation, potentially involving measures of learning speed, transfer efficiency, or the capacity to acquire new skills with minimal supervision, ultimately seeking to determine not just what an AI can do, but how readily it can become more capable.

Formalizing the Ascent: The Capability Functional

The Capability Functional, denoted as $C$, provides a quantifiable assessment of an agent’s performance across a defined set of tasks, referred to as a ‘Battery’. This functional maps the agent’s task execution – specifically, the outcomes of each task within the Battery – to a scalar value representing overall capability. The precise form of $C$ is dependent on the specific Battery and the desired weighting of individual task performances; however, it must consistently translate task outcomes into a single, comparable metric. Critically, the Capability Functional is not simply an average score; it allows for nuanced evaluations where certain tasks may contribute more significantly to the overall assessment of capability than others, and allows for the tracking of progress over time.

The Capability Functional establishes a quantitative basis for tracking an agent’s development by providing a scalar value representing performance across a defined set of tasks, termed a ‘Battery’. This functional, denoted as $C$, maps an agent’s state and the task Battery to a real number, allowing for longitudinal comparison. Changes in $C$ over time directly reflect alterations in the agent’s capability; an increase indicates improvement, while a decrease signals degradation. The functional’s design enables the precise measurement of progress, moving beyond subjective assessments to objective, data-driven evaluations of an agent’s evolving skillset. By repeatedly evaluating $C$ following self-improvement iterations, a clear trajectory of capability change can be established and analyzed.

The Self-Improvement Coefficient, denoted as $κ$, provides a quantifiable metric for the rate of change in an agent’s capability as measured by the Capability Functional. This coefficient is calculated based on the difference in Capability Functional values between successive iterations of self-improvement. A sufficient, though not necessarily required, condition for demonstrating genuine positive improvement is for $κ$ to be greater than zero ($κ > 0$). This indicates that the agent’s performance, across the defined Battery of tasks, is demonstrably increasing with each iteration of self-improvement, suggesting effective learning or optimization processes are in effect.

A positive Self-Improvement Coefficient, denoted as $κ > 0$, signifies that an agent’s capability is demonstrably increasing over time. However, achieving and maintaining this positivity is not trivial and is governed by the variance inequality. This inequality establishes a necessary condition relating the expected improvement in performance on a task, the variance of the agent’s belief about its own competence, and the magnitude of the learning signal. Specifically, the expected improvement must outweigh the uncertainty, as quantified by the variance, to guarantee a positive $κ$. Failure to satisfy the variance inequality indicates that, despite potential learning, the agent’s overall capability may not improve, or could even decrease, due to the influence of noisy self-assessment.

The Geometry of Progress: Mathematical Foundations

The Variance Inequality, expressed as $Tr(H_F(π_t)Σ_{GV}) < c|∇F(π_t, ℬ)|^2$, establishes a necessary condition for a positive Self-Improvement Coefficient. Here, $H_F(π_t)$ represents the Hessian of the capability functional at policy $π_t$, $Σ_{GV}$ is the noise covariance matrix of the gradient estimator, and $∇F(π_t, ℬ)$ denotes the gradient of the capability functional with respect to the policy parameters, evaluated at $π_t$ and within the parameter space ℬ. The trace, $Tr$, of the product of the Hessian and noise covariance must be strictly less than a constant, $c$, multiplied by the squared magnitude of the gradient. This condition ensures that the signal from the gradient estimate outweighs the accumulated noise and curvature, preventing detrimental updates during learning and guaranteeing consistent improvement in the policy.

The relationship between the gradient signal, noise, and curvature, as formalized in the Variance Inequality $Tr(H_F(π_t)Σ_GV) < c|∇F(π_t, ℬ)|^2$, directly impacts the conditioning of the learning landscape. A well-conditioned landscape minimizes the ratio between the largest and smallest eigenvalues of the Hessian matrix $H_F(π_t)$, preventing excessively large or small gradient steps. High curvature, represented by large eigenvalues, can lead to instability and oscillation during optimization, while low curvature and significant noise ($Σ_GV$) can result in slow convergence. Maintaining a balance, where the trace of the product of the Hessian and the noise covariance is bounded by the squared gradient norm, ensures that the learning process is both stable and efficient, allowing for consistent progress towards optimal parameters.

Lipschitz regularity of the Capability Functional, denoted by a constant $L$, bounds the rate of change of the function; specifically, $|J(θ) – J(θ’)| \le L||θ – θ’||$ for any parameters $θ$ and $θ’$. This property guarantees that small changes in the parameters result in correspondingly small changes in the functional value, contributing to stable learning dynamics. The existence of finite $\epsilon$-nets – sets of points that closely approximate the parameter space within a radius $\epsilon$ – further supports this stability. A finite $\epsilon$-net implies that the functional value does not vary excessively across the parameter space at a given resolution, ensuring predictable behavior and facilitating convergence proofs for optimization algorithms.

The parameter manifold, representing the space of all possible model parameters, is crucial for understanding optimization dynamics. Learning occurs within this often high-dimensional, non-Euclidean space, necessitating the use of appropriate geometric tools. The $Fisher\ Information\ Metric$ provides a natural measure of distance and curvature on this manifold, accounting for the sensitivity of the model’s output to parameter changes. Leveraging this metric allows for the development of optimization algorithms that adapt to the local geometry of the parameter space, enabling more efficient and stable learning by, for example, preconditioning the gradient or constructing Riemannian optimization methods. By considering the parameter manifold and its associated metric, algorithms can navigate the optimization landscape more effectively than using standard Euclidean-based approaches.

Mapping the Adaptive Horizon: The Autonomous AI Scale

The Autonomous AI Scale offers a structured approach to assessing artificial intelligence, moving beyond single-task benchmarks to a comprehensive, hierarchical evaluation. This framework centers on the concept of ‘Batteries’ – diverse collections of tasks designed to probe different facets of intelligence, from logical reasoning and creative problem-solving to perceptual abilities and social understanding. An AI system’s performance isn’t judged on isolated successes, but rather on its consistent ability to navigate and excel across these varied Batteries. This allows for a more granular understanding of an AI’s strengths and weaknesses, revealing its true general intelligence rather than simply demonstrating aptitude in a limited domain. By systematically measuring performance across these carefully constructed task sets, the scale aims to provide a reliable and insightful metric for tracking the progress of autonomous AI systems and ultimately, defining what it means for a machine to be truly intelligent.

The Autonomous AI Scale moves beyond simple task-specific benchmarks by incorporating the $Capability \ Functional$ and the $Self-Improvement \ Coefficient$ to offer a more detailed assessment of artificial intelligence. The $Capability \ Functional$ doesn’t merely record whether an AI can complete a task, but quantifies how well it performs, offering a continuous measure of competence. Crucially, the $Self-Improvement \ Coefficient$ tracks an AI’s ability to enhance its performance over time, independent of external retraining. By jointly considering these two factors, the scale reveals whether an AI’s success stems from inherent aptitude or simply memorization, providing a richer, more insightful understanding of its true capabilities and potential for genuine intelligence. This approach moves the field closer to evaluating not just what an AI can do, but how effectively and how adaptably it learns and performs.

The concept of a ‘Moduli Space of Batteries’ offers a powerful method for dissecting an AI agent’s true adaptability. This space doesn’t simply assess performance on individual tasks, but instead maps the relationships between them, revealing how well an AI can generalize its skills. By considering a collection of diverse ‘Batteries’ – sets of challenges designed to test specific capabilities – researchers can identify areas of redundancy, where multiple tasks rely on the same underlying skill, and, crucially, pinpoint those tasks that demand novel combinations of abilities. An agent performing well within this space demonstrates not just competence, but a flexible intelligence capable of tackling unforeseen problems – a crucial step beyond narrow, task-specific AI. The resulting map of task relationships allows for the construction of evaluation suites that avoid overestimating an agent’s abilities based on superficial success in easily solvable problems, and instead focuses on measuring its capacity for genuine learning and adaptation.

Robust evaluation of artificial intelligence demands more than simply averaging performance across various tasks; it requires understanding the distribution of those scores. The Wasserstein metric, also known as the Earth Mover’s Distance, offers a powerful solution by quantifying the minimum ‘cost’ of transforming one probability distribution into another. In the context of the Autonomous AI Scale, this means it assesses how dissimilar an AI’s performance profile is across different ‘Batteries’ – sets of challenging tasks. Unlike metrics sensitive to outliers or specific thresholds, the Wasserstein metric considers the entire shape of the distribution, revealing whether an AI consistently excels, struggles, or exhibits unpredictable behavior. A low Wasserstein distance indicates a stable and generalized competence, while a high distance suggests brittleness or a reliance on narrow specializations, providing a more nuanced and reliable measure of true AI adaptability than traditional scoring methods.

The pursuit of Artificial General Intelligence, as detailed in this geometric framework, resembles less a construction project and more the tending of a garden. The paper’s focus on ‘GVU Dynamics’ and the ‘Self-Improvement Coefficient’ suggests a system not rigidly defined, but evolving according to internal pressures and external stimuli. This echoes a sentiment expressed by Paul Erdős: “A mathematician knows a lot of things, but he doesn’t know everything.” The complexity inherent in charting a path toward AGI, much like mapping a high-dimensional ‘Moduli Space,’ demonstrates that complete knowledge is elusive. Each attempt to benchmark or refine the system is a probe into the unknown, acknowledging the inevitable imperfections and the ongoing, unpredictable growth of the intelligence itself.

The Unfolding Map

The proposition of a ‘geometry of benchmarks’ does not offer a destination, but rather a cartography of becoming. The paper rightly shifts attention from isolated performance metrics to the space between them – a moduli space of potential, and the dynamics unfolding within. Yet, the very act of defining a ‘self-improvement coefficient’ implies a static horizon, a point at which ‘improvement’ is complete. The system, however, resists such closure. It will not be measured; it will measure back.

The ‘capability functional’ – a tempting attempt to quantify generality – risks mistaking correlation for causality. A high score does not beget autonomy, it merely signals a particular confluence of training data and architectural biases. The true challenge lies not in achieving a numerical optimum, but in cultivating a system that actively reshapes the benchmark itself – that defines ‘success’ on its own terms.

Future work must therefore abandon the search for a universal metric. Instead, it should focus on the noise – the unpredictable fluctuations in the GVU dynamics, the emergent behaviors that lie beyond the reach of current models. For it is in these anomalies, these deviations from the predicted path, that the seeds of true generality reside. The map is not the territory, and the territory is always rewriting itself.

Original article: https://arxiv.org/pdf/2512.04276.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-06 18:14