Ilya Levin

Ilya Levin, Maksim Shuklin, Eric Moulines et al.

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.

Ilya Levin

This paper examines the role of threshold logic in understanding generative artificial intelligence. Threshold functions, originally studied in the 1960s in digital circuit synthesis, provide a structurally transparent model of neural computation: a weighted sum of inputs compared to a threshold, geometrically realized as a hyperplane partitioning a space. The paper shows that this operation undergoes a qualitative transition as dimensionality increases. In low dimensions, the perceptron acts as a determinate logical classifier, separating classes when possible, as decided by linear programming. In high dimensions, however, a single hyperplane can separate almost any configuration of points (Cover, 1965); the space becomes saturated with potential classifiers, and the perceptron shifts from a logical device to a navigational one, functioning as an indexical indicator in the sense of Peirce. The limitations of the perceptron identified by Minsky and Papert (1969) were historically addressed by introducing multilayer architectures. This paper considers an alternative path: increasing dimensionality while retaining a single threshold element. It argues that this shift has equally significant implications for understanding neural computation. The role of depth is reinterpreted as a mechanism for the sequential deformation of data manifolds through iterated threshold operations, preparing them for linear separability already afforded by high-dimensional geometry. The resulting triadic account - threshold function as ontological unit, dimensionality as enabling condition, and depth as preparatory mechanism - provides a unified perspective on generative AI grounded in established mathematics.

Ilya Levin

Generative AI systems produce meaning with a quality indistinguishable from - and occasionally surpassing - human performance, yet the epistemic mechanism through which this occurs remains poorly understood. This paper argues that generative AI instantiates a fundamentally new mode of knowledge production: geometric navigation through high-dimensional manifolds, grounded in indexical rather than symbolic signification. Drawing on the structural properties of high-dimensional spaces, we demonstrate that meaning in generative AI is constituted through positional relation and orientation rather than through symbolic convention. This shift corresponds precisely to what Peirce identified as indexical signification: a mode of meaning in which the sign is constituted by its real causal connection to its object, not by arbitrary assignment. We develop the pedagogical implications of this shift through a geometrized reading of Papert's constructionism, reconceptualizing the generative AI system as a new kind of microworld - high-dimensional, non-visualizable, and indexical - in which knowledge is constructed through navigation rather than symbolic programming. From this analysis, we derive the concept of Navigational Thinking: a mode of knowing characterized by positional, enactive, and bounded engagement with geometrically structured spaces. We argue that Navigational Thinking and Computational Thinking are not alternatives, but two sequential phases of the same cognitive process: while a problem remains indexical, Navigational Thinking is operative; when the problem space stabilizes into symbolizable form, Computational Thinking becomes applicable. Vibe-coding is merely the visible tip of an iceberg - the iceberg being a new cognitive ecology in which these two modes coexist as the necessary phases of problem-solving in the age of generative AI.

Ilya Levin

The encounter between human reasoning and generative artificial intelligence (GenAI) cannot be adequately described by inherited metaphors of tool use, augmentation, or collaborative partnership. This article argues that such interactions produce a qualitatively distinct cognitive-epistemic formation, designated here as the Third Entity: an emergent, transient structure that arises from the transductive coupling of two ontologically incommensurable modes of cognition. Drawing on Peirce semiotics, Polanyi theory of tacit knowledge, Simondon philosophy of individuation, Ihde postphenomenology, and Morin complexity theory, we develop a multi-layered theoretical account of this formation. We introduce the concept of vibe-creation to designate the pre-reflective cognitive mode through which the Third Entity navigates high-dimensional semantic space and argue that this mode constitutes the automation of tacit knowledge - a development with far-reaching consequences for epistemology, the philosophy of mind, and educational theory. We further propose the notion of asymmetric emergence to characterize the agency of the Third Entity: genuinely novel and irreducible, yet anchored in human intentional responsibility. The article concludes by examining the implications of this theoretical framework for the transformation of educational institutions and the redefinition of intellectual competence in the age of GenAI.

Ilya Levin, Alexei L. Semenov, Mikael Gorsky

This article explores the evolution of constructionism as an educational framework, tracing its relevance and transformation across three pivotal eras: the advent of personal computing, the networked society, and the current era of generative AI. Rooted in Seymour Papert constructionist philosophy, this study examines how constructionist principles align with the expanding role of digital technology in personal and collective learning. We discuss the transformation of educational environments from hierarchical instructionism to constructionist models that emphasize learner autonomy and interactive, creative engagement. Central to this analysis is the concept of an expanded personality, wherein digital tools and AI integration fundamentally reshape individual self-perception and social interactions. By integrating constructionism into the paradigm of smart education, we propose it as a foundational approach to personalized and democratized learning. Our findings underscore constructionism enduring relevance in navigating the complexities of technology-driven education, providing insights for educators and policymakers seeking to harness digital innovations to foster adaptive, student-centered learning experiences.

Paul Mangold, Sergey Samsonov, Safwan Labbi et al.

In this paper, we analyze the sample and communication complexity of the federated linear stochastic approximation (FedLSA) algorithm. We explicitly quantify the effects of local training with agent heterogeneity. We show that the communication complexity of FedLSA scales polynomially with the inverse of the desired accuracy $ε$. To overcome this, we propose SCAFFLSA a new variant of FedLSA that uses control variates to correct for client drift, and establish its sample and communication complexities. We show that for statistically heterogeneous agents, its communication complexity scales logarithmically with the desired accuracy, similar to Scaffnew. An important finding is that, compared to the existing results for Scaffnew, the sample complexity scales with the inverse of the number of agents, a property referred to as linear speed-up. Achieving this linear speed-up requires completely new theoretical arguments. We apply the proposed method to federated temporal difference learning with linear function approximation and analyze the corresponding complexity improvements.

Ilya Levin

Generative AI presents an unprecedented challenge to our understanding of knowledge and its production. Unlike previous technological transformations, where engineering understanding preceded or accompanied deployment, generative AI operates through mechanisms whose epistemic character remains obscure, and without such understanding, its responsible integration into science, education, and institutional life cannot proceed on a principled basis. This paper argues that the missing account must begin with a paradigmatic break that has not yet received adequate philosophical attention. In the Turing-Shannon-von Neumann tradition, information enters the machine as encoded binary vectors, and semantics remains external to the process. Neural network architectures rupture this regime: symbolic input is instantly projected into a high-dimensional space where coordinates correspond to semantic parameters, transforming binary code into a position in a geometric space of meanings. It is this space that constitutes the active epistemic condition shaping generative production. Drawing on four structural properties of high-dimensional geometry concentration of measure, near-orthogonality, exponential directional capacity, and manifold regularity the paper develops an Indexical Epistemology of High-Dimensional Spaces. Building on Peirce semiotics and Papert constructionism, it reconceptualizes generative models as navigators of learned manifolds and proposes navigational knowledge as a third mode of knowledge production, distinct from both symbolic reasoning and statistical recombination.

Ilya Levin

The emergence of generative artificial intelligence (GenAI) represents not an incremental technological advance but a qualitative epistemological shift that challenges foundational assumptions of computer science. Whereas machine learning has been described as the automation of automation, generative AI operates by navigating contextual, semantic, and stylistic coherence rather than optimizing predefined objective metrics. This paper introduces the concept of Vibe-Automation to characterize this transition. The central claim is that the significance of GenAI lies in its functional access to operationalized tacit regularities: context-sensitive patterns embedded in practice that cannot be fully specified through explicit algorithmic rules. Although generative systems do not possess tacit knowledge in a phenomenological sense, they operationalize sensitivities to tone, intent, and situated judgment encoded in high-dimensional latent representations. On this basis, the human role shifts from algorithmic problem specification toward Vibe-Engineering, understood as the orchestration of alignment and contextual judgment in generative systems. The paper connects this epistemological shift to educational and institutional transformation by proposing a conceptual framework structured across three analytical levels and three domains of action: faculty worldview, industry relations, and curriculum design. The risks of mode collapse and cultural homogenization are briefly discussed, emphasizing the need for deliberate engagement with generative systems to avoid regression toward synthetic uniformity.

Mikael Gorsky, Ilya Levin

Artificial intelligence (AI) is rapidly reshaping educational practice, challenging long held assumptions about teaching and learning. This article integrates conceptual perspectives from recent books (Genesis by Eric Schmidt, Henry Kissinger and Craig Mundie, CoIntelligence by Ethan Mollick, and The Inevitable by Kevin Kelly) with empirical insights from popular AI podcasts and Anthropic public releases. We examine seven key domains: emotional support, creativity, contextual understanding, student engagement, problem solving, ethics and morality, and collaboration. For each domain, we explore AI capabilities, opportunities for transformative change, and emerging best practices, drawing equally from theoretical analysis and real world observations. Overall, we find that AI, when used thoughtfully, can complement and enhance human educators in fostering richer learning experiences across cognitive, social, and emotional dimensions. We emphasize an optimistic yet responsible outlook: educators and students should actively shape AI integration to amplify human potential in creativity, ethical reasoning, collaboration, and beyond, while maintaining a focus on human centric values.

Ilya Levin, Alexey Naumov, Sergey Samsonov

In this paper, we study the bias and high-order error bounds of the Linear Stochastic Approximation (LSA) algorithm with Polyak-Ruppert (PR) averaging under Markovian noise. We focus on the version of the algorithm with constant step size $α$ and propose a novel decomposition of the bias via a linearization technique. We analyze the structure of the bias and show that the leading-order term is linear in $α$ and cannot be eliminated by PR averaging. To address this, we apply the Richardson-Romberg (RR) extrapolation procedure, which effectively cancels the leading bias term. We derive high-order moment bounds for the RR iterates and show that the leading error term aligns with the asymptotically optimal covariance matrix of the vanilla averaged LSA iterates.

Denis Belomestny, Ilya Levin, Alexey Naumov et al.

Policy evaluation is an important instrument for the comparison of different algorithms in Reinforcement Learning (RL). However, even a precise knowledge of the value function $V^π$ corresponding to a policy $π$ does not provide reliable information on how far the policy $π$ is from the optimal one. We present a novel model-free upper value iteration procedure ({\sf UVIP}) that allows us to estimate the suboptimality gap $V^{\star}(x) - V^π(x)$ from above and to construct confidence intervals for \(V^\star\). Our approach relies on upper bounds to the solution of the Bellman optimality equation via the martingale approach. We provide theoretical guarantees for {\sf UVIP} under general assumptions and illustrate its performance on a number of benchmark RL problems.