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Researcher profile

Mikhail Belkin

Mikhail Belkin contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.OCComputation and LanguageComputer Visionmath.STStatistics Theory

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Published work

11 published item(s)

preprint2026arXiv

PersonalAI 2.0: Enhancing knowledge graph traversal/retrieval with planning mechanism for Personalized LLM Agents

We introduce PersonalAI 2.0 (PAI-2), a novel framework, designed to enhance large language model (LLM) based systems through integration of external knowledge graphs (KG). The proposed approach addresses key limitations of existing Graph Retrieval-Augmented Generation (GraphRAG) methods by incorporating a dynamic, multistage query processing pipeline. The central point of PAI-2 design is its ability to perform adaptive, iterative information search, guided by extracted entities, matched graph vertices and generated clue-queries. Conducted evaluation over six benchmarks (Natural Questions, TriviaQA, HotpotQA, 2WikiMultihopQA, MuSiQue and DiaASQ) demonstrates improvement in factual correctness of generating answers compared to analogues methods (LightRAG, RAPTOR, and HippoRAG 2). PAI-2 achieves 4% average gain by LLM-as-a-Judge across four benchmarks, reflecting its effectiveness in reducing hallucination rates and increasing precision. We show that use of graph traversal algorithms (e.g. BeamSearch, WaterCircles) gain superior results compared to standard flatten retriever on average 6%, while enabled search plan enhancement mechanism gain 18% boost compared to disabled one by LLM-as-a-Judge across six datasets. In addition, ablation study reveals that PAI-2 achieves the SOTA result on MINE-1 benchmark, achieving 89% information-retention score, using LLMs from 7-14B tiers. Collectively, these findings underscore the potential of PAI-2 to serve as a foundational model for next-generation personalized AI applications, requiring scalable, context-aware knowledge representation and reasoning capabilities.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

A note on Linear Bottleneck networks and their Transition to Multilinearity

Randomly initialized wide neural networks transition to linear functions of weights as the width grows, in a ball of radius $O(1)$ around initialization. A necessary condition for this result is that all layers of the network are wide enough, i.e., all widths tend to infinity. However, the transition to linearity breaks down when this infinite width assumption is violated. In this work we show that linear networks with a bottleneck layer learn bilinear functions of the weights, in a ball of radius $O(1)$ around initialization. In general, for $B-1$ bottleneck layers, the network is a degree $B$ multilinear function of weights. Importantly, the degree only depends on the number of bottlenecks and not the total depth of the network.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Benign Overfitting in Two-layer Convolutional Neural Networks

Modern neural networks often have great expressive power and can be trained to overfit the training data, while still achieving a good test performance. This phenomenon is referred to as "benign overfitting". Recently, there emerges a line of works studying "benign overfitting" from the theoretical perspective. However, they are limited to linear models or kernel/random feature models, and there is still a lack of theoretical understanding about when and how benign overfitting occurs in neural networks. In this paper, we study the benign overfitting phenomenon in training a two-layer convolutional neural network (CNN). We show that when the signal-to-noise ratio satisfies a certain condition, a two-layer CNN trained by gradient descent can achieve arbitrarily small training and test loss. On the other hand, when this condition does not hold, overfitting becomes harmful and the obtained CNN can only achieve a constant level test loss. These together demonstrate a sharp phase transition between benign overfitting and harmful overfitting, driven by the signal-to-noise ratio. To the best of our knowledge, this is the first work that precisely characterizes the conditions under which benign overfitting can occur in training convolutional neural networks.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Limitations of Neural Collapse for Understanding Generalization in Deep Learning

The recent work of Papyan, Han, & Donoho (2020) presented an intriguing "Neural Collapse" phenomenon, showing a structural property of interpolating classifiers in the late stage of training. This opened a rich area of exploration studying this phenomenon. Our motivation is to study the upper limits of this research program: How far will understanding Neural Collapse take us in understanding deep learning? First, we investigate its role in generalization. We refine the Neural Collapse conjecture into two separate conjectures: collapse on the train set (an optimization property) and collapse on the test distribution (a generalization property). We find that while Neural Collapse often occurs on the train set, it does not occur on the test set. We thus conclude that Neural Collapse is primarily an optimization phenomenon, with as-yet-unclear connections to generalization. Second, we investigate the role of Neural Collapse in feature learning. We show simple, realistic experiments where training longer leads to worse last-layer features, as measured by transfer-performance on a downstream task. This suggests that neural collapse is not always desirable for representation learning, as previously claimed. Finally, we give preliminary evidence of a "cascading collapse" phenomenon, wherein some form of Neural Collapse occurs not only for the last layer, but in earlier layers as well. We hope our work encourages the community to continue the rich line of Neural Collapse research, while also considering its inherent limitations.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures

Modern machine learning systems such as deep neural networks are often highly over-parameterized so that they can fit the noisy training data exactly, yet they can still achieve small test errors in practice. In this paper, we study this "benign overfitting" phenomenon of the maximum margin classifier for linear classification problems. Specifically, we consider data generated from sub-Gaussian mixtures, and provide a tight risk bound for the maximum margin linear classifier in the over-parameterized setting. Our results precisely characterize the condition under which benign overfitting can occur in linear classification problems, and improve on previous work. They also have direct implications for over-parameterized logistic regression.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Simple, Fast, and Flexible Framework for Matrix Completion with Infinite Width Neural Networks

Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications, but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). In particular, we derive the NTK for fully connected and convolutional neural networks for matrix completion. The flexibility stems from a feature prior, which allows encoding relationships between coordinates of the target matrix, akin to semi-supervised learning. The effectiveness of our framework is demonstrated through competitive results for virtual drug screening and image inpainting/reconstruction. We also provide an implementation in Python to make our framework accessible on standard hardware to a broad audience.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Transition to Linearity of Wide Neural Networks is an Emerging Property of Assembling Weak Models

Wide neural networks with linear output layer have been shown to be near-linear, and to have near-constant neural tangent kernel (NTK), in a region containing the optimization path of gradient descent. These findings seem counter-intuitive since in general neural networks are highly complex models. Why does a linear structure emerge when the networks become wide? In this work, we provide a new perspective on this "transition to linearity" by considering a neural network as an assembly model recursively built from a set of sub-models corresponding to individual neurons. In this view, we show that the linearity of wide neural networks is, in fact, an emerging property of assembling a large number of diverse "weak" sub-models, none of which dominate the assembly.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Local Quadratic Convergence of Stochastic Gradient Descent with Adaptive Step Size

Establishing a fast rate of convergence for optimization methods is crucial to their applicability in practice. With the increasing popularity of deep learning over the past decade, stochastic gradient descent and its adaptive variants (e.g. Adagrad, Adam, etc.) have become prominent methods of choice for machine learning practitioners. While a large number of works have demonstrated that these first order optimization methods can achieve sub-linear or linear convergence, we establish local quadratic convergence for stochastic gradient descent with adaptive step size for problems such as matrix inversion.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

On the linearity of large non-linear models: when and why the tangent kernel is constant

The goal of this work is to shed light on the remarkable phenomenon of transition to linearity of certain neural networks as their width approaches infinity. We show that the transition to linearity of the model and, equivalently, constancy of the (neural) tangent kernel (NTK) result from the scaling properties of the norm of the Hessian matrix of the network as a function of the network width. We present a general framework for understanding the constancy of the tangent kernel via Hessian scaling applicable to the standard classes of neural networks. Our analysis provides a new perspective on the phenomenon of constant tangent kernel, which is different from the widely accepted "lazy training". Furthermore, we show that the transition to linearity is not a general property of wide neural networks and does not hold when the last layer of the network is non-linear. It is also not necessary for successful optimization by gradient descent.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Overparameterized Neural Networks Implement Associative Memory

Identifying computational mechanisms for memorization and retrieval of data is a long-standing problem at the intersection of machine learning and neuroscience. Our main finding is that standard overparameterized deep neural networks trained using standard optimization methods implement such a mechanism for real-valued data. Empirically, we show that: (1) overparameterized autoencoders store training samples as attractors, and thus, iterating the learned map leads to sample recovery; (2) the same mechanism allows for encoding sequences of examples, and serves as an even more efficient mechanism for memory than autoencoding. Theoretically, we prove that when trained on a single example, autoencoders store the example as an attractor. Lastly, by treating a sequence encoder as a composition of maps, we prove that sequence encoding provides a more efficient mechanism for memory than autoencoding.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Reconciling modern machine learning practice and the bias-variance trade-off

Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias-variance trade-off, appears to be at odds with the observed behavior of methods used in the modern machine learning practice. The bias-variance trade-off implies that a model should balance under-fitting and over-fitting: rich enough to express underlying structure in data, simple enough to avoid fitting spurious patterns. However, in the modern practice, very rich models such as neural networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered over-fit, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This "double descent" curve subsumes the textbook U-shaped bias-variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine learning models delineates the limits of classical analyses, and has implications for both the theory and practice of machine learning.

0 citations0 reviewsNo code linkedOpen access