Leo Kozachkov

Mitchell Ostrow, Adam Eisen, Leo Kozachkov et al.

How can we tell whether two neural networks utilize the same internal processes for a particular computation? This question is pertinent for multiple subfields of neuroscience and machine learning, including neuroAI, mechanistic interpretability, and brain-machine interfaces. Standard approaches for comparing neural networks focus on the spatial geometry of latent states. Yet in recurrent networks, computations are implemented at the level of dynamics, and two networks performing the same computation with equivalent dynamics need not exhibit the same geometry. To bridge this gap, we introduce a novel similarity metric that compares two systems at the level of their dynamics, called Dynamical Similarity Analysis (DSA). Our method incorporates two components: Using recent advances in data-driven dynamical systems theory, we learn a high-dimensional linear system that accurately captures core features of the original nonlinear dynamics. Next, we compare different systems passed through this embedding using a novel extension of Procrustes Analysis that accounts for how vector fields change under orthogonal transformation. In four case studies, we demonstrate that our method disentangles conjugate and non-conjugate recurrent neural networks (RNNs), while geometric methods fall short. We additionally show that our method can distinguish learning rules in an unsupervised manner. Our method opens the door to comparative analyses of the essential temporal structure of computation in neural circuits.

Michaela Ennis, Leo Kozachkov, Jean-Jacques Slotine

To push forward the important emerging research field surrounding multi-area recurrent neural networks (RNNs), we expand theoretically and empirically on the provably stable RNNs of RNNs introduced by Kozachkov et al. in "RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks". We prove relaxed stability conditions for salient special cases of this architecture, most notably for a global workspace modular structure. We then demonstrate empirical success for Global Workspace Sparse Combo Nets with a small number of trainable parameters, not only through strong overall test performance but also greater resilience to removal of individual subnetworks. These empirical results for the global workspace inter-area topology are contingent on stability preservation, highlighting the relevance of our theoretical work for enabling modular RNN success. Further, by exploring sparsity in the connectivity structure between different subnetwork modules more broadly, we improve the state of the art performance for stable RNNs on benchmark sequence processing tasks, thus underscoring the general utility of specialized graph structures for multi-area RNNs.

Franz A. Heinsen, Leo Kozachkov

The most widely used artificial intelligence (AI) models today are Transformers employing self-attention. In its standard form, self-attention incurs costs that increase with context length, driving demand for storage, compute, and energy that is now outstripping society's ability to provide them. To help address this issue, we show that self-attention is efficiently computable to arbitrary precision with constant cost per token, achieving orders-of-magnitude reductions in memory use and computation. We derive our formulation by decomposing the conventional formulation's Taylor expansion into expressions over symmetric chains of tensor products. We exploit their symmetry to obtain feed-forward transformations that efficiently map queries and keys to coordinates in a minimal polynomial-kernel feature basis. Notably, cost is fixed inversely in proportion to head size, enabling application over a greater number of heads per token than otherwise feasible. We implement our formulation and empirically validate its correctness. Our work enables unbounded token generation at modest fixed cost, substantially reducing the infrastructure and energy demands of large-scale Transformer models. The mathematical techniques we introduce are of independent interest.

Lucas Shoji, Kenta Suzuki, Leo Kozachkov

This paper shows that a wide class of effective learning rules -- those that improve a scalar performance measure over a given time window -- can be rewritten as natural gradient descent with respect to a suitably defined loss function and metric. Specifically, we show that parameter updates within this class of learning rules can be expressed as the product of a symmetric positive definite matrix (i.e., a metric) and the negative gradient of a loss function. We also demonstrate that these metrics have a canonical form and identify several optimal ones, including the metric that achieves the minimum possible condition number. The proofs of the main results are straightforward, relying only on elementary linear algebra and calculus, and are applicable to continuous-time, discrete-time, stochastic, and higher-order learning rules, as well as loss functions that explicitly depend on time.

Reece Keller, Alyn Kirsch, Felix Pei et al.

Autonomy is a hallmark of animal intelligence, enabling adaptive and intelligent behavior in complex environments without relying on external reward or task structure. Existing reinforcement learning approaches to exploration in reward-free environments, including a class of methods known as model-based intrinsic motivation, exhibit inconsistent exploration patterns and do not converge to an exploratory policy, thus failing to capture robust autonomous behaviors observed in animals. Moreover, systems neuroscience has largely overlooked the neural basis of autonomy, focusing instead on experimental paradigms where animals are motivated by external reward rather than engaging in ethological, naturalistic and task-independent behavior. To bridge these gaps, we introduce a novel model-based intrinsic drive explicitly designed after the principles of autonomous exploration in animals. Our method (3M-Progress) achieves animal-like exploration by tracking divergence between an online world model and a fixed prior learned from an ecological niche. To the best of our knowledge, we introduce the first autonomous embodied agent that predicts brain data entirely from self-supervised optimization of an intrinsic goal -- without any behavioral or neural training data -- demonstrating that 3M-Progress agents capture the explainable variance in behavioral patterns and whole-brain neural-glial dynamics recorded from autonomously behaving larval zebrafish, thereby providing the first goal-driven, population-level model of neural-glial computation. Our findings establish a computational framework connecting model-based intrinsic motivation to naturalistic behavior, providing a foundation for building artificial agents with animal-like autonomy.

Franz A. Heinsen, Leo Kozachkov

Many domains, from deep learning to finance, require compounding real numbers over long sequences, often leading to catastrophic numerical underflow or overflow. We introduce generalized orders of magnitude (GOOMs), a principled extension of traditional orders of magnitude that incorporates floating-point numbers as a special case, and which in practice enables stable computation over significantly larger dynamic ranges of real numbers than previously possible. We implement GOOMs, along with an efficient custom parallel prefix scan, to support native execution on parallel hardware such as GPUs. We demonstrate that our implementation of GOOMs outperforms traditional approaches with three representative experiments, all of which were previously considered impractical or impossible, and now become possible and practical: (1) compounding real matrix products far beyond standard floating-point limits; (2) estimating spectra of Lyapunov exponents in parallel, orders of magnitude faster than with previous methods, applying a novel selective-resetting method to prevent state colinearity; and (3) capturing long-range dependencies in deep recurrent neural networks with non-diagonal recurrent states, computed in parallel via a prefix scan, without requiring any form of stabilization. Our results show that our implementation of GOOMs, combined with efficient parallel scanning, offers a scalable and numerically robust alternative to conventional floating-point numbers for high-dynamic-range applications.

Leo Kozachkov, Patrick M. Wensing, Jean-Jacques Slotine

We prove that Riemannian contraction in a supervised learning setting implies generalization. Specifically, we show that if an optimizer is contracting in some Riemannian metric with rate $λ> 0$, it is uniformly algorithmically stable with rate $\mathcal{O}(1/λn)$, where $n$ is the number of labelled examples in the training set. The results hold for stochastic and deterministic optimization, in both continuous and discrete-time, for convex and non-convex loss surfaces. The associated generalization bounds reduce to well-known results in the particular case of gradient descent over convex or strongly convex loss surfaces. They can be shown to be optimal in certain linear settings, such as kernel ridge regression under gradient flow.

Leo Kozachkov, Michaela Ennis, Jean-Jacques Slotine

Recurrent neural networks (RNNs) are widely used throughout neuroscience as models of local neural activity. Many properties of single RNNs are well characterized theoretically, but experimental neuroscience has moved in the direction of studying multiple interacting areas, and RNN theory needs to be likewise extended. We take a constructive approach towards this problem, leveraging tools from nonlinear control theory and machine learning to characterize when combinations of stable RNNs will themselves be stable. Importantly, we derive conditions which allow for massive feedback connections between interacting RNNs. We parameterize these conditions for easy optimization using gradient-based techniques, and show that stability-constrained "networks of networks" can perform well on challenging sequential-processing benchmark tasks. Altogether, our results provide a principled approach towards understanding distributed, modular function in the brain.

Leo Kozachkov, Konstantinos P. Michmizos

Finding the origin of slow and infra-slow oscillations could reveal or explain brain mechanisms in health and disease. Here, we present a biophysically constrained computational model of a neural network where the inclusion of astrocytes introduced slow and infra-slow-oscillations, through two distinct mechanisms. Specifically, we show how astrocytes can modulate the fast network activity through their slow inter-cellular calcium wave speed and amplitude and possibly cause the oscillatory imbalances observed in diseases commonly known for such abnormalities, namely Alzheimer's disease, Parkinson's disease, epilepsy, depression and ischemic stroke. This work aims to increase our knowledge on how astrocytes and neurons synergize to affect brain function and dysfunction.