Complex time-varying systems are often studied by abstracting away from the dynamics of individual components to build a model of the population-level dynamics from the start. However, when building a population-level description, it can be easy to lose sight of each individual and how they contribute to the larger picture. In this paper, we present a novel transformer architecture for learning from time-varying data that builds descriptions of both the individual as well as the collective population dynamics. Rather than combining all of our data into our model at the onset, we develop a separable architecture that operates on individual time-series first before passing them forward; this induces a permutation-invariance property and can be used to transfer across systems of different size and order. After demonstrating that our model can be applied to successfully recover complex interactions and dynamics in many-body systems, we apply our approach to populations of neurons in the nervous system. On neural activity datasets, we show that our model not only yields robust decoding performance, but also provides impressive performance in transfer across recordings of different animals without any neuron-level correspondence. By enabling flexible pre-training that can be transferred to neural recordings of different size and order, our work provides a first step towards creating a foundation model for neural decoding.

In many contexts, simpler models are preferable to more complex models and the control of this model complexity is the goal for many methods in machine learning such as regularization, hyperparameter tuning and architecture design. In deep learning, it has been difficult to understand the underlying mechanisms of complexity control, since many traditional measures are not naturally suitable for deep neural networks. Here we develop the notion of geometric complexity, which is a measure of the variability of the model function, computed using a discrete Dirichlet energy. Using a combination of theoretical arguments and empirical results, we show that many common training heuristics such as parameter norm regularization, spectral norm regularization, flatness regularization, implicit gradient regularization, noise regularization and the choice of parameter initialization all act to control geometric complexity, providing a unifying framework in which to characterize the behavior of deep learning models.

Language models employ a very large number of trainable parameters. Despite being highly overparameterized, these networks often achieve good out-of-sample test performance on the original task and easily fine-tune to related tasks. Recent observations involving, for example, intrinsic dimension of the objective landscape and the lottery ticket hypothesis, indicate that often training actively involves only a small fraction of the parameter space. Thus, a question remains how large a parameter space needs to be in the first place -- the evidence from recent work on model compression, parameter sharing, factorized representations, and knowledge distillation increasingly shows that models can be made much smaller and still perform well. Here, we focus on factorized representations of matrices that underpin dense, embedding, and self-attention layers. We use low-rank factorized representation of a reshaped and rearranged original matrix to achieve space efficient and expressive linear layers. We prove that stacking such low-rank layers increases their expressiveness, providing theoretical understanding for their effectiveness in deep networks. In Transformer models, our approach leads to more than tenfold reduction in the number of total trainable parameters, including embedding, attention, and feed-forward layers, with little degradation in on-task performance. The approach operates out-of-the-box, replacing each parameter matrix with its compact equivalent while maintaining the architecture of the network.

Recent works have derived neural networks with online correlation-based learning rules to perform kernel similarity matching. These works applied existing linear similarity matching algorithms to nonlinear features generated with random Fourier methods. In this paper we attempt to perform kernel similarity matching by directly learning the nonlinear features. Our algorithm proceeds by deriving and then minimizing an upper bound for the sum of squared errors between output and input kernel similarities. The construction of our upper bound leads to online correlation-based learning rules which can be implemented with a 1 layer recurrent neural network. In addition to generating high-dimensional linearly separable representations, we show that our upper bound naturally yields representations which are sparse and selective for specific input patterns. We compare the approximation quality of our method to neural random Fourier method and variants of the popular but non-biological "Nyström" method for approximating the kernel matrix. Our method appears to be comparable or better than randomly sampled Nyström methods when the outputs are relatively low dimensional (although still potentially higher dimensional than the inputs) but less faithful when the outputs are very high dimensional.

In all environments, we use a Transformer architecture with four layers and four self-attention heads. The total input vocabulary of the model is V (N + M +2) to account for states, actions, rewards, and rewards-to-go, but the output linear layer produces logits only over a vocabulary of size V; output tokens can be interpreted unambiguously because their offset is uniquely determined by that of the previous input. The dimension of each token embedding is 128. Dropout is applied at the end of each block with probability 0.1. We follow the learning rate scheduling of (Radford et al., 2018), increasing linearly from 0 to 2.5 10 4 over the course of 2000 updates.