Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon.

This view of the data generating process is also adopted and promoted by popular existing works.

Efficient coding provides a direct link between prior and likelihood in perceptual Bayesian inference.

We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be \emph{exactly} equivalent to continuous-time mirror descent by Amid and Warmuth, but theory for the analogous discrete time algorithms is left as an open problem. We prove an $O(T^{\frac{2}{3}})$ regret bound for non-convex online gradient descent in this setting, answering this open problem. Our analysis is based on a new and simple algorithmic equivalence method.

Large language models (LLMs) have shown impressive performance on language tasks but face challenges when deployed on resource-constrained devices due to their extensive parameters and reliance on dense multiplications, resulting in high memory demands and latency bottlenecks. Shift-and-add reparameterization offers a promising solution by replacing costly multiplications with hardware-friendly primitives in both the attention and multi-layer perceptron (MLP) layers of an LLM. However, current reparameterization techniques require training from scratch or full parameter fine-tuning to restore accuracy, which is resource-intensive for LLMs. To address this, we propose accelerating pretrained LLMs through post-training shift-and-add reparameterization, creating efficient multiplication-free models, dubbed ShiftAddLLM. Specifically, we quantize each weight matrix into binary matrices paired with group-wise scaling factors. The associated multiplications are reparameterized into (1) shifts between activations and scaling factors and (2) queries and adds according to the binary matrices. To reduce accuracy loss, we present a multi-objective optimization method to minimize both weight and output activation reparameterization errors. Additionally, based on varying sensitivity across layers to reparameterization, we develop an automated bit allocation strategy to further reduce memory usage and latency. Experiments on five LLM families and eight tasks consistently validate the effectiveness of ShiftAddLLM, achieving average perplexity reductions of 5.6 and 22.7 points at comparable or lower latency compared to the most competitive quantized LLMs at 3-and 2-bit precision, respectively, and more than 80% memory and energy reductions over the original LLMs.

By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time. We demonstrate the usefulness of our method on applications in supervised image recognition, generative modelling, and deep reinforcement learning.

Behavioral experiments on humans and animals suggest that the brain performs probabilistic inference to interpret its environment. Here we present a new general-purpose, biologically-plausible neural implementation of approximate inference. The neural network represents uncertainty using Probabilistic Population Codes (PPCs), which are distributed neural representations that naturally encode probability distributions, and support marginalization and evidence integration in a biologically-plausible manner. By connecting multiple PPCs together as a probabilistic graphical model, we represent multivariate probability distributions. Approximate inference in graphical models can be accomplished by message-passing algorithms that disseminate local information throughout the graph. An attractive and often accurate example of such an algorithm is Loopy Belief Propagation (LBP), which uses local marginalization and evidence integration operations to perform approximate inference efficiently even for complex models.

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V ariational inference is increasingly being addressed with stochastic optimization. In this setting, the gradient's variance plays a crucial role in the optimization procedure, since high variance gradients lead to poor convergence.

However, it is not applicable to a number of important continuous distributions.