For iterative algorithms, implicit differentiation alleviates this issue but requires custom implementation of Jacobian evaluation. In this paper, we study one-step differentiation, also known as Jacobian-free backpropagation, a method as easy as automatic differentiation and as efficient as implicit differentiation for fast algorithms (e.g., superlinear

AIDE: Fastand Communication Efficient Distributed Optimization.arXive-prints,

However, wecanmake(9)moretractable (bothforpredictionandtraining) byreducingittoa1-dimensional root-findingproblem: Find ?

Consider GofthelossL(w), meaning g 2G, L(w)= L(g w).

They derived a parallel form of Newton's method to solve the fixed-point problem and achieved significant speedups over sequential evaluation.

Here are three smart tricks, based on an understanding of frictional forces, to beat a slippery slope. I don't know who invented this crazy challenge, but the idea is to put someone in a carved-out ice bowl and see if they can get out. The bowl is shaped like the inside of a sphere, so the higher up the sides you go, the steeper it gets. If you think an icy sidewalk is slippery, try going uphill on an icy sidewalk. What do you do when faced with a problem like this?

We develop a distributed second order optimization algorithm that is communication-efficient as well as robust against Byzantine failures of the worker machines. We propose an iterative approximate Newton-type algorithm, where the worker machines communicate \emph{only once} per iteration with the central machine. This is in sharp contrast with the state-of-the-art distributed second order algorithms like GIANT \cite{giant}, DINGO\cite{dingo}, where the worker machines send (functions of) local gradient and Hessian sequentially; thus ending up communicating twice with the central machine per iteration. Furthermore, we employ a simple norm based thresholding rule to filter-out the Byzantine worker machines. We establish the linear-quadratic rate of convergence of our proposed algorithm and establish that the communication savings and Byzantine resilience attributes only correspond to a small statistical error rate for arbitrary convex loss functions. To the best of our knowledge, this is the first work that addresses the issue of Byzantine resilience in second order distributed optimization. Furthermore, we validate our theoretical results with extensive experiments on synthetically generated and benchmark LIBSVM \cite{libsvm} data-set and demonstrate convergence guarantees.

Transformers excel at *in-context learning* (ICL)---learning from demonstrations without parameter updates---but how they do so remains a mystery. Recent work suggests that Transformers may internally run Gradient Descent (GD), a first-order optimization method, to perform ICL. In this paper, we instead demonstrate that Transformers learn to approximate second-order optimization methods for ICL. For in-context linear regression, Transformers share a similar convergence rate as *Iterative Newton's Method*, both *exponentially* faster than GD. Empirically, predictions from successive Transformer layers closely match different iterations of Newton's Method linearly, with each middle layer roughly computing 3 iterations; thus, Transformers and Newton's method converge at roughly the same rate.