We present a universal theoretical framework for understanding long-context language modeling based on a bipartite mutual information scaling law that we rigorously verify in natural language. We demonstrate that bipartite mutual information captures multi-token interactions distinct from and scaling independently of conventional two-point mutual information, and show that this provides a more complete characterization of the dependencies needed for accurately modeling long sequences. Leveraging this scaling law, we formulate the Long-context Language Modeling (L2M) condition, which lower bounds the necessary scaling of a model's history state--the latent variables responsible for storing past information--for effective long-context modeling.

This work establishes a novel link between the problem of PAC-learning high-dimensional graphical models and the task of (efficient) counting and sampling of graph structures, using an online learning framework. The problem of efficiently counting and sampling graphical structures, such as spanning trees and acyclic orientations, has been a vibrant area of research in algorithms. We show that this rich algorithmic foundation can be leveraged to develop new algorithms for learning high-dimensional graphical models. We present the first efficient algorithm for (both realizable and agnostic) learning of Bayes nets with a chordal skeleton. In particular, we present an algorithm that, given integers $k,d > 0$, error parameter $\varepsilon > 0$, an undirected chordal graph $G$ on $n$ vertices, and sample access to a distribution $P^\ast$ on $[k]^n$; (1) returns a Bayes net $\widehat{P}$ with skeleton $G$ and indegree $d$, whose KL-divergence from $P^\ast$ is at most $\varepsilon$ more than the optimal KL-divergence between $P^\ast$ and any Bayes net with skeleton $G$ and indegree $d$, (2) uses $\widetilde{O}(n^3k^{d+1}/\varepsilon^2)$ samples from $P^\ast$ and runs in time $\mathrm{poly}(n,k,\varepsilon^{-1})$ for constant $d$. Prior results in this spirit were for only for trees ($d=1$, tree skeleton) via Chow-Liu, and in the realizable setting for polytrees (arbitrary $d$ but tree skeleton). Thus, our result significantly extends the state-of-the-art in learning Bayes net distributions. We also establish new results for learning tree and polytree distributions.

Guided policy search algorithms can be used to optimize complex nonlinear policies, such as deep neural networks, without directly computing policy gradients in the high-dimensional parameter space. Instead, these methods use supervised learning to train the policy to mimic a "teacher" algorithm, such as a trajectory optimizer or a trajectory-centric reinforcement learning method. Guided policy search methods provide asymptotic local convergence guarantees by construction, but it is not clear how much the policy improves within a small, finite number of iterations. We show that guided policy search algorithms can be interpreted as an approximate variant of mirror descent, where the projection onto the constraint manifold is not exact. We derive a new guided policy search algorithm that is simpler and provides appealing improvement and convergence guarantees in simplified convex and linear settings, and show that in the more general nonlinear setting, the error in the projection step can be bounded. We provide empirical results on several simulated robotic navigation and manipulation tasks that show that our method is stable and achieves similar or better performance when compared to prior guided policy search methods, with a simpler formulation and fewer hyperparameters.

Neural Information Processing Systems http://nips.cc/

We mitigate the usual worst-case nature of minimax analysis by showing that our bounds are tight for any given31 hypothesis class, and, tight in any noise regime (Theorems 1 and 2).

Ensemble approaches for uncertainty estimation have recently been applied to the tasks of misclassification detection, out-of-distribution input detection and adversarial attack detection. Prior Networks have been proposed as an approach to efficiently emulate an ensemble of models for classification by parameteris-ing a Dirichlet prior distribution over output distributions.