Although current large language models are complex, the most basic specifications of the underlying language generation problem itself are simple to state: given a finite set of training samples from an unknown language, produce valid new strings from the language that don't already appear in the training data. Here we ask what we can conclude about language generation using only this specification, without further assumptions. In particular, suppose that an adversary enumerates the strings of an unknown target language L that is known only to come from one of a possibly infinite list of candidates. A computational agent is trying to learn to generate from this language; we say that the agent generates from $L$ in the limit if after some finite point in the enumeration of $L$, the agent is able to produce new elements that come exclusively from $L$ and that have not yet been presented by the adversary. Our main result is that there is an agent that is able to generate in the limit for every countable list of candidate languages. This contrasts dramatically with negative results due to Gold and Angluin in a well-studied model of language learning where the goal is to identify an unknown language from samples; the difference between these results suggests that identifying a language is a fundamentally different problem than generating from it.

Fine-tuning foundation models often compromises their robustness to distribution shifts. To remedy this, most robust fine-tuning methods aim to preserve the pre-trained features. However, not all pre-trained features are robust and those methods are largely indifferent to which ones to preserve. We propose dual risk minimization (DRM), which combines empirical risk minimization with worst-case risk minimization, to better preserve the core features of downstream tasks. In particular, we utilize core-feature descriptions generated by LLMs to induce core-based zero-shot predictions which then serve as proxies to estimate the worst-case risk. DRM balances two crucial aspects of model robustness: expected performance and worst-case performance, establishing a new state of the art on various real-world benchmarks. DRM significantly improves the out-of-distribution performance of CLIP ViT-L/14@336 on ImageNet (75.9$\to$77.1),

Langevin Dynamics is a Stochastic Differential Equation (SDE) central to sampling and generative modeling and is implemented via time discretization. Langevin Monte Carlo (LMC), based on the Euler-Maruyama discretization, is the simplest and most studied algorithm. LMC can suffer from slow convergence - requiring a large number of steps of small step-size to obtain good quality samples. This becomes stark in the case of diffusion models where a large number of steps gives the best samples, but the quality degrades rapidly with smaller number of steps. Randomized Midpoint Method has been recently proposed as a better discretization of Langevin dynamics for sampling from strongly log-concave distributions. However, important applications such as diffusion models involve non-log concave densities and contain time varying drift. We propose its variant, the Poisson Midpoint Method, which approximates a small step-size LMC with large step-sizes. We prove that this can obtain a quadratic speed up of LMC under very weak assumptions. We apply our method to diffusion models for image generation and show that it maintains the quality of DDPM with 1000 neural network calls with just 50-80 neural network calls and outperforms ODE based methods with similar compute.

Predictive combinatorial optimization, where the parameters of combinatorial optimization (CO) are unknown at the decision-making time, is the precise modeling of many real-world applications, including energy cost-aware scheduling and budget allocation on advertising. Tackling such a problem usually involves a prediction model and a CO solver. These two modules are integrated into the predictive CO pipeline following two design principles: ''Predict-then-Optimize (PtO)'', which learns predictions by supervised training and subsequently solves CO using predicted coefficients, while the other, named ''Predict-and-Optimize (PnO)'', directly optimizes towards the ultimate decision quality and claims to yield better decisions than traditional PtO approaches. However, there lacks a systematic benchmark of both approaches, including the specific design choices at the module level, as well as an evaluation dataset that covers representative real-world scenarios. To this end, we develop a modular framework to benchmark 11 existing PtO/PnO methods on 8 problems, including a new industrial dataset for combinatorial advertising that will be released. Our study shows that PnO approaches are better than PtO on 7 out of 8 benchmarks, but there is no silver bullet found for the specific design choices of PnO. A comprehensive categorization of current approaches and integration of typical scenarios are provided under a unified benchmark. Therefore, this paper could serve as a comprehensive benchmark for future PnO approach development and also offer fast prototyping for application-focused development.

General Value Functions (GVFs) (Sutton et al., 2011) represent predictive knowledge in reinforcement learning. Each GVF computes the expected return for a given policy, based on a unique reward. Existing methods relying on fixed behavior policies or pre-collected data often face data efficiency issues when learning multiple GVFs in parallel using off-policy methods. To address this, we introduce, which adaptively learns a single behavior policy that efficiently collects data for evaluating multiple GVFs in parallel.

We consider the problem of estimating a structured multivariate density, subject to Markov conditions implied by an undirected graph. In the worst case, without Markovian assumptions, this problem suffers from the curse of dimensionality. Our main result shows how the curse of dimensionality can be avoided or greatly alleviated under the Markov property, and applies to arbitrary graphs. While existing results along these lines focus on sparsity or manifold assumptions, we introduce a new graphical quantity called ``graph resilience'' and show that it dictates the optimal sample complexity. Surprisingly, although one might expect the sample complexity of this problem to scale with local graph parameters such as the degree, this turns out not to be the case. Through explicit examples, we compute uniform deviation bounds and illustrate how the curse of dimensionality in density estimation can thus be circumvented. Notable examples where the rate improves substantially include sequential, hierarchical, and spatial data.

Heterophily, or the tendency of connected nodes in networks to have different class labels or dissimilar features, has been identified as challenging for many Graph Neural Network (GNN) models. While the challenges of applying GNNs for node classification when class labels display strong heterophily are well understood, it is unclear how heterophily affects GNN performance in other important graph learning tasks where class labels are not available. In this work, we focus on the link prediction task and systematically analyze the impact of heterophily in node features on GNN performance. We first introduce formal definitions of homophilic and heterophilic link prediction tasks, and present a theoretical framework that highlights the different optimizations needed for the respective tasks. We then analyze how different link prediction encoders and decoders adapt to varying levels of feature homophily and introduce designs for improved performance. Based on our definitions, we identify and analyze six real-world benchmarks spanning from homophilic to heterophilic link prediction settings, with graphs containing up to 30M edges. Our empirical analysis on a variety of synthetic and real-world datasets confirms our theoretical insights and highlights the importance of adopting learnable decoders and GNN encoders with ego-and neighbor-embedding separation in message passing for link prediction tasks beyond homophily.

Rare event sampling in dynamical systems is a fundamental problem arising in the natural sciences, which poses significant computational challenges due to an exponentially large space of trajectories. For settings where the dynamical system of interest follows a Brownian motion with known drift, the question of conditioning the process to reach a given endpoint or desired rare event is definitively answered by Doob's $h$-transform. However, the naive estimation of this transform is infeasible, as it requires simulating sufficiently many forward trajectories to estimate rare event probabilities. In this work, we propose a variational formulation of Doob's $h$-transform as an optimization problem over trajectories between a given initial point and the desired ending point. To solve this optimization, we propose a simulation-free training objective with a model parameterization that imposes the desired boundary conditions by design. Our approach significantly reduces the search space over trajectories and avoids expensive trajectory simulation and inefficient importance sampling estimators which are required in existing methods. We demonstrate the ability of our method to find feasible transition paths on real-world molecular simulation and protein folding tasks.

This paper uses chess, a landmark planning problem in AI, to assess transformers' performance on a planning task where memorization is futile -- even at a large scale. To this end, we release ChessBench, a large-scale benchmark dataset of 10 million chess games with legal move and value annotations (15 billion data points) provided by Stockfish 16, the state-of-the-art chess engine. We train transformers with up to 270 million parameters on ChessBench via supervised learning and perform extensive ablations to assess the impact of dataset size, model size, architecture type, and different prediction targets (state-values, action-values, and behavioral cloning). Our largest models learn to predict action-values for novel boards quite accurately, implying highly non-trivial generalization. Despite performing no explicit search, our resulting chess policy solves challenging chess puzzles and achieves a surprisingly strong Lichess blitz Elo of 2895 against humans (grandmaster level). We also compare to Leela Chess Zero and AlphaZero (trained without supervision via self-play) with and without search. We show that, although a remarkably good approximation of Stockfish's search-based algorithm can be distilled into large-scale transformers via supervised learning, perfect distillation is still beyond reach, thus making ChessBench well-suited for future research.