It is vital to infer a signed distance function (SDF) in multi-view based surface reconstruction.

Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the expressivity of LLMs with CoT in solving fundamental mathematical and decisionmaking problems. By using circuit complexity theory, we first give impossibility results showing that bounded-depth Transformers are unable to directly produce correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of constant size suffice to solve both tasks by generating CoT derivations using a commonly used math language format. Moreover, we show LLMs with CoT can handle a general class of decision-making problems known as Dynamic Programming, thus justifying their power in tackling complex real-world tasks. Finally, an extensive set of experiments show that, while Transformers always fail to directly predict the answers, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.

Graph neural networks (GNNs) are the de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erd os-Rényi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or to one. This class includes the popular graph convolutional network architecture. The result establishes'zero-one laws' for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.

Aligning foundation models is essential for their safe and trustworthy deployment. However, traditional fine-tuning methods are computationally intensive and require updating billions of model parameters. A promising alternative, alignment via decoding, adjusts the response distribution directly without model updates to maximize a target reward r, thus providing a lightweight and adaptable framework for alignment. However, principled decoding methods rely on oracle access to an optimal Q-function (Q), which is often unavailable in practice. Hence, prior SoTA methods either approximate this Q using Qπsft (derived from the reference SFTmodel) or rely on short-term rewards, resulting in sub-optimal decoding performance. In this work, we propose Transfer Q, which implicitly estimates the optimal value function for a target reward r through a baseline model ρBL aligned with a baseline reward rBL (which can be different from the target reward r). Theoretical analyses of Transfer Q provide a rigorous characterization of its optimality, deriving an upper bound on the sub-optimality gap and identifying a hyperparameter to control the deviation from the pre-trained reference SFTmodel based on user needs. Our approach significantly reduces the sub-optimality gap observed in prior SoTA methods and demonstrates superior empirical performance across key metrics such as coherence, diversity, and quality in extensive tests on several synthetic and real datasets.

Given a policy of a Markov Decision Process, we define a SAFEZONE as a subset of states, such that most of the policy's trajectories are confined to this subset. The quality of a SAFEZONE is parameterized by the number of states and the escape probability, i.e., the probability that a random trajectory will leave the subset. SAFEZONES are especially interesting when they have a small number of states and low escape probability. We study the complexity of finding optimal SAFEZONES, and show that in general, the problem is computationally hard. Our main result is a bi-criteria approximation learning algorithm with a factor of almost 2 approximation for both the escape probability and SAFEZONE size, using a polynomial size sample complexity.

An accurate environment dynamics model is crucial for various downstream tasks in sequential decision-making, such as counterfactual prediction, off-policy evaluation, and offline reinforcement learning. Currently, these models were learned through empirical risk minimization (ERM) by step-wise fitting of historical transition data. This way was previously believed unreliable over long-horizon rollouts because of the compounding errors, which can lead to uncontrollable inaccuracies in predictions. In this paper, we find that the challenge extends beyond just longterm prediction errors: we reveal that even when planning with one step, learned dynamics models can also perform poorly due to the selection bias of behavior policies during data collection.