Inference-time scaling has recently emerged as a powerful paradigm for improving the reasoning capability of large language models. Among various approaches, Sequential Monte Carlo (SMC) has become a particularly important framework, enabling iterative generation, evaluation, rejection, and resampling of intermediate reasoning trajectories. A central component in this process is the reward model, which evaluates partial solutions and guides the allocation of computation during inference. However, in practice, true reward models are never available. All deployed systems rely on approximate reward models, raising a fundamental question: Why and when do approximate reward models suffice for effective inference-time scaling? In this work, we provide a theoretical answer. We identify the Bellman error of the approximate reward model as the key quantity governing the effectiveness of SMC-based inference-time scaling. For a reasoning process of length $T$, we show that if the Bellman error of the approximate reward model is bounded by $O(1/T)$, then combining this reward model with SMC reduces the computational complexity of reasoning from exponential in $T$ to polynomial in $T$. This yields an exponential improvement in inference efficiency despite using only approximate rewards.

The explicit incorporation of task-specific inductive biases through symmetry has emerged as a general design precept in the development of high-performance machine learning models. For example, group equivariant neural networks have demonstrated impressive performance across various domains and applications such as protein and drug design. A prevalent intuition about such models is that the integration of relevant symmetry results in enhanced generalization. Moreover, it is posited that when the data and/or the model exhibits only approximate or partial symmetry, the optimal or best-performing model is one where the model symmetry aligns with the data symmetry. In this paper, we conduct a formal unified investigation of these intuitions. To begin, we present quantitative bounds that demonstrate how models capturing task-specific symmetries lead to improved generalization. Utilizing this quantification, we examine the more general question of dealing with approximate/partial symmetries. We establish, for a given symmetry group, a quantitative comparison between the approximate equivariance of the model and that of the data distribution, precisely connecting model equivariance error and data equivariance error. Our result delineates the conditions under which the model equivariance error is optimal, thereby yielding the best-performing model for the given task and data.

In the first-order query model for zero-sum $K\times K$ matrix games, players observe the expected pay-offs for all their possible actions under the randomized action played by their opponent.

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error $O_p(m^{-1/2})$ for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.

Many problems in machine learning can be solved by rounding the solution of an appropriate linear program. We propose a scheme that is based on a quadratic program relaxation which allows us to use parallel stochastic-coordinate-descent to approximately solve large linear programs efficiently. Our software is an order of magnitude faster than Cplex (a commercial linear programming solver) and yields similar solution quality. Our results include a novel perturbation analysis of a quadratic-penalty formulation of linear programming and a convergence result, which we use to derive running time and quality guarantees.

The explicit incorporation of task-specific inductive biases through symmetry has emerged as a general design precept in the development of high-performance machine learning models. For example, group equivariant neural networks have demonstrated impressive performance across various domains and applications such as protein and drug design. A prevalent intuition about such models is that the integration of relevant symmetry results in enhanced generalization. Moreover, it is posited that when the data and/or the model exhibits only approximate or partial symmetry, the optimal or best-performing model is one where the model symmetry aligns with the data symmetry. In this paper, we conduct a formal unified investigation of these intuitions.

In the first-order query model for zero-sum K\times K matrix games, players observe the expected pay-offs for all their possible actions under the randomized action played by their opponent. Surprisingly, the optimal number of such queries, as a function of both \epsilon and K, is not known. We make progress on this question on two fronts. First, we fully characterise the query complexity of learning exact equilibria ( \epsilon 0), by showing that they require a number of queries that is linear in K, which means that it is essentially as hard as querying the whole matrix, which can also be done with K queries. We argue that, unfortunately, obtaining a matching lower bound is not possible with existing techniques: we prove that no lower bound can be derived by constructing hard matrices whose entries take values in a known countable set, because such matrices can be fully identified by a single query.

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error O_p(m {-1/2}) for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.