Solving mathematical problems requires advanced reasoning abilities and presents notable challenges for large language models. Previous works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results. However, our analysis of these datasets reveals severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries. Hypothesizing that difficult queries are crucial to learning complex reasoning, we propose Difficulty-Aware Rejection Tuning (DART), a method that allocates difficult queries more trials during the synthesis phase, enabling more extensive training on difficult samples. Utilizing DART, we have created new datasets for mathematical problem-solving that focus more on difficult queries and are substantially smaller than previous ones. Remarkably, our synthesis process solely relies on a 7B-sized open-weight model, without reliance on the commonly used proprietary GPT-4. We fine-tune various base models on our datasets ranging from 7B to 70B in size, resulting in a series of strong models called DART-Math. In comprehensive in-domain and out-of-domain evaluation on 6 mathematical benchmarks, DART-Math outperforms vanilla rejection tuning significantly, being superior or comparable to previous arts, despite using much smaller datasets and no proprietary models.

Let us begin by recalling how SPADE works, and study where its defects come from. These statistics are calculated via averages over examples and all spatial dimensions. To clarify, the subtraction and division in (3) are broadcasted on non-channel dimensions, and the pointwise multiplication and addition are broadcasted over examples. SPADE layers are remarkably similar to the Adaptive Instance Normalization (AdaIN) layers that are used in StyleGAN to condition on z. Finally, the conditioning of the generator's output y = g(z) (StyleGAN is an unconditional generative model) is done via AdaIN layers conditioned on s(z).

We introduce a nonparametric approach for estimating drift functions in systems of stochastic differential equations from sparse observations of the state vector. Using a Gaussian process prior over the drift as a function of the state vector, we develop an approximate EM algorithm to deal with the unobserved, latent dynamics between observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the MAP estimation of the drift is facilitated by a sparse Gaussian process regression.

Empirical risk minimization (ERM) is a fundamental learning rule for statistical learning problems where the data is generated according to some unknown distribution P and returns a hypothesis f chosen from a fixed class F with small loss l. In the parametric setting, depending upon (l, F, P) ERM can have slow (1/ n) or fast (1/n) rates of convergence of the excess risk as a function of the sample size n. There exist several results that give sufficient conditions for fast rates in terms of joint properties of l, F, and P, such as the margin condition and the Bernstein condition. In the non-statistical prediction with expert advice setting, there is an analogous slow and fast rate phenomenon, and it is entirely characterized in terms of the mixability of the loss l (there being no role there for F or P). The notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents a direct proof of fast rates for ERM in terms of stochastic mixability of (l, F, P), and in so doing provides new insight into the fast-rates phenomenon.

Given the damping factor α and precision tolerance ϵ, Andersen et al. [2] introduced Approximate Personalized PageRank (APPR), the de facto local method for approximating the PPR vector, with runtime bounded by Θ(1/(αϵ)) independent of the graph size. Recently, Fountoulakis & Yang [12] asked whether faster local algorithms could be developed using Õ(1/( αϵ)) operations. By noticing that APPR is a local variant of Gauss-Seidel, this paper explores the question of whether standard iterative solvers can be effectively localized. We propose to use the locally evolving set process, a novel framework to characterize the algorithm locality, and demonstrate that many standard solvers can be effectively localized.

Hypothesis selection, also known as density estimation, is a fundamental problem in statistics and learning theory.

Aligning neural dynamics with movements is a fundamental goal in neuroscience and brain-machine interfaces. However, there is still a lack of dimensionality reduction methods that can effectively align low-dimensional latent dynamics with movements. To address this gap, we propose Neural Embeddings Rank (NER), a technique that embeds neural dynamics into a 3D latent space and contrasts the embeddings based on movement ranks. NER learns to regress continuous representations of neural dynamics (i.e., embeddings) on continuous movements. We apply NER and six other dimensionality reduction techniques to neurons in the primary motor cortex (M1), dorsal premotor cortex (PMd), and primary somatosensory cortex (S1) as monkeys perform reaching tasks.

Algorithms for bilevel optimization often encounter Hessian computations, which are prohibitive in high dimensions. While recent works offer first-order methods for unconstrained bilevel problems, the constrained setting remains relatively underexplored. We present first-order linearly constrained optimization methods with finite-time hypergradient stationarity guarantees.

We introduce DynaMITE-RL, a meta-reinforcement learning (meta-RL) approach to approximate inference in environments where the latent state evolves at varying rates. We model episode sessions--parts of the episode where the latent state is fixed--and propose three key modifications to existing meta-RL methods: (i) consistency of latent information within sessions, (ii) session masking, and (iii) prior latent conditioning. We demonstrate the importance of these modifications in various domains, ranging from discrete Gridworld environments to continuouscontrol and simulated robot assistive tasks, illustrating the efficacy of DynaMITE-RL over state-of-the-art baselines in both online and offline RL settings.

Differential privacy is the dominant standard for formal and quantifiable privacy and has been used in major deployments that impact millions of people. Many differentially private algorithms for query release and synthetic data contain steps that reconstruct answers to queries from answers to other queries that have been measured privately. Reconstruction is an important subproblem for such mechanisms to economize the privacy budget, minimize error on reconstructed answers, and allow for scalability to high-dimensional datasets. In this paper, we introduce a principled and efficient postprocessing method ReM (Residuals-to-Marginals) for reconstructing answers to marginal queries. Our method builds on recent work on efficient mechanisms for marginal query release, based on making measurements using a residual query basis that admits efficient pseudoinversion, which is an important primitive used in reconstruction. An extension GReM-LNN (Gaussian Residuals-to-Marginals with Local Non-negativity) reconstructs marginals under Gaussian noise satisfying consistency and non-negativity, which often reduces error on reconstructed answers. We demonstrate the utility of ReM and GReM-LNN by applying them to improve existing private query answering mechanisms.