The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$--potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

The remarkable success of transformers in sequence modeling tasks, spanning various applications in natural language processing and computer vision, is attributed to the critical role of self-attention. Similar to the development of most deep learning models, the construction of these attention mechanisms relies on heuristics and experience. In our work, we derive self-attention from kernel principal component analysis (kernel PCA) and show that self-attention projects its query vectors onto the principal component axes of its key matrix in a feature space. We then formulate the exact formula for the value matrix in self-attention, theoretically and empirically demonstrating that this value matrix captures the eigenvectors of the Gram matrix of the key vectors in self-attention. Leveraging our kernel PCA framework, we propose Attention with Robust Principal Components (RPC-Attention), a novel class of robust attention that is resilient to data contamination. We empirically demonstrate the advantages of RPC-Attention over softmax attention on the ImageNet-1K object classification, WikiText-103 language modeling, and ADE20K image segmentation task.

Data serves as the fundamental basis for advancing deep learning.

Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs--namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.

Interest in bilevel optimization has grown in recent years, partially due to its relevance for challenging machine-learning problems. Several exciting recent works have been centered around developing efficient gradient-based algorithms that can solve bilevel optimization problems with provable guarantees. However, the existing literature mainly focuses on bilevel problems either without constraints, or featuring only simple constraints that do not couple variables across the upper and lower levels, excluding a range of complex applications. Our paper studies this challenging but less explored scenario and develops a (fully) first-order algorithm, which we term BLOCC, to tackle BiLevel Optimization problems with Coupled Constraints. We establish rigorous convergence theory for the proposed algorithm and demonstrate its effectiveness on two well-known real-world applications - support vector machine (SVM) - based model training and infrastructure planning in transportation networks.

Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries -- a broad subclass of continuous symmetries present in the model architecture or loss function -- interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a ``Noether flow) of the parameters $\theta$ along the degenerate direction to a unique initialization-independent fixed point $\theta^*$. These points are referred to as the noise equilibria because, at these points, noise contributions from different directions are balanced and aligned. Then, we show that the balance and alignment of gradient noise can serve as a novel alternative mechanism for explaining important phenomena such as progressive sharpening/flattening and representation formation within neural networks and have practical implications for understanding techniques like representation normalization and warmup.

While the Transformer architecture has achieved remarkable success across various domains, a thorough theoretical foundation explaining its optimization dynamics is yet to be fully developed. In this study, we aim to bridge this understanding gap by answering the following two core questions: (1) Which types of Transformer architectures allow Gradient Descent (GD) to achieve guaranteed convergence?

We study the problem of retrieving accurate correspondence through multi-scale consistency (MSC) for robust point cloud registration. Existing works in a coarse-to-fine manner either suffer from severe noisy correspondences caused by unreliable coarse matching or struggle to form outlier-free coarse-level correspondence sets. To tackle this, we present Hierarchical Sinkhorn Tree (HST), a pruned tree structure designed to hierarchically measure the local consistency of each coarse correspondence across multiple feature scales, thereby filtering out the local dissimilar ones. In this way, we convert the modeling of MSC for each correspondence into a BFS traversal with pruning of a K-ary tree rooted at the superpoint, with its K nearest neighbors in the feature pyramid serving as child nodes. To achieve efficient pruning and accurate vicinity characterization, we further propose a novel overlap-aware Sinkhorn Distance, which retains only the most likely overlapping points for local measurement and next level exploration. The modeling process essentially involves traversing a pair of HSTs synchronously and aggregating the consistency measures of corresponding tree nodes. Extensive experiments demonstrate HST consistently outperforms the state-of-the-art methods on both indoor and outdoor benchmarks.

Previous visual object tracking methods employ image-feature regression models or coordinate autoregression models for bounding box prediction. Image-feature regression methods heavily depend on matching results and do not utilize positional prior, while the autoregressive approach can only be trained using bounding boxes available in the training set, potentially resulting in suboptimal performance during testing with unseen data. Inspired by the diffusion model, denoising learning enhances the model's robustness to unseen data. Therefore, We introduce noise to bounding boxes, generating noisy boxes for training, thus enhancing model robustness on testing data. We propose a new paradigm to formulate the visual object tracking problem as a denoising learning process. However, tracking algorithms are usually asked to run in real-time, directly applying the diffusion model to object tracking would severely impair tracking speed.

Auto-regressive large language models (LLMs) show impressive capacities to solve many complex reasoning tasks while struggling with some simple logical reasoning tasks such as inverse search: when trained on ''$A \to B$'' (e.g., *Tom is the parent of John*), LLM fails to directly conclude ''$B \gets A$'' (e.g., *John is the child of Tom*) during inference even if the two sentences are semantically identical, which is known as the ''reversal curse''. In this paper, we theoretically analyze the reversal curse via the training dynamics of (stochastic) gradient descent for two auto-regressive models: (1) a bilinear model that can be viewed as a simplification of a one-layer transformer; (2) one-layer transformers under certain assumptions. Our analysis reveals that for both models, the reversal curse is a consequence of the (effective) model weights *asymmetry*, i.e., the increase of weights from a token $A$ to token $B$ during training does not necessarily cause the increase of the weights from $B$ to $A$, which is caused by the training dynamics under certain choice of loss function and the optimization space of model parameters. Moreover, our analysis can be naturally applied to other logical reasoning tasks such as chain-of-thought (COT), which provides a new perspective different from previous work that focuses on expressivity. Finally, we conduct experiments to validate our theory on multi-layer transformers under different settings.