Transformers have achieved great success in recent years. Interestingly, transformers have shown particularly strong in-context learning capability -- even without fine-tuning, they are still able to solve unseen tasks well purely based on task-specific prompts. In this paper, we study the capability of one-layer transformers in learning the one-nearest neighbor prediction rule. Under a theoretical framework where the prompt contains a sequence of labeled training data and unlabeled test data, we show that, although the loss function is nonconvex, when trained with gradient descent, a single softmax attention layer can successfully learn to behave like a one-nearest neighbor classifier. Our result gives a concrete example on how transformers can be trained to implement nonparametric machine learning algorithms, and sheds light on the role of softmax attention in transformer models.

We consider the problem of learning a target function corresponding to a singlehidden layer neural network, with a quadratic activation function after the first layer,and random weights. We consider the asymptotic limit where the input dimensionand the network width are proportionally large. Recent work [Cui et al., 2023]established that linear regression provides Bayes-optimal test error to learn sucha function when the number of available samples is only linear in the dimension.That work stressed the open challenge of theoretically analyzing the optimal testerror in the more interesting regime where the number of samples is quadratic inthe dimension. In this paper, we solve this challenge for quadratic activations andderive a closed-form expression for the Bayes-optimal test error. We also provide analgorithm, that we call GAMP-RIE, which combines approximate message passingwith rotationally invariant matrix denoising, and that asymptotically achieves theoptimal performance. Technically, our result is enabled by establishing a linkwith recent works on optimal denoising of extensive-rank matrices and on theellipsoid fitting problem. We further show empirically that, in the absence ofnoise, randomly-initialized gradient descent seems to sample the space of weights,leading to zero training loss, and averaging over initialization leads to a test errorequal to the Bayes-optimal one.

While 3D Gaussian Splatting has recently become popular for neural rendering, current methods rely on carefully engineered cloning and splitting strategies for placing Gaussians, which does not always generalize and may lead to poor-quality renderings. For many real-world scenes this leads to their heavy dependence on good initializations. In this work, we rethink the set of 3D Gaussians as a random sample drawn from an underlying probability distribution describing the physical representation of the scene--in other words, Markov Chain Monte Carlo (MCMC) samples. Under this view, we show that the 3D Gaussian updates can be converted as Stochastic Gradient Langevin Dynamics (SGLD) update by simply introducing noise. We then rewrite the densification and pruning strategies in 3D Gaussian Splatting as simply a deterministic state transition of MCMC samples, removing these heuristics from the framework. To do so, we revise the'cloning' of Gaussians into a relocalization scheme that approximately preserves sample probability. To encourage efficient use of Gaussians, we introduce an L1-regularizer on the Gaussians. On various standard evaluation scenes, we show that our method provides improved rendering quality, easy control over the number of Gaussians, and robustness to initialization. The project website is available at https://3dgs-mcmc.github.io/.

Graph regression is a fundamental task that has gained significant attention invarious graph learning tasks. However, the inference process is often not easilyinterpretable. Current explanation techniques are limited to understanding GraphNeural Network (GNN) behaviors in classification tasks, leaving an explanation gapfor graph regression models. In this work, we propose a novel explanation methodto interpret the graph regression models (XAIG-R). Our method addresses thedistribution shifting problem and continuously ordered decision boundary issuesthat hinder existing methods away from being applied in regression tasks. Weintroduce a novel objective based on the graph information bottleneck theory (GIB)and a new mix-up framework, which can support various GNNs and explainersin a model-agnostic manner. Additionally, we present a self-supervised learningstrategy to tackle the continuously ordered labels in regression tasks. We evaluateour proposed method on three benchmark datasets and a real-life dataset introducedby us, and extensive experiments demonstrate its effectiveness in interpreting GNNmodels in regression tasks.

This paper investigates a class of stochastic bilevel optimization problems where the upper-level function is nonconvex with potentially unbounded smoothness and the lower-level problem is strongly convex. These problems have significant applications in sequential data learning, such as text classification using recurrent neural networks. The unbounded smoothness is characterized by the smoothness constant of the upper-level function scaling linearly with the gradient norm, lacking a uniform upper bound. Existing state-of-the-art algorithms require $\widetilde{O}(\epsilon^{-4})$ oracle calls of stochastic gradient or Hessian/Jacobian-vector product to find an $\epsilon$-stationary point. However, it remains unclear if we can further improve the convergence rate when the assumptions for the function in the population level also hold for each random realization almost surely (e.g., Lipschitzness of each realization of the stochastic gradient).

Transformers can efficiently learn in-context from example demonstrations. Most existing theoretical analyses studied the in-context learning (ICL) ability of transformers for linear function classes, where it is typically shown that the minimizer of the pretraining loss implements one gradient descent step on the least squares objective. However, this simplified linear setting arguably does not demonstrate the statistical efficiency of ICL, since the pretrained transformer does not outperform directly solving linear regression on the test prompt.

In this paper, we consider the uncertainty quantification problem for regression models. Specifically, we consider an individual calibration objective for characterizing the quantiles of the prediction model. While such an objective is well-motivated from downstream tasks such as newsvendor cost, the existing methods have been largely heuristic and lack of statistical guarantee in terms of individual calibration. We show via simple examples that the existing methods focusing on population-level calibration guarantees such as average calibration or sharpness can lead to harmful and unexpected results. We propose simple nonparametric calibration methods that are agnostic of the underlying prediction model and enjoy both computational efficiency and statistical consistency.

The largest eigenvalue of the Hessian, or sharpness, of neural networks is a key quantity to understand their optimization dynamics. In this paper, we study the sharpness of deep linear networks for univariate regression. Minimizers can have arbitrarily large sharpness, but not an arbitrarily small one. Indeed, we show a lower bound on the sharpness of minimizers, which grows linearly with depth. We then study the properties of the minimizer found by gradient flow, which is the limit of gradient descent with vanishing learning rate.

We develop a Mean-Field (MF) view of the learning dynamics of overparametrized Artificial Neural Networks (NN) under distributional symmetries of the data w.r.t. the action of a general compact group $G$. We consider for this a class of generalized shallow NNs given by an ensemble of $N$ multi-layer units, jointly trained using stochastic gradient descent (SGD) and possibly symmetry-leveraging (SL) techniques, such as Data Augmentation (DA), Feature Averaging (FA) or Equivariant Architectures (EA). We introduce the notions of weakly and strongly invariant laws (WI and SI) on the parameter space of each single unit, corresponding, respectively, to $G$-invariant distributions, and to distributions supported on parameters fixed by the group action (which encode EA). This allows us to define symmetric models compatible with taking $N\to\infty$ and give an interpretation of the asymptotic dynamics of DA, FA and EA in terms of Wasserstein Gradient Flows describing their MF limits. When activations respect the group action, we show that, for symmetric data, DA, FA and freely-trained models obey the exact same MF dynamic, which stays in the space of WI parameter laws and attains therein the population risk's minimizer. We also provide a counterexample to the general attainability of such an optimum over SI laws.Despite this, and quite remarkably, we show that the space of SI laws is also preserved by these MF distributional dynamics even when freely trained. This sharply contrasts the finite-$N$ setting, in which EAs are generally not preserved by unconstrained SGD. We illustrate the validity of our findings as $N$ gets larger, in a teacher-student experimental setting, training a student NN to learn from a WI, SI or arbitrary teacher model through various SL schemes. We lastly deduce a data-driven heuristic to discover the largest subspace of parameters supporting SI distributions for a problem, that could be used for designing EA with minimal generalization error.

Stochastic gradient descent (SGD) is perhaps the most prevalent optimization method in modern machine learning. Contrary to the empirical practice of sampling from the datasets \emph{without replacement} and with (possible) reshuffling at each epoch, the theoretical counterpart of SGD usually relies on the assumption of \emph{sampling with replacement}. It is only very recently that SGD using sampling without replacement -- shuffled SGD -- has been analyzed with matching upper and lower bounds. However, we observe that those bounds are too pessimistic to explain often superior empirical performance of data permutations (sampling without replacement) over vanilla counterparts (sampling with replacement) on machine learning problems. Through fine-grained analysis in the lens of primal-dual cyclic coordinate methods and the introduction of novel smoothness parameters, we present several results for shuffled SGD on smooth and non-smooth convex losses, where our novel analysis framework provides tighter convergence bounds over all popular shuffling schemes (IG, SO, and RR). Notably, our new bounds predict faster convergence than existing bounds in the literature -- by up to a factor of O(\sqrt{n}), mirroring benefits from tighter convergence bounds using component smoothness parameters in randomized coordinate methods.