Finding the k largest or smallest elements from a collection of scores, i.e., top-k operation, is an important model component widely used in information retrieval, machine learning, and data mining. However, if the top-k operation is implemented in an algorithmic way, e.g., using bubble algorithm, the resulted model cannot be trained in an end-to-end way using prevalent gradient descent algorithms. This is because these implementations typically involve swapping indices, whose gradient cannot be computed. Moreover, the corresponding mapping from the input scores to the indicator vector of whether this element belongs to the top-k set is essentially discontinuous. To address the issue, we propose a smoothed approximation, namely SOFT (Scalable Optimal transport-based diFferenTiable) top-k operator. Specifically, our SOFT top-k operator approximates the output of top-k operation as the solution of an Entropic Optimal Transport (EOT) problem. The gradient of the SOFT operator can then be efficiently approximated based on the optimality conditions of EOT problem. We then apply the proposed operator to k-nearest neighbors algorithm and beam search algorithm. The numerical experiment demonstrates their achieve improved performance.

Motivated from the theory of non-parametric regression, we introduce a \emph{new variational constraint} that enforces the comparator sequence to belong to a discrete $k^{th}$ order Total Variation ball of radius $C_n$. This variational constraint models comparators that have piece-wise polynomial structure which has many relevant practical applications [Tibshirani2015]. By establishing connections to the theory of wavelet based non-parametric regression, we design a \emph{polynomial time} algorithm that achieves the nearly \emph{optimal dynamic regret} of $\tilde{O}(n^{\frac{1}{2k+3}}C_n^{\frac{2}{2k+3}})$. The proposed policy is \emph{adaptive to the unknown radius} $C_n$. Further, we show that the same policy is minimax optimal for several other non-parametric families of interest.

In this paper we tackle the problem of Generalized Category Discovery (GCD). Specifically, given a dataset with labelled and unlabelled images, the task is to cluster all images in the unlabelled subset, whether or not they belong to the labelled categories. Our first contribution is to recognise that most existing GCD benchmarks only contain labels for a single clustering of the data, making it difficult to ascertain whether models are leveraging the available labels to solve the GCD task, or simply solving an unsupervised clustering problem. As such, we present a synthetic dataset, named'Clevr-4', for category discovery. Clevr-4 contains four equally valid partitions of the data, i.e based on object'shape', 'texture' or'color' or'count'.

In recent years we see a rapidly growing line of research which shows learnability of various models via common neural network algorithms. Yet, besides a very few outliers, these results show learnability of models that can be learned using linear methods. Namely, such results show that learning neural-networks with gradient-descent is competitive with learning a linear classifier on top of a data-independent representation of the examples. This leaves much to be desired, as neural networks are far more successful than linear methods. Furthermore, on the more conceptual level, linear models don't seem to capture the``deepness of deep networks. In this paper we make a step towards showing leanability of models that are inherently non-linear. We show that under certain distributions, sparse parities are learnable via gradient decent on depth-two network. On the other hand, under the same distributions, these parities cannot be learned efficiently by linear methods.

Learning curve extrapolation aims to predict model performance in later epochs of training, based on the performance in earlier epochs.In this work, we argue that, while the inherent uncertainty in the extrapolation of learning curves warrants a Bayesian approach, existing methods are (i) overly restrictive, and/or (ii) computationally expensive. We describe the first application of prior-data fitted neural networks (PFNs) in this context. A PFN is a transformer, pre-trained on data generated from a prior, to perform approximate Bayesian inference in a single forward pass. We propose LC-PFN, a PFN trained to extrapolate 10 million artificial right-censored learning curves generated from a parametric prior proposed in prior art using MCMC. We demonstrate that LC-PFN can approximate the posterior predictive distribution more accurately than MCMC, while being over 10 000 times faster. We also show that the same LC-PFN achieves competitive performance extrapolating a total of 20 000 real learning curves from four learning curve benchmarks (LCBench, NAS-Bench-201, Taskset, and PD1) that stem from training a wide range of model architectures (MLPs, CNNs, RNNs, and Transformers) on 53 different datasets with varying input modalities (tabular, image, text, and protein data). Finally, we investigate its potential in the context of model selection and find that a simple LC-PFN based predictive early stopping criterion obtains 2 - 6x speed-ups on 45 of these datasets, at virtually no overhead.

Theoretically, sDBSCAN preserves the DBSCAN's clustering structure under mild conditions with high probability. To facilitate sDBSCAN, we present sOPTICS, a scalable visual tool to guide the parameter setting of sDBSCAN. We also extend sDBSCAN and sOPTICS to L2, L1, χ2, and Jensen-Shannon distances via random kernel features. Empirically, sDBSCAN is significantly faster and provides higher accuracy than competitive DBSCAN variants on real-world million-point data sets. On these data sets, sDBSCAN and sOPTICS run in a few minutes, while the scikit-learn counterparts and other clustering competitors demand several hours orcannot run on our hardware due to memory constraints.

We initiate the study of stochastic optimization with oblivious noise, broadly generalizing the standard heavy-tailed noise setup.In our setting, in addition to random observation noise, the stochastic gradient may be subject to independent \emph{oblivious noise}, which may not have bounded moments and is not necessarily centered. Specifically, we assume access to a noisy oracle for the stochastic gradient of $f$ at $x$, which returns a vector $\nabla f(\gamma, x) + \xi$, where $\gamma$ is the bounded variance observation noise and $\xi$ is the oblivious noise that is independent of $\gamma$ and $x$. The only assumption we make on the oblivious noise $\xi$ is that $\Pr[\xi = 0] \ge \alpha$, for some $\alpha \in (0, 1)$.In this setting, it is not information-theoretically possible to recover a single solution close to the target when the fraction of inliers $\alpha$ is less than $1/2$. Our main result is an efficient {\em list-decodable} learner that recovers a small list of candidates at least one of which is close to the true solution. On the other hand, if $\alpha = 1-\epsilon$, where $0 < \epsilon < 1/2$ is sufficiently smallconstant, the algorithm recovers a single solution.Along the way, we develop a rejection-sampling-based algorithm to perform noisy location estimation, which may be of independent interest.

Neural Architecture Search (NAS) has fostered the automatic discovery of state-of-the-art neural architectures. Despite the progress achieved with NAS, so far there is little attention to theoretical guarantees on NAS. In this work, we study the generalization properties of NAS under a unifying framework enabling (deep) layer skip connection search and activation function search. To this end, we derive the lower (and upper) bounds of the minimum eigenvalue of the Neural Tangent Kernel (NTK) under the (in)finite-width regime using a certain search space including mixed activation functions, fully connected, and residual neural networks. We use the minimum eigenvalue to establish generalization error bounds of NAS in the stochastic gradient descent training. Importantly, we theoretically and experimentally show how the derived results can guide NAS to select the top-performing architectures, even in the case without training, leading to a train-free algorithm based on our theory. Accordingly, our numerical validation shed light on the design of computationally efficient methods for NAS. Our analysis is non-trivial due to the coupling of various architectures and activation functions under the unifying framework and has its own interest in providing the lower bound of the minimum eigenvalue of NTK in deep learning theory.

Graph convolutional networks (GCN) with one or two hidden layers have been widely used in handling graph data that are prevalent in various disciplines. Many studies showed that the gain of making GCNs deeper is tiny or even negative. This implies that the complexity of graph data is often limited and shallow models are often sufficient to extract expressive features for various tasks such as node classification. Therefore, in this work, we present a framework called graph convolutional kernel machine (GCKM) for graph-based machine learning. GCKMs are built upon kernel functions integrated with graph convolution.

We consider stochastic optimization when one only has access to biased stochastic oracles of the objective, and obtaining stochastic gradients with low biases comes at high costs. This setting captures a variety of optimization paradigms widely used in machine learning, such as conditional stochastic optimization, bilevel optimization, and distributionally robust optimization. We examine a family of multi-level Monte Carlo (MLMC) gradient methods that exploit a delicate trade-off among the bias, the variance, and the oracle cost. We provide a systematic study of their convergences and total computation complexities for strongly convex, convex, and nonconvex objectives, and demonstrate their superiority over the naive biased stochastic gradient method. Moreover, when applied to conditional stochastic optimization, the MLMC gradient methods significantly improve the best-known sample complexity in the literature.