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This paper will describe and analyze a new phenomenon that was not known before, which we call "Early Transferability". Its essence is that the adversarial perturbations transfer among different networks even at extremely early stages in their training. In fact, one can initialize two networks with two different independent choices of random weights and measure the angle between their adversarial perturbations after each step of the training. What we discovered was that these two adversarial directions started to align with each other already after the first few training steps (which typically use only a small fraction of the available training data), even though the accuracy of the two networks hadn't started to improve from their initial bad values due to the early stage of the training. The purpose of this paper is to present this phenomenon experimentally and propose plausible explanations for some of its properties.

Gradient Descent (GD) is a powerful workhorse of modern machine learning thanks to its scalability and efficiency in high-dimensional spaces. Its ability to find local minimisers is only guaranteed for losses with Lipschitz gradients, where it can be seen as a 'bona-fide' discretisation of an underlying gradient flow. Yet, many ML setups involving overparametrised models do not fall into this problem class, which has motivated research beyond the so-called "Edge of Stability", where the step-size crosses the admissibility threshold inversely proportional to the Lipschitz constant above. Perhaps surprisingly, GD has been empirically observed to still converge regardless of local instability. In this work, we study a local condition for such an unstable convergence around a local minima in a low dimensional setting. We then leverage these insights to establish global convergence of a two-layer single-neuron ReLU student network aligning with the teacher neuron in a large learning rate beyond the Edge of Stability under population loss. Meanwhile, while the difference of norms of the two layers is preserved by gradient flow, we show that GD above the edge of stability induces a balancing effect, leading to the same norms across the layers.

Gaussian processes provide an elegant framework for specifying prior and posterior distributions over functions. They are, however, also computationally expensive, and limited by the expressivity of their covariance function. We propose Neural Diffusion Processes (NDPs), a novel approach based upon diffusion models, that learn to sample from distributions over functions. Using a novel attention block, we can incorporate properties of stochastic processes, such as exchangeability, directly into the NDP's architecture. We empirically show that NDPs are able to capture functional distributions that are close to the true Bayesian posterior of a Gaussian process. This enables a variety of downstream tasks, including hyperparameter marginalisation and Bayesian optimisation.

Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting where the two factors are generated from known component-wise independent prior distributions, and the statistician observes a (possibly noisy) component-wise function of their matrix product. In the limit where the dimensions of the matrices tend to infinity, but their ratios remain fixed, we expect to be able to derive closed form expressions for the optimal mean squared error on the estimation of the two factors. However, this remains a very involved mathematical and algorithmic problem. A related, but simpler, problem is extensive-rank matrix denoising, where one aims to reconstruct a matrix with extensive but usually small rank from noisy measurements. In this paper, we approach both these problems using high-temperature expansions at fixed order parameters. This allows to clarify how previous attempts at solving these problems failed at finding an asymptotically exact solution. We provide a systematic way to derive the corrections to these existing approximations, taking into account the structure of correlations particular to the problem. Finally, we illustrate our approach in detail on the case of extensive-rank matrix denoising. We compare our results with known optimal rotationally-invariant estimators, and show how exact asymptotic calculations of the minimal error can be performed using extensive-rank matrix integrals.

Many future technologies rely on neural networks, but verifying the correctness of their behavior remains a major challenge. It is known that neural networks can be fragile in the presence of even small input perturbations, yielding unpredictable outputs. The verification of neural networks is therefore vital to their adoption, and a number of approaches have been proposed in recent years. In this paper we focus on semidefinite programming (SDP) based techniques for neural network verification, which are particularly attractive because they can encode expressive behaviors while ensuring a polynomial time decision. Our starting point is the DeepSDP framework proposed by Fazlyab et al, which uses quadratic constraints to abstract the verification problem into a large-scale SDP. When the size of the neural network grows, however, solving this SDP quickly becomes intractable. Our key observation is that by leveraging chordal sparsity and specific parametrizations of DeepSDP, we can decompose the primary computational bottleneck of DeepSDP -- a large linear matrix inequality (LMI) -- into an equivalent collection of smaller LMIs. Our parametrization admits a tunable parameter, allowing us to trade-off efficiency and accuracy in the verification procedure. We call our formulation Chordal-DeepSDP, and provide experimental evaluation to show that it can: (1) effectively increase accuracy with the tunable parameter and (2) outperform DeepSDP on deeper networks.

We study the problem of reconstructing solutions of inverse problems with neural networks when only noisy data is available. We assume that the problem can be modeled with an infinite-dimensional forward operator that is not continuously invertible. Then, we restrict this forward operator to finite-dimensional spaces so that the inverse is Lipschitz continuous. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the operator. In addition, we show that these neural networks can be learned from appropriately perturbed training data. We demonstrate the admissibility of this approach to a wide range of inverse problems of practical interest. Numerical examples are given that support the theoretical findings.

We study to what extent may stochastic gradient descent (SGD) be understood as a "conventional" learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex optimization framework, where (one pass, without-replacement) SGD is classically known to minimize the population risk at rate $O(1/\sqrt n)$, and prove that, surprisingly, there exist problem instances where the SGD solution exhibits both empirical risk and generalization gap of $\Omega(1)$. Consequently, it turns out that SGD is not algorithmically stable in any sense, and its generalization ability cannot be explained by uniform convergence or any other currently known generalization bound technique for that matter (other than that of its classical analysis). We then continue to analyze the closely related with-replacement SGD, for which we show that an analogous phenomenon does not occur and prove that its population risk does in fact converge at the optimal rate. Finally, we interpret our main results in the context of without-replacement SGD for finite-sum convex optimization problems, and derive upper and lower bounds for the multi-epoch regime that significantly improve upon previously known results.

Much of the literature on optimal design of bandit algorithms is based on minimization of expected regret. It is well known that designs that are optimal over certain exponential families can achieve expected regret that grows logarithmically in the number of arm plays, at a rate governed by the Lai-Robbins lower bound. In this paper, we show that when one uses such optimized designs, the regret distribution of the associated algorithms necessarily has a very heavy tail, specifically, that of a truncated Cauchy distribution. Furthermore, for $p>1$, the $p$'th moment of the regret distribution grows much faster than poly-logarithmically, in particular as a power of the total number of arm plays. We show that optimized UCB bandit designs are also fragile in an additional sense, namely when the problem is even slightly mis-specified, the regret can grow much faster than the conventional theory suggests. Our arguments are based on standard change-of-measure ideas, and indicate that the most likely way that regret becomes larger than expected is when the optimal arm returns below-average rewards in the first few arm plays, thereby causing the algorithm to believe that the arm is sub-optimal. To alleviate the fragility issues exposed, we show that UCB algorithms can be modified so as to ensure a desired degree of robustness to mis-specification. In doing so, we also provide a sharp trade-off between the amount of UCB exploration and the tail exponent of the resulting regret distribution.

This work presents a new procedure for obtaining predictive distributions in the context of Gaussian process (GP) modeling, with a relaxation of the interpolation constraints outside some ranges of interest: the mean of the predictive distributions no longer necessarily interpolates the observed values when they are outside ranges of interest, but are simply constrained to remain outside. This method called relaxed Gaussian process (reGP) interpolation provides better predictive distributions in ranges of interest, especially in cases where a stationarity assumption for the GP model is not appropriate. It can be viewed as a goal-oriented method and becomes particularly interesting in Bayesian optimization, for example, for the minimization of an objective function, where good predictive distributions for low function values are important. When the expected improvement criterion and reGP are used for sequentially choosing evaluation points, the convergence of the resulting optimization algorithm is theoretically guaranteed (provided that the function to be optimized lies in the reproducing kernel Hilbert spaces attached to the known covariance of the underlying Gaussian process). Experiments indicate that using reGP instead of stationary GP models in Bayesian optimization is beneficial.

Many scientific problems require to process data in the form of geometric graphs. Unlike generic graph data, geometric graphs exhibit symmetries of translations, rotations, and/or reflections. Researchers have leveraged such inductive bias and developed geometrically equivariant Graph Neural Networks (GNNs) to better characterize the geometry and topology of geometric graphs. Despite fruitful achievements, it still lacks a survey to depict how equivariant GNNs are progressed, which in turn hinders the further development of equivariant GNNs. To this end, based on the necessary but concise mathematical preliminaries, we analyze and classify existing methods into three groups regarding how the message passing and aggregation in GNNs are represented. We also summarize the benchmarks as well as the related datasets to facilitate later researches for methodology development and experimental evaluation. The prospect for future potential directions is also provided.