Current theoretical results on optimization trajectories of neural networks trained by gradient descent typically have the form of rigorous but potentially loose bounds on the loss values. In the present work we take a different approach and show that the learning trajectory of a wide network in a lazy training regime can be characterized by an explicit asymptotic at large training times. Specifically, the leading term in the asymptotic expansion of the loss behaves as a power law L(t) Ct ξ with exponent ξ expressed only through the data dimension, the smoothness of the activation function, and the class of function being approximated. Our results are based on spectral analysis of the integral operator representing the linearized evolution of a large network trained on the expected loss. Importantly, the techniques we employ do not require a specific form of the data distribution, for example Gaussian, thus making our findings sufficiently universal.

Tyler's M-estimator is a well known procedure for robust and heavy-tailed covariance estimation. Tyler himself suggested an iterative fixed-point algorithm for computing his estimator however, it requires super-linear (in the size of the data) runtime per iteration, which maybe prohibitive in large scale. In this work we propose, to the best of our knowledge, the first Frank-Wolfe-based algorithms for computing Tyler's estimator. One variant uses standard Frank-Wolfe steps, the second also considers away-steps (AFW), and the third is a geodesic version of AFW (GAFW). AFW provably requires, up to a log factor, only linear time per iteration, while GAFW runs in linear time (up to a log factor) in a large n(number of data-points) regime. All three variants are shown to provably converge to the optimal solution with sublinear rate, under standard assumptions, despite the fact that the underlying optimization problem is not convex nor smooth. Under an additional fairly mild assumption, that holds with probability 1 when the (normalized) data-points are i.i.d.

Current vision systems are trained on huge datasets, and these datasets come with costs: curation is expensive, they inherit human biases, and there are concerns over privacy and usage rights. To counter these costs, interest has surged in learning from cheaper data sources, such as unlabeled images. In this paper, we go a step further and ask if we can do away with real image datasets entirely, by learning from procedural noise processes. We investigate a suite of image generation models that produce images from simple random processes. These are then used as training data for a visual representation learner with a contrastive loss. In particular, we study statistical image models, randomly initialized deep generative models, and procedural graphics models. Our findings show that it is important for the noise to capture certain structural properties of real data but that good performance can be achieved even with processes that are far from realistic. We also find that diversity is a key property for learning good representations.

Self-supervised Learning (SSL) including the mainstream contrastive learning has achieved great success in learning visual representations without data annotations. However, most of methods mainly focus on the instance level information (i.e., the different augmented images of the same instance should have the same feature or cluster into the same class), but there is a lack of attention on the relationships between different instances. In this paper, we introduced a novel SSL paradigm, which we term as relational self-supervised learning (ReSSL) framework that learns representations by modeling the relationship between different instances. Specifically, our proposed method employs sharpened distribution of pairwise similarities among different instances as relation metric, which is thus utilized to match the feature embeddings of different augmentations. Moreover, to boost the performance, we argue that weak augmentations matter to represent a more reliable relation, and leverage momentum strategy for practical efficiency. Experimental results show that our proposed ReSSL significantly outperforms the previous stateof-the-art algorithms in terms of both performance and training efficiency.

Following [2], we extend our method into multi-domain translation on AFHQ dataset, where the source domain includes Cat and Wild and the target domain is Dog. In this setting, similar to two-domain unpaired I2I, the EGSDE also employs an energy function pretrained on both the source and target domains to guide the inference process of a pretrained SDE. The only difference is the domain-specific feature extractor Es(,) involved in the energy function is the all but the last eliminate layer of randomness.

Score-based diffusion models (SBDMs) have achieved the SOTAFID results in unpaired image-to-image translation (I2I). However, we notice that existing methods totally ignore the training data in the source domain, leading to sub-optimal solutions for unpaired I2I. To this end, we propose energy-guided stochastic differential equations (EGSDE) that employs an energy function pretrained on both the source and target domains to guide the inference process of a pretrained SDE for realistic and faithful unpaired I2I. Building upon two feature extractors, we carefully design the energy function such that it encourages the transferred image to preserve the domain-independent features and discard domain-specific ones. Further, we provide an alternative explanation of the EGSDE as a product of experts, where each of the three experts (corresponding to the SDE and two feature extractors) solely contributes to faithfulness or realism. Empirically, we compare EGSDE to a large family of baselines on three widely-adopted unpaired I2I tasks under four metrics. EGSDE not only consistently outperforms existing SBDMs-based methods in almost all settings but also achieves the SOTA realism results without harming the faithful performance. Furthermore, EGSDE allows for flexible trade-offs between realism and faithfulness and we improve the realism results further (e.g., FID of 51.04 in Cat Dog and FID of 50.43 in Wild Dog on AFHQ) by tuning hyper-parameters. The code is available at https://github.com/ML-GSAI/EGSDE.