Neural Information Processing Systems http://nips.cc/

Low-rank matrix factorization (LRMF) is a canonical problem in non-convex optimization, the objective function to be minimized is non-convex and even non-smooth, which makes the global convergence guarantee of gradient-based algorithm quite challenging.

The phenomenon that stochastic gradient descent (SGD) favors flat minima has played a critical role in understanding the implicit regularization of SGD. In this paper, we provide an explanation of this striking phenomenon by relating the particular noise structure of SGD to its linear stability (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum θ is linearly stable for SGD, then it must satisfy H(θ) F O( B/η), where H(θ) F,B,η denote the Frobenius norm of Hessian at θ, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum exponentially fast. Hence, for minima accessible to SGD, the sharpness--as measured by the Frobenius norm of the Hessian--is bounded independently of the model size and sample size. The key to obtaining these results is exploiting the particular structure of SGD noise: The noise concentrates in sharp directions of local landscape and the magnitude is proportional to loss value. This alignment property of SGD noise provably holds for linear networks and random feature models (RFMs), and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are also justified by extensive experiments on CIFAR-10 dataset.

We study the population loss landscape of two-layer ReLU networks of the form $\sum_{k=1}^K \mathrm{ReLU}(w_k^\top x)$ in a realisable teacher-student setting with Gaussian covariates. We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space. This perspective reveals a hierarchical structure of minima: they are typically isolated in the well-specified regime, but become connected by flat directions as network width increases. In this overparameterised regime, global minima become increasingly accessible, attracting the dynamics and reducing convergence to spurious solutions. Overall, our results reveal intrinsic limitations of common simplifying assumptions, which may miss essential features of the loss landscape even in minimal neural network models.

It is generally accepted that starting neural networks training with large learning rates (LRs) improves generalization. Following a line of research devoted to understanding this effect, we conduct an empirical study in a controlled setting focusing on two questions: 1) how large an initial LR is required for obtaining optimal quality, and 2) what are the key differences between models trained with different LRs? We discover that only a narrow range of initial LRs slightly above the convergence threshold lead to optimal results after fine-tuning with a small LR or weight averaging. By studying the local geometry of reached minima, we observe that using LRs from this optimal range allows for the optimization to locate a basin that only contains high-quality minima. Additionally, we show that these initial LRs result in a sparse set of learned features, with a clear focus on those most relevant for the task. In contrast, starting training with too small LRs leads to unstable minima and attempts to learn all features simultaneously, resulting in poor generalization. Conversely, using initial LRs that are too large fails to detect a basin with good solutions and extract meaningful patterns from the data.

How to balance the learning'sensitivity-stability' upon new task training and memory preserving is critical in CL to resolve catastrophic forgetting. Improving model generalization ability within each learning phase is one solution to help CL learning overcome the gap in the joint knowledge space. Zeroth-order loss landscape sharpness-aware minimization is a strong training regime improving model generalization in transfer learning compared with optimizer like SGD. It has also been introduced into CL to improve memory representation or learning efficiency. However, zeroth-order sharpness alone could favors sharper over flatter minima in certain scenarios, leading to a rather sensitive minima rather than a global optima. To further enhance learning stability, we propose a Continual Flatness (C-Flat) method featuring a flatter loss landscape tailored for CL. C-Flat could be easily called with only one line of code and is plug-and-play to any CL methods. A general framework of C-Flat applied to all CL categories and a thorough comparison with loss minima optimizer and flat minima based CL approaches is presented in this paper, showing that our method can boost CL performance in almost all cases. Code is available at https://github.com/WanNaa/C-Flat.

We study the set of continuous functions that admit no spurious local optima (i.e.