Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries - a broad subclass of continuous symmetries present in the model architecture or loss function - interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a "Noether flow") of the parameters θ along the degenerate direction to a unique initialization-independent fixed point θ

We show that GWG exhibits strong convergence guarantees.

The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been associated with favorable features. While prior work often relies on linearization to determine stability, it remains unclear whether linearized dynamics faithfully capture the full nonlinear behavior. Recent work has shown that GD may stably oscillate near a linearly unstable minimum and still converge once the step size decays, indicating that linear analysis can be misleading. In this work, we explicitly study the effect of nonlinear terms. Specifically, we derive an exact criterion for stable oscillations of GD near minima in the multivariate setting. Our condition depends on high-order derivatives, generalizing existing results. Extending the analysis to stochastic gradient descent (SGD), we show that nonlinear dynamics can diverge in expectation even if a single batch is unstable. This implies that stability can be dictated by a single batch that oscillates unstably, rather than an average effect, as linear analysis suggests. Finally, we prove that if all batches are linearly stable, the nonlinear dynamics of SGD are stable in expectation.

Accordingly, extensive evaluations demonstrate the significance of our proposal and its scalability to multimodal large models in boosting robustness.

Effective resistances have a broad range of algorithmic implications.

Our estimators assume that there is no interference between the units, i.e., that the treatment of one The first assumption is quite realistic as in most peer review systems the reviewers cannot see other reviews until they submit their own. The second assumption is important to understand, as there could be "batch effects": a Monte Carlo methods to tightly estimate these covariances. AAAI datasets, we sampled 1 million assignments and computed the empirical covariance. In our setting, small amounts of attrition (relative to the number of policy-induced positivity violations) mean that the fraction of data that is missing is not exactly known before assignment, but almost. To get more robust estimates of the performance, we repeat this process 10 times.

In our case, it boils down to: ' The smoothing effect induced by the average helps to reduce the visual noise, and hence improves the explanations. For the experiment, m and are the same as SmoothGrad. We start by deriving the closed form of Saliency (SA) and naturally Gradient-Input (GI): ' The case of V arGrad is specific, as the gradient of a linear system being constant, its variance is null. W We recall that for Gradient Input, Integrated Gradients, Occlusion, ' It was quickly realized that they unified properties of various domains such as graph theory, linear algebra or geometry. Later, in the '60s, a connection was made At each step, the insertion metric selects the concepts of maximum score given a cardinality constraint.