Sharpness-A ware Minimization (SAM) is a recently proposed gradient-based optimizer (Foret et al., ICLR 2021) that greatly improves the prediction performance of deep neural networks.

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.

In the setting of Section 2.1, we assume The proof is based on the following lemma, that we state clearly for another use below. Lemma 2. Let θ H . Then for all β This lemma follows from Hölder's inequality with Applying Hölder's inequality, we get E We start with a few preliminary remarks. By summing for k = 1,...,n and using the bound (17), ϕ We continue the proof of Theorem 1 to prove Theorem 3. By the log-convexity Property 1, for all This proves conclusion 1 of the theorem. Both terms of the equality can be infinite: here we are using the convention stated in Section 2.1 that We can assume that (a) is satisfied, i.e., Thus the theorem below extends Theorem 5. Theorem 6. This theorem is proved at the end of this section.

We study linear approximate value iteration (LAVI) with a generative model. While linear models may accurately represent the optimal value function using a few parameters, several empirical and theoretical studies show the combination of least-squares projection with the Bellman operator may be expansive, thus leading LAVI to amplify errors over iterations and eventually diverge. We introduce an algorithm that approximates value functions by combining Q-values estimated at a set of \textit{anchor} states. Our algorithm tries to balance the generalization and compactness of linear methods with the small amplification of errors typical of interpolation methods. We prove that if the features at any state can be represented as a convex combination of features at the anchor points, then errors are propagated linearly over iterations (instead of exponentially) and our method achieves a polynomial sample complexity bound in the horizon and the number of anchor points. These findings are confirmed in preliminary simulations in a number of simple problems where a traditional least-square LAVI method diverges.

In machine learning, stochastic gradient descent (SGD) is widely deployed to train models using highly non-convex objectives with equally complex noise models. Unfortunately, SGD theory often makes restrictive assumptions that fail to capture the non-convexity of real problems, and almost entirely ignore the complex noise models that exist in practice. In this work, we demonstrate the restrictiveness of these assumptions using three canonical models in machine learning. Then, we develop novel theory to address this shortcoming in two ways. First, we establish that SGD's iterates will either globally converge to a stationary point or diverge under nearly arbitrary nonconvexity and noise models. Under a slightly more restrictive assumption on the joint behavior of the non-convexity and noise model that generalizes current assumptions in the literature, we show that the objective function cannot diverge, even if the iterates diverge. As a consequence of our results, SGD can be applied to a greater range of stochastic optimization problems with confidence about its global convergence behavior and stability.

Recurrent neural networks (RNNs) are wide-spread machine learning tools for modeling sequential and time series data. They are notoriously hard to train because their loss gradients backpropagated in time tend to saturate or diverge during training. This is known as the exploding and vanishing gradient problem. Previous solutions to this issue either built on rather complicated, purpose-engineered architectures with gated memory buffers, or - more recently - imposed constraints that ensure convergence to a fixed point or restrict (the eigenspectrum of) the recurrence matrix. Such constraints, however, convey severe limitations on the expressivity of the RNN.