The recipe behind the success of deep learning has been the combination of neural networks and gradient-based optimization. Understanding the behavior of gradient descent however, and particularly its instability, has lagged behind its empirical success. To add to the theoretical tools available to study gradient descent we propose the principal flow (PF), a continuous time flow that approximates gradient descent dynamics. To our knowledge, the PF is the only continuous flow that captures the divergent and oscillatory behaviors of gradient descent, including escaping local minima and saddle points. Through its dependence on the eigendecomposition of the Hessian the PF sheds light on the recently observed edge of stability phenomena in deep learning. Using our new understanding of instability we propose a learning rate adaptation method which enables us to control the trade-off between training stability and test set evaluation performance.

Bilevel optimization enjoys a wide range of applications in hyper-parameter optimization, meta-learning and reinforcement learning. However, bilevel optimization problems are difficult to solve. Recent progress on scalable bilevel algorithms mainly focuses on bilevel optimization problems where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle the bilevel problem through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the solutions of the original bilevel problem. Further, we propose the penalty-based bilevel gradient descent (PBGD) algorithm and establish its finite-time convergence for the constrained bilevel problem without lower-level strong convexity. Experiments showcase the efficiency of the proposed PBGD algorithm.

We present a novel, fast (exponential rate adaption), ab initio (hyper-parameter-free) gradient based optimizer algorithm. The main idea of the method is to adapt the learning rate $\alpha$ by situational awareness, mainly striving for orthogonal neighboring gradients. The method has a high success and fast convergence rate and does not rely on hand-tuned parameters giving it greater universality. It can be applied to problems of any dimensions n and scales only linearly (of order O(n)) with the dimension of the problem. It optimizes convex and non-convex continuous landscapes providing some kind of gradient. In contrast to the Ada-family (AdaGrad, AdaMax, AdaDelta, Adam, etc.) the method is rotation invariant: optimization path and performance are independent of coordinate choices. The impressive performance is demonstrated by extensive experiments on the MNIST benchmark data-set against state-of-the-art optimizers. We name this new class of optimizers after its core idea Exponential Learning Rate Adaption - ELRA. We present it in two variants c2min and p2min with slightly different control. The authors strongly believe that ELRA will open a completely new research direction for gradient descent optimize.

Solving linear inverse problems plays a crucial role in numerous applications. Algorithm unfolding based, model-aware data-driven approaches have gained significant attention for effectively addressing these problems. Learned iterative soft-thresholding algorithm (LISTA) and alternating direction method of multipliers compressive sensing network (ADMM-CSNet) are two widely used such approaches, based on ISTA and ADMM algorithms, respectively. In this work, we study optimization guarantees, i.e., achieving near-zero training loss with the increase in the number of learning epochs, for finite-layer unfolded networks such as LISTA and ADMM-CSNet with smooth soft-thresholding in an over-parameterized (OP) regime. We achieve this by leveraging a modified version of the Polyak-Lojasiewicz, denoted PL$^*$, condition. Satisfying the PL$^*$ condition within a specific region of the loss landscape ensures the existence of a global minimum and exponential convergence from initialization using gradient descent based methods. Hence, we provide conditions, in terms of the network width and the number of training samples, on these unfolded networks for the PL$^*$ condition to hold. We achieve this by deriving the Hessian spectral norm of these networks. Additionally, we show that the threshold on the number of training samples increases with the increase in the network width. Furthermore, we compare the threshold on training samples of unfolded networks with that of a standard fully-connected feed-forward network (FFNN) with smooth soft-thresholding non-linearity. We prove that unfolded networks have a higher threshold value than FFNN. Consequently, one can expect a better expected error for unfolded networks than FFNN.

Model-free Reinforcement Learning (RL) generally suffers from poor sample complexity, mostly due to the need to exhaustively explore the state-action space to find well-performing policies. On the other hand, we postulate that expert knowledge of the system often allows us to design simple rules we expect good policies to follow at all times. In this work, we hence propose a simple yet effective modification of continuous actor-critic frameworks to incorporate such rules and avoid regions of the state-action space that are known to be suboptimal, thereby significantly accelerating the convergence of RL agents. Concretely, we saturate the actions chosen by the agent if they do not comply with our intuition and, critically, modify the gradient update step of the policy to ensure the learning process is not affected by the saturation step. On a room temperature control case study, it allows agents to converge to well-performing policies up to 6-7x faster than classical agents without computational overhead and while retaining good final performance.

Robust Markov Decision Processes (MDPs) are receiving much attention in learning a robust policy which is less sensitive to environment changes. There are an increasing number of works analyzing sample-efficiency of robust MDPs. However, there are two major barriers to applying robust MDPs in practice. First, most works study robust MDPs in a model-based regime, where the transition probability needs to be estimated and requires a large amount of memories $\mathcal{O}(|\mathcal{S}|^2|\mathcal{A}|)$. Second, prior work typically assumes a strong oracle to obtain the optimal solution as an intermediate step to solve robust MDPs. However, in practice, such an oracle does not exist usually. To remove the oracle, we transform the original robust MDPs into an alternative form, which allows us to use stochastic gradient methods to solve the robust MDPs. Moreover, we prove the alternative form still plays a similar role as the original form. With this new formulation, we devise a sample-efficient algorithm to solve the robust MDPs in a model-free regime, which does not require an oracle and trades off a lower storage requirement $\mathcal{O}(|\mathcal{S}||\mathcal{A}|)$ with being able to generate samples from a generative model or Markovian chain. Finally, we validate our theoretical findings via numerical experiments, showing the efficiency with the alternative form of robust MDPs.

Is Stochastic Gradient Descent (SGD) substantially different from Glauber dynamics? This is a fundamental question at the time of understanding the most used training algorithm in the field of Machine Learning, but it received no answer until now. Here we show that in discrete optimization and inference problems, the dynamics of an SGD-like algorithm resemble very closely that of Metropolis Monte Carlo with a properly chosen temperature, which depends on the mini-batch size. This quantitative matching holds both at equilibrium and in the out-of-equilibrium regime, despite the two algorithms having fundamental differences (e.g.\ SGD does not satisfy detailed balance). Such equivalence allows us to use results about performances and limits of Monte Carlo algorithms to optimize the mini-batch size in the SGD-like algorithm and make it efficient at recovering the signal in hard inference problems.

Distributed stochastic gradient descent (SGD) with gradient compression has become a popular communication-efficient solution for accelerating distributed learning. One commonly used method for gradient compression is Top-K sparsification, which sparsifies the gradients by a fixed degree during model training. However, there has been a lack of an adaptive approach to adjust the sparsification degree to maximize the potential of the model's performance or training speed. This paper proposes a novel adaptive Top-K in SGD framework that enables an adaptive degree of sparsification for each gradient descent step to optimize the convergence performance by balancing the trade-off between communication cost and convergence error. Firstly, an upper bound of convergence error is derived for the adaptive sparsification scheme and the loss function. Secondly, an algorithm is designed to minimize the convergence error under the communication cost constraints. Finally, numerical results on the MNIST and CIFAR-10 datasets demonstrate that the proposed adaptive Top-K algorithm in SGD achieves a significantly better convergence rate compared to state-of-the-art methods, even after considering error compensation.

In recent years, there has been an intense debate about how learning in biological neural networks (BNNs) differs from learning in artificial neural networks. It is often argued that the updating of connections in the brain relies only on local information, and therefore a stochastic gradient-descent type optimization method cannot be used. In this paper, we study a stochastic model for supervised learning in BNNs. We show that a (continuous) gradient step occurs approximately when each learning opportunity is processed by many local updates. This result suggests that stochastic gradient descent may indeed play a role in optimizing BNNs.

Physics-informed Neural Network (PINN) is one of the most preeminent solvers of Navier-Stokes equations, which are widely used as the governing equation of blood flow. However, current approaches, relying on full Navier-Stokes equations, are impractical for ultrafast Doppler ultrasound, the state-of-the-art technique for depiction of complex blood flow dynamics \emph{in vivo} through acquired thousands of frames (or, timestamps) per second. In this article, we first propose a novel training framework of PINN for solving Navier-Stokes equations by discretizing Navier-Stokes equations into steady state and sequentially solving steady-state Navier-Stokes equations with transfer learning. The novel training framework is coined as SeqPINN. Upon the success of SeqPINN, we adopt the idea of averaged constant stochastic gradient descent (SGD) as initialization and propose a parallel training scheme for all timestamps. To ensure an initialization that generalizes well, we borrow the concept of Stochastic Weight Averaging Gaussian to perform uncertainty estimation as an indicator of generalizability of the initialization. This algorithm, named SP-PINN, further expedites training of PINN while achieving comparable accuracy with SeqPINN. Finite-element simulations and \emph{in vitro} phantoms of single-branch and trifurcate blood vessels are used to evaluate the performance of SeqPINN and SP-PINN. Results show that both SeqPINN and SP-PINN are manyfold faster than the original design of PINN, while respectively achieving Root Mean Square Errors (RMSEs) of 1.01 cm/s and 1.26 cm/s on the straight vessel and 1.91 cm/s and 2.56 cm/s on the trifurcate blood vessel when recovering blood flow velocities.