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.

Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex nonconcave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the "hidden" convex-concave game. We provide conditions under which vanilla GDA provably converges not merely to local Nash, but the actual von-Neumann solution. If the hidden game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence results are non-local despite working in the setting of non-convex non-concave games. Critically, under proper assumptions we combine the Center-Stable Manifold Theorem along with novel type of initialization dependent Lyapunov functions to prove that almost all initial conditions converge to the solution. Finally, we discuss diverse applications of our framework ranging from generative adversarial networks to evolutionary biology.

Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex nonconcave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the "hidden" convex-concave game. We provide conditions under which vanilla GDA provably converges not merely to local Nash, but the actual von-Neumann solution. If the hidden game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence results are non-local despite working in the setting of non-convex non-concave games. Critically, under proper assumptions we combine the Center-Stable Manifold Theorem along with novel type of initialization dependent Lyapunov functions to prove that almost all initial conditions converge to the solution. Finally, we discuss diverse applications of our framework ranging from generative adversarial networks to evolutionary biology.

A.1 Connection to online learning In Section 2 we motivated the update (2) as a way to adjust the size of our prediction sets in response to the realized historical miscoverage frequency. Alternatively, one could also derive (2) as an online gradient descent algorithm with respect to the pinball loss. To be more precise let t:= sup{: Yt 2 Cˆt()}, where we remark that Cˆt( t) can be thought of as the smallest prediction set containing Yt. Because the pinball loss is convex, this gradient descent update falls within a well understood class of algorithms that have been extensively studied in the online learning literature (see e.g. Unfortunately, this notion of regret fails to capture our intuition that t is adaptively tracking the moving target .

This is a canonical problem that admits two difficulties in optimization: 1) non-convexity and 2) non-smoothness (due to unbalancedness of U and V). This is also a prototype for more complex problems such as asymmetric matrix sensing and matrix completion. Despite being non-convex and non-smooth, it has been observed empirically that the randomly initialized gradient descent algorithm can solve this problem in polynomial time. Existing theories to explain this phenomenon all require artificial modifications of the algorithm, such as adding noise in each iteration and adding a balancing regularizer to balance the U and V. This paper presents the first proof that shows randomly initialized gradient descent converges to a global minimum of the asymmetric low-rank factorization problem with a polynomial rate. For the proof, we develop 1) a new symmetrization technique to capture the magnitudes of the symmetry and asymmetry, and 2) a quantitative perturbation analysis to approximate matrix derivatives. We believe both are useful for other related non-convex problems.

We disprove this conjecture and show, for two notions of algorithmic stability (including uniform stability), that the stability of Nesterov's accelerated method in fact deteriorates exponentially fast with the number of gradient steps.

Softmax policy gradient is a popular algorithm for policy optimization in singleagent reinforcement learning, particularly since projection is not needed for each gradient update. However, in multi-agent systems, the lack of central coordination introduces significant additional difficulties in the convergence analysis. Even for a stochastic game with identical interest, there can be multiple Nash Equilibria (NEs), which disables proof techniques that rely on the existence of a unique global optimum. Moreover, the softmax parameterization introduces non-NE policies with zero gradient, making it difficult for gradient-based algorithms in seeking NEs. In this paper, we study the finite time convergence of decentralized softmax gradient play in a special form of game, Markov Potential Games (MPGs), which includes the identical interest game as a special case. We investigate both gradient play and natural gradient play, with and without log-barrier regularization. The established convergence rates for the unregularized cases contain a trajectory dependent constant that can be arbitrarily large, whereas the log-barrier regularization overcomes this drawback, with the cost of slightly worse dependence on other factors such as the action set size. An empirical study on an identical interest matrix game confirms the theoretical findings.

Neural networks are known to be susceptible to adversarial samples: small variations of natural examples crafted to deliberately mislead the models. While they can be easily generated using gradient-based techniques in digital and physical scenarios, they often differ greatly from the actual data distribution of natural images, resulting in a trade-off between strength and stealthiness. In this paper, we propose a novel framework dubbed Diffusion-Based Projected Gradient Descent (Diff-PGD) for generating realistic adversarial samples. By exploiting a gradient guided by a diffusion model, Diff-PGD ensures that adversarial samples remain close to the original data distribution while maintaining their effectiveness. Moreover, our framework can be easily customized for specific tasks such as digital attacks, physical-world attacks, and style-based attacks. Compared with existing methods for generating natural-style adversarial samples, our framework enables the separation of optimizing adversarial loss from other surrogate losses (e.g., content/smoothness/style loss), making it more stable and controllable. Finally, we demonstrate that the samples generated using Diff-PGD have better transferability and anti-purification power than traditional gradient-based methods.

Estimating the per-state expected cumulative rewards is a critical aspect of reinforcement learning approaches, however the experience is obtained, but standard deep neural-network function-approximation methods are often inefficient in this setting. An alternative approach, exemplified by value iteration networks, is to learn transition and reward models of a latent Markov decision process whose value predictions fit the data. This approach has been shown empirically to converge faster to a more robust solution in many cases, but there has been little theoretical study of this phenomenon. In this paper, we explore such implicit representations of value functions via theory and focused experimentation. We prove that, for a linear parametrization, gradient descent converges to global optima despite nonlinearity and non-convexity introduced by the implicit representation. Furthermore, we derive convergence rates for both cases which allow us to identify conditions under which stochastic gradient descent (SGD) with this implicit representation converges substantially faster than its explicit counterpart. Finally, we provide empirical results in some simple domains that illustrate the theoretical findings.

Estimating the per-state expected cumulative rewards is a critical aspect of reinforcement learning approaches, however the experience is obtained, but standard deep neural-network function-approximation methods are often inefficient in this setting. An alternative approach, exemplified by value iteration networks, is to learn transition and reward models of a latent Markov decision process whose value predictions fit the data. This approach has been shown empirically to converge faster to a more robust solution in many cases, but there has been little theoretical study of this phenomenon. In this paper, we explore such implicit representations of value functions via theory and focused experimentation. We prove that, for a linear parametrization, gradient descent converges to global optima despite nonlinearity and non-convexity introduced by the implicit representation. Furthermore, we derive convergence rates for both cases which allow us to identify conditions under which stochastic gradient descent (SGD) with this implicit representation converges substantially faster than its explicit counterpart. Finally, we provide empirical results in some simple domains that illustrate the theoretical findings.