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We consider the case of derivative-free algorithms for non-convex optimization, also known as zero order algorithms, that use only function evaluations rather than gradients. For a wide variety of gradient approximators based on finite…
In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper-…
We address the problem of computing stationary points for non-smooth, non-convex optimization problems. While this topic is well studied in the smooth setting, fewer algorithmic and theoretical results exist for the non-smooth case. Within…
We prove the local convergence to minima and estimates on the rate of convergence for the stochastic gradient descent method in the case of not necessarily globally convex nor contracting objective functions. In particular, the results are…
In this paper, we consider nonconvex minimax optimization, which is gaining prominence in many modern machine learning applications such as GANs. Large-scale edge-based collection of training data in these applications calls for…
We contribute the first provable guarantees of global convergence to Nash equilibria (NE) in two-player zero-sum convex Markov games (cMGs) by using independent policy gradient methods. Convex Markov games, recently defined by Gemp et al.…
We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the…
Many emerging applications - such as adversarial training, AI alignment, and robust optimization - can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While…
Many problems in high-dimensional statistics and optimization involve minimization over nonconvex constraints-for instance, a rank constraint for a matrix estimation problem-but little is known about the theoretical properties of such…
Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are…
This paper introduces a class of two-stage stochastic minimax problems where the first-stage objective function is nonconvex-concave while the second-stage objective function is strongly convex-concave. We establish properties of the…
In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The…
Algorithmic reproducibility measures the deviation in outputs of machine learning algorithms upon minor changes in the training process. Previous work suggests that first-order methods would need to trade-off convergence rate (gradient…
This paper presents a comprehensive analysis of a broad range of variations of the stochastic proximal point method (SPPM). Proximal point methods have attracted considerable interest owing to their numerical stability and robustness…
This work aims to solve a stochastic nonconvex nonsmooth composite optimization problem. Previous works on composite optimization problem requires the major part to satisfy Lipschitz smoothness or some relaxed smoothness conditions, which…
Minimax problems of the form $\min_x \max_y \Psi(x,y)$ have attracted increased interest largely due to advances in machine learning, in particular generative adversarial networks. These are typically trained using variants of stochastic…
This paper presents a Successive Convexification ($ \texttt{SCvx} $) algorithm to solve a class of non-convex optimal control problems with certain types of state constraints. Sources of non-convexity may include nonlinear dynamics and…
Policy gradients methods apply to complex, poorly understood, control problems by performing stochastic gradient descent over a parameterized class of polices. Unfortunately, even for simple control problems solvable by standard dynamic…
We propose local symplectic surgery, a two-timescale procedure for finding local Nash equilibria in two-player zero-sum games. We first show that previous gradient-based algorithms cannot guarantee convergence to local Nash equilibria due…
Recently, the problem of local minima in very high dimensional non-convex optimization has been challenged and the problem of saddle points has been introduced. This paper introduces a dynamic type of normalization that forces the system to…