Benjamin Grimmer

Benjamin Grimmer

This work establishes new convergence guarantees for gradient descent in smooth convex optimization via a computer-assisted analysis technique. Our theory allows nonconstant stepsize policies with frequent long steps potentially violating descent by analyzing the overall effect of many iterations at once rather than the typical one-iteration inductions used in most first-order method analyses. We show that long steps, which may increase the objective value in the short term, lead to provably faster convergence in the long term. A conjecture towards proving a faster $O(1/T\log T)$ rate for gradient descent is also motivated along with simple numerical validation.

Ning Liu, Benjamin Grimmer

We consider feasibility and constrained optimization problems defined over smooth and/or strongly convex sets. These notions mirror their popular function counterparts but are much less explored in the first-order optimization literature. We propose new scalable, projection-free, accelerated first-order methods in these settings. Our methods avoid linear optimization or projection oracles, only using cheap one-dimensional linesearches and normal vector computations. Despite this, we derive optimal accelerated convergence guarantees of $O(1/T)$ for strongly convex problems, $O(1/T^2)$ for smooth problems, and accelerated linear convergence given both. Our algorithms and analysis are based on novel characterizations of the Minkowski gauge of smooth and/or strongly convex sets, which may be of independent interest: although the gauge is neither smooth nor strongly convex, we show the gauge squared inherits any structure present in the set.

Thabo Samakhoana, Benjamin Grimmer

We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least $\sim 0.8145\ln(d)$. Hence, LogSumExp is optimal up to small constant factors. However, in small dimensions, we provide stronger, exactly optimal smoothings attaining our lower bound, showing that the entropy-based LogSumExp approach to smoothing is not exactly optimal.

Benjamin Grimmer, Danlin Li

We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a not previously analyzed dual gap for strongly convex optimization. Consequently, our theory provides these classic methods with simple, optimal stopping criteria and optimality certificates at no added computational cost. Our results apply to a wide range of stepsize selections and of non-Lipschitz ill-conditioned problems where the early iterations of the subgradient method may diverge exponentially quickly (a phenomenon which, to the best of our knowledge, no prior works address). Even in the presence of such undesirable behaviors, our theory still ensures and bounds eventual convergence.

Benjamin Grimmer, Haihao Lu, Pratik Worah et al.

Unlike nonconvex optimization, where gradient descent is guaranteed to converge to a local optimizer, algorithms for nonconvex-nonconcave minimax optimization can have topologically different solution paths: sometimes converging to a solution, sometimes never converging and instead following a limit cycle, and sometimes diverging. In this paper, we study the limiting behaviors of three classic minimax algorithms: gradient descent ascent (GDA), alternating gradient descent ascent (AGDA), and the extragradient method (EGM). Numerically, we observe that all of these limiting behaviors can arise in Generative Adversarial Networks (GAN) training and are easily demonstrated for a range of GAN problems. To explain these different behaviors, we study the high-order resolution continuous-time dynamics that correspond to each algorithm, which results in the sufficient (and almost necessary) conditions for the local convergence by each method. Moreover, this ODE perspective allows us to characterize the phase transition between these different limiting behaviors caused by introducing regularization as Hopf Bifurcations.

Benjamin Grimmer, Haihao Lu, Pratik Worah et al.

Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties this poses. In this paper, we study the classic proximal point method (PPM) applied to nonconvex-nonconcave minimax problems. We find that a classic generalization of the Moreau envelope by Attouch and Wets provides key insights. Critically, we show this envelope not only smooths the objective but can convexify and concavify it based on the level of interaction present between the minimizing and maximizing variables. From this, we identify three distinct regions of nonconvex-nonconcave problems. When interaction is sufficiently strong, we derive global linear convergence guarantees. Conversely when the interaction is fairly weak, we derive local linear convergence guarantees with a proper initialization. Between these two settings, we show that PPM may diverge or converge to a limit cycle.

Lijun Ding, Benjamin Grimmer

In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimization problems and from the analysis of bundle methods. Under strong duality and strict complementarity of SDP, our algorithm produces primal and the dual sequences converging in feasibility at a rate of $\tilde{O}(1/ε)$ and in optimality at a rate of $\tilde{O}(1/ε^2)$. Moreover, our algorithm outputs a low rank representation of its approximate solution with distance to the optimal solution at most $O(\sqrtε)$ within $\tilde{O}(1/ε^2)$ iterations.

Benjamin Grimmer

We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global $O(1/\sqrt{T})$ convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor's classic subgradient analysis and implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or Hölder continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergence rate for the stochastic projected subgradient method on convex functions with at most quadratic growth, which improves to $O(1/T)$ under either strong convexity or a weaker quadratic lower bound condition.

Damek Davis, Benjamin Grimmer

In this paper, we introduce a stochastic projected subgradient method for weakly convex (i.e., uniformly prox-regular) nonsmooth, nonconvex functions---a wide class of functions which includes the additive and convex composite classes. At a high-level, the method is an inexact proximal point iteration in which the strongly convex proximal subproblems are quickly solved with a specialized stochastic projected subgradient method. The primary contribution of this paper is a simple proof that the proposed algorithm converges at the same rate as the stochastic gradient method for smooth nonconvex problems. This result appears to be the first convergence rate analysis of a stochastic (or even deterministic) subgradient method for the class of weakly convex functions.