Fast Rates for Online Gradient Descent Without Strong Convexity via Hoffman's Bound

by   Dan Garber, et al.

Hoffman's classical result gives a bound on the distance of a point from a convex and compact polytope in terms of the magnitude of violation of the constraints. Recently, several results showed that Hoffman's bound can be used to derive strongly-convex-like rates for first-order methods for convex optimization of curved, though not strongly convex, functions, over polyhedral sets. In this work, we use this classical result for the first time to obtain faster rates for online convex optimization over polyhedral sets with curved convex, though not strongly convex, loss functions. Mainly, we show that under several reasonable assumptions on the data, the standard Online Gradient Descent (OGD) algorithm guarantees logarithmic regret. To the best of our knowledge, the only previous algorithm to achieve logarithmic regret in the considered settings is the Online Newton Step algorithm which requires quadratic (in the dimension) memory and to solve a linear system on each iteration, which greatly limits its applicability to large-scale problems. We also show that in the corresponding stochastic convex optimization setting, Stochastic Gradient Descent achieves convergence rate of 1/t, matching the strongly-convex case.


page 1

page 2

page 3

page 4


Online Strongly Convex Optimization with Unknown Delays

We investigate the problem of online convex optimization with unknown de...

A Linearly Convergent Conditional Gradient Algorithm with Applications to Online and Stochastic Optimization

Linear optimization is many times algorithmically simpler than non-linea...

Less Regret via Online Conditioning

We analyze and evaluate an online gradient descent algorithm with adapti...

Variants of RMSProp and Adagrad with Logarithmic Regret Bounds

Adaptive gradient methods have become recently very popular, in particul...

Dropping Convexity for Faster Semi-definite Optimization

We study the minimization of a convex function f(X) over the set of n× n...

Curvature of Feasible Sets in Offline and Online Optimization

It is known that the curvature of the feasible set in convex optimizatio...

Second-Order Kernel Online Convex Optimization with Adaptive Sketching

Kernel online convex optimization (KOCO) is a framework combining the ex...

Please sign up or login with your details

Forgot password? Click here to reset