DeepAI

# Quasi-Newton Steps for Efficient Online Exp-Concave Optimization

The aim of this paper is to design computationally-efficient and optimal algorithms for the online and stochastic exp-concave optimization settings. Typical algorithms for these settings, such as the Online Newton Step (ONS), can guarantee a O(dln T) bound on their regret after T rounds, where d is the dimension of the feasible set. However, such algorithms perform so-called generalized projections whenever their iterates step outside the feasible set. Such generalized projections require Ω(d^3) arithmetic operations even for simple sets such a Euclidean ball, making the total runtime of ONS of order d^3 T after T rounds, in the worst-case. In this paper, we side-step generalized projections by using a self-concordant barrier as a regularizer to compute the Newton steps. This ensures that the iterates are always within the feasible set without requiring projections. This approach still requires the computation of the inverse of the Hessian of the barrier at every step. However, using the stability properties of the Newton steps, we show that the inverse of the Hessians can be efficiently approximated via Taylor expansions for most rounds, resulting in a O(d^2 T +d^ω√(T)) total computational complexity, where ω is the exponent of matrix multiplication. In the stochastic setting, we show that this translates into a O(d^3/ϵ) computational complexity for finding an ϵ-suboptimal point, answering an open question by Koren 2013. We first show these new results for the simple case where the feasible set is a Euclidean ball. Then, to move to general convex set, we use a reduction to Online Convex Optimization over the Euclidean ball. Our final algorithm can be viewed as a more efficient version of ONS.

• 14 publications
• 8 publications
11/10/2021

### Efficient Projection-Free Online Convex Optimization with Membership Oracle

In constrained convex optimization, existing methods based on the ellips...
05/23/2022

### Exploiting the Curvature of Feasible Sets for Faster Projection-Free Online Learning

In this paper, we develop new efficient projection-free algorithms for O...
02/27/2022

### Thinking Outside the Ball: Optimal Learning with Gradient Descent for Generalized Linear Stochastic Convex Optimization

We consider linear prediction with a convex Lipschitz loss, or more gene...
06/22/2020

### Beyond O(√(T)) Regret for Constrained Online Optimization: Gradual Variations and Mirror Prox

We study constrained online convex optimization, where the constraints c...
11/10/2019

### A unified approach for projections onto the intersection of ℓ_1 and ℓ_2 balls or spheres

This paper focuses on designing a unified approach for computing the pro...
11/04/2019

### Generalized Self-concordant Hessian-barrier algorithms

Many problems in statistical learning, imaging, and computer vision invo...
06/22/2021

### Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes

Optimization algorithms such as projected Newton's method, FISTA, mirror...