Gradientless Descent: High-Dimensional Zeroth-Order Optimization
This work addresses optimization problems in high-dimensional spaces where gradient information is unavailable, offering a method with poly-logarithmic dimensionality dependence and monotone invariance, which is incremental but improves upon existing zeroth-order approaches.
The paper tackles high-dimensional zeroth-order optimization by introducing GradientLess Descent (GLD) algorithms that avoid gradient estimation, achieving convergence to an ε-ball of the optimum in O(kQ log(n) log(R/ε)) evaluations for smooth and strongly convex objectives with latent dimension k < n, as validated on BBOB and MuJoCo benchmarks.
Zeroth-order optimization is the process of minimizing an objective $f(x)$, given oracle access to evaluations at adaptively chosen inputs $x$. In this paper, we present two simple yet powerful GradientLess Descent (GLD) algorithms that do not rely on an underlying gradient estimate and are numerically stable. We analyze our algorithm from a novel geometric perspective and present a novel analysis that shows convergence within an $ε$-ball of the optimum in $O(kQ\log(n)\log(R/ε))$ evaluations, for any monotone transform of a smooth and strongly convex objective with latent dimension $k < n$, where the input dimension is $n$, $R$ is the diameter of the input space and $Q$ is the condition number. Our rates are the first of its kind to be both 1) poly-logarithmically dependent on dimensionality and 2) invariant under monotone transformations. We further leverage our geometric perspective to show that our analysis is optimal. Both monotone invariance and its ability to utilize a low latent dimensionality are key to the empirical success of our algorithms, as demonstrated on BBOB and MuJoCo benchmarks.