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Title: Gradientless Descent: High-Dimensional Zeroth-Order Optimization

Paper content:
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 we show that for {\it any monotone transform} of a smooth and strongly convex objective with latent dimension $k \ge n$, we present a novel analysis that shows convergence within an $\epsilon$-ball of the optimum in $O(kQ\log(n)\log(R/\epsilon))$ evaluations, 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 synthetic and MuJoCo benchmarks.
We consider the problem of zeroth-order optimization (also known as gradient-free optimization, or bandit optimization), where our goal is to minimize an objective function f : R n → R with as few evaluations of f (x) as possible.
For many practical and interesting objective functions, gradients are difficult to compute and there is still a need for zeroth-order optimization in applications such as reinforcement learning (Mania et al., 2018; Salimans et al., 2017; Choromanski et al., 2018) , attacking neural networks Papernot et al., 2017) , hyperparameter tuning of deep networks (Snoek et al., 2012) , and network control (Liu et al., 2017) .
The standard approach to zeroth-order optimization is, ironically, to estimate the gradients from function values and apply a first-order optimization algorithm (Flaxman et al., 2005) .
Nesterov & Spokoiny (2011) analyze this class of algorithms as gradient descent on a Gaussian smoothing of the objective and gives an accelerated O(n √ Q log((LR 2 + F )/ )) iteration complexity for an LLipschitz convex function with condition number Q and R = x 0 − x * and F = f (x 0 ) − f (x * ).
They propose a two-point evaluation scheme that constructs gradient estimates from the difference between function values at two points that are close to each other.
This scheme was extended by (Duchi et al., 2015) for stochastic settings, by (Ghadimi & Lan, 2013) for nonconvex settings, and by (Shamir, 2017) for non-smooth and non-Euclidean norm settings.
Since then, first-order techniques such as variance reduction (Liu et al., 2018) , conditional gradients (Balasubramanian & Ghadimi, 2018) , and diagonal preconditioning (Mania et al., 2018) have been successfully adopted in this setting.
This class of algorithms are also known as stochastic search, random search, or (natural) evolutionary strategies and have been augmented with a variety of heuristics, such as the popular CMA-ES (Auger & Hansen, 2005) .
These algorithms, however, suffer from high variance due to non-robust local minima or highly non-smooth objectives, which are common in the fields of deep learning and reinforcement learn-ing.
Mania et al. (2018) notes that gradient variance increases as training progresses due to higher variance in the objective functions, since often parameters must be tuned precisely to achieve reasonable models.
Therefore, some attention has shifted into direct search algorithms that usually finds a descent direction u and moves to x + δu, where the step size is not scaled by the function difference.
The first approaches for direct search were based on deterministic approaches with a positive spanning set and date back to the 1950s (Brooks, 1958) .
Only recently have theoretical bounds surfaced, with Gratton et al. (2015) giving an iteration complexity that is a large polynomial of n and Dodangeh & Vicente (2016) giving an improved O(n 2 L 2 / ).
Stochastic approaches tend to have better complexities: Stich et al. (2013) uses line search to give a O(nQ log(F/ )) iteration complexity for convex functions with condition number Q and most recently, Gorbunov et al. (2019) uses importance sampling to give a O(nQ log(F/ )) complexity for convex functions with average condition numberQ, assuming access to sampling probabilities.
Stich et al. (2013) notes that direct search algorithms are invariant under monotone transforms of the objective, a property that might explain their robustness in high-variance settings.
In general, zeroth order optimization suffers an at least linear dependence on input dimension n and recent works have tried to address this limitation when n is large but f (x) admits a low-dimensional structure.
Some papers assume that f (x) depends only on k coordinates and Wang et al. (2017) applies Lasso to find the important set of coordinates, whereas Balasubramanian & Ghadimi (2018) simply change the step size to achieve an O(k(log(n)/ )
2 ) iteration complexity.
Other papers assume more generally that f (x) = g(P A x) only depends on a k-dimensional subspace given by the range of P A and Djolonga et al. (2013) apply low-rank approximation to find the low-dimensional subspace while Wang et al. (2013) use random embeddings.
Hazan et al. (2017) assume that f (x) is a sparse collection of k-degree monomials on the Boolean hypercube and apply sparse recovery to achieve a O(n k ) runtime bound.
We will show that under the case that f (x) = g(P A x), our algorithm will inherently pick up any low-dimensional structure in f (x) and achieve a convergence rate that depends on k log(n).
This initial convergence rate survives, even if we perturb f (x) = g(P A x) + h(x), so long as h(x) is sufficiently small.
We will not cover the whole variety of black-box optimization methods, such as Bayesian optimization or genetic algorithms.
In general, these methods attempt to solve a broader problem (e.g. multiple optima), have weaker theoretical guarantees and may require substantial computation at each step: e.g. Bayesian optimization generally has theoretical iteration complexities that grow exponentially in dimension, and CMA-ES lacks provable complexity bounds beyond convex quadratic functions.
In addition to the slow runtime and weaker guarantees, Bayesian optimization assumes the success of an inner optimization loop of the acquisition function.
This inner optimization is often implemented with many iterations of a simpler zeroth-order methods, justifying the need to understand gradient-less descent algorithms within its own context.

assistant: text: Gradientless Descent is a provably efficient gradient-free algorithm that is monotone-invariant and fast for high-dimensional zero-th order optimization.