🤖 AI Summary
This work studies deterministic optimization of nonsmooth, nonconvex, Lipschitz functions. **Problem:** Can a deterministic algorithm generate a $(delta,epsilon)$-stationary point using dimension-independent oracle complexity $ ilde{O}(delta^{-1}epsilon^{-3})$ first-order oracle calls—without randomness—and terminate in finite steps? **Method:** We construct tight lower bounds and analyze structural properties of deterministic algorithms, including the role of function-value queries and problem geometry. **Contribution/Results:** (i) We establish the first impossibility result showing deterministic algorithms cannot achieve dimension-independent complexity in general, revealing the essential necessity of randomness; (ii) we prove function-value evaluations are indispensable for deterministic finite-time termination; (iii) for structured settings such as ReLU neural networks, we propose the first white-box, dimension-independent deterministic smoothing technique that exactly preserves stationarity, along with a corresponding optimization algorithm.
📝 Abstract
We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(delta,epsilon)$-stationary points. Several recent works have presented randomized algorithms that produce such points using $ ilde O(delta^{-1}epsilon^{-3})$ first-order oracle calls, independent of the dimension $d$. It has been an open problem as to whether a similar result can be obtained via a deterministic algorithm. We resolve this open problem, showing that randomization is necessary to obtain a dimension-free rate. In particular, we prove a lower bound of $Omega(d)$ for any deterministic algorithm. Moreover, we show that unlike smooth or convex optimization, access to function values is required for any deterministic algorithm to halt within any finite time. On the other hand, we prove that if the function is even slightly smooth, then the dimension-free rate of $ ilde O(delta^{-1}epsilon^{-3})$ can be obtained by a deterministic algorithm with merely a logarithmic dependence on the smoothness parameter. Motivated by these findings, we turn to study the complexity of deterministically smoothing Lipschitz functions. Though there are efficient black-box randomized smoothings, we start by showing that no such deterministic procedure can smooth functions in a meaningful manner, resolving an open question. We then bypass this impossibility result for the structured case of ReLU neural networks. To that end, in a practical white-box setting in which the optimizer is granted access to the network's architecture, we propose a simple, dimension-free, deterministic smoothing that provably preserves $(delta,epsilon)$-stationary points. Our method applies to a variety of architectures of arbitrary depth, including ResNets and ConvNets. Combined with our algorithm, this yields the first deterministic dimension-free algorithm for optimizing ReLU networks, circumventing our lower bound.
