arXiv subjects

Nadav Hallak

Publications and source records attributed to Nadav Hallak.

6 recordsLinked to original sources

arXiv2026

Randomized and Oracle-Based Feasible Directions Methods for Nonsmooth Nonconvex Constrained Optimization

Minimizing nonsmooth nonconvex objectives over general convex sets using no derivatives of the objective, together with structural access to the constraint set, poses a fundamental geometric difficulty: at boundary points, the feasible direction cone may fail to be closed, preventing the construction of finite sets of search directions that capture all feasible descent directions. We develop a feasible-directions framework for problems structured as the difference between a continuously differentiable function and a convex function over a closed convex set, and access the objective without knowledge of its decomposition, through exact line searches along feasible segments. To address the geometric obstruction, we introduce $η$-feasible directions, which remain uniformly separated from the boundary of the feasible direction cone, enabling finite-direction search even in the presence of nonlinear constraints. We present two algorithmic realizations of this framework. The first assumes access to an $η$-direction oracle and extends feasible-directions methods from linear to general convex constraints. The second replaces the oracle by a noise-augmented randomization scheme that constructs spanning directions from the affine-hull inner normal and almost surely avoids nonsmooth boundary points when initialized outside that set. Under curvature bounds and Lipschitz continuity of the smooth component's gradient, we jointly control the spanning coefficients and feasible step lengths of the oracle construction. This yields finite-time approximate $d$-stationarity along every unit feasible direction admitting a prescribed positive step length. For both methods, we also establish a rate in the proximal directional minimization (PDM) measure, assuming uniformly bounded direction-set sizes. Randomized exploration ensures that, almost surely, every accumulation point is $d$-stationary.

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arXiv2026

Achieving Directional-Stationarity from a Single Random Direction Step

This paper addresses the challenge of obtaining strong optimality guarantees in constrained nonsmooth nonconvex optimization under mild regularity conditions, namely local Lipschitz continuity and existence of directional derivatives. While standard methods typically ensure weak stationarity notions, achieving directional (d-)stationarity remains nontrivial. We show that a random direction exploration step is sufficient to attain d-stationarity. The proposed approach augments any feasible base method with one additional randomized exploration step per iteration, which samples a direction and step size and accepts the candidate based on a function value comparison. The resulting scheme guarantees that all accumulation points are d-stationary almost surely, independently of the behavior of the underlying method. Moreover, under conditions on the base method, it preserves convergence rates of the base method, as established for the difference-of-convex algorithm (DCA), prox-linear-type schemes, and interpolated normalized gradient descent (INGD). The theoretical results are complemented by numerical experiments illustrating the effect and guarantees of the exploration step.

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arXiv2024

An Augmented Lagrangian Approach to Composite Problems with a Random Linear Operator

We consider the minimization of a sum of a smooth function with a nonsmooth composite function, where the composition is applied on a random linear mapping. This random composite model encompasses many problems, and can especially capture realistic scenarios in which the data is sampled during the optimization process. We propose and analyze a method that combines the classical Augmented Lagrangian framework with a sampling mechanism and adaptive update of the penalty parameter. We show that every accumulation point of the sequence produced by our algorithm is almost surely a critical point.

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arXiv2020

Regret minimization in stochastic non-convex learning via a proximal-gradient approach

Motivated by applications in machine learning and operations research, we study regret minimization with stochastic first-order oracle feedback in online constrained, and possibly non-smooth, non-convex problems. In this setting, the minimization of external regret is beyond reach for first-order methods, so we focus on a local regret measure defined via a proximal-gradient mapping. To achieve no (local) regret in this setting, we develop a prox-grad method based on stochastic first-order feedback, and a simpler method for when access to a perfect first-order oracle is possible. Both methods are min-max order-optimal, and we also establish a bound on the number of prox-grad queries these methods require. As an important application of our results, we also obtain a link between online and offline non-convex stochastic optimization manifested as a new prox-grad scheme with complexity guarantees matching those obtained via variance reduction techniques.

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arXiv2020

Efficient Proximal Mapping of the 1-path-norm of Shallow Networks

We demonstrate two new important properties of the 1-path-norm of shallow neural networks. First, despite its non-smoothness and non-convexity it allows a closed form proximal operator which can be efficiently computed, allowing the use of stochastic proximal-gradient-type methods for regularized empirical risk minimization. Second, when the activation functions is differentiable, it provides an upper bound on the Lipschitz constant of the network. Such bound is tighter than the trivial layer-wise product of Lipschitz constants, motivating its use for training networks robust to adversarial perturbations. In practical experiments we illustrate the advantages of using the proximal mapping and we compare the robustness-accuracy trade-off induced by the 1-path-norm, L1-norm and layer-wise constraints on the Lipschitz constant (Parseval networks).

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arXiv2020

On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems

This paper analyzes the trajectories of stochastic gradient descent (SGD) to help understand the algorithm's convergence properties in non-convex problems. We first show that the sequence of iterates generated by SGD remains bounded and converges with probability $1$ under a very broad range of step-size schedules. Subsequently, going beyond existing positive probability guarantees, we show that SGD avoids strict saddle points/manifolds with probability $1$ for the entire spectrum of step-size policies considered. Finally, we prove that the algorithm's rate of convergence to Hurwicz minimizers is $\mathcal{O}(1/n^{p})$ if the method is employed with a $Θ(1/n^p)$ step-size schedule. This provides an important guideline for tuning the algorithm's step-size as it suggests that a cool-down phase with a vanishing step-size could lead to faster convergence; we demonstrate this heuristic using ResNet architectures on CIFAR.

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