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Matthew Hale

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2 papers

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AAAI Conference 2026 Conference Paper

Differentially Private Linear Programming: Reduced Sub-Optimality and Guaranteed Constraint Satisfaction

Alexander Benvenuti
Brendan Bialy
Miriam Dennis
Matthew Hale

Linear programming is a fundamental tool in a wide range of decision systems. However, without privacy protections, sharing the solution to a linear program may reveal information about the underlying data used to formulate it, which may be sensitive. Therefore, in this paper we introduce an approach for protecting sensitive data while formulating and solving a linear program. First, we prove that this method perturbs objectives and constraints in a way that makes them differentially private. Then, we show that (i) privatized problems always have solutions, and (ii) their solutions satisfy the constraints in their corresponding original, non-private problems. The latter result solves an open problem in the literature. Next, we analytically bound the expected sub-optimality of solutions that is induced by privacy. Numerical simulations show that, under a typical privacy setup, the solution produced by our method yields a 65% reduction in sub-optimality compared to the state of the art.

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AAAI Conference 2023 Conference Paper

Linear Regularizers Enforce the Strict Saddle Property

Matthew Ubl
Matthew Hale
Kasra Yazdani

Satisfaction of the strict saddle property has become a standard assumption in non-convex optimization, and it ensures that many first-order optimization algorithms will almost always escape saddle points. However, functions exist in machine learning that do not satisfy this property, such as the loss function of a neural network with at least two hidden layers. First-order methods such as gradient descent may converge to non-strict saddle points of such functions, and there do not currently exist any first-order methods that reliably escape non-strict saddle points. To address this need, we demonstrate that regularizing a function with a linear term enforces the strict saddle property, and we provide justification for only regularizing locally, i.e., when the norm of the gradient falls below a certain threshold. We analyze bifurcations that may result from this form of regularization, and then we provide a selection rule for regularizers that depends only on the gradient of an objective function. This rule is shown to guarantee that gradient descent will escape the neighborhoods around a broad class of non-strict saddle points, and this behavior is demonstrated on numerical examples of non-strict saddle points common in the optimization literature.

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