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Newton's method for polynomial root finding is one of mathematics' most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge…
Meta-learning methods aim to build learning algorithms capable of quickly adapting to new tasks in low-data regime. One of the most difficult benchmarks of such algorithms is a one-shot learning problem. In this setting many algorithms face…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
Stochastic Bilevel optimization usually involves minimizing an upper-level (UL) function that is dependent on the arg-min of a strongly-convex lower-level (LL) function. Several algorithms utilize Neumann series to approximate certain…
Deep learning is formulated as a discrete-time optimal control problem. This allows one to characterize necessary conditions for optimality and develop training algorithms that do not rely on gradients with respect to the trainable…
This work analyzes the stochastic approximation algorithm with non-decaying gains as applied in time-varying problems. The setting is to minimize a sequence of scalar-valued loss functions $f_k(\cdot)$ at sampling times $\tau_k$ or to…
In the first part of this dissertation research, we develop a modular framework that can serve as a recipe for constructing and analyzing iterative algorithms for convex optimization. Specifically, our work casts optimization as iteratively…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
Nonconvex regularization has been popularly used in low-rank matrix learning. However, extending it for low-rank tensor learning is still computationally expensive. To address this problem, we develop an efficient solver for use with a…
We develop and analyze DASHA: a new family of methods for nonconvex distributed optimization problems. When the local functions at the nodes have a finite-sum or an expectation form, our new methods, DASHA-PAGE and DASHA-SYNC-MVR, improve…
During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as…
We propose an inexact variable-metric proximal point algorithm to accelerate gradient-based optimization algorithms. The proposed scheme, called QNing can be notably applied to incremental first-order methods such as the stochastic…
Topology optimization under uncertainty or reliability-based topology optimization is usually numerically very expensive. This is mainly due to the fact that an accurate evaluation of the probabilistic model requires the system to be…
We consider $d$-dimensional linear stochastic approximation algorithms (LSAs) with a constant step-size and the so called Polyak-Ruppert (PR) averaging of iterates. LSAs are widely applied in machine learning and reinforcement learning…
Stochastic momentum methods have been widely adopted in training deep neural networks. However, their theoretical analysis of convergence of the training objective and the generalization error for prediction is still under-explored. This…
This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general…
Random reshuffling with momentum (RRM) corresponds to the SGD optimizer with momentum option enabled, as found in many machine learning libraries like PyTorch and TensorFlow. Despite its widespread use, the convergence properties of RRM do…
We study accelerated optimization methods in the Gaussian phase retrieval problem. In this setting, we prove that gradient methods with Polyak or Nesterov momentum have similar implicit regularization to gradient descent. This implicit…
We study the problem of solving strongly convex and smooth unconstrained optimization problems using stochastic first-order algorithms. We devise a novel algorithm, referred to as Recursive One-Over-T SGD (ROOT-SGD), based on an easily…
We introduce a new second-order inertial optimization method for machine learning called INNA. It exploits the geometry of the loss function while only requiring stochastic approximations of the function values and the generalized…