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We provide tight upper and lower bounds on the complexity of minimizing the average of $m$ convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs…
We establish new lower-bounds for the information complexity of mixed-integer convex optimization under two "bit-wise" oracles. The first oracle provides bits of first-order information in the standard coordinate model, and the second…
In this paper, we initiate a systematic investigation of differentially private algorithms for convex empirical risk minimization. Various instantiations of this problem have been studied before. We provide new algorithms and matching lower…
In this paper, we leverage an information-theoretic upper bound on the maximum admissible level of noise (MALN) in convex Lipschitz-continuous zeroth-order optimisation to establish corresponding upper bounds for classes of strongly convex…
Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show for any Lipschitz, weakly convex objectives and…
We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension…
Numerical evaluations have definitively shown that, for deep learning optimizers such as stochastic gradient descent, momentum, and adaptive methods, the number of steps needed to train a deep neural network halves for each doubling of the…
We show that any memory-constrained, first-order algorithm which minimizes $d$-dimensional, $1$-Lipschitz convex functions over the unit ball to $1/\mathrm{poly}(d)$ accuracy using at most $d^{1.25 - \delta}$ bits of memory must make at…
The regret bound of an optimization algorithms is one of the basic criteria for evaluating the performance of the given algorithm. By inspecting the differences between the regret bounds of traditional algorithms and adaptive one, we…
Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these…
This paper discusses several (sub)gradient methods attaining the optimal complexity for smooth problems with Lipschitz continuous gradients, nonsmooth problems with bounded variation of subgradients, weakly smooth problems with H\"older…
We propose an accelerated meta-algorithm, which allows to obtain accelerated methods for convex unconstrained minimization in different settings. As an application of the general scheme we propose nearly optimal methods for minimizing…
We prove lower bounds for higher-order methods in smooth non-convex finite-sum optimization. Our contribution is threefold: We first show that a deterministic algorithm cannot profit from the finite-sum structure of the objective, and that…
First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds has been shown to increase both the theoretical guarantees and the practical performance. In…
We prove lower bounds on the complexity of finding $\epsilon$-stationary points (points $x$ such that $\|\nabla f(x)\| \le \epsilon$) of smooth, high-dimensional, and potentially non-convex functions $f$. We consider oracle-based complexity…
In this paper some adaptive mirror descent algorithms for problems of minimization convex objective functional with several convex Lipschitz (generally, non-smooth) functional constraints are considered. It is shown that the methods are…
We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An…
Algorithmic reproducibility measures the deviation in outputs of machine learning algorithms upon minor changes in the training process. Previous work suggests that first-order methods would need to trade-off convergence rate (gradient…
We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e.\ problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected)…
We consider a class of popular distributed non-convex optimization problems, in which agents connected by a network $\mathcal{G}$ collectively optimize a sum of smooth (possibly non-convex) local objective functions. We address the…