168 research outputs found
Bilevel Optimization without Lower-Level Strong Convexity from the Hyper-Objective Perspective
Bilevel optimization reveals the inner structure of otherwise oblique
optimization problems, such as hyperparameter tuning and meta-learning. A
common goal in bilevel optimization is to find stationary points of the
hyper-objective function. Although this hyper-objective approach is widely
used, its theoretical properties have not been thoroughly investigated in cases
where the lower-level functions lack strong convexity. In this work, we take a
step forward and study the hyper-objective approach without the typical
lower-level strong convexity assumption. Our hardness results show that the
hyper-objective of general convex lower-level functions can be intractable
either to evaluate or to optimize. To tackle this challenge, we introduce the
gradient dominant condition, which strictly relaxes the strong convexity
assumption by allowing the lower-level solution set to be non-singleton. Under
the gradient dominant condition, we propose the Inexact Gradient-Free Method
(IGFM), which uses the Switching Gradient Method (SGM) as the zeroth order
oracle, to find an approximate stationary point of the hyper-objective. We also
extend our results to nonsmooth lower-level functions under the weak sharp
minimum condition
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
An Evolutionary Algorithm Using Duality-Base-Enumerating Scheme for Interval Linear Bilevel Programming Problems
Interval bilevel programming problem is hard to solve due to its hierarchical structure as well as the uncertainty of coefficients. This paper is focused on a class of interval linear bilevel programming problems, and an evolutionary algorithm based on duality bases is proposed. Firstly, the objective coefficients of the lower level and the right-hand-side vector are uniformly encoded as individuals, and the relative intervals are taken as the search space. Secondly, for each encoded individual, based on the duality theorem, the original problem is transformed into a single level program simply involving one nonlinear equality constraint. Further, by enumerating duality bases, this nonlinear equality is deleted, and the single level program is converted into several linear programs. Finally, each individual can be evaluated by solving these linear programs. The computational results of 7 examples show that the algorithm is feasible and robust
On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation
In this work, we study first-order algorithms for solving Bilevel
Optimization (BO) where the objective functions are smooth but possibly
nonconvex in both levels and the variables are restricted to closed convex
sets. As a first step, we study the landscape of BO through the lens of penalty
methods, in which the upper- and lower-level objectives are combined in a
weighted sum with penalty parameter . In particular, we establish a
strong connection between the penalty function and the hyper-objective by
explicitly characterizing the conditions under which the values and derivatives
of the two must be -close. A by-product of our analysis is the
explicit formula for the gradient of hyper-objective when the lower-level
problem has multiple solutions under minimal conditions, which could be of
independent interest. Next, viewing the penalty formulation as
-approximation of the original BO, we propose first-order algorithms
that find an -stationary solution by optimizing the penalty
formulation with . When the perturbed lower-level problem
uniformly satisfies the small-error proximal error-bound (EB) condition, we
propose a first-order algorithm that converges to an -stationary
point of the penalty function, using in total and
accesses to first-order (stochastic) gradient oracles when
the oracle is deterministic and oracles are noisy, respectively. Under an
additional assumption on stochastic oracles, we show that the algorithm can be
implemented in a fully {\it single-loop} manner, i.e., with samples per
iteration, and achieves the improved oracle-complexity of
and , respectively
Efficient Learning of Decision-Making Models: A Penalty Block Coordinate Descent Algorithm for Data-Driven Inverse Optimization
Decision-making problems are commonly formulated as optimization problems,
which are then solved to make optimal decisions. In this work, we consider the
inverse problem where we use prior decision data to uncover the underlying
decision-making process in the form of a mathematical optimization model. This
statistical learning problem is referred to as data-driven inverse
optimization. We focus on problems where the underlying decision-making process
is modeled as a convex optimization problem whose parameters are unknown. We
formulate the inverse optimization problem as a bilevel program and propose an
efficient block coordinate descent-based algorithm to solve large problem
instances. Numerical experiments on synthetic datasets demonstrate the
computational advantage of our method compared to standard commercial solvers.
Moreover, the real-world utility of the proposed approach is highlighted
through two realistic case studies in which we consider estimating risk
preferences and learning local constraint parameters of agents in a multiplayer
Nash bargaining game
A regularized variance-reduced modified extragradient method for stochastic hierarchical games
The theory of learning in games has so far focused mainly on games with
simultaneous moves. Recently, researchers in machine learning have started
investigating learning dynamics in games involving hierarchical
decision-making. We consider an -player hierarchical game in which the th
player's objective comprises of an expectation-valued term, parametrized by
rival decisions, and a hierarchical term. Such a framework allows for capturing
a broad range of stochastic hierarchical optimization problems, Stackelberg
equilibrium problems, and leader-follower games. We develop an iteratively
regularized and smoothed variance-reduced modified extragradient framework for
learning hierarchical equilibria in a stochastic setting. We equip our analysis
with rate statements, complexity guarantees, and almost-sure convergence
claims. We then extend these statements to settings where the lower-level
problem is solved inexactly and provide the corresponding rate and complexity
statements
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