408 research outputs found
Optimal Dynamic Portfolio with Mean-CVaR Criterion
Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are popular risk
measures from academic, industrial and regulatory perspectives. The problem of
minimizing CVaR is theoretically known to be of Neyman-Pearson type binary
solution. We add a constraint on expected return to investigate the Mean-CVaR
portfolio selection problem in a dynamic setting: the investor is faced with a
Markowitz type of risk reward problem at final horizon where variance as a
measure of risk is replaced by CVaR. Based on the complete market assumption,
we give an analytical solution in general. The novelty of our solution is that
it is no longer Neyman-Pearson type where the final optimal portfolio takes
only two values. Instead, in the case where the portfolio value is required to
be bounded from above, the optimal solution takes three values; while in the
case where there is no upper bound, the optimal investment portfolio does not
exist, though a three-level portfolio still provides a sub-optimal solution
Constrained Convex Neyman-Pearson Classification Using an Outer Approximation Splitting Method
This paper presents an algorithm for Neyman-Pearson classification. While empirical riskminimization approaches focus on minimizing a global risk, the Neyman-Pearson frameworkminimizes the type II risk under an upper bound constraint on the type I risk. Sincethe 0=1 loss function is not convex, optimization methods employ convex surrogates thatlead to tractable minimization problems. As shown in recent work, statistical bounds canbe derived to quantify the cost of using such surrogates instead of the exact 1/0 loss.However, no specific algorithm has yet been proposed to actually solve the resulting minimizationproblem numerically. The contribution of this paper is to propose an efficientsplitting algorithm to address this issue. Our method alternates a gradient step on the objectivesurrogate risk and an approximate projection step onto the constraint set, which isimplemented by means of an outer approximation subgradient projection algorithm. Experimentson both synthetic data and biological data show the efficiency of the proposed method
GBM-based Bregman Proximal Algorithms for Constrained Learning
As the complexity of learning tasks surges, modern machine learning
encounters a new constrained learning paradigm characterized by more intricate
and data-driven function constraints. Prominent applications include
Neyman-Pearson classification (NPC) and fairness classification, which entail
specific risk constraints that render standard projection-based training
algorithms unsuitable. Gradient boosting machines (GBMs) are among the most
popular algorithms for supervised learning; however, they are generally limited
to unconstrained settings. In this paper, we adapt the GBM for constrained
learning tasks within the framework of Bregman proximal algorithms. We
introduce a new Bregman primal-dual method with a global optimality guarantee
when the learning objective and constraint functions are convex. In cases of
nonconvex functions, we demonstrate how our algorithm remains effective under a
Bregman proximal point framework. Distinct from existing constrained learning
algorithms, ours possess a unique advantage in their ability to seamlessly
integrate with publicly available GBM implementations such as XGBoost (Chen and
Guestrin, 2016) and LightGBM (Ke et al., 2017), exclusively relying on their
public interfaces. We provide substantial experimental evidence to showcase the
effectiveness of the Bregman algorithm framework. While our primary focus is on
NPC and fairness ML, our framework holds significant potential for a broader
range of constrained learning applications. The source code is currently freely
available at
https://github.com/zhenweilin/ConstrainedGBM}{https://github.com/zhenweilin/ConstrainedGBM
Proximally Constrained Methods for Weakly Convex Optimization with Weakly Convex Constraints
Optimization models with non-convex constraints arise in many tasks in
machine learning, e.g., learning with fairness constraints or Neyman-Pearson
classification with non-convex loss. Although many efficient methods have been
developed with theoretical convergence guarantees for non-convex unconstrained
problems, it remains a challenge to design provably efficient algorithms for
problems with non-convex functional constraints. This paper proposes a class of
subgradient methods for constrained optimization where the objective function
and the constraint functions are are weakly convex. Our methods solve a
sequence of strongly convex subproblems, where a proximal term is added to both
the objective function and each constraint function. Each subproblem can be
solved by various algorithms for strongly convex optimization. Under a uniform
Slater's condition, we establish the computation complexities of our methods
for finding a nearly stationary point
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