Optimization of utility-based shortfall risk: A non-asymptotic viewpoint

• URL: http://arxiv.org/abs/2310.18743v2
• Date: Sat, 30 Mar 2024 08:53:49 GMT
• Title: Optimization of utility-based shortfall risk: A non-asymptotic viewpoint
• Authors: Sumedh Gupte, Prashanth L. A., Sanjay P. Bhat,
• Abstract summary: We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR) In the context of UBSR estimation, we derive a non-asymptotic bound on the mean-squared error of the classical sample average approximation (SAA) of UBSR. We derive non-asymptotic bounds that quantify the rate of the gradient algorithm for UBSR optimization.
• Score: 11.907026010541674
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR), which is a popular risk measure in finance. In the context of UBSR estimation, we derive a non-asymptotic bound on the mean-squared error of the classical sample average approximation (SAA) of UBSR. Next, in the context of UBSR optimization, we derive an expression for the UBSR gradient under a smooth parameterization. This expression is a ratio of expectations, both of which involve the UBSR. We use SAA for the numerator as well as denominator in the UBSR gradient expression to arrive at a biased gradient estimator. We derive non-asymptotic bounds on the estimation error, which show that our gradient estimator is asymptotically unbiased. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm for UBSR optimization. Finally, we derive non-asymptotic bounds that quantify the rate of convergence of our SG algorithm for UBSR optimization.

Related papers

• Gradient Normalization with(out) Clipping Ensures Convergence of Nonconvex SGD under Heavy-Tailed Noise with Improved Results [60.92029979853314]
  This paper investigates Gradient Normalization without (NSGDC) its gradient reduction variant (NSGDC-VR) We present significant improvements in the theoretical results for both algorithms.
  arXiv  Detail & Related papers  (2024-10-21T22:40:42Z)
• Accelerated stochastic approximation with state-dependent noise [7.4648480208501455]
  We consider a class of smooth convex optimization problems under general assumptions on the quadratic noise in the gradient observation. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate.
  arXiv  Detail & Related papers  (2023-07-04T06:06:10Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• Online Estimation and Optimization of Utility-Based Shortfall Risk [0.9999629695552195]
  We consider the problem of estimating Utility-Based Shortfall Risk (UBSR) We cast the UBSR estimation problem as a root finding problem, and propose approximation-based estimations schemes. We derive non-asymptotic bounds on the estimation error in the number of samples.
  arXiv  Detail & Related papers  (2021-11-16T22:16:03Z)
• Heavy-tailed Streaming Statistical Estimation [58.70341336199497]
  We consider the task of heavy-tailed statistical estimation given streaming $p$ samples. We design a clipped gradient descent and provide an improved analysis under a more nuanced condition on the noise of gradients.
  arXiv  Detail & Related papers  (2021-08-25T21:30:27Z)
• Zeroth-Order Hybrid Gradient Descent: Towards A Principled Black-Box Optimization Framework [100.36569795440889]
  This work is on the iteration of zero-th-order (ZO) optimization which does not require first-order information. We show that with a graceful design in coordinate importance sampling, the proposed ZO optimization method is efficient both in terms of complexity as well as as function query cost.
  arXiv  Detail & Related papers  (2020-12-21T17:29:58Z)
• ROOT-SGD: Sharp Nonasymptotics and Near-Optimal Asymptotics in a Single Algorithm [71.13558000599839]
  We study the problem of solving strongly convex and smooth unconstrained optimization problems using first-order algorithms. We devise a novel, referred to as Recursive One-Over-T SGD, based on an easily implementable, averaging of past gradients. We prove that it simultaneously achieves state-of-the-art performance in both a finite-sample, nonasymptotic sense and an sense.
  arXiv  Detail & Related papers  (2020-08-28T14:46:56Z)
• Outlier Robust Mean Estimation with Subgaussian Rates via Stability [46.03021473600576]
  We study the problem of robust outlier high-dimensional mean estimation. We obtain first computationally efficient rate with subgaussian for outlier-robust mean estimation.
  arXiv  Detail & Related papers  (2020-07-30T17:33:03Z)
• Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis [102.29671176698373]
  We address the problem of policy evaluation in discounted decision processes, and provide Markov-dependent guarantees on the $ell_infty$error under a generative model. We establish both and non-asymptotic versions of local minimax lower bounds for policy evaluation, thereby providing an instance-dependent baseline by which to compare algorithms.
  arXiv  Detail & Related papers  (2020-03-16T17:15:28Z)
• Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles [8.655294504286635]
  We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error. Our proposed oracles are in practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed simulation.
  arXiv  Detail & Related papers  (2020-02-26T12:53:04Z)
• On Low-rank Trace Regression under General Sampling Distribution [9.699586426043885]
  We show that cross-validated estimators satisfy near-optimal error bounds on general assumptions. We also show that the cross-validated estimator outperforms the theory-inspired approach of selecting the parameter.
  arXiv  Detail & Related papers  (2019-04-18T02:56:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.