Efficient Smoothed Online Learning for Piecewise Continuous Decision Making at MIT

Temel Kavramlar

Efficient algorithms for piecewise continuous functions in online learning.

Özet

Introduction

Smoothed online learning mitigates statistical and computational challenges.
Regret bounds are crucial for efficient algorithms in adversarial settings.

Formal Setting and Notation

Lipschitz assumptions on adversaries and loss functions.
Empirical Risk Minimization (ERM) oracle usage for optimization.

Follow the Perturbed Leader and Generalized Brackets

FTPL algorithm application with random path sampling.
Control of stability term crucial for regret analysis.

Exponential Perturbations and Piecewise Continuous Functions

Efficient algorithms for piecewise continuous prediction.
Use of bracketing numbers to control regret scaling.

Piecewise Continuous Prediction with Generalized Affine Boundaries

Affine decision boundaries in PWA functions.
Regret bounds based on bracketing numbers and smoothness assumptions.

Piecewise Continuous Prediction with Polynomial Boundaries

Polynomial decision boundaries and regret bounds.
Application of polynomial smoothness assumptions for efficient learning.

Smoothed Multi-Step Planning

Multi-step planning in hybrid dynamical systems.
Regret analysis for discontinuous systems with affine boundaries.

Further Work

Extension to polynomial boundaries in planning scenarios.

İstatistikler

Recent works show regret scaling polynomially in 1/σ for oracle-efficient algorithms.
Theoretical framework combines bracketing numbers, pseudo-metrics, and directional smoothness.
Algorithms achieve low regret through careful control of stability terms and complexity measures.

Alıntılar

"Recent works have demonstrated strong computational-statistical tradeoffs in smoothed online learning."
"A natural question remains: under which types of smoothing is it possible to design oracle-efficient algorithms?"

Önemli Bilgiler Şuradan Elde Edildi

Oracle-Efficient Smoothed Online Learning for Piecewise Continuous Decision Making

by Adam Block,A... : arxiv.org 03-20-2024

https://arxiv.org/pdf/2302.05430.pdf

Daha Derin Sorular

How does the use of generalized bracketing numbers impact the efficiency of smoothed online learning

The use of generalized bracketing numbers plays a crucial role in determining the efficiency of smoothed online learning. These numbers provide a measure of complexity that combines constraints on the adversary with the complexity of the space. By bounding these generalized bracketing numbers, it becomes possible to design practical algorithms that experience low regret in online learning scenarios. Specifically, when these bracketing numbers are small, they lead to practical algorithms with provably small regret. This means that by controlling and minimizing these bracketing numbers, we can achieve optimal performance in terms of regret while still ensuring computational efficiency.

What are the implications of the exponential perturbations on regret bounds in piecewise continuous functions

Exponential perturbations have significant implications on regret bounds in piecewise continuous functions within the context of smoothed online learning. When dealing with piecewise continuous functions and utilizing exponential perturbations as part of the algorithmic strategy, it is essential to consider how these perturbations affect the overall performance and efficiency of the learning process. In this scenario, leveraging exponential perturbations can help mitigate challenges related to discontinuities across different regions or modes within the piecewise continuous functions. By appropriately tuning parameters such as noise levels based on exponential distributions, one can optimize regret bounds and improve overall algorithmic performance for handling complex decision-making processes involving piecewise continuous functions.

How can the concept of polynomial smoothness be extended to improve multi-step planning strategies

The concept of polynomial smoothness can be extended to enhance multi-step planning strategies by providing a structured framework for managing decision boundaries between different regions or modes within dynamic systems. By incorporating polynomial boundaries into planning models where decisions are made over multiple steps or time horizons, one can introduce more flexibility and adaptability into the planning process. This extension allows for smoother transitions between different states or modes based on polynomial relationships defined by specific degrees or coefficients.
In practice, extending polynomial smoothness principles to multi-step planning enables more sophisticated control strategies that take into account not only immediate decisions but also longer-term objectives and constraints within dynamic environments. This approach enhances robustness and optimality in decision-making processes across various applications such as robotics, economics modeling, or system optimization tasks where multi-step planning is critical for achieving desired outcomes efficiently.