Non-Stationary Bandits with Periodic Behavior: Harnessing Ramanujan Periodicity Transforms to Conquer Time-Varying Challenges

P Thaker, V Gattani, V Tirukkonda et al.
April 2024
ICASSP 2024
Non-stationary BanditsRamanujan TransformsSignal ProcessingPeriodic BehaviorTime-varying Systems

Abstract

We address the challenging problem of non-stationary multi-armed bandits where the reward distributions change over time in a periodic manner. Traditional approaches struggle with time-varying environments due to the exploration-exploitation trade-off becoming more complex when the optimal actions shift cyclically. We propose a novel framework that harnesses Ramanujan Periodicity Transforms to detect and exploit periodic patterns in reward structures. Our approach leverages the mathematical elegance of Ramanujan sums to efficiently identify co-resonant frequencies in the time-varying reward signals, enabling more effective adaptation to periodic non-stationarity. The method provides theoretical guarantees on regret bounds while demonstrating superior empirical performance on both synthetic and real-world datasets with periodic reward patterns.

What Excited Me

This paper represents the perfect fusion of classical number theory and modern machine learning that absolutely thrills me! The idea of using Ramanujan Periodicity Transforms for non-stationary bandits is pure genius - it's like discovering that 100-year-old mathematics holds the key to solving cutting-edge AI problems. What excites me most is how Ramanujan sums, originally developed for number theory, turn out to be the perfect tool for detecting periodicities in time-varying reward structures. The elegance is breathtaking: instead of brute-force learning that rewards change, we use mathematical insights about co-resonant frequencies to efficiently identify when and how the environment shifts. It's the kind of interdisciplinary breakthrough that shows how fundamental mathematics continues to unlock new possibilities in machine learning. Plus, the practical implications for real-world systems with cyclical patterns - from traffic optimization to financial markets - are enormous!

Problem Motivation

Non-stationary multi-armed bandits present one of the most challenging problems in online learning, where the fundamental assumption of stationary reward distributions is violated. When rewards change over time, traditional bandit algorithms can become ineffective, continuously chasing shifting optima without leveraging underlying patterns.

The Periodic Challenge

Many real-world systems exhibit periodic non-stationarity:

  • Web traffic follows daily and weekly patterns
  • Financial markets have cyclical behaviors tied to trading hours and market cycles
  • Energy consumption varies with predictable daily, seasonal, and weekly patterns
  • User engagement on platforms follows time-of-day and day-of-week trends

Technical Innovation

1. Ramanujan Periodicity Transforms

Mathematical Foundation

Ramanujan sums provide a powerful framework for periodic analysis:

  • Co-resonant frequencies: Focus on frequencies that are harmonically related
  • Computational efficiency: Faster than Fourier transforms for periodicity detection
  • Number-theoretic elegance: Leverages deep mathematical properties for signal analysis

Application to Bandit Problems

The key insight is treating time-varying rewards as signals that can be decomposed using Ramanujan transforms:

  • Periodic decomposition: Identifying dominant periodic components in reward structures
  • Frequency analysis: Detecting which arms have similar periodic behaviors
  • Pattern exploitation: Using discovered periodicities to predict future reward patterns

2. Adaptive Bandit Framework

Periodicity-Aware Exploration

  • Dynamic arm selection: Balancing exploration with exploitation based on detected periods
  • Phase-aware decisions: Considering where we are in the periodic cycle when choosing actions
  • Confidence intervals: Adjusting uncertainty estimates based on periodic patterns

Multi-Scale Analysis

  • Hierarchical periods: Detecting periods at different time scales simultaneously
  • Nested periodicities: Handling complex periodic structures with multiple overlapping cycles
  • Adaptive window sizing: Adjusting analysis windows based on detected period lengths

3. Theoretical Contributions

Regret Analysis

  • Bounded regret: Theoretical guarantees on performance relative to optimal periodic policy
  • Adaptation rates: Analysis of how quickly the algorithm adapts to new periodic patterns
  • Robustness: Performance guarantees when periodic assumptions are violated

Computational Complexity

  • Efficient algorithms: Leveraging Ramanujan transform efficiency for real-time adaptation
  • Scalability: Performance with large numbers of arms and long time horizons
  • Memory requirements: Compact representation of periodic patterns

Algorithmic Framework

1. Periodicity Detection Phase

  • Signal preprocessing: Converting reward sequences into suitable formats for Ramanujan analysis
  • Transform computation: Efficient calculation of Ramanujan Periodicity Transforms
  • Period identification: Detecting dominant periodic components and their characteristics

2. Adaptive Decision Making

  • Periodic models: Building predictive models based on detected periodicities
  • Confidence estimation: Quantifying uncertainty in periodic predictions
  • Action selection: Choosing arms based on predicted rewards and confidence levels

3. Online Adaptation

  • Continuous monitoring: Real-time detection of changes in periodic patterns
  • Model updates: Adapting periodic models as new data becomes available
  • Robustness mechanisms: Handling periods when periodic assumptions break down

Applications and Impact

1. Digital Advertising

  • Ad placement optimization: Timing advertisements based on predicted user engagement patterns
  • Budget allocation: Distributing advertising spend across time periods optimally
  • Campaign scheduling: Leveraging periodic user behavior patterns

2. Energy Grid Management

  • Demand prediction: Forecasting energy consumption based on periodic patterns
  • Supply optimization: Adjusting energy generation to match predicted demand cycles
  • Storage management: Optimizing battery charging/discharging based on cyclical patterns

3. Financial Markets

  • Algorithmic trading: Exploiting periodic patterns in market behavior
  • Portfolio rebalancing: Timing trades based on detected market cycles
  • Risk management: Adjusting exposure based on periodic volatility patterns

4. Network Resource Allocation

  • Traffic management: Optimizing network resources based on usage patterns
  • Server scaling: Predicting and preparing for periodic load variations
  • Quality of service: Maintaining performance during predictable traffic cycles

Technical Advantages

1. Mathematical Rigor

  • Number-theoretic foundation: Leveraging deep mathematical insights about periodicity
  • Provable guarantees: Theoretical bounds on algorithm performance
  • Optimal complexity: Efficient algorithms with favorable computational characteristics

2. Practical Robustness

  • Noise tolerance: Performance maintained under realistic noise conditions
  • Partial periodicity: Effective even when only some arms exhibit periodic behavior
  • Graceful degradation: Reasonable performance when periodic assumptions fail

3. Scalability

  • Large-scale deployment: Efficient algorithms suitable for high-dimensional problems
  • Real-time operation: Fast enough for online decision making
  • Memory efficiency: Compact representation of learned periodic patterns

Future Research Directions

1. Extensions to Complex Environments

  • Multi-agent settings: Periodic bandits with multiple competing learners
  • Contextual bandits: Incorporating contextual information into periodic analysis
  • Structured arms: Leveraging arm similarities in periodic environments

2. Advanced Periodicity Models

  • Non-stationary periods: Handling periods that change over time
  • Stochastic periodicities: Dealing with approximate or noisy periodic patterns
  • Hierarchical structures: Multiple nested periodicities at different scales

3. Practical Deployments

  • A/B testing: Improved experimental design for time-varying treatments
  • Recommendation systems: Personalized recommendations based on user activity cycles
  • Autonomous systems: Periodic optimization for robotic and IoT applications

Interdisciplinary Impact

This work demonstrates the power of cross-pollination between classical mathematics and modern machine learning:

  • Historical mathematics: 100-year-old number theory solving contemporary AI problems
  • Signal processing insights: Transferring signal analysis techniques to decision-making problems
  • Theoretical computer science: Bridging computational complexity with practical algorithms

The research showcases how fundamental mathematical insights can unlock new capabilities in machine learning, providing both theoretical guarantees and practical performance improvements for challenging real-world problems.