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Adaptive Regularization of Representation Rank as an Implicit Constraint of Bellman Equation

Qiang He, Tianyi Zhou, Meng Fang, Setareh Maghsudi

TL;DR

This work identifies representation rank as a crucial yet potentially harmful lever in deep RL when maximized without restraint. By deriv­ing a Bellman-equation–based bound on the cosine similarity between adjacent state-action representations, the authors derive BEER, an adaptive regularizer that modulates representation rank during learning. BEER caps excessive similarity and allows rank to adapt to the learning dynamics, and it can be plugged into standard value-based and policy-gradient algorithms. Empirical results on Lunar Lander and 12 challenging DMControl tasks show that BEER improves value function approximation and overall performance with competitive computational cost, offering a principled path to balancing expressiveness and stability in DRL.

Abstract

Representation rank is an important concept for understanding the role of Neural Networks (NNs) in Deep Reinforcement learning (DRL), which measures the expressive capacity of value networks. Existing studies focus on unboundedly maximizing this rank; nevertheless, that approach would introduce overly complex models in the learning, thus undermining performance. Hence, fine-tuning representation rank presents a challenging and crucial optimization problem. To address this issue, we find a guiding principle for adaptive control of the representation rank. We employ the Bellman equation as a theoretical foundation and derive an upper bound on the cosine similarity of consecutive state-action pairs representations of value networks. We then leverage this upper bound to propose a novel regularizer, namely BEllman Equation-based automatic rank Regularizer (BEER). This regularizer adaptively regularizes the representation rank, thus improving the DRL agent's performance. We first validate the effectiveness of automatic control of rank on illustrative experiments. Then, we scale up BEER to complex continuous control tasks by combining it with the deterministic policy gradient method. Among 12 challenging DeepMind control tasks, BEER outperforms the baselines by a large margin. Besides, BEER demonstrates significant advantages in Q-value approximation. Our code is available at https://github.com/sweetice/BEER-ICLR2024.

Adaptive Regularization of Representation Rank as an Implicit Constraint of Bellman Equation

TL;DR

This work identifies representation rank as a crucial yet potentially harmful lever in deep RL when maximized without restraint. By deriv­ing a Bellman-equation–based bound on the cosine similarity between adjacent state-action representations, the authors derive BEER, an adaptive regularizer that modulates representation rank during learning. BEER caps excessive similarity and allows rank to adapt to the learning dynamics, and it can be plugged into standard value-based and policy-gradient algorithms. Empirical results on Lunar Lander and 12 challenging DMControl tasks show that BEER improves value function approximation and overall performance with competitive computational cost, offering a principled path to balancing expressiveness and stability in DRL.

Abstract

Representation rank is an important concept for understanding the role of Neural Networks (NNs) in Deep Reinforcement learning (DRL), which measures the expressive capacity of value networks. Existing studies focus on unboundedly maximizing this rank; nevertheless, that approach would introduce overly complex models in the learning, thus undermining performance. Hence, fine-tuning representation rank presents a challenging and crucial optimization problem. To address this issue, we find a guiding principle for adaptive control of the representation rank. We employ the Bellman equation as a theoretical foundation and derive an upper bound on the cosine similarity of consecutive state-action pairs representations of value networks. We then leverage this upper bound to propose a novel regularizer, namely BEllman Equation-based automatic rank Regularizer (BEER). This regularizer adaptively regularizes the representation rank, thus improving the DRL agent's performance. We first validate the effectiveness of automatic control of rank on illustrative experiments. Then, we scale up BEER to complex continuous control tasks by combining it with the deterministic policy gradient method. Among 12 challenging DeepMind control tasks, BEER outperforms the baselines by a large margin. Besides, BEER demonstrates significant advantages in Q-value approximation. Our code is available at https://github.com/sweetice/BEER-ICLR2024.
Paper Structure (25 sections, 3 theorems, 33 equations, 7 figures, 7 tables, 1 algorithm)

This paper contains 25 sections, 3 theorems, 33 equations, 7 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Method
  4. Upper bounds on similarity measures for adjacent representations
  5. Adaptive regularization of representation rank
  6. An illustrative example
  7. Experiments
  8. Effectiveness of BEER on simple environments
  9. Scaling up to complex control tasks
  10. BEER reduces approximation error.
  11. Related Work
  12. RL with NNs
  13. NNs-centric DRL
  14. Conclusion
  15. Proof of \ref{['theorem: dot similarity']}
  16. ...and 10 more sections

Key Result

Theorem 1

Under assumption: function and weight, given Q value $Q(s,a)=\phi(s,a)^\top w$, where $\langle \cdot, \cdot \rangle$ represents the inner product, $\phi$ is a state-action representation vector, $w$ is a weight, and $\overline{\phi(s',a')} = \mathbb{E}_{s',a'} \phi(s',a')$ denotes the expectation of the representation of the next state action pair.

Figures (7)

  • Figure 1: Illustrative experiments on the Lunar Lander environment, with results averaged over ten random seeds. The shaded area represents half a standard deviation. (a) A snapshot of the Lunar Lander environment. (b) Comparison of representation ranks. BEER exhibits a more balanced rank compared to InFeR and DQN. (c) Approximation errors of different algorithms. BEER displays a lower approximation error compared to both DQN and InFeR in the latter stage (0.9 to 1 $\times 40K$ time steps). (d) Performance curves substantiating the superiority of BEER.
  • Figure 2: Illustrative experiments on the Grid World task. We report the results over twenty random seeds. The shaded area represents a half standard deviation. (a) The grid world task. The initial state follows a uniform distribution over state space. The objective is to arrive at state $S_T$, which results in a reward (=10); Otherwise, it is zero. (b) Representation rank of tested algorithms. Our proposal, BEER, has the highest representation rank compared to InFeR and DQN. (c) Approximation error. The error of BEER is lower than that of InFeR and DQN. (d) The BEER algorithm requires the least time to reach $S_T$, i.e., it learns faster than the benchmarks.
  • Figure 3: Performance curves for OpenAI gym continuous control tasks on DeepMind Control suite. The proposed algorithm, BEER, outperforms other tested algorithms significantly. The shaded region represents half of the standard deviation of the average evaluation over 10 seeds. The curves are smoothed with a moving average window of size ten.
  • Figure 4: Approximation error curves. The results demonstrate that the approximation error of BEER is empirically minimal when compared to other algorithms.
  • Figure 5: Relationship between cosine similarity and representation rank when minimizing cosine similarity. Our experiment demonstrates the effective control of cosine similarity for representation rank.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 1: Numerical representation rank
  • Theorem 1
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • Definition 2: Stop Gradient