DiCE: The Infinitely Differentiable Monte-Carlo Estimator (1802.05098v3)

Abstract: The score function estimator is widely used for estimating gradients of stochastic objectives in stochastic computation graphs (SCG), eg, in reinforcement learning and meta-learning. While deriving the first-order gradient estimators by differentiating a surrogate loss (SL) objective is computationally and conceptually simple, using the same approach for higher-order derivatives is more challenging. Firstly, analytically deriving and implementing such estimators is laborious and not compliant with automatic differentiation. Secondly, repeatedly applying SL to construct new objectives for each order derivative involves increasingly cumbersome graph manipulations. Lastly, to match the first-order gradient under differentiation, SL treats part of the cost as a fixed sample, which we show leads to missing and wrong terms for estimators of higher-order derivatives. To address all these shortcomings in a unified way, we introduce DiCE, which provides a single objective that can be differentiated repeatedly, generating correct estimators of derivatives of any order in SCGs. Unlike SL, DiCE relies on automatic differentiation for performing the requisite graph manipulations. We verify the correctness of DiCE both through a proof and numerical evaluation of the DiCE derivative estimates. We also use DiCE to propose and evaluate a novel approach for multi-agent learning. Our code is available at https://www.github.com/alshedivat/lola.

• The paper introduces DiCE for computing higher-order gradient estimators in stochastic computation graphs, overcoming the limitations of surrogate loss methods.
• The methodology leverages automatic differentiation and the MagicBox operator to ensure accurate dependency handling in gradient computations.
• Empirical evaluations in multi-agent reinforcement learning demonstrate DiCE’s potential to improve convergence and optimization performance.

An Analysis of DiCE: The Infinitely Differentiable Monte Carlo Estimator

The paper "DiCE: The Infinitely Differentiable Monte Carlo Estimator" contributes a novel method for computing higher-order gradient estimators within stochastic computation graphs (SCGs). This work focuses on addressing the limitations inherent in existing techniques and proposes a unified framework that ensures correct estimation of higher-order derivatives, which is crucial for optimizing algorithms in fields such as reinforcement learning and meta-learning.

The central innovation presented is DiCE, the Infinitely Differentiable Monte Carlo Estimator. The method combines insights from stochastic gradient estimation with automatic differentiation (auto-diff) to facilitate gradient computation of any order in SCGs. DiCE is crafted to be compatible with auto-diff frameworks commonly used in today’s machine learning environments like TensorFlow and PyTorch, overcoming the computational and conceptual challenges posed by traditional techniques.

One of the primary drawbacks of surrogate loss (SL) approaches, commonly used in existing literature, is their inadequate handling of dependencies when calculating higher-order gradients. The SL method treats certain components of the cost as fixed samples, resulting in missing and incorrect terms when extended beyond first-order gradient estimation. The authors of this paper address these issues by introducing the MagicBox operator as part of the DiCE framework. This operator ensures that the derivative estimations retain all necessary dependencies and are integrated correctly across multiple differentiation orders.

The implications of DiCE are significant, particularly for applications where the computational graph includes non-differentiable nodes or the need for higher-order derivatives is critical. Such applications are prevalent in reinforcement learning where higher-order techniques can accelerate convergence and improve policy optimization by correctly embodying the influence of earlier stochastic decisions on subsequent ones.

DiCE is validated through both theoretical proofs and empirical evaluations, with the authors focusing on multi-agent reinforcement learning scenarios as a testbed. The results suggest that DiCE not only matches the theoretical predictions but enhances the practicality of computing higher-order gradient estimates, which traditionally would demand cumbersome analytical derivations.

The potential for broader application of DiCE is manifold. By guaranteeing correct gradient estimations through a differentiable objective, DiCE can aid in the deployment of second-order optimizers that rely on accurate Hessian-vector products, effectively expanding the toolkit available for tackling optimization problems in stochastic settings.

Future directions for research may involve the integration of variance reduction techniques within the DiCE framework to further refine the efficiency of the estimator. Additionally, the open-source release of the DiCE implementation invites community exploration into its usage across different environments and problem specifications.

In summary, the DiCE method stands out as a significant step towards bridging conceptual gaps in higher-order gradient estimation within stochastic computation graphs, facilitating improvements in both theoretical understanding and practical implementations of AI algorithms. This contribution is poised to support ongoing advancements in reinforcement learning and meta-learning paradigms by enabling accurate, efficient gradient computations across a range of applications.