Backpropagation through the Void: Optimizing control variates for black-box gradient estimation (1711.00123v3)

Abstract: Gradient-based optimization is the foundation of deep learning and reinforcement learning. Even when the mechanism being optimized is unknown or not differentiable, optimization using high-variance or biased gradient estimates is still often the best strategy. We introduce a general framework for learning low-variance, unbiased gradient estimators for black-box functions of random variables. Our method uses gradients of a neural network trained jointly with model parameters or policies, and is applicable in both discrete and continuous settings. We demonstrate this framework for training discrete latent-variable models. We also give an unbiased, action-conditional extension of the advantage actor-critic reinforcement learning algorithm.

• The paper presents a novel framework for optimizing control variates that significantly reduces the variance in black-box gradient estimation.
• It leverages techniques like the score-function estimator and reparameterization trick to create differentiable surrogates for improved learning stability.
• Experimental results show enhanced sample efficiency and robust performance in reinforcement learning and discrete variational autoencoder applications.

Overview of "Backpropagation through the Void: Optimizing Control Variates for Black-Box Gradient Estimation"

Gradient-based optimization methods are indispensable in deep learning and reinforcement learning. However, direct application of gradient-based techniques encounters difficulties when dealing with black-box functions or non-differentiable objectives, which is often the case in real-world applications such as reinforcement learning scenarios with unknown dynamics and stochastic environments. The paper "Backpropagation through the Void: Optimizing Control Variates for Black-Box Gradient Estimation" addresses this issue by introducing a general framework for learning low-variance, unbiased gradient estimators using control variates.

The methodology presented capitalizes on constructing a differentiable surrogate model using neural networks to form a control variate, which is then optimized alongside the primary model parameters. This approach allows for unbiased gradient estimation without necessitating a differentiable objective, a feature that becomes highly advantageous in black-box function scenarios or when dealing with discrete variables where traditional backpropagation fails.

Key Components of the Method

The paper introduces a comprehensive gradient estimation framework that integrates well-known variance reduction techniques such as the score-function gradient estimator (REINFORCE) and the reparameterization trick. The central innovation is the development and optimization of a control variate parameterized by a neural network, dubbed LAX (Learning Action-dependent baselines for eXploration). This includes an extension, RELAX, aimed at discrete scenarios with conditional input reparameterization to further refine variance reduction.

1. Score-Function Estimator: This estimator provides a path for unbiased gradient estimation, albeit at potentially high variance when the expectation depends sharply on the parameters.
2. Reparameterization Trick: Enables differentiation under the integral sign, but only applicable when a continuous path through the parameters exists.
3. Control Variates: Introduced to reduce variance, they subtract a function with known expectations from the primary gradient estimation process, thereby refining it. Their variance is minimized using gradient-based optimization.
4. LAX and RELAX Algorithms: They integrate control variates into both continuous and discrete settings, introducing surrogate functions with differential properties to enable gradient propagation through otherwise non-differentiable functions.

Experimental Evaluation

The paper demonstrates the efficacy of the proposed framework through several experimental setups:

• Toy Problems: Highlighted the reduced variance comparative performance of RELAX against other estimators such as REINFORCE and REBAR by optimizing smooth approximations of otherwise discontinuous functions.
• Discrete Variational Autoencoders (VAE): Implementation on VAEs with Bernoulli latent variables showed more rapid convergence and robust optimization compared to baselines.
• Reinforcement Learning: Applied in both discrete (e.g., Cart Pole) and continuous (e.g., Inverted Pendulum) environments, the proposed methods showed significant improvements in sample efficiency and policy performance over conventional Advantage Actor-Critic (A2C) algorithms.

Implications and Future Directions

The introduction of backpropagation methods that incorporate optimized control variates extends the toolkit for tackling black-box optimization problems and non-differentiable domains in AI research. The framework potentially paves the way for more effective strategies in reinforcement learning, particularly in actor-critic methods where variance reduction is crucial for stability and convergence.

Looking ahead, the capability to handle stochastic or non-differentiable objectives broadens the applicability to tasks such as model-based reinforcement learning, hierarchical decision making, and optimization problems encountered in fine-tuning hyperparameters in neural architectures. Further exploration into hybrid methods that integrate these proposed techniques with off-policy learning and advanced sampling strategies could enhance the robustness and efficiency of such frameworks in practical, high-dimensional settings.

The paper provides a solid foundation and a versatile technique that might see future developments focusing on scalability and integration with emerging architectures in the AI landscape.