REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models (1703.07370v4)

Abstract: Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work (Jang et al. 2016, Maddison et al. 2016) has taken a different approach, introducing a continuous relaxation of discrete variables to produce low-variance, but biased, gradient estimates. In this work, we combine the two approaches through a novel control variate that produces low-variance, \emph{unbiased} gradient estimates. Then, we introduce a modification to the continuous relaxation and show that the tightness of the relaxation can be adapted online, removing it as a hyperparameter. We show state-of-the-art variance reduction on several benchmark generative modeling tasks, generally leading to faster convergence to a better final log-likelihood.

An Analytical Overview of "REBAR: Low-variance, Unbiased Gradient Estimates for Discrete Latent Variable Models"

The paper "REBAR: Low-variance, Unbiased Gradient Estimates for Discrete Latent Variable Models," authored by Tucker et al., addresses a significant challenge in the domain of learning models with discrete latent variables: the high variance of gradient estimators. Traditional approaches have been reliant on control variates to mitigate the variance inherent in the REINFORCE estimator. Recent advances proposed continuous relaxations of discrete variables using the Concrete or Gumbel-Softmax distribution, which, while reducing variance, introduce a bias in gradient estimates. In response, this paper proposes a novel control variate technique, REBAR, that achieves low-variance and unbiased gradient estimations, without the need for tuning additional hyperparameters.

Background and Prior Work

Discrete latent variable models are foundational within machine learning applications including mixture models, reinforcement learning frameworks such as Markov Decision Processes, and generative models in structured prediction. These models encounter difficulties when discrete latent variables cannot be analytically marginalized out, leading reliance on REINFORCE-like methods, which suffer from high variance due to sampling. Past research efforts focused on crafting intelligent control variates to alleviate this variance. Furthermore, the advent of the Concrete distribution, introduced by Jang et al. and Maddison et al., facilitated the continuous relaxation of discrete random variables, thereby allowing the application of the reparameterization trick. Although successful in reducing variance, the resulting gradient estimates bear a bias reflecting the continuous model.

Key Contributions

The authors bridge the gap between control variates and continuous relaxation through REBAR, an innovative control variate which uniquely combines REINFORCE gradients with gradients derived from the Concrete relaxation. The paper’s main contribution is showcasing a method that guarantees low variance and unbiasedness in gradient estimation. Notably, a key insight reveals that it's feasible to conditionally marginalize control variates, significantly enhancing their effectiveness. Additionally, the paper shows how to modify the Concrete relaxation to connect REBAR with MuProp in the high-temperature limit, further demonstrating that the tightness of the relaxation can optimize online, effectively nullifying the necessity of setting an additional hyperparameter.

Experimental Validation and Results

Empirical evaluations demonstrate REBAR’s proficiency in variance reduction across several benchmark generative modeling tasks, including training generative sigmoid belief networks on MNIST and Omniglot datasets. The research reveals REBAR's superior performance in variance reduction, which correlates with faster convergence and better final log-likelihood scores. The experimental framework also includes a development and examination of optimization on a toy problem, illustrating the deficiencies of biased gradient estimators in landing suboptimal stochastic solutions.

Practical and Theoretical Implications

REBAR’s introduction has noteworthy implications both practically and theoretically. Practically, REBAR facilitates more efficient training of models reliant on discrete latent variables by achieving faster optimization and improved performance stability without increasing computational burden through hyperparameter tuning. Theoretically, the approach suggests pathways for further exploration into structured variational autoencoders and optimizing multi-sample variational bounds. The continuous blending of discrete gradient estimators with a novel control variate could offer deeper insights into enhancing learning algorithms within reinforcement learning by leveraging learned differentiable functions analogous to $Q$-functions within REBAR’s framework.

Potential Future Work

Future developments may pursue extending REBAR’s methodology into multi-sample scenarios, optimize its application within reinforcement learning by generalizing to $Q$-functions, and explore the hybrid utilization of multi-layer dependencies without additional network evaluations. Moreover, further exploration into adaptive temperature optimization strategies could probe the dynamics of unbiased estimation under various configurations, potentially offering more robust training mechanisms in the broader context of artificial intelligence research.

In sum, "REBAR: Low-variance, Unbiased Gradient Estimates for Discrete Latent Variable Models" offers substantial improvements over existing methods by integrating low-variance with unbiasedness in an elegant, computationally efficient manner, laying a foundational framework for future advancements within machine learning domains reliant on discrete latent structures.