[Reinforcement Learning] The log-derivative trick and its application in policy gradient

The log-derivative trick

The “log-derivative trick” is a powerful mathematical technique widely employed in statistics and machine learning to simplify the differentiation of functions, especially when these involve the product of many variables, such as probabilities and likelihood functions. This trick is instrumental because it helps transform complex multiplicative forms into more manageable additive forms, facilitating easier differentiation and computational handling.

How the Log-Derivative Trick Works

1. Convert to Logarithm: The first step is to take the natural logarithm of the function you’re dealing with. This step is beneficial because the logarithm transforms products into sums:

$$\log \left(\prod _{i=1}^n{g}_i(x)\right)=\sum _{i=1}^n\log g_i(x)$$

This transformation simplifies many operations, especially differentiation, as sums are generally simpler to differentiate than products.
2. Differentiation: After converting to a logarithmic form, you differentiate the resulting expression. The derivative of the logarithm of a function $f(x)$ is:

$$\frac{d}{dx}\log f(x)=\frac{f^{\prime }(x)}{f(x)}$$

This simplifies differentiation because it decomposes complex product rules into simpler sum rules, making it particularly useful when dealing with powers and products.
3. Apply the Chain Rule: If the function includes compositions of different functions, the chain rule is applied within the logarithmic differentiation to handle these compositions correctly.

Applications of the Log-Derivative Trick

• Maximum Likelihood Estimation (MLE): In statistics, MLE often involves products of probability densities. Using the log-derivative trick, these products are converted into sums of logarithms, making the differentiation straightforward and more tractable computationally.
• Bayesian Inference: Similar to MLE but includes combining prior distributions with likelihood functions, often leading to integrations over complex products.
• Gradient-Based Optimization: In machine learning, algorithms such as neural networks use gradient descent methods where derivatives of loss functions are essential. The log-derivative trick simplifies the computation of these derivatives, especially when the functions involve products of terms.

Practical Example

Consider a function $f(x)$ that is the product of multiple terms $g_i(x)$:

$$f(x)=\prod _{i=1}^n{g}_i(x)$$

Taking the logarithm of $f(x)$ gives:

$$\log f(x)=\sum _{i=1}^n\log g_i(x)$$

Differentiating this logarithmic form with respect to $x$ results in:

$$\frac{d}{dx}\log f(x)=\sum _{i=1}^n\frac{g_i^{\prime }(x)}{g_i(x)}$$

This expression greatly simplifies gradient computations in optimization algorithms or other analytic scenarios where handling the original product form directly would be cumbersome and computationally expensive.

The log-derivative trick in Policy Gradient

The log-derivative trick is particularly important in the field of reinforcement learning, especially when it comes to implementing policy gradient methods. This trick facilitates the computation of gradients of expected rewards with respect to the parameters of a policy, which is usually a non-trivial task due to the stochastic nature of policies and rewards in reinforcement learning environments.

Understanding the Policy Gradient

In reinforcement learning, a policy ${\pi }_{\theta }(a|s)$ defines the probability of taking action $a$ in state $s$ given parameters $\theta$. The goal is often to adjust $\theta$ so as to maximize some notion of long-term reward. The objective function $J(\theta )$ is typically defined as the expected return:

$$J(\theta )={\mathbb{E}}_{\tau \sim {\pi }_{\theta }}[R(\tau )]$$

where $\tau$ represents a trajectory of states and actions, and $R(\tau )$ is the total reward for the trajectory.

Application of the Log-Derivative Trick

To optimize $J(\theta )$ using gradient ascent, we need to compute the gradient ${\nabla }_{\theta }J(\theta )$. This is where the log-derivative trick comes into play:

$${\nabla }_{\theta }J(\theta )={\nabla }_{\theta }{\mathbb{E}}_{\tau \sim {\pi }_{\theta }}[R(\tau )]$$

Expanding this using the expectation, we get:

$${\nabla }_{\theta }J(\theta )={\nabla }_{\theta }\int {\pi }_{\theta }(\tau )R(\tau )d\tau$$

However, differentiating ${\pi }_{\theta }(\tau )$ directly with respect to $\theta$ is complex. The log-derivative trick simplifies this by expressing the gradient of the probability in terms of the gradient of its logarithm:

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$${\nabla }_{\theta }{\pi }_{\theta }(\tau )={\pi }_{\theta }(\tau ){\nabla }_{\theta }\log {\pi }_{\theta }(\tau )$$

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Using this, we can rewrite the gradient of the objective as:

$${\nabla }_{\theta }J(\theta )=\int {\pi }_{\theta }(\tau ){\nabla }_{\theta }\log {\pi }_{\theta }(\tau )R(\tau )d\tau$$

which simplifies to:

$${\nabla }_{\theta }J(\theta )={\mathbb{E}}_{\tau \sim {\pi }_{\theta }}[{\nabla }_{\theta }\log {\pi }_{\theta }(\tau )R(\tau )]$$

Practical Implementation

In practice, especially in policy gradient algorithms like REINFORCE, the gradient of the log of the policy is computed with respect to the actions taken:

$${\nabla }_{\theta }\log {\pi }_{\theta }(a_t|s_t)={\nabla }_{\theta }\log {\pi }_{\theta }\text{ for action }{a}_t\text{ in state }{s}_t$$

And the expectation is approximated using sampled trajectories. Thus, the update rule for the parameters in a policy gradient method becomes:

$$\theta \leftarrow \theta +\alpha \sum _{t=0}^T{\nabla }_{\theta }\log {\pi }_{\theta }(a_t|s_t)R_t$$

where $\alpha$ is the learning rate and $R_t$ is the cumulative reward from time $t$ onwards.

This formulation makes it feasible to use gradient-based optimization techniques in environments where the reward and action dynamics are only partially known, leveraging the stochasticity of the policy to explore and optimize behavior effectively.