Soft Q-Learning

Soft q-learning is a variation of q-learning that it replaces the max function by its soft equivalent:

\[\text{max}^{(\tau)}_i x_i\ =\ \tau\,\log\sum_i \exp\left( x_i / \tau \right)\]

The temperature parameter \(\tau>0\) determines the softness of the operation. We recover the ordinary (hard) max function in the limit \(\tau\to0\).

The \(n\)-step bootstrapped target is thus computed as

\[G^{(n)}_t\ =\ R^{(n)}_t + I^{(n)}_t\,\tau\,\log\sum_a \exp\Bigl( q(S_{t+n}, a) / \tau \Bigr)\]


\[\begin{split}R^{(n)}_t\ =\ \sum_{k=0}^{n-1}\gamma^kR_{t+k}\ , \qquad I^{(n)}_t\ =\ \left\{\begin{matrix} 0 & \text{if $S_{t+n}$ is a terminal state} \\ \gamma^n & \text{otherwise} \end{matrix}\right.\end{split}\]

Soft q-learning (partially) mitigates over-estimation in the Bellman error. What’s more, there is a natural connection between soft q-learning and actor-critic methods (see paper).

Open in Google Colab
import gymnasium
import coax
import haiku as hk
import jax
import jax.numpy as jnp
from optax import adam

# pick environment
env = gymnasium.make(...)
env = coax.wrappers.TrainMonitor(env)

def func_type1(S, A, is_training):
    # custom haiku function: s,a -> q(s,a)
    value = hk.Sequential([...])
    X = jax.vmap(jnp.kron)(S, A)  # or jnp.concatenate((S, A), axis=-1) or whatever you like
    return value(X)  # output shape: (batch_size,)

def func_type2(S, is_training):
    # custom haiku function: s -> q(s,.)
    value = hk.Sequential([...])
    return value(S)  # output shape: (batch_size, num_actions)

# function approximator
func = ...  # func_type1 or func_type2
q = coax.Q(func, env)
pi = coax.BoltzmannPolicy(q, temperature=0.1)

# specify how to update q-function
qlearning = coax.td_learning.SoftQLearning(q, optimizer=adam(0.001), temperature=pi.temperature)

# specify how to trace the transitions
cache = coax.reward_tracing.NStep(n=1, gamma=0.9)

for ep in range(100):
    pi.epsilon = ...  # exploration schedule
    s, info = env.reset()

    for t in range(env.spec.max_episode_steps):
        a = pi(s)
        s_next, r, done, truncated, info = env.step(a)

        # add transition to cache
        cache.add(s, a, r, done)

        # update
        while cache:
            transition_batch = cache.pop()
            metrics = qlearning.update(transition_batch)

        if done or truncated:

        s = s_next