C51

Overview

C51 was first proposed in A Distributional Perspective on Reinforcement Learning, different from previous works, C51 evaluates the complete distribution of a q-value rather than only the expectation. The authors designed a distributional Bellman operator, which preserves multimodality in value distributions and is believed to achieve more stable learning and mitigates the negative effects of learning from a non-stationary policy.

Quick Facts

1. C51 is a model-free and value-based RL algorithm.

2. C51 only support discrete action spaces.

3. C51 is an off-policy algorithm.

4. Usually, C51 use eps-greedy or multinomial sample for exploration.

5. C51 can be equipped with RNN.

Pseudo-code

Note

C51 models the value distribution using a discrete distribution, whose support set are N atoms: \(z_i = V_\min + i * delta, i = 0,1,...,N-1\) and \(delta = (V_\max - V_\min) / N\). Each atom \(z_i\) has a parameterized probability \(p_i\). The Bellman update of C51 projects the distribution of \(r + \gamma * z_j^{\left(t+1\right)}\) onto the distribution \(z_i^t\).

Key Equations or Key Graphs

The Bellman target of C51 is derived by projecting the returned distribution \(r + \gamma * z_j\) onto the current distribution \(z_i\). Given a sample transition \((x, a, r, x')\), we compute the Bellman update \(Tˆz_j := r + \gamma z_j\) for each atom \(z_j\), then distribute its probability \(p_{j}(x', \pi(x'))\) to the immediate neighbors \(p_{i}(x, \pi(x))\):

\[\left(\Phi \hat{T} Z_{\theta}(x, a)\right)_{i}=\sum_{j=0}^{N-1}\left[1-\frac{\left|\left[\hat{\mathcal{T}} z_{j}\right]_{V_{\mathrm{MIN}}}^{V_{\mathrm{MAX}}}-z_{i}\right|}{\Delta z}\right]_{0}^{1} p_{j}\left(x^{\prime}, \pi\left(x^{\prime}\right)\right)\]

Extensions

• C51s can be combined with:

  • PER (Prioritized Experience Replay)

  • Multi-step TD-loss

  • Double (target) network

  • Dueling head

  • RNN

Implementation

Tip

Our benchmark result of C51 uses the same hyper-parameters as DQN except the exclusive n_atom of C51, which is empirically set as 51.

The default config of C51 is defined as follows:

class ding.policy.c51.C51Policy(cfg: EasyDict, model: Module | None = None, enable_field: List[str] | None = None)

Overview:

Policy class of C51 algorithm.

Config:

ID	Symbol	Type	Default Value	Description	Other(Shape)
1	type	str	c51	RL policy register name, refer to registry POLICY_REGISTRY	this arg is optional, a placeholder
2	cuda	bool	False	Whether to use cuda for network	this arg can be diff- erent from modes
3	on_policy	bool	False	Whether the RL algorithm is on-policy or off-policy
4	priority	bool	False	Whether use priority(PER)	priority sample, update priority
5	model.v_min	float	-10	Value of the smallest atom in the support set.
6	model.v_max	float	10	Value of the largest atom in the support set.
7	model.n_atom	int	51	Number of atoms in the support set of the value distribution.
8	other.eps .start	float	0.95	Start value for epsilon decay.
9	other.eps .end	float	0.1	End value for epsilon decay.
10	discount_ factor	float	0.97, [0.95, 0.999]	Reward’s future discount factor, aka. gamma	may be 1 when sparse reward env
11	nstep	int	1,	N-step reward discount sum for target q_value estimation
12	learn.update per_collect	int	3	How many updates(iterations) to train after collector’s one collection. Only valid in serial training	this args can be vary from envs. Bigger val means more off-policy

The network interface C51 used is defined as follows:

class ding.model.template.q_learning.C51DQN(obs_shape: int | SequenceType, action_shape: int | SequenceType, encoder_hidden_size_list: SequenceType = [128, 128, 64], head_hidden_size: int | None = None, head_layer_num: int = 1, activation: Module | None = ReLU(), norm_type: str | None = None, v_min: float | None = -10, v_max: float | None = 10, n_atom: int | None = 51)

Overview:

The neural network structure and computation graph of C51DQN, which combines distributional RL and DQN. You can refer to https://arxiv.org/pdf/1707.06887.pdf for more details. The C51DQN is composed of encoder and head. encoder is used to extract the feature of observation, and head is used to compute the distribution of Q-value.

Interfaces:

__init__, forward

Note

Current C51DQN supports two types of encoder: FCEncoder and ConvEncoder.

forward(x: Tensor) → Dict

Overview:

C51DQN forward computation graph, input observation tensor to predict q_value and its distribution.

Arguments:

• x (torch.Tensor): The input observation tensor data.

Returns:

• outputs (Dict): The output of DQN’s forward, including q_value, and distribution.

ReturnsKeys:

• logit (torch.Tensor): Discrete Q-value output of each possible action dimension.

• distribution (torch.Tensor): Q-Value discretized distribution, i.e., probability of each uniformly spaced atom Q-value, such as dividing [-10, 10] into 51 uniform spaces.

Shapes:

• x (torch.Tensor): \((B, N)\), where B is batch size and N is head_hidden_size.

• logit (torch.Tensor): \((B, M)\), where M is action_shape.

• distribution(torch.Tensor): \((B, M, P)\), where P is n_atom.

Examples:

>>> model = C51DQN(128, 64)  # arguments: 'obs_shape' and 'action_shape'
>>> inputs = torch.randn(4, 128)
>>> outputs = model(inputs)
>>> assert isinstance(outputs, dict)
>>> # default head_hidden_size: int = 64,
>>> assert outputs['logit'].shape == torch.Size([4, 64])
>>> # default n_atom: int = 51
>>> assert outputs['distribution'].shape == torch.Size([4, 64, 51])

Note

For consistency and compatibility, we name all the outputs of the network which are related to action selections as logit.

Note

For convenience, we recommend that the number of atoms should be odd, so that the middle atom is exactly the value of the Q-value.

Benchmark

Benchmark and comparison of c51 algorithm

environment	best mean reward	evaluation results	config link	comparison
Pong (PongNoFrameskip-v4)	20.6		config_link_p	Tianshou(20)
Qbert (QbertNoFrameskip-v4)	20006		config_link_q	Tianshou(16245)
SpaceInvaders (SpaceInvadersNoFrame skip-v4)	2766		config_link_s	Tianshou(988.5)

P.S.：

1. The above results are obtained by running the same configuration on five different random seeds (0, 1, 2, 3, 4)

2. For the discrete action space algorithm like DQN, the Atari environment set is generally used for testing (including sub-environments Pong), and Atari environment is generally evaluated by the highest mean reward training 10M env_step. For more details about Atari, please refer to Atari Env Tutorial .

Other Public Implementations

• Tianshou