Datasets:

sci-modeling-bench
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design-bench

A feedforward neural policy maps an observation through affine transformations and nonlinear activations to an action or to parameters of an action distribution. Its parameter vector consists of weight matrices and bias vectors, but parameter values are not a unique representation of the computed function: hidden-unit permutations and, for some activations, coordinated sign changes can preserve the input–output mapping.

Scope

Covered

Dense layers, weight matrices, biases, and activation functions.
Multilayer policies that produce action-distribution parameters.
Flattened parameter counts and the distinction between blocks and matrices.
Hidden-unit permutation and sign symmetries.

Not covered

Exact offsets or byte order for a particular saved controller.
A universal matrix-flattening convention.
A method for predicting policy quality from weights.

Key concepts and notation

Symbol or term	Meaning
(x\in\mathbb{R}^{d_{\mathrm{in}}})	Layer input
(W\in\mathbb{R}^{d_{\mathrm{in}}\times d_{\mathrm{out}}})	Weight matrix under a row-vector convention
(b\in\mathbb{R}^{d_{\mathrm{out}}})	Bias vector
(\phi)	Elementwise activation function
(h=\phi(xW+b))	Dense-layer output
hidden unit	Intermediate coordinate without a fixed external physical identity

Core knowledge

Affine transformations and nonlinearities

Under a row-vector convention, a dense layer computes

[ z=xW+b,\qquad h=\phi(z). ]

An equivalent column-vector convention writes (z=Wx+b) and stores the transpose-shaped matrix. Both are common. Matrix shape and multiplication convention must therefore be known before individual flattened weights can be assigned input and output meanings [1].

For an (m)-input, (n)-output dense layer, the number of scalar parameters is

[ mn+n, ]

including the bias. Layer widths and the declared order of parameter blocks therefore determine block lengths, even though they do not by themselves determine how each matrix was flattened.

Multilayer neural policy

A two-hidden-layer policy can be written as

[ h_1=\phi_1(oW_1+b_1), ]

[ h_2=\phi_2(h_1W_2+b_2), ]

[ u=h_2W_3+b_3. ]

The output (u) can be an action directly or a parameter such as the mean of a stochastic action distribution. Additional standalone parameters can represent quantities such as action log standard deviations.

Hyperbolic tangent activation

The hyperbolic tangent is

[ \tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}}, ]

with range ((-1,1)). It is odd:

[ \tanh(-z)=-\tanh(z). ]

Large-magnitude preactivations lie in saturated regions where the output is close to (+1) or (-1), while values near zero are approximately linear. These are properties of the activation, not guarantees about the behavior of a complete closed-loop controller.

Hidden-unit permutation symmetry

Hidden coordinates do not have an externally fixed order. If a permutation is applied to the outputs of one hidden layer and the inverse-compatible permutation is applied to the inputs of the following layer, the overall network function is unchanged [2].

Consequently, two parameter vectors can be far apart in ordinary Euclidean distance while computing the same function. Conversely, similar weight statistics do not guarantee similar functions.

Sign symmetry for odd activations

For an odd activation such as tanh, changing the sign of all incoming weights and the bias of one hidden unit changes the sign of its activation. Changing the signs of that unit's outgoing weights at the same time cancels the change. This gives another family of functionally equivalent parameter representations.

Such symmetries concern coordinated transformations. Arbitrarily changing individual signs generally changes the network.

Flattening is serialization

A matrix can be flattened row by row, column by column, or according to a framework-specific tensor layout. Concatenating multiple weights and biases adds a second ordering choice. These are serialization conventions rather than mathematical properties of a feedforward network.

Conditions, limitations, and uncertainty

Layer widths determine parameter counts but not the storage order inside a matrix block.
Biases, observation preprocessing, and output transformations are part of the policy function and cannot be ignored.
Weight magnitude is not a coordinate-invariant measure of policy behavior.
Neural-network symmetries create exact equivalences, while many other different parameterizations may be only approximately behaviorally similar.
Closed-loop behavior also depends on the environment dynamics and the distribution of observations encountered.

Related knowledge resources

continuous_control_and_gaussian_policies: interpretation of policy outputs.
proximal_policy_optimization_and_policy_checkpoints: how policy parameters may be learned.
hopper_locomotion_and_mujoco_dynamics: an example of closed-loop physical control.

References

Goodfellow I, Bengio Y, Courville A. Deep feedforward networks. In: Deep Learning. MIT Press; 2016. https://www.deeplearningbook.org/contents/mlp.html. [Textbook]
Brea J, Simsek B, Illing B, Gerstner W. Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape. arXiv. 2019. https://arxiv.org/abs/1907.02911. [Primary theory]