The canonical deep learning approach for learning requires computing a gradient term at each layer by back-propagating the error signal from the output towards each learnable parameter. Given the stacked structure of neural networks, where each layer builds on the representation of the layer below, this approach leads to hierarchical representations. More abstract features live on the top layers of the model, while features on lower layers are expected to be less abstract. In contrast to this, we introduce a new learning method named NoProp, which does not rely on either forward or backwards propagation. Instead, NoProp takes inspiration from diffusion and flow matching methods, where each layer independently learns to denoise a noisy target. We believe this work takes a first step towards introducing a new family of gradient-free learning methods, that does not learn hierarchical representations – at least not in the usual sense. NoProp needs to fix the representation at each layer beforehand to a noised version of the target, learning a local denoising process that can then be exploited at inference. We demonstrate the effectiveness of our method on MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks. Our results show that NoProp is a viable learning algorithm which achieves superior accuracy, is easier to use and computationally more efficient compared to other existing back-propagation-free methods. By departing from the traditional gradient based learning paradigm, NoProp alters how credit assignment is done within the network, enabling more efficient distributed learning as well as potentially impacting other characteristics of the learning process.

Let \( x\) and \( y\) be a sample input-label pair in our classification dataset, assumed drawn from a data distribution \( q_0(x,y)\) , and \( z_0,z_{1},…,z_T \in \mathbb{R}^{d}\) be corresponding stochastic intermediate activations of a neural network with \( T\) blocks, which we aim to train to estimate \( q_0(y|x)\) .

We define two distributions \( p\) and \( q\) decomposed in the following ways:

The distribution \( p\) can be interpreted as a stochastic forward propagation process which iteratively computes the next activation \( z_t\) given the previous one \( z_{t-1}\) and the input \( x\) . In fact, we shall see that it can be explicitly given as a residual network with Gaussian noise added to the activations:

where \( \mathcal{N}_d(\cdot|0, 1)\) is a \( d\) -dimensional Gaussian density with mean vector 0 and identity covariance matrix, \( a_t, b_t, c_t\) are scalars (given below), \( b_tz_{t-1}\) is a weighted skip connection, and \( \hat{u}_{\theta_t}(z_{t-1},x)\) being a residual block parameterised by \( \theta_t\) . Note that this computation structure is different from a standard deep neural network which does not have direct connections from the input \( x\) into each block.

Following the variational diffusion model approach, we can alternatively interpret \( p\) as a conditional latent variable model for \( y\) given \( x\) , with the \( z_t\) ’s being a sequence of latent variables. We can learn the forward process \( p\) using a variational formulation, with the \( q\) distribution serving as the variational posterior. The objective of interest is then the ELBO, a lower bound of the log likelihood \( \log p(y|x)\) (a.k.athe evidence):

See Appendix A.1 for the derivation of Equation 3. Following [20, 24], we fix the variational posterior \( q\) to a tractable Gaussian distribution; here we use the variance preserving Ornstein-Uhlenbeck process:

where \( u_y\) is an embedding of the class label \( y\) in \( \mathbb{R}^d\) , defined by a trainable embedding matrix \( W_{\mathrm{Embed}} \in \mathbb{R}^{m \times d}\) , with \( m\) as the number of classes. The embedding is given by \( u_y = \{W_{\text{Embed}}\}_{y}\) . Using standard properties of the Gaussian, we can obtain

with \( \bar{\alpha}_t = \prod_{s=t}^T \alpha_s \) , \( \mu_t(z_{t-1},u_y) = a_t u_y + b_t z_{t-1}\) , \( a_t=\frac{\sqrt{\bar{\alpha}_{t}}(1-\alpha_{t-1})}{1-\bar{\alpha}_{t-1}}\) , \( b_t=\frac{\sqrt{\alpha_{t-1}}(1-\bar{\alpha}_{t})}{1-\bar{\alpha}_{t-1}}\) , and \( c_t = \frac{(1-\bar{\alpha}_{t})(1-\alpha_{t-1})}{1-\bar{\alpha}_{t-1}} \) . See Appendix A.2 and A.3 for a full derivation. To optimise the ELBO, we parameterise \( p\) to match the form for \( q\) :

where \( p(z_0)\) has been chosen to be the stationary distribution of the Ornstein-Uhlenbeck process, and \( \hat{u}_{\theta_t}(z_{t-1},x)\) is a neural network block parameterised by \( \theta_t\) . The resulting computation to sample \( z_{t}\) given \( z_{t-1}\) and \( x\) is as given in by the residual architecture (Equation 2), with \( a_t, b_t, c_t\) as specified above. Finally, plugging this parameterisation into the ELBO (Equation 3) and simplifying, we obtain the NoProp objective,

where \( \textrm{SNR}(t) = \frac{\bar{\alpha}_{t}}{1 - \bar{\alpha}_{t}}\) is the signal-to-noise ratio, \( \eta\) is a hyperparameter, and \( \mathcal{U}\{1, T\}\) is the uniform distribution on the integers \( 1, …, T\) . See Appendix A.4 for a full derivation of the NoProp objective.

We see that each \( \hat{u}_{\theta_t}(z_{t-1},x)\) is trained to directly predict \( u_y\) given \( z_{t-1}\) and \( x\) with an L2 loss, while \( \hat{p}_{\theta_{\textrm{out}}}(y|z_T)\) is trained to minimise cross-entropy loss. Each block \( \hat{u}_{\theta_t}(z_{t-1},x)\) is trained independently, and this is achieved without forward- or back-propagation across the network.

2.2.1 Fixed and learnable \( W_{\mathrm{Embed}}\)

2.2.2 Continuous-time diffusion models and neural ODEs

2.2.3 Flow matching

2.3.1 Architecture

2.3.2 Training procedure

Forward-forward algorithm [19] replaces the traditional forward and backward passes with two forward passes in image classification: one using images paired with correct labels and the other with images paired with incorrect labels.

Target propagation and its variant [e.g. 2, 30] work by learning an inverse mapping from the output layer to the hidden layers, producing local targets for weight updates without back-propagation. However, their performance heavily depends on the quality of the learned inverse mappings, and inaccuracies in the autoencoders can lead to suboptimal updates, limiting their effectiveness in deep networks.

Zero-order methods [10, 11, 14] is a broad family of algorithms that estimate gradients without explicit differentiation, making them useful for black-box and non-differentiable functions.

Evolution strategies [15, 16, 18], inspired by natural evolution, optimize parameters by perturbing them with random noise and updating in the direction of higher rewards using a population-based optimization approach. However, they require a large number of function evaluations, leading to high sample complexity, and effective exploration remains a key challenge for high-dimensional parameter spaces.

Several works in diffusion and flow matching are closely related to our method. [36] introduced classification and regression diffusion models, [37] proposed flow matching for paired data, and [29] applied flow matching to conditional text generation. In contrast, our paper explores the implications of these ideas within the framework of back-propagation-free learning.

Generally speaking, most alternative methods to back-propagation, whether they simply approximate gradients differently, or utilize a different search scheme — as it is the case of evolution strategies — still rely on learning intermediate representations that build on top of each other [31]. This allows learning more abstract representations as we look at deeper layers of the model, and is thought to be of fundamental importance for deep learning and for representing cognitive processes [32]. Indeed, initial successes of deep learning have been attributed to the ability to train deep architectures to learn hierarchical representations [33, 31], while early interpretability work focused on visualising these increasingly more complex features [34, 35].

By getting each layer to learn to denoise labels, with the label noise distribution chosen by the user, we can argue that NoProp does not learn representations. Rather, it relies on representations designed by the user (specifically, the representations in intermediate layers are Gaussian noised embeddings of target labels for diffusion, and an interpolation between Gaussian noise and embeddings of target labels for flow matching). This is perhaps unsurprising: forward- and back-propagation can be understood as information being disseminated across the layers of a neural network in order to enable the representation of each layer to “fit in” with those of neighbouring layers and such that the target label can be easily predicted from the representation at the last layer. So for NoProp to work without forward- or back-propagation, these layer representations have to be fixed beforehand, i.eto be designed by the user.

The fact that NoProp achieves good performance without representation learning leads to the question of whether representation learning is in fact necessary for deep learning, and whether by designing representations we can enable alternative approaches to deep learning with different characteristics. Specifically, note that representations fixed in NoProp are not those one might think of as being more abstract in later layers, which opens the door to revisiting the role of hierarchical representations in modelling complex behaviour [32]. These questions can become increasingly important as core assumptions of back-propagation based learning, like the i.i.dassumption and sequential propagation of information and error signal through the network, are proving to be limiting.

We use three benchmark datasets: MNIST, CIFAR-10, and CIFAR-100. The MNIST dataset consists of 70,000 grayscale images of handwritten digits (28x28 pixels), spanning 10 classes (digits 0-9). CIFAR-10 contains 60,000 color images (32x32 pixels) across 10 different object classes. CIFAR-100, similar in structure to CIFAR-10, includes 60,000 color images, but it is divided into 100 fine-grained classes. We use the standard training/test set splits. In our experiments, we do not apply any data augmentation techniques to these datasets.

We refer to this method as NoProp with Discrete-Time Diffusion (NoProp-DT). We fix \( T = 10 \) and, during each epoch, update the parameters for each time step sequentially, as outlined in Algorithm 1. While sequential updates are not strictly necessary for the algorithm to work — as time steps can also be sampled — we choose this approach to align with other back-propagation-free methods, ensuring consistency with existing approaches in the literature. In terms of parameterisation of the class probabilities \( \hat{p}_{\theta_{\textrm{out}}}(y|z_T)\) in Equation 7, we explore two approaches. The first, as described earlier, involves using softmax on the outputs of a fully connected layer \( f\) with parameters \( \theta_{\textrm{out}}\) , such that

where \( m\) is the number of classes. As an additional exploration, we investigate an alternative approach that relies on radial distance to parameterise the class probabilities. In this approach, we estimate a reconstructed label \( \tilde{y}\) from \( z_T\) by applying softmax to the output of the same fully connected layer \( f_{\theta_{\mathrm{out}}}\) , followed by a projection onto the class embedding matrix \( W_{\mathrm{Embed}} \), such that \( \tilde{y}\) is a weighted embedding. The resulting probability of class \( y \) given \( z_T\) is then based on the squared Euclidean distance between \( \tilde{y} \) and the true class embedding \( u_y = (W_{\mathrm{Embed}})_y\):

This formulation can be interpreted as computing the posterior class probability assuming equal prior class probabilities and a normal likelihood \( p(\tilde{y}|y) = \mathcal{N}_d(\tilde{y} | u_y, \sigma^2)\) .

We also fix \( T = 10\) and the forward pass is given by

where \( \alpha_1, \dots, \alpha_T \) are learnable parameters constrained to the range \( (-1,1) \), defined as \( \alpha_t = \tanh(w_t) \) with learnable \( w_1, \dots, w_T \). This design closely resembles the forward pass of NoProp-DT, but with the network trained by standard back-propagation. The \( \hat{u}_{\theta_t}(z_{t-1}, x)\) and \( \hat{p}_{\theta_{\mathrm{out}}}(y | z_T)\) have identical model structures to those in NoProp-DT.

We refer to the continuous-time variants as NoProp with Continuous-Time Diffusion (NoProp-CT) and NoProp with Flow Matching (NoProp-FM). During training, the time variable \( t \) is randomly sampled from \( [0,1] \). During inference, we set \( T=1000 \) steps to simulate the diffusion process for NoProp-CT and the probability flow ODE for NoProp-FM. Detailed training procedures are provided in Algorithms 2 and 3.

We also evaluate the adjoint sensitivity method [21], which trains a neural ODE using gradients estimated by solving a related adjoint equation backward in time. Including this method provides a meaningful baseline to assess the efficiency and accuracy of our approach in the continuous-time case. For fair comparison, we fix \( T=1000 \) during both training and inference and use a linear layer for final classification.

Our results, summarised in Table 1, demonstrate that NoProp-DT achieves performance comparable to or better than back-propagation on MNIST, CIFAR-10, and CIFAR-100 in the discrete-time setting. Moreover, NoProp-DT outperforms prior back-propagation-free methods, including the Forward-Forward Algorithm, Difference Target Propagation [2], and a forward gradient method referred to as Local Greedy Forward Gradient Activity-Perturbed [14]. While these methods use different architectures and do not condition layers explicitly on image inputs as NoProp does—making direct comparisons challenging—our method offers the distinct advantage of not requiring a forward pass. Additionally, NoProp demonstrates reduced GPU memory consumption during training, as shown in Table 2. To illustrate the learned class embeddings, Figure 2 visualises both the initializations and the final class embeddings for CIFAR-10 learned by NoProp-DT, where the embedding dimension matches the image dimension.

In the continuous setting, NoProp-CT and NoProp-FM achieve lower accuracy than NoProp-DT, likely due to the additional conditioning on time \( t \). However, they generally outperform the adjoint sensitivity method on CIFAR-10 and CIFAR-100, both in terms of accuracy and computational efficiency. While the adjoint method achieves similar accuracy to NoProp-CT and NoProp-FM on MNIST, it does so much slower, as shown in Figure 3.

For CIFAR-100 with one-hot embeddings, NoProp-FM fails to learn effectively, resulting in very slow accuracy improvement. In contrast, NoProp-CT still outperforms the adjoint method. However, once label embeddings are learned jointly, the performance of NoProp-FM improves significantly.

We also conducted ablation studies on the parameterisations of class probabilities, \( \hat{p}_{\theta_{\textrm{out}}}(y|z_T)\) , and the initializations of the class embedding matrix, \( W_{\text{Embed}}\) , with results shown in Figure 7 and Figure 11, respectively. The ablation results reveal no consistent advantage between the class probability parameterisations, with performance varying across datasets. For class embedding initializations, both orthogonal and prototype initializations are generally comparable to or outperform random initializations.

The last step follows from Jensen’s inequality, yielding a lower bound on \( \log p(y | x)\) . This bound is commonly referred to as the evidence lower bound (ELBO).

We will use the reparameterization trick to derive the expressions for \( q(z_t|y)\) and \( q(z_t|z_s)\) . The reparameterization trick allows us to rewrite a random variable sampled from a distribution in terms of a deterministic function of a noise variable. Specifically, for a Gaussian random variable \( z \sim \mathcal{N}_d(z|\mu, \sigma^2)\) , we can reparameterize it as:

Let \( \{\epsilon_t^*, \epsilon_t\}_{t=0}^T, \epsilon_y^*, \epsilon_y \sim \mathcal{N}_d(0, 1)\) . Then, for any sample \( z_t \sim q(z_t | z_s)\) for any \( 0 \le t < s \le T\) , it can be expressed as

Hence,

In particular,

Similarly, it can be shown that

Applying the Bayes’ theorem, we can obtain the posterior density

Hence we have \( q(z_t | z_{t-1}, y) = \mathcal{N}_d(z_t | \mu_t(z_{t-1}, u_y), c_t)\) , where

Starting from the ELBO derived in Appendix A.1, we have

Since \( q(z_t | z_{t-1}, y)\) and \( p(z_{t}|z_{t-1},x)\) are both Gaussian, with \( q(z_t | z_{t-1}, y) = \mathcal{N}_d(z_t | \mu_t(z_{t-1}, u_y), c_t)\) and \( p(z_{t}|z_{t-1},x) = \mathcal{N}_d(z_t | \mu_t(z_{t-1}, \hat{u}_{\theta_t}(z_{t-1},x)), c_t)\) , their KL divergence is available in closed form:

Using \( \hat{p}_{\theta_{\textrm{out}}}(y|z_T)\) to estimate \( p(y|z_T)\) , the ELBO becomes

Instead of computing all \( T\) terms in the sum, we replace it with an unbiased estimator, which gives

With an additional hyperparameter \( \eta\) in Equation 7, this yields the NoProp objective in the discrete-time case. However, note that in our experiments, we choose to iterate over the values of \( t\) from 1 to \( T\) instead of sampling random values. Details can be found in Algorithm 1.

In the continuous-time case where \( t\) is scaled to the range (0, 1), let \( \tau = 1/T\) . If we rewrite the term involving SNR in the NoProp objective as

As \( T \to \infty\) , this becomes

Again, with an additional hyperparameter \( \eta\) , this gives the NoProp objective in the continuous-time case in Equation 8.

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