Symmetries have proven useful in machine learning models, improving generalisation and overall performance.

Now, let us derive the differential equation that gives a solution to the variational problem. This condition yields the Euler-Lagrange equation, d dt @ L @ q = @ L @q . Here, we derive the Noether's learning dynamics by applying Noether's theorem to the A general form of the Noether's theorem relates the dynamics of Noether By evaluating the right hand side of Eq. 23, we get e Now, we harness the covariant property of the Lagrangian formulation, i.e., it preserves the form Plugging this expression obtained from the steady-state condition of Eq.27 Here, we ignore the inertia term in Eq. 16, assuming that the mass (learning rate) is finite but small All the experiments were run using the PyTorch code base. We used Tiny ImageNet dataset to generate all the empirical figures in this work. The key hyperparameters we used are listed with each figure.

Useful biases often exploit symmetries intheprediction problem, suchasconvolutional networks relying on translation equivariance.

Symmetries have proven useful in machine learning models, improving generalisation and overall performance. At the same time, recent advancements in learning dynamical systems rely on modelling the underlying Hamiltonian to guarantee the conservation of energy.These approaches can be connected via a seminal result in mathematical physics: Noether's theorem, which states that symmetries in a dynamical system correspond to conserved quantities.This work uses Noether's theorem to parameterise symmetries as learnable conserved quantities. We then allow conserved quantities and associated symmetries to be learned directly from train data through approximate Bayesian model selection, jointly with the regular training procedure. As training objective, we derive a variational lower bound to the marginal likelihood. The objective automatically embodies an Occam's Razor effect that avoids collapse of conversation laws to the trivial constant, without the need to manually add and tune additional regularisers. We demonstrate a proof-of-principle on n-harmonic oscillators and n-body systems. We find that our method correctly identifies the correct conserved quantities and U(n) and SE(n) symmetry groups, improving overall performance and predictive accuracy on test data.

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle to efficiently capture regularities in the world, but the role of symmetry breaking is not well understood. Here, we develop a theoretical framework to study the geometry of learning dynamics in neural networks, and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. To build this understanding, we model the discrete learning dynamics of gradient descent using a continuous-time Lagrangian formulation, in which the learning rule corresponds to the kinetic energy and the loss function corresponds to the potential energy. Then, we identify kinetic symmetry breaking (KSB), the condition when the kinetic energy explicitly breaks the symmetry of the potential function. We generalize Noether's theorem known in physics to take into account KSB and derive the resulting motion of the Noether charge: Noether's Learning Dynamics (NLD). Finally, we apply NLD to neural networks with normalization layers and reveal how KSB introduces a mechanism of implicit adaptive optimization, establishing an analogy between learning dynamics induced by normalization layers and RMSProp. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

Dimensionality reduction algorithms like principal component analysis (PCA) are workhorses of machine learning and neuroscience, but each has well-known limitations. Variants of PCA are simple and interpretable, but not flexible enough to capture nonlinear data manifold structure. More flexible approaches have other problems: autoencoders are generally difficult to interpret, and graph-embedding-based methods can produce pathological distortions in manifold geometry. Motivated by these shortcomings, we propose a variational framework that casts dimensionality reduction algorithms as solutions to an optimal manifold embedding problem. By construction, this framework permits nonlinear embeddings, allowing its solutions to be more flexible than PCA. Moreover, the variational nature of the framework has useful consequences for interpretability: each solution satisfies a set of partial differential equations, and can be shown to reflect symmetries of the embedding objective. We discuss these features in detail and show that solutions can be analytically characterized in some cases. Interestingly, one special case exactly recovers PCA.

The continuous-time model of Nesterov's momentum provides a thought-provoking perspective for understanding the nature of the acceleration phenomenon in convex

Symmetries have proven useful in machine learning models, improving generalisation and overall performance.

Now, let us derive the differential equation that gives a solution to the variational problem. This condition yields the Euler-Lagrange equation, d dt @ L @ q = @ L @q . Here, we derive the Noether's learning dynamics by applying Noether's theorem to the A general form of the Noether's theorem relates the dynamics of Noether By evaluating the right hand side of Eq. 23, we get e Now, we harness the covariant property of the Lagrangian formulation, i.e., it preserves the form Plugging this expression obtained from the steady-state condition of Eq.27 Here, we ignore the inertia term in Eq. 16, assuming that the mass (learning rate) is finite but small All the experiments were run using the PyTorch code base. We used Tiny ImageNet dataset to generate all the empirical figures in this work. The key hyperparameters we used are listed with each figure.