Summary Conservation Laws for Gradient Flows arxiv.org
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The article examines the geometric aspects of gradient descent in machine learning, focusing on conservation laws and the preservation of functions during optimization.
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Key Points
- Conservation laws are sets of independent quantities that are conserved during the gradient flows of a model.
- A factorization of the cost function E is proposed, which is valid for optimization by gradient descent.
- Conservation laws in a finite-dimensional space can be found by projecting the equations in a basis.
- The dimension of the trace of Lie(V) is locally constant and equal to the dimension of Vi.
- The existence of a symmetric semi-definite matrix that satisfies an ODE equation is discussed.
- Numerical comparison confirms that there are no more conservation laws than the ones already known for deeper linear networks and ReLU networks.
- The text excerpt contains a list of references and citations related to gradient flows and implicit bias in machine learning.
Summaries
34 word summary
This article explores the geometric properties of gradient descent in machine learning. It introduces "conservation laws" as independent quantities conserved during gradient flows. The paper analyzes the functions preserved during optimization by gradient descent.
44 word summary
This article focuses on understanding the geometric properties of gradient descent dynamics in machine learning models. It introduces the concept of "conservation laws" as sets of independent quantities conserved during gradient flows. The paper analyzes the functions preserved during optimization by gradient descent and
677 word summary
This article focuses on understanding the geometric properties of gradient descent dynamics in machine learning models. It introduces the concept of "conservation laws", which are sets of independent quantities that are conserved during the gradient flows of a model. The article explains how to
The paper aims to analyze the functions that are preserved during optimization by gradient descent. It proposes a factorization of the cost function E, where E is a function of the mapping ? and the data fidelity f X,Y. The factorization is valid for
A function h is conserved through a subset V if and only if ?h(?) is in the linear space V. The set of functions conserved during all flows defined by the ODE corresponds to the functions conserved through a specific subset of
Conservation laws in a prescribed finite-dimensional space can be found by projecting the equations in a basis. For linear and ReLU cases, known conservation laws are polynomial "balancedness-type conditions." By focusing on the corresponding subspace of polynomials, a
The document discusses conservation laws for gradient flows. It states that the dimension of the trace of Lie(V) is locally constant and equal to the dimension of Vi. The number of conservation laws is characterized by the Lie algebra generated by V? and can be
Assuming certain conditions are met, the document discusses the existence of a symmetric semi-definite matrix that satisfies an ODE equation. Analytic examples are provided to illustrate these concepts. The document then explores the conservation laws for linear and ReLU neural networks
We conducted a numerical comparison to confirm that there are no more conservation laws than the ones already known for deeper linear networks and ReLU networks. Our code is open-sourced and available on GitHub. Our theory can be applied to any space of displacements
This excerpt contains a list of references to various articles and papers related to the topic of gradient flows and implicit bias in machine learning. The references cover a range of topics including functional dependence, algorithmic regularization, optimization geometry, implicit regularization, nonlinear system control
The summary of the text excerpt is as follows:
The excerpt contains a list of references and citations related to the topic of conservation laws for gradient flows. The references include papers on implicit regularization in deep learning, exact solutions to the nonlinear dynamics of learning,
The document discusses conservation laws for gradient flows in the context of linear and ReLU networks. The main goal is to show that CL(V ? [C ? ]) = CL(V ? [F ? ]) under certain assumptions. The document introduces Assumption B
The proof shows that the activation status of neurons is locally constant in a neighborhood, which implies that the conclusion follows from the fact that g ? (x) = C ?,x (? ReLU (?)) for all ?, x.
Lemma B.8
The text excerpt discusses conservation laws for gradient flows. It presents a proof of Theorem 3.3 and introduces the fundamental result of Frobenius, which states that if the dimension of a vector space is constant on a domain, then two conditions
The text discusses conservation laws for gradient flows. It introduces a condition (23) of Frobenius theorem and proves that it holds for each block of coordinates. It also shows that the dimension of V ? (?) is n + m - 1 and
The text excerpt discusses conservation laws for gradient flows. It presents two cases and shows that the condition of Frobenius Theorem is satisfied. It also provides a proof of Proposition 3.8 and an additional example. The example demonstrates how the function
The document discusses conservation laws for gradient flows. It states that if certain conditions are met, there are linearly dependent functions that have already been obtained. It also mentions the existence of independent conserved functions based on the values of n, m, and
If (U ; V ) has full rank, all conserved functions are given by ? : (U, V ) 7? U ? U ? V ? V and no more conserved functions exist. The dimension of Lie(V ? )(U