Model Architecture Controls Gradient Descent Dynamics: A Combinatorial Path-Based Formula
Recently, there has been a growing interest in automatically exploring neural network architecture design space with the goal of finding an architecture that improves performance (characterized as improved accuracy, speed of training, or resource requirements). However, our theoretical understanding of how model architecture affects performance or accuracy is limited. In this paper, we study the impact of model architecture on the speed of training in the context of gradient descent optimization. We model gradient descent as a first-order ODE and use ODE*s coefficient matrix H to characterize the convergence rate. We introduce a simple analysis technique that enumerates H in terms of all possible ``paths** in the network. We show that changes in model architecture parameters reflect as changes in the number of paths and the properties of each path, which jointly control the speed of convergence. We believe our analysis technique is useful in reasoning about more complex model architecture modifications.
This paper studies the training dynamics of a neural network model as a dynamical system. The authors proposed a path-based approach to compute the derivatives that would appear in the H matrix which governs the learning dynamics. They further utilized this formulation to (1) simplify the analysis of convergence rate of 2-layer neural networks w.r.t. width; and (2) presented an argument for the similarity between added depth in the network and momentum-based optimization; and (3) also argued about the importance of the number of paths for fast convergence. The dynamical systems view of the learning process is quite new to me, even though I’m aware of a few recent papers exploring this. I found the paper to be mostly clear and not too hard to follow, and the dynamical systems perspective interesting. The path gradient is quite intuitive, but I’m a bit surprised there’s no prior work (at least not discussed in this paper) studying the relationship between gradients and the paths in the network. It is a bit hard for me to judge the significance of this work because of my lack of context. The main implications of the path gradients were presented in section 5. The first part shows that using their theorem 4.1 can simplify the derivation of the linear relationship between convergence rate and the width of 2-layer nets. This is a simplification but the original derivation is not complicated either. The second part tried to draw a relationship between added depth in a network and momentum-based optimization, which I found to be a bit hand-wavy. In particular, I found the argument that, du/dt prefers the direction of u itself which carries information about past directions therefore this is similar to momentum, to be not convincing. The third implication argues that we need H to be full-rank for fast convergence, and in order to achieve this we want to increase the number of paths in the network, which is interesting. There are a few other things that could be clarified: - the terminology in section 4.3 is a bit confusing, w_s seems to be associated with an edge in the graph, but is also referred to as a node, “we denote the activation value at node w_s with activation(w_s, X_i) ”. - on page 6 there is a reference to Equation 5.2 which does not exist - the results in Figure 4a is counter-intuitive - is this showing wider networks actually converge slower? Isn’t this against the argument of this paper? - it is unclear how the convergence rate lambda values are computed in Figure 4b, the curves in Figure 4a clearly doesn’t follow an exponential decay pattern. Overall I found this paper presented some interesting ideas, but may need a bit more work to be ready to be published. Happy to change my judgement however, if other more experienced reviewers can comment better on the significance of this work.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Model Architecture Controls Gradient Descent Dynamics: A Combinatorial Path-Based Formula
Recently, there has been a growing interest in automatically exploring neural network architecture design space with the goal of finding an architecture that improves performance (characterized as improved accuracy, speed of training, or resource requirements). However, our theoretical understanding of how model architecture affects performance or accuracy is limited. In this paper, we study the impact of model architecture on the speed of training in the context of gradient descent optimization. We model gradient descent as a first-order ODE and use ODE*s coefficient matrix H to characterize the convergence rate. We introduce a simple analysis technique that enumerates H in terms of all possible ``paths** in the network. We show that changes in model architecture parameters reflect as changes in the number of paths and the properties of each path, which jointly control the speed of convergence. We believe our analysis technique is useful in reasoning about more complex model architecture modifications.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



This paper studies the training dynamics of a neural network model as a dynamical system. The authors proposed a path-based approach to compute the derivatives that would appear in the H matrix which governs the learning dynamics. They further utilized this formulation to (1) simplify the analysis of convergence rate of 2-layer neural networks w.r.t. width; and (2) presented an argument for the similarity between added depth in the network and momentum-based optimization; and (3) also argued about the importance of the number of paths for fast convergence. The dynamical systems view of the learning process is quite new to me, even though I’m aware of a few recent papers exploring this. I found the paper to be mostly clear and not too hard to follow, and the dynamical systems perspective interesting. The path gradient is quite intuitive, but I’m a bit surprised there’s no prior work (at least not discussed in this paper) studying the relationship between gradients and the paths in the network. It is a bit hard for me to judge the significance of this work because of my lack of context. The main implications of the path gradients were presented in section 5. The first part shows that using their theorem 4.1 can simplify the derivation of the linear relationship between convergence rate and the width of 2-layer nets. This is a simplification but the original derivation is not complicated either. The second part tried to draw a relationship between added depth in a network and momentum-based optimization, which I found to be a bit hand-wavy. In particular, I found the argument that, du/dt prefers the direction of u itself which carries information about past directions therefore this is similar to momentum, to be not convincing. The third implication argues that we need H to be full-rank for fast convergence, and in order to achieve this we want to increase the number of paths in the network, which is interesting. There are a few other things that could be clarified: - the terminology in section 4.3 is a bit confusing, w_s seems to be associated with an edge in the graph, but is also referred to as a node, “we denote the activation value at node w_s with activation(w_s, X_i) ”. - on page 6 there is a reference to Equation 5.2 which does not exist - the results in Figure 4a is counter-intuitive - is this showing wider networks actually converge slower? Isn’t this against the argument of this paper? - it is unclear how the convergence rate lambda values are computed in Figure 4b, the curves in Figure 4a clearly doesn’t follow an exponential decay pattern. Overall I found this paper presented some interesting ideas, but may need a bit more work to be ready to be published. Happy to change my judgement however, if other more experienced reviewers can comment better on the significance of this work.
assistant
This paper studies the training dynamics of a neural network model as a dynamical system. The authors proposed a path-based approach to compute the derivatives that would appear in the H matrix which governs the learning dynamics. They further utilized this formulation to (1) simplify the analysis of convergence rate of 2-layer neural networks w.r.t. width; and (2) presented an argument for the similarity between added depth in the network and momentum-based optimization; and (3) also argued about the importance of the number of paths for fast convergence. The dynamical systems view of the learning process is quite new to me, even though I’m aware of a few recent papers exploring this. I found the paper to be mostly clear and not too hard to follow, and the dynamical systems perspective interesting. The path gradient is quite intuitive, but I’m a bit surprised there’s no prior work (at least not discussed in this paper) studying the relationship between gradients and the paths in the network. It is a bit hard for me to judge the significance of this work because of my lack of context. The main implications of the path gradients were presented in section 5. The first part shows that using their theorem 4.1 can simplify the derivation of the linear relationship between convergence rate and the width of 2-layer nets. This is a simplification but the original derivation is not complicated either. The second part tried to draw a relationship between added depth in a network and momentum-based optimization, which I found to be a bit hand-wavy. In particular, I found the argument that, du/dt prefers the direction of u itself which carries information about past directions therefore this is similar to momentum, to be not convincing. The third implication argues that we need H to be full-rank for fast convergence, and in order to achieve this we want to increase the number of paths in the network, which is interesting. There are a few other things that could be clarified: - the terminology in section 4.3 is a bit confusing, w_s seems to be associated with an edge in the graph, but is also referred to as a node, “we denote the activation value at node w_s with activation(w_s, X_i) ”. - on page 6 there is a reference to Equation 5.2 which does not exist - the results in Figure 4a is counter-intuitive - is this showing wider networks actually converge slower? Isn’t this against the argument of this paper? - it is unclear how the convergence rate lambda values are computed in Figure 4b, the curves in Figure 4a clearly doesn’t follow an exponential decay pattern. Overall I found this paper presented some interesting ideas, but may need a bit more work to be ready to be published. Happy to change my judgement however, if other more experienced reviewers can comment better on the significance of this work.