# A Linear Interpolation and Curvature-Controlled Gradient Optimization Strategy Based on [PERSON]
## 2 LCMAdam Algorithm Design
### LCMAdam Algorithm  
The selection of the learning rate is a critical aspect of neural network training. The Adam algorithm adjusts the learning rate by adaptive gradients and uses exponentially weighted averaging to balance historical and current gradients for smoother parameter updates. However, during the initial stages of training, the Adam algorithm may cause rapid convergence and oscillations, as well as the issue of being trapped in a local optimum point, which can affect the stability and speed of training. To address the above problems, an LCMAdam algorithm is proposed, which is improved on the basis of the Adam algorithm, and the specific steps of the algorithm are as follows.  
To initialize, set the initial learning rate to \(lr\), the maximum learning rate to \(initial\_lr\), the minimum learning rate to \(min\_lr\), the maximum change in the learning rate to max \(lr\_change\), the relative position, \(t\), in the cycle and the cycle length to \(period\), and initialize the first-order momentum to \(m_{t}\) and the second-order momentum to \(v_{t}\) as zero.  
Curvature adjustment stage: The curvature of the gradient was calculated, a correction factor, \(GR\), was calculated based on the curvature value of the gradient, and it was ensured that the correction factor did not exceed 1.0. Then, the gradient was multiplied by this correction factor, \(GR\), so that the magnitude of the gradient was regulated by the curvature. Finally, the gradient was processed, i.e., the gradient vector was divided by its parameter to ensure stability and convergence during the gradient update process.  
Learning rate adjustment phase: After obtaining the value set by initialization, the value of the current learning rate was determined by calculating the relative position, t, of the current step in the cycle, and the learning rate was adjusted according to the global number of steps by linear interpolation.  
Parameter update: By controlling the variation range of the learning rate, it ensures that the learning rate does not exceed the specified maximum variation value during the training process, and that the learning rate does not fall below the specified minimum value.  
The LCMAdam optimizer introduces a linear interpolation adjustment and a curvature control gradient strategy. This allows the model to adaptively adjust the learning rate based on the actual situation, accelerating convergence speed, and improving training efficiency. This section comprehensively discusses the learning rate and gradient adjustment, which makes the LCMAdam algorithm more flexible in coping with different data situations when training deep learning models. The algorithm for this approach is shown in Algorithm 1.  
```
1: Input : initial point \(\theta_{0}\), first - moment decay \(\beta_{1}\), second - moment decay \(\beta_{2}\), regularization constant \(\epsilon\)
2: Initialize \(m_{0}=0\) and \(v_{0}=0\), \(lr\), \(initial\_lr\), \(min\_lr\), \(t\), period
3: For 0 to \(period\) perform
4:\(g_{t}=\
abla f_{t}(\theta_{t-1})\)
5:\(m_{t}=\beta_{1}m_{t-1}+(1-\beta_{1})g_{t}\)
6:\(v_{t}=\beta_{2}v_{t-1}+(1-\beta_{2})g_{t}^{2}\)
7:\(\hat{m}_{t}=m/(1-\beta_{1}^{t})\)
8:\(\hat{v}_{t}=v/(1-\beta_{2}^{t})\)
9:\(curvature=v_{t}-m_{t}^{2}\)
10:\(momentum=\min(|curvature|,1.0)\)
11:\(t=step^{\otimes}period\)
12:\(lr=initial\_lr-(initial\_lr-\min\_lr)\cdot(t/period)\)
13:\(lr=\max(\min\_lr,lr-\max\_lr\_change)\)
14:\(\theta_{t+1}=\theta_{t}-lr\cdot\hat{m}_{t}/(\sqrt{\phi}+\epsilon)\) Endfor Return \(\theta_{t}\)
```  
**Algorithm 1**LCMAdam