This paper investigates the potential and limitations of warm starts in variational quantum algorithms, focusing on an iterative variational method for learning shorter-depth circuits for quantum real and imaginary time evolution. The study aims to understand how warm starts, which initialize the algorithm closer to a solution, can help overcome the barren plateau phenomenon, a challenge in scaling variational quantum algorithms due to exponentially vanishing gradients. The authors prove that for short enough time-steps, the loss variance decreases at worst polynomially in the number of parameters, and that the loss landscape is approximately convex in a region around the initialization. This suggests that training is possible within this region. However, the study also highlights scenarios where a good minimum may shift outside this region, raising questions about whether optimization across barren plateaus is necessary or if fertile valleys with substantial gradients exist. The paper presents theoretical and numerical results showing that for certain parameter settings, the loss landscape exhibits non-vanishing gradients and approximate convexity, enabling training. It also demonstrates that even in the presence of barren plateaus, gradient flows can exist between minima, allowing for successful training without crossing into the most barren regions. The analysis shows that by decreasing the time-step appropriately, it is possible to train to a new minimum. However, the study leaves open the question of whether the region with polynomially vanishing gradients strictly decreases with the number of parameters or if a larger region with substantial gradients exists. The results suggest that the $1/\sqrt{M}$ scaling is reasonable for the problems considered, but further research is needed to confirm this. The paper also discusses the implications of these findings for the broader field of variational quantum algorithms, highlighting the importance of understanding the structure of loss landscapes and the potential of warm starts in overcoming the challenges posed by barren plateaus.This paper investigates the potential and limitations of warm starts in variational quantum algorithms, focusing on an iterative variational method for learning shorter-depth circuits for quantum real and imaginary time evolution. The study aims to understand how warm starts, which initialize the algorithm closer to a solution, can help overcome the barren plateau phenomenon, a challenge in scaling variational quantum algorithms due to exponentially vanishing gradients. The authors prove that for short enough time-steps, the loss variance decreases at worst polynomially in the number of parameters, and that the loss landscape is approximately convex in a region around the initialization. This suggests that training is possible within this region. However, the study also highlights scenarios where a good minimum may shift outside this region, raising questions about whether optimization across barren plateaus is necessary or if fertile valleys with substantial gradients exist. The paper presents theoretical and numerical results showing that for certain parameter settings, the loss landscape exhibits non-vanishing gradients and approximate convexity, enabling training. It also demonstrates that even in the presence of barren plateaus, gradient flows can exist between minima, allowing for successful training without crossing into the most barren regions. The analysis shows that by decreasing the time-step appropriately, it is possible to train to a new minimum. However, the study leaves open the question of whether the region with polynomially vanishing gradients strictly decreases with the number of parameters or if a larger region with substantial gradients exists. The results suggest that the $1/\sqrt{M}$ scaling is reasonable for the problems considered, but further research is needed to confirm this. The paper also discusses the implications of these findings for the broader field of variational quantum algorithms, highlighting the importance of understanding the structure of loss landscapes and the potential of warm starts in overcoming the challenges posed by barren plateaus.