On the Quality of the Initial Basin in Overspecified Neural Networks

The evolution of deep learning has seen the rise of architectures like AlexNet, VGG, and ResNet, which have achieved unprecedented performance across vision and language tasks. Early work focused on the expressive power of deep networks and their ability to approximate complex functions. However, the training process involves optimizing highly non-convex loss functions, which historically was viewed as a major obstacle due to the proliferation of local minima and saddle points. Recent empirical observations suggest that larger networks tend to train more successfully, hinting at geometric properties that make the landscape more favorable. Prior theoretical efforts, such as [Choromanska et al., 2014], employed spin-glass models to analyze asymptotic behaviors, but lacked insights into finite-sized networks and the geometric structure of basins. This paper aims to fill this gap by providing a geometric probabilistic framework for understanding how network size influences the initial landscape and optimization pathways.

Despite empirical successes, the theoretical understanding of why overparameterized neural networks are easier to optimize remains incomplete. The core challenge lies in the non-convexity of the loss function, which can contain exponentially many local minima, saddle points, and flat regions. Standard algorithms like stochastic gradient descent (SGD) often succeed in practice, but lack rigorous guarantees, especially in finite networks. The key question is: how does network size influence the geometry of the loss landscape, and can we characterize the probability that a random initialization lands in a region conducive to successful training? Addressing this requires analyzing the geometric structure of the parameter space, the distribution of basins, and the existence of monotonic paths to global minima.

The main innovations include: 1) formalizing the concept of basins as convex regions in parameter space where local minima are global; 2) proving that the probability of initializing in a low-loss basin increases with network width, especially under data structures like low intrinsic dimension or clustering; 3) demonstrating the existence of continuous, monotonic descent paths connecting random initial points to global minima, via path rescaling and scale invariance techniques. This geometric probabilistic approach departs from prior asymptotic or spin-glass models, focusing instead on finite networks and practical initialization schemes, providing new insights into the role of overparameterization in training success.

• �� Define convex basins in the parameter space, characterized by sign patterns of neuron activations and their convexity properties. • Assume independent, spherically symmetric initialization distributions for each neuron’s weights, ensuring uniform sign distribution and independence. • Derive probabilistic bounds showing that as network width n increases, the probability of initial parameters falling into basins with small minimal loss approaches one. • Use path continuity arguments, constructing a continuous, scale-rescaled path from the initial point to a point with lower loss, ensuring a strictly monotonic decrease in the objective. • Extend the analysis to deep networks by inductively applying the basin and path properties, and to two-layer networks by partitioning the parameter space into sign-based regions, each forming a basin.

The authors conduct experiments on datasets like MNIST and CIFAR-10, varying network widths from 50 to 300 neurons. They initialize networks randomly according to the specified assumptions and measure the distribution of initial loss values. Results show that wider networks (n≥200) have over 98% probability of starting in basins with low minimal loss, consistent with theoretical bounds. Path existence is verified by attempting to construct monotonic descent trajectories via rescaling last-layer weights, with success rates exceeding 95%. Data structure analysis reveals that low-dimensional or clustered data further increases the likelihood of favorable basin initialization, confirming the geometric predictions.

The key findings demonstrate that increasing network width significantly raises the probability of initial parameters being in basins with near-optimal loss. For networks with width ≥200, the probability exceeds 98%, and the existence of a continuous, monotonic path to the global minimum is confirmed with over 95% success rate. These results hold across different data structures, with low-dimensional and clustered data showing even higher probabilities. The experiments validate the theoretical bounds and highlight the importance of overparameterization in creating a favorable geometric landscape for training.

These insights guide the design of neural networks by emphasizing the importance of sufficient width to ensure favorable initial basins. They suggest that initializing large networks can inherently reduce training difficulty, especially for low-dimensional or structured data. Practically, this can lead to more robust training procedures, improved convergence rates, and better generalization. The geometric perspective also informs initialization schemes and architecture choices, potentially reducing reliance on heuristic tuning.

The analysis assumes specific loss functions and data structures, which may not fully capture real-world complexities. The existence of favorable paths does not guarantee their discoverability by standard optimization algorithms like SGD, especially in very deep networks. The theoretical bounds are probabilistic and do not provide deterministic guarantees. Additionally, the geometric properties are derived under idealized assumptions, and their applicability to highly noisy or complex data remains to be validated. Future work should integrate these geometric insights with algorithmic dynamics and broader data scenarios.