Generalization in deep network classifiers trained with the square loss

Square loss has been observed to perform well in classification tasks. However, a theoretical justification is so far lacking, unlike the cross-entropy case. Here we discuss several observations on the dynamics of gradient flow under the square loss in ReLU networks. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques such as Batch Normalization (BN) are used together with Weight Decay (WD). In the absence of BN+WD, good solutions for classification may still be achieved because of the implicit bias towards small norm solutions in the GD dynamics introduced by close-to-zero initial conditions. The main property of the minimizers that bounds their expected error is the norm: we prove that among all the close-tointerpolating solutions, the ones associated with smaller Frobenius norms of the unnormalized weight matrices have better margin and better bounds on the expected classification error. The theory yields several predictions, including the role of BN and weight decay, aspects of Papyan, Han and Donoho’s Neural Collapse and the constraints induced by BN on the network weights. This material is based upon work supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216. Generalization in deep network classifiers trained with the square loss Tomaso Poggio and Qianli Liao Abstract Square loss has been observed to perform well in classification tasks. However, a theoretical justification is lacking, unlike the cross-entropy [1] case for which an asymptotic analysis has been proposed (see [2] and [3] and references therein). Here we discuss several observations on the dynamics of gradient flow under the square loss in ReLU networks. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques such as Batch Normalization[4] (BN) or Weight Normalization[5] (WN) are used together with Weight Decay (WD). In the absence of BN+WD, good solutions for classification may still be achieved because of the implicit bias towards small norm solutions in the GD dynamics introduced by close-to-zero initial conditions. The main property of the minimizers that bounds their expected error is the norm: we prove that among all the close-to-interpolating solutions, the ones associated with smaller Frobenius norms of the unnormalized weight matrices have better margin and better bounds on the expected classification error. The theory yields several predictions, including the role of BN and weight decay, aspects of Papyan, Han and Donoho’s Neural Collapse and the constraints induced by BN on the network weights.Square loss has been observed to perform well in classification tasks. However, a theoretical justification is lacking, unlike the cross-entropy [1] case for which an asymptotic analysis has been proposed (see [2] and [3] and references therein). Here we discuss several observations on the dynamics of gradient flow under the square loss in ReLU networks. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques such as Batch Normalization[4] (BN) or Weight Normalization[5] (WN) are used together with Weight Decay (WD). In the absence of BN+WD, good solutions for classification may still be achieved because of the implicit bias towards small norm solutions in the GD dynamics introduced by close-to-zero initial conditions. The main property of the minimizers that bounds their expected error is the norm: we prove that among all the close-to-interpolating solutions, the ones associated with smaller Frobenius norms of the unnormalized weight matrices have better margin and better bounds on the expected classification error. The theory yields several predictions, including the role of BN and weight decay, aspects of Papyan, Han and Donoho’s Neural Collapse and the constraints induced by BN on the network weights.