One of the most common machine learning setups is logistic regression. In many classification models, including neural networks, the final prediction is obtained by applying a logistic link function to a linear score. In binary logistic regression, the feedback can be either soft labels, corresponding to the true conditional probability of the data (as in distillation), or sampled hard labels (taking values $\pm 1$). We point out a fundamental problem that arises even in a particularly favorable setting, where the goal is to learn a noise-free soft target of the form $σ(\mathbf{x}^{\top}\mathbf{w}^{\star})$. In the over-constrained case (i.e. the number of samples $n$ exceeds the input dimension $d$) with examples $(\mathbf{x}_i,σ(\mathbf{x}_i^{\top}\mathbf{w}^{\star}))$, it is sufficient to recover $\mathbf{w}^{\star}$ and hence achieve the Bayes risk. However, we prove that when the examples are labeled by hard labels $y_i$ sampled from the same conditional distribution $σ(\mathbf{x}_i^{\top}\mathbf{w}^{\star})$ and $\mathbf{w}^{\star}$ is $s$-sparse, then rotation-invariant algorithms are provably suboptimal: they incur an excess risk $Ω\!\left(\frac{d-1}{n}\right)$, while there are simple non-rotation invariant algorithms with excess risk $O(\frac{s\log d}{n})$. The simplest rotation invariant algorithm is gradient descent on the logistic loss (with early stopping). A simple non-rotation-invariant algorithm for sparse targets that achieves the above upper bounds uses gradient descent on the weights $u_i,v_i$, where now the linear weight $w_i$ is reparameterized as $u_iv_i$.

It is well known that the class of rotation invariant algorithms are suboptimal even for learning sparse linear problems when the number of examples is below the "dimension" of the problem. This class includes any gradient descent trained neural net with a fully-connected input layer (initialized with a rotationally symmetric distribution). The simplest sparse problem is learning a single feature out of $d$ features. In that case the classification error or regression loss grows with $1-k/n$ where $k$ is the number of examples seen. These lower bounds become vacuous when the number of examples $k$ reaches the dimension $d$. We show that when noise is added to this sparse linear problem, rotation invariant algorithms are still suboptimal after seeing $d$ or more examples. We prove this via a lower bound for the Bayes optimal algorithm on a rotationally symmetrized problem. We then prove much lower upper bounds on the same problem for simple non-rotation invariant algorithms. Finally we analyze the gradient flow trajectories of many standard optimization algorithms in some simple cases and show how they veer toward or away from the sparse targets. We believe that our trajectory categorization will be useful in designing algorithms that can exploit sparse targets and our method for proving lower bounds will be crucial for analyzing other families of algorithms that admit different classes of invariances.