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Learning to optimize is an approach that leverages training data to accelerate the solution of optimization problems. Many approaches use unrolling to parametrize the update step and learn optimal parameters. Although L2O has shown empirical advantages over classical optimization algorithms, memory restrictions often greatly limit the unroll length and learned algorithms usually do not provide convergence guarantees. In contrast, we introduce a novel method employing a greedy strategy that learns iteration-specific parameters by minimizing the function value at the next iteration. This enables training over significantly more iterations while maintaining constant GPU memory usage. We parameterize the update such that parameter learning corresponds to solving a convex optimization problem at each iteration. In particular, we explore preconditioned gradient descent with multiple parametrizations including a novel convolutional preconditioner. With our learned algorithm, convergence in the training set is proved even when the preconditioner is neither symmetric nor positive definite. Convergence on a class of unseen functions is also obtained, ensuring robust performance and generalization beyond the training data. We test our learned algorithms on two inverse problems, image deblurring and Computed Tomography, on which learned convolutional preconditioners demonstrate improved empirical performance over classical optimization algorithms such as Nesterov's Accelerated Gradient Method and the quasi-Newton method L-BFGS.