Note that f(βt) = gt is computed either as in line 5 or line 9 of the algorithm and one can use these computations for any gradient based algorithm (e.g. Note also that this is simply gradient descent on a smooth function, and one can apply typical methods to choosing the stepsize γk, such as the Barzilai-Borwein stepsize [Barzilai and Borwein, 1988]. Algorithm 1: Gradient descent implementation of Ncvx-Pro for solving Lasso. 1 initialization v0 Rn (with no zero entries), stepsize γt > 0; Result: βt 2 while not converged do 3 if n6 mand λ>0 then 4 ut = diag(vt)X>Xdiag(vt) + λId To show that i) implies ii), recall that a convex, proper and lower semicontinuous function ϕ can be written in terms of its convex conjugate which has domain Rd . For the expression of ψwhen Ris a norm,from the above, we know that ψ = ( ϕ) ( z), and recall that for any norm, R(β) = maxR (w)61hw, βi. We derive some properties of the function h: Lemma 1.

We start by fleshing out the connection between strong convexity and smoothness charted in Lemma 1: Lemma 1. If F is -strongly convex w.r.t.

Given the intractably large size of the space of proofs, any model that is capable of general deductive reasoning must generalize to proofs of greater complexity. Recent studies have shown that large language models (LLMs) possess some abstract deductive reasoning ability given chain-of-thought prompts. However, they have primarily been tested on proofs using modus ponens or of a specific size, and from the same distribution as the in-context examples. To measure the general deductive reasoning ability of LLMs, we test on a broad set of deduction rules and measure their ability to generalize to more complex proofs from simpler demonstrations from multiple angles: depth-, width-, and compositional generalization. To facilitate systematic exploration, we construct a new synthetic and programmable reasoning dataset that enables control over deduction rules and proof complexity. Our experiments on four LLMs of various sizes and training objectives show that they are able to generalize to compositional proofs. However, they have difficulty generalizing to longer proofs, and they require explicit demonstrations to produce hypothetical subproofs, specifically in proof by cases and proof by contradiction.