I've seen the same *ahem* discussions.
Faith is simply trusting in Jesus Christ for the atonement of your sins. When God Regenerates a Man, He creates that Trusting Faith within that Man. Men do not give Faith to God; God gives Faith to men. Think of the "exercise of faith" as the "exercise of breathing" -- God breathes spiritual life into the spiritually dead, and they do live and breathe.
Was Adam alive before God breathed life into him? Was he ever alive apart from breathing, or breathing apart from being alive? No, of course not.
In the sovereign act of Regeneration, God creates Faith within a spiritually dead Man, bringing that Man to spiritual life by His own purely monergistic-grace, through the creation of Faith within that Man.
After a spiritually-dead Man has been brought to spiritual life by God's purely-monergistic creation of Faith within him, he does "exercise faith" as the living exercise breath. But the initiation of that Man's Faith, the creation of his spiritual breath -- and thus the entire transaction of his salvation -- is purely by the monergistic Regeneration of God alone.
For the Bible adamantly teaches -- Faith is God-pleasing; and no man, while yet Unregenerate, ever chooses that which is God-pleasing.