WASHINGTON — Defense Secretary Mark T. Esper said on Monday that the United States had begun withdrawing troops from Afghanistan, the first steps in what could be a complete exit from the country within the next 14 months.

Speaking to reporters at a Pentagon news conference, Mr. Esper said the initial troop drawdown — which would reduce the American presence in Afghanistan to 8,600 from the current 12,000 — was required to begin within 10 days of the peace agreement signed Saturday between the United States and the Taliban in Doha, Qatar.

“We are going to show good faith and begin withdrawing our troops,” Mr. Esper said.

The announcement came as the Taliban declared renewed attacks on Afghan forces. The bloodshed quickly dashed hopes of a prolonged cease-fire or a further reduction in violence after a weeklong drop in episodes across the country. The renewed violence spread fear among Afghans that the war would continue with little respite.

An Afghan commando was killed Monday by a Taliban sniper in Badghis Province in northwestern Afghanistan, according to Hasibullah Massoud, an Afghan army commander there. In Balkh Province in the north, the Taliban attacked several checkpoints in a three-hour-long gun battle. Three Afghan soldiers were wounded, said Mohammad Hakim, a local police commander. And in the east, a motorcycle rigged with explosives killed three civilians and wounded seven during a soccer match in Khost Province. Taliban officials denied responsibility for the attack in Khost.