Rosenbaum was born in Toledo, Ohio and attended Whitmer High School. His father, Marvin Rosenbaum, was a furniture salesman and later the owner of a premiums and incentives distributor, and his mother, Edith Millman Rosenbaum, was a vice president at a local advertising agency. Rosenbaum graduated from Harvard College in 1974 with an A.B. in Government. He received a joint master in public policy and juris doctor from the Harvard Kennedy School and Harvard Law School in 1978. At Harvard Law he was a member of the winning team in the 1976 Ames Moot Court Competition. Rosenbaum is married to Marti Radlo Rosenbaum, and has three grown children and one grandchild.
At Harvard, Rosenbaum was a member of the winning team at the 1974 collegiate National Debate Tournament. In 1979, he was one of the coaches for Harvard's National Debate Tournament-winning team. Rosenbaum and his debate partner Charles Garvin were voted the second-best policy debate team of the 1970s in a poll of contemporary debate coaches and participants.
Rosenbaum serves on the board of directors of Harvard Debate, Inc, and is a former trustee and chair of the National Debate Tournament. In 2011, Rosenbaum was named to the board of directors of the National Association for Urban Debate Leagues.
Rosenbaum is currently the president of Palisades Associates, Inc., a Bethesda, Maryland-based private equity firm. He was previously a consultant with the Boston Consulting Group and a vice president of buyout firm Dyson-Kissner-Moran.
From 2003 to 2012 Rosenbaum was the Chairman of Empire Kosher Poultry, Inc., the largest producer of kosher poultry in the United States, and from 2006 to 2012 he was the company's chief executive officer.
In 2013, President Barack Obama appointed Rosenbaum to the United States Holocaust Memorial Council, the governing body of the United States Holocaust Memorial Museum. In 2011, Rosenbaum was named to the board of directors of the National Jewish Democratic Council (NJDC), and in 2014 was elected as the NJDC's chair. In the 2012 election cycle, Rosenbaum was a co-chair of NJDC's political action committee.
Rosenbaum is a charter member of the Clinton Global Initiative and was listed as a donor in reports released by the Clinton Foundation. At the Harvard Kennedy School, Rosenbaum is co-chair of the Littauer Society and a member of the Dean's Council. 
In 2011, the Metropolitan Council on Jewish Poverty granted Rosenbaum its Humanitarian Award. As the CEO of Empire Kosher, Rosenbaum oversaw the donation of kosher food products to community food pantries.
In 2009, the Philadelphia Jewish Labor Committee granted Rosenbaum, along with U.S. Senator Bob Casey, a Labor Human Rights Award. In 2007, the Jewish Labor Committee granted Rosenbaum a National Trade Union Council for Human Rights Award.
Rosenbaum is currently a trustee of the United States Olympic and Paralympic Foundation, which supports the United States Olympic and Paralympic teams. Rosenbaum and his family are also supporters of the USA Swimming Foundation.
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